compressible flow

Compressible Flow

Compressible FLow

We know that fluids are classified as Incompressible and Compressible fluids Incompressible fluids do not undergo significant changes in density as they flow In general liquids are incompressiblewater being an excellent example In contrast compressible fluids do undergo density changes Gases are generally compressibleair being the most common compressible fluid we can find Compressibility of gases leads to many interesting features such as shocks which are absent for incompressible fluids Gasdynamics is the discipline that studies the flow of compressible fluids and forms an important branch of Fluid Mechanics In this book we give a broad introduction to the basics of compressible fluid flow

Figure 11 Classification of Fluids

Though gases are compressible the density changes they undergo at low speeds may not be considerable Take air for instance Fig 12 shows the

density changes plotted as a function of Mach Number Density change is represented as where is the air density at zero speed (ie Zero

Mach Number)

httpwwwaesuozauaerogasdynnode1html (1 of 75)9172007 31356 PM

Compressible Flow

Figure 12 Density change as a function of Mach Number

We observe that for Mach numbers up to 03 density changes are within about 5 of So for all practical purposes one can ignore density changes

in this region But as the Mach Number increases beyond 03 changes do become appreciable and at a Mach Number of 1 it is 365 it is interesting to note that at a Mach Number of 2 the density changes are as high as 77 It follows that air flow can be considered incompressible for Mach Numbers below 03

Another important difference between incompressible and compressible flows is due to temperature changes For an incompressible flow temperature is generally constant But in a compressible flow one will see a significant change in temperature and an exchange between the modes of energy Consider a flow at a Mach Number of 2 It has two important modes of energy-Kinetic and Internal At this Mach Number these are of magnitudes 23 x 105 Joules and 2 x 105 Joules You will recognise that these are of the same order of magnitude This is in sharp contrast to incompressible flows where only the kinetic energy is important In addition when the Mach 2 flow is brought to rest as happens at a stagnation point all the kinetic energy gets converted into internal energy according to the principle of conservation of energy Consequently the temperature increases at the stagnation point When the flow Mach number is 2 at a temperature of 200C the stagnation temperature is as high as 260 0C as indicated in Fig 13


Compressible Flow

Figure 13 stagnation Temperature

A direct consequence of these facts is that while calculating compressible flows energy equation has to be considered (not done for incompressible flows) Further to handle the exchange in modes of energy one has to understand the thermodynamics of the flow Accordingly we begin with a review of the concepts in thermodynamics

Thermodynamics is a vast subject Many great books have been written describing the concepts in it and their application It is not the intention here to give a detailed treatise than it is to review the basic concepts which hep us understand gasdynamics Reader is referred to exclusive books on thermodynamics for details

System Surroundings and Control Volume

Concepts in Thermodynamics are developed with the help of systems and control volumes We define a System as an entity of fixed mass and concentrate on what happens to this fixed mass Its boundary is not fixed and is allowed to vary depending upon the changes taking place within it Consider the system sketched namely water in a container placed on a heater We are allowed to chose the system as is convenient to us We could have system as defined in (a) or (b) or (c) as in Fig 14

Everything outside of a system becomes the Surroundings

Properties of the system are usually measured by noting the changes it makes in the surrounding For example temperature of water in system (a) is measured by a the raise of the mercury column in a thermometer which is not a part of the system

Sometimes the system and the surroundings are together called the Universe

Figure 14 Definition of a System

Control Volume should now be familiar to you Most of Integral Approach to Fluid Dynamics exploits control volumes which can be defined as a window in a flow with a fixed boundary Mass momentum and energy can cross its boundary

Density pressure temperature etc become properties of a given system Note that these are all measurable quantities In addition these properties


Compressible Flow

also a characterise a system To define the state of a system (Fig 15) uniquely we need to specify two properties say (pT) (p ) (Ts) etc where p

T s are pressure temperature density and specific entropy respectively

Figure 15 State of a System

Properties can be Extensive or Intensive Extensive properties depend on the mass of the system On the other hand Intensive properties are

independent of the mass Volume Energy E Entropy S Enthalpy H are Extensive properties Corresponding intensive properties are Specific Volume v Specific Energy e Specific Entropy s and Specific Enthalpy h and are obtained by considering extensive properties per unit mass In other words

(11)

Laws of Thermodynamics

Thermodynamics centers around a few laws We will consider them briefly so that the concepts in gasdynamics can be easily developed

Zeroth Law of Thermodynamics


Compressible Flow

This laws helps define Temperature It states - Two systems which are in thermal equilibrium with a third system are themselves in thermal equilibrium

Figure 16 Zeroth Law of Thermodynamics

When in thermal equilibrium we say that the two systems are at the same temperature In the figure 16 system A and Bare independently in equilibrium with system C It follows that Aand B are themselves in thermal equilibrium and they are at the same temperature

First Law of Thermodynamics

The first law of Thermodynamics is a statement of the principle of conservation of energy It is simply stated as Energy of a system and surroundings is conserved Consider a system S If one adds dq amount of heat per unit mass into the system and the work done by the system is dw per unit mass we have the change in internal energy of the system du given by

du = dq - dw (12)

where u is Internal Energy Bringing in Specific Enthalpy defined as

(13)

the statement for the first law can also be written as

dh = dq + vdp (14)


Compressible Flow

While writing Eqn14 we have included only one form of energy namely internal Other forms such as the kinetic energy have been ignored Of course it is possible to account for all the forms of energy

Second Law of Thermodynamics

Second Law of Thermodynamics has been a subject of extensive debate and explanation Its realm ranges from physics chemistry to biology life and even philosophy There are numerous websites and books which discuss these topics They form an exciting reading in their own right Our application however is restricted to gasdynamics

The first law is just a statement that energy is conserved during a process (The term Process stands for the mechanism which changes the state of a system) It does not worry about the direction of the process whereas the second law does It determines the direction of a process In addition it involves another property - Entropy

Figure 17 Second Law of Thermodynamics

There are numerous statements of the Second Law Consider a Reversible Process Suppose a system at state A undergoes changes say by an addition of heat Q and attains state B While doing so the surroundings change from A to B Let us try to bring the state of the system back to A by removing an amount of heat equal to Q In doing so if we can bring the surroundings also back to state A then the process is said to be reversible This is possible only under ideal conditions In any real process there is friction which dissipates heat Consequently it is not possible to bring the system back to state A and at the same time surroundings back to A


Compressible Flow

Assuming the process to be reversible the second law defines entropy such that

(15)

where s is Specific Entropy For small changes the above equation is written as

(16)

Generalising the equation 16 we have

(17)

where an = sign is used for reversible processes and gt is used for ireversibe processes

Thus with any natural process entropy of the system and universe increases In the event the process is reversible entropy remains constant Such a process is called an Isentropic process

Perfect Gas Law

It is well known that a perfect gas obeys

(18)

where R is the Gas constant For any given gas R is given by

(19)

where is known as the Universal Gas Constant with the same value for all gases Its numerical value is 83135 Jkg-mol K M is the molecular weight of the gas The following table gives the value of the gas constant (along with other important constants) for some of the gases

Gas Molecular Weight Gas Constant R cp


Compressible Flow

Jkg K Jkg K cpcv

Air 2897 2870 1004 14

Ammonia 1703 4882 2092 13

Argon 3994 2081 519 167

Carbon dioxide 44 1889 845 13

Helium 4003 2077 5200 167

Hydrogen 2016 4124 14350 14

Oxygen 32 2598 916 14

Consequences of First Law for a Perfect Gas

For a perfect gas internal energy and enthalpy are functions of temperature alone Hence

u = u(T) h = h(T) (110)

Specific Heat of a gas depends upon how heat is added - at constant pressure or at constant volume We have two specific heats cp specific heat at

constant pressure and cv specific heat at constant volume It can be shown that

(111)

Then introducing it follows that

(112)

A Calorically Perfect Gas is one for which cp and cv are constants Accordingly


Compressible Flow

(113)

Consequences of Second Law for a Perfect Gas

We have shown in Eqn 14 in First Law of Thermodynamics

Now assuming a perfect gas and a reversible process we have

Integrating between states 1 and 2 we can show that

(114)

If we assume that the process is isentropic that is adiabatic(implying no heat transfer) and reversible we can show that the above equation leads to

(115)

A familiar form of equation for an isentropic flow is

(116)

Equations of Motion for a Compressible Flow

We now write the equations of motion for a compressible flow Recall that for an incompressible flow one calculates velocity from continuity and other considerations Pressure is obtained through the Bernoulli Equation Such a simple approach is not possible for a compressible flow where temperature is not a constant One needs to solve the energy equation in addition to the continuity and momentum equations The latter equations have already been derived for incompressible flows Of course one has to account for the changes in density We focuss here on the energy equation and briefly outline the other two We restrict ourselves to an Integral Approach and write the equations for a control volume


Compressible Flow

Equations are derived under the following assumptions

Flow is one-dimensional Viscosity and Heat Transfer are neglected Behaviour of flow as a consequence of area changes only considered The flow is steady

We consider a one-dimensional control volume as shown

Figure 18 Control Volume for a Compressible Flow

Continuity Equation

For a steady flow it is obvious that the mass flow rates at entry (1) and exit (2) of the control volume must be equal Hence

(117)

Written in a differential form the above equation becomes

(118)

At this stage it is usual to consider some applications of the above equation But we note that this equation has always to be solved with momentum and energy equations while calculating any flow Accordingly we skip any worked example at this stage

Momentum Equation


Compressible Flow

The derivation of Momentum Equation closely follows that for incompressible flows Basically it equates net force on the control volume to the rate of change of momentum Defining pm as the average pressure between the entry (1) and exit (2)(see Fig 18) we have for a steady flow

(119)

For a steady flow through a duct of constant area the momentum equation assumes a simple form

(120)

It is to be noted that the above equations can be applied even for the cases where frictional and viscous effects prevail between (1) and (2) But it is necessary that these effects be absent at (1) and (2)

Energy Equation

From the first law of Thermodynamics it follows that for a unit mass

q + work done = increase in energy (121)

where q is the heat added Work done is given by

p1v1 - p2v2 (122)

We consider only internal and kinetic energies Accordingly we have

Change of energy = (123)

Substituting Eqns 122 and 123 into Eqn 121 we have as the energy equation for a gas flow as

(124)

Noting that enthalpy h = e + pv we have


Compressible Flow

Considering an adiabatic process q = 0 we have

(125)

This equation demands that the states (1) and (2) be in equilibrium but does allow non-equilibrium conditions between (1) and (2)

If the flow is such that equilibrium exists all along the path from (1) to (2) then we have at any location along 1-2

= constant (126)

Differentiating the above equation we have

(127)

For a thermally perfect gas ie enthalpy h depends only on temperature T (h =cpT) the above equation becomes

(128)

Further for a calorically perfect gas ie cp is constant we have

(129)

Stagnation Conditions

What should be the constant on the RHS of Eqn 129 equal to We have left it as an open question It appears that stagnation conditions and sonic conditions are good candidates to provide the required constant We now discuss the consequences of each of these choices

Constant from Stagnation Conditions


Compressible Flow

Stagnation conditions are reached when the flow is brought to restie u = 0 Temperature pressure density entropy and enthalpy become equal to

Stagnation Temperature T0 Stagnation Pressure p0 Stagnation Density Stagnation Entropy s0 and Stagnation Enthalpy s0 These are

also known as Total conditions This is in contrast to incompressible flows where we have only the Stagnation Pressure

Rewrite Eqn126 as

(130)

Recalling that h = cpT for a calorically perfect gas we have

(131)

The constant we have arrived at is h0 or cpT0

It is to be noted that there does not have to be a stagnation point in a flow in order to use the above equations Stagnation or Total conditions are only reference conditions Further it is apparent that there can be only one stagnation condition for a given flow Such a statement is to be qualified and is true only for isentropic flows In a non-isentropic flow every point can have its own stagnation conditions meaning if the flow is brought to rest locally at every point one can have a series of stagnation points

In an adiabatic flow (a flow where heat is not added or taken away) the stagnation or total temperatureT0 does not change This is true even in

presence of a shock as we will see later But the total pressure p0 can change from point to point Consider again the control volume shown in Fig 18

As long as the flow is adiabatic we have

(132)

To deduce the conditions for total (stagnation) pressure we consider the Second Law of Thermodynamics

(133)

For a perfect gas the above equation becomes


Compressible Flow

(134)

Since we have

(135)

Equals sign applies when the flow is isentropic and Greater Than sign applies for any non-isentropic flow Thus for any natural process involving dissipation total pressure drops It is preserved for an isentropic flow A very good example of a non-isentropic flow is that of a shock Across a shock there is a reduction of total pressure

Area-Velocity Relation

We are all used to the trends of an incompressible flow where velocity changes inversely with area changes - as the area offered to the flow increases velocity decreases and vice versa This seems to be the commonsense But a compressible flow at supersonic speeds does beat this commonsense Let us see how

Consider the Continuity Equation Eqn118 which reads

Consider also the Euler Equation (derived before for incompressible flows)

Rewriting the equation

(136)

Where we have brought in speed of sound which is given by Further introducing Mach Number M the above equation becomes


Compressible Flow

(137)

Upon substituting this in the continuity equation 118 we have

(138)

Figure 19 Response of Subsonic and Supersonic Flows to Area Changes


Compressible Flow

Studying Eqn138 and Fig19 one can observe the following

For incompressible flows M = 0 As the area of cross section for a flow decreases velocity increases and vice versa For subsonic flows Mlt1 the behaviour resembles that for incompressible flows For supersonic flowsM gt 1 as the area decreases velocity also decreases and as the area increases velocity also increases We can explain this

behaviour like this In response to an area change all the static properties change At subsonic speeds changes in density are smaller The velocity decreases when there is an increased area offered (and vice versa) But in case of a supersonic flow with increasing area density decreases at a faster rate than velocity In order to preserve continuity velocity now increases (and decreases when area is reduced) vice versa)

Not apparent from the above equation is another important property If the geometry of the flow involves a throat then mathematically it can shown that if a sonic point occurs in the flow it occurs only at the Throat But the converse - The flow is always sonic at throat is not true

Isentropic Relations

For an isentropic flow all the static properties such as and s when expressed as a ration of their stagnation values become functions of Mach

Number M and alone This can be shown as below Recall the energy equation

(139)

Eliminating T using the equation for speed of sound (still not proved) we have

(140)

where a0 is the stagnation speed of sound The above equation simplifies to

(141)

Multiplying throughout by yields

(142)


Compressible Flow

Thus we have a relationship which connects temperature ratio with Mach Number Assuming isentropy and using the relation (see

Eqn116 we can derive expressions for pressure and density as

(143)

(144)

The relations just developed prove very useful in calculating isentropic flows Once Mach Number is known it is easy now to calculate pressure density and temperature as ratios of their stagnation values These are tabulated as functions of Mach number in tables in the Appendix There are also calculation scripts found in the Appendix for compressible flow and the aerodynamics calculator at httpwwwaoevteduaoe3114calchtml is very useful in this regard

Sonic Point as Reference

The preceding relations were arrived at with stagnation point as the reference As stated before it is also possible to choose sonic point the position where M = 1 as the reference At this point let u = u and a = a Since M = 1 we have u = a As a consequence the energy equation 140 Isentropic Relations) becomes

(145)

Comparing with the energy equation Eqn 140 we obtain

(146)

As a result for air with we have


Compressible Flow

(147)

It may be pointed out a sonic point need not be present in the flow for the above equations to be applicable

Mass Flow Rate

Now derive an equation for mass flow rate in terms of Mach Number of flow Denote the area in an isentropic flow where the Mach Number becomes 1 as A We have for mass flow rate

For an isentropic flow

It follows by noting that

Substituting for terms such as etc and simplifying one obtains

(148)

This is a very useful relation in Gasdynamics connecting the local area and local Mach Number Tables in Appendix also list this function ie AA as a function of Mach Number It helps one to determine changes in Mach Number as area changes


Compressible Flow

Figure 110 Mach Number as a function of area

Figure 110 shows the area function AA plotted as a function of Mach Number We again see what was found under Area Velocity rule before The difference is that now we have a relationship between area and Mach Number For subsonic flows Mach Number increases as the area decreases and it decreases as the area increases While with supersonic flows Mach Number decreases as area decreases and it increases as area increases

Equations of Motion in absence of Area Changes

We can now gather the equations that we have derived for mass momentum and energy If we ignore any area change these become


Compressible Flow

(149)

Wave Propagation

Waves carry information in a flow These waves travel at the local speed of sound This brings a sharp contrast between incompressible and compressible flows For an incompressible fluid the speed of sound is infinite (Mach number is zero) Consequently information in the form of pressure density and velocity changes is conveyed to all parts of the flow instantaneously The flow too therefore changes instantaneously Take any incompressible flow and study its flow pattern You will notice that the stream lines are continuous and the flow changes smoothly to accommodate the presence of a body placed in the flow It seems to change far upstream of the body This may not happen in compressible flows The reason is that in compressible medium any disturbance travels only at finite speeds Signals generate at a point in the flow take a finite time to reach other parts of the flow ( See annimation given below) The smooth streamline pattern noticed for incompressible flow may now be absent In fact this stems from the compressibility effects which make a compressible flow far more interesting than an incompressible flow In this section we derive an expression for the speed of sound Then we study the difference in the way a supersonic flow responds to the presence of an obstacle in comparison to a subsonic flow We then find out how shocks are formed in compressible flows

Press RED to start

Speed of Sound


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Compressible Flow

We first derive an expression for the speed of sound Consider a sound wave propagating to the right as shown (Fig21) with a speed a The medium to

the right is at rest and has pressure density and temperature respectively As a consequence of wave motion the medium gets compressed

and the gas is set to motion Let the speed of gas behind the wave be du Pressure Density and temperature will now be

respectively

Figure 21 Speed of Sound

For analysis it is better to make the wave stationary Accordingly we superpose an equal and opposite speed a (equal to the wave speed) everywhere in the flow This is shown in part (b) of Fig 21

Applying the equations of mass and momentum to the control volume indicated we have

(21)

Simplifying the continuity equation and ignoring the products such as we have

(22)


Compressible Flow

Replacing the term in the momentum equation we have

which simplifies to (23)

Combining Eqns 22 and 23 one gets

(24)

If we now consider the propagation of sound wave to be isentropic the above equation becomes

(25)

For a perfect gas the above expression reduces to

(26)

As stated before any change in flow conditions is transmitted to other parts of the flow by means of waves travelling at the local speed of sound The ratio of flow speed to the speed of sound called Mach Number is a very important parameter for a given flow Writing it out as an equation we have

(27)

Mach Number also indicates the relative importance of compressibility effects for a given flow

Propagation of a Source of Sound


Compressible Flow

Let us now consider a point source of sound and the changes that occur when it moves at different speeds

Stationary Source

First consider a stationary source This source emits a sound wave at every second say The waves travels in the form of a circle with its centre at the location of the source as shown in Fig 22 After 3 seconds we will see three concentric circles as shown The effect of the sound source is felt within the largest circle

Source moving at Subsonic Speeds

Let the source move to the left at half the speed of sound ieM = 05 The source occupies various position shown Sound produced reaches out as far as a distance of 3a being the distance travelled by the sound emitted at t = 0 In this case as well as the one above sound travels faster than the particle which is in some sense left behind

Figure 22 Propagation of a Source of Sound at different speeds


Compressible Flow

Source moving at the Speed of Sound

Now consider the situation where the source moves at the speed of sound iea This is sketched in the same figure (Fig 22) Now sound travels with the particle speed and it does not outrun it Consequently the circles representing wave motion touch each other as shown One can draw a line which is tangential to each of these circles Any effect of the sound wave is felt only to the right of this line In the region to the left of the line one does not see any effect of the source These regions are designated Zone of Action and Zone of Silence respectively

Source moving at Supersonic Speeds

The situation becomes dramatic when the source moves at speeds greater than that of sound Now the boundary between the Zone of Silence and the Zone of Action is not a single straight line but it two lines meeting at the present position of the source In addition the Zone of Action is now a more restricted region An observer watching the flight of the source does not hear any sound till he is within the Zone of Action This is a common experience when one watches a supersonic aircraft fly past The observer on ground first sees the aircraft but hears nothing He has to wait till the aircraft flies past him and immerses him in the Zone of Action But in case of a subsonic aircraft the observer always hears the sound These are again illustrated in Fig 22

The boundary between the two zones is called a Mach Wave and is a straight line when the source moves at the speed of sound and is a wedge when it

moves at supersonic speeds The half angle of the wedge is called the Mach Angle

(28)

Mach Angle is a function of the Mach Number of the flow and is in this sense a property of the flow It is an important one for supersonic flows and is listed in Tables in the Appendix


Compressible Flow

The animation shows the effect of propagation of the source of sound at different speeds

Response of Subsonic and Supersonic Flows to an Obstacle

We now consider an obstacle placed in subsonic and supersonic flows and study its effect upon the flow


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Compressible Flow

Figure 23 A Body placed in Subsonic and Supersonic Flows

Subsonic Flow

Consider a body placed in a subsonic stream As the flow interacts with the body several disturbances are created These propagate at the speed of sound The question is whether these disturbances can propagate upstream The answer is a YES Since the incoming flow is slower than sound these disturbances can propagate upstream As they propagate upstream they modify the incoming flow Consequently the flow adjusts itself to the presence of the body sufficiently upstream and flows past the body smoothly This is also what happens with incompressible flows where the speed of sound is infinite

Supersonic Flow

With a supersonic flow too disturbances are formed as a result of flow interacting with the body But now they cannot propagate upstream because the incoming flow is faster Any signal that tries to go upstream is pushed back towards the body These signals unable to go upstream are piled up closed to the body (Fig 23) The incoming flow is therefore not warned of the presence of the body It flows as if the body is absent and encounters the


Compressible Flow

region where the disturbances are piled up Then it suddenly modifies itself to accommodate the presence of the body This marks a sharp difference between subsonic and supersonic flows

The example indicates that in a supersonic flow disturbances cannot propagate upstream This technically stated as In a Supersonic flow there is no upstream influence Further the region where the disturbances have piled up is a Shock Wave These are regions of infinitesimally small thickness across which flow properties such as pressure density and temperature can jump orders of magnitude sometimes depending upon the Mach Number of the flow

Shock Waves

We had a small introduction to the formation of a Shock Wave in the previous section Now we consider shock waves in detail their formation and the equations that connect properties across a shock We restrict ourselves to one-dimensional flow now But in a later chapter we do consider two-dimensional flows

Formation of a Shock Wave

Consider a piston-cylinder arrangement as shown in Fig24 We have a gas at rest in front of the piston with pressure density and temperature given by

and Let the piston be given a jerk at time t = 0 The jerk disturbs the flow A weak wave is emitted (a in the figure) The wave moves to right

at a speed a ie the speed of sound The wave as it propagates sets the gas into motion Accordingly behind it we will have a medium which is slightly

compressed Its properties are given by and Now let the piston be given second jerk (b in the figure) One more is

generated Its speed however is not a but is a + da1 This waves has a higher speed because it is generated in a medium of higher temperature We

have the second wave chasing the other with a higher speed Will the second wave overtake the first one No what happens is that the second wave merges with the first one and becomes a stronger wave The pressure jump across the stronger wave is not dp1 but is dp1+dp2 This phenomenon where

the waves merge is called Coalescence


Compressible Flow

Figure 24 Formation of a Shock Wave

Now imagine the situation where the piston is a series of jerks (abc and d in Fig24 ) or the piston is pushed continuously We will have a train of waves where each wave is stronger and faster than the one before Very soon we sill see these waves coalesce into a strong wave with pressure and temperature jumping across it This is a Shock Wave


Compressible Flow

Waves in piston-cylinder arrangement

Shock Formation - Animation

Normal Shock Waves


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Compressible Flow

We have seen the formation of shock waves in two different situations- (1) when a supersonic flow negotiates an obstacle and (2) When a piston is pushed inside a cylinder Now we analyse a shock and see how flow properties change across it For this we choose shocks which are stationary and normal to flow and called Normal Shock Waves As stated before these are very thin surfaces across which properties can jump by big magnitudes A normal shock in a flow with a Mach number of 3 will have a pressure which jumps 10333 times across it This jump occurs in a distance of about 10-6 cm The shock thickness is so small that for all practical purposes it is a discontinuity Of course in reality a shock is not a discontinuity many interesting processes of heat transfer and dissipation do take place in this narrow thickness But in practice one is interested only in the changes in flow properties that occur across a shock wave It is for these changes that we derive expressions here

Figure 25 Shock Wave

Consider a shock wave as shown across which pressure density temperature velocity jump from to Let us

put a control volume around it For no change in area the governing equations are

(29)

(210)

(211)

Noting that the term in the momentum equation is


Compressible Flow

(212)

The momentum equation now becomes

(213)

Energy equation written above is a statement of the fact that total enthalpy is constant across the shock Accordingly we have

h01 = h02

which for a perfect gas becomes

T01 = T 02 (214)

Substituting from Eqn142 we have

(215)

We have just derived expressions for temperature (Eqn215) and pressure (Eqn213) ratios Density relation follows from the perfect gas equation

(216)

Substituting for as the continuity equation becomes

(217)

Substituting for pressure ratio from Eqn213 we have


Compressible Flow

(218)

Substituting for temperature ratio from Eqn215 we have

(219)

This is the equation which connects Mach Numbers across a normal shock We see that Mach number downstream of the shock is a function of

Mach Number upstream of the shock and Equation 219 has two solutions given by

(220)

(221)

We rule out the imaginary solutions for Eqns220 and 221 One of the possible solutions M1 = M2 is trivial telling that there is no shock

Now that we have a relation that connects M2 with M1 (Eqn221) we can write down the relationships that connect pressure density and other

variables across the shock This is done by substituting for M2 in the equations derived before namely 213 215 and 216 The final form of the

equations are

(222)

(223)


Compressible Flow

(224)

(225)

Considering now the Total properties we have

(226)

(227)

Change in entropy across the shock is given by

(228)

and

(229)

which in terms of M1 alone becomes

(230)

Tables in the Appendix have the values of the ratios of pressure density temperature Mach Numbers etc tabulated for different inlet Mach Numbers i


Compressible Flow

eM1 The aerodynamics calculator at httpwwwaoevteduaoe3114calchtml is very useful in this regard

Important Characteristics of a Normal Shock

We plot Eqn221 in Fig 26 to reveal an important property of normal shocks It is evident from the plot that -

If M1 gt 1 then M2lt 1 ie if the incoming flow is supersonic the outgoing flow is subsonic

If M1 lt 1 then M2gt 1 ie if the incoming flow is subsonic the outgoing flow is supersonic

It appears that both the solutions are mathematically possible Is it so physically This question has to be investigated from entropy considerations Accordingly we plot the entropy change across the shock s2 - s2 given by Eqn230 as a function of Mach Number in the same figure

Figure 26 Downstream Mach Number and Entropy rise across a shock

It comes out that if M1 lt 1 then we have a decrease in entropy across the shock which is a violation of the second law of thermodynamics and

therefore a physical impossibilityIn fact this solution gives a shock which tries to expand a flow and decrease pressure It is clear that expansion shocks are ruled out Further if M1 gt 1 there is an increase in entropy which is physically possible This is a compressive shock across which pressure

increases So the traffic rules for compressible flows are such that shocks are always compressive incoming flow is always supersonic and the outgoing flow is always subsonic


Compressible Flow

Figure 27 Traffic Rules for Compressible Flow

Flow through Nozzles and Ducts

We now consider application of the theory of compressible flows that we have developed so far What happens when we have a gas flow through area changes - is the first question we ask

We have noted already that a subsonic flow responds to area changes in the same manner as an incompressible flow A supersonic flow behaves in an opposite manner in that when there is an area decrease Mach Number decreases while for an area increase Mach Number increases We have also stated that a sonic flow can occur only at a throat a section where area is the minimum With this background we can explore the phenomena of gas flow through nozzles

Flow through a Converging Nozzle

Consider a converging nozzle connected to a reservoir where stagnation conditions prevailp = p0 T = T0 u = 0 By definition reservoirs are such that

no matter how much the fluid flows out of them the conditions in them do not change In other words pressure temperature density etc remain the same always

Pressure level pb at the exit of the nozzle is referred to as the Back Pressure and it is this pressure that determines the flow in the nozzle Let us now

study how the flow responds to changes in Back Pressure

When the Back Pressure pb is equal to the reservoir pressurep0 there is no flow in the nozzle This is condition (1) in Fig31 Let us reduce pb slightly

to p2(condition (2) in the Figure) Now a flow is induced in the nozzle For relatively high values of pb the flow is subsonic throughout A further

reduction in Back Pressure results in still a subsonic flow but of a higher Mach Number at the exit (condition (3)) Note that the mass flow rate increases As pbis reduced we have an increased Mach Number at the exit along with an increased mass flow rate How long can this go on At a

particular Back Pressure value the flow reaches sonic conditions (4) This value of Back pressure follows from Eqn147 For air it is given by


Compressible Flow

(31)

What happens when the Back Pressure is further reduced (56 etc) is interesting Now the Mach Number at the exit tries to increase It demands an increased mass flow from the reservoir But as the condition at the exit is sonic signals do not propagate upstream The Reservoir is unaware of the conditions downstream and it does not send any more mass flow Consequently the flow pattern remains unchanged in the nozzle Any adjustment to the Back Pressure takes place outside of the nozzle The nozzle is now said to be choked The mass flow rate through the nozzle has reached its maximum possible value choked value From the Fig 31 we see that there is an increase in mass flow rate only till choking condition (4) is reached Thereafter mass flow rate remains constant

Figure 31 Flow through a Converging Nozzle

It is to be noted that for a non-choked flow the Back Pressure and the pressure at the exit plane are equal No special adjustment is necessary on the part of the flow But when the nozzle is choked the two are different The flow need to adjust Usually this take place by means of expansion waves which help to reduce the pressure further


Compressible Flow

Flow through a Converging-diverging nozzle

A converging-diverging nozzle is an important tool in aerodynamics Also called a de Laval nozzle it is an essential element of a supersonic wind tunnel In this application the nozzle draws air from a reservoir which is at atmospheric conditions or contains compressed air Back pressure at the end of the diverging section is such that air reaches sonic conditions at throat This flow is then led through the diverging section As we have seen before the flow Mach Number increases in this section Area ratio and the back pressure are such that required Mach Number is obtained at the end of the diverging section where the test section is located Different area ratios give different Mach Numbers

We study here the effect of Back Pressure on the flow through a given converging-diverging nozzle The flow is somewhat more complicated than that for a converging nozzle Flow configurations for various back pressures and the corresponding pressure and Mach Number distributions are given in Fig 32 Let us discuss now the events for various back pressures abc

(a) Back Pressure is equal to the reservoir pressure pb= p0 There is no flow through the nozzle

(b) Back Pressure slightly reducedpblt p0 A flow is initiated in the nozzle but the condition at throat is still subsonic The flow is subsonic and

isentropic through out

(c) The Back Pressure is reduced sufficiently to make the flow reach sonic conditions at the throatpb = pc The flow in the diverging section is

still subsonic as the back pressure is still high The nozzle has reached choking conditions As the Back Pressure is further reduced flow in the converging section remains unchanged

We now change the order deliberately to facilitate an easy understanding of the figure 32

(i) We can now think of a back Pressure pb = pi which is small enough to render the flow in the diverging section supersonic For this Back

Pressure the flow is everywhere isentropic and shock-free

(d) When the Back Pressure is pd the flow follows the supersonic path But the Back Pressure is higher than pi Consequently the flow meets

the Back Pressure through a shock in the diverging section The location and strength of the shock depends upon the Back Pressure Decreasing the Back Pressure moves the shock downstream


Compressible Flow

Figure 32 Pressure and Mach Number Distribution for the Flow through a Converging-Diverging Nozzle

(e) One can think of a Back Pressure pf when the shock formed is found at the exit plane pf p0 is the smallest pressure ratio required for the

operation of this nozzle

(f) A further reduction in Back Pressure results in the shocks being formed outside of the nozzle These are not Normal Shocks They are Oblique Shocks Implication is that the flow has reduced the pressure to low values Additional shocks are required to compress the flow further Such a nozzle is termed Overexpanded

(g) The other interesting situation is where the Back Pressure is less than pi Even now the flow adjustment takes place outside of the nozzle not

through shocks but through Expansion Waves Here the implication is that the flow could not expand to reach the back Pressure It required further expansion to finish the job Such a nozzle is termed Underexpanded


Compressible Flow

This is an annimation to illustrate the flow through a converging-diverging nozzle

Two-Dimensional Compressible Flow

Hither too we have only been considering one-dimensional flows While it is difficult to find a true one-dimensional flow in nature it helps to build various concepts Using these concepts we now try to understand some of the typical examples of two-dimensional flows in particular supersonic flows Main features of interest here are

Oblique Shock Waves Prandtl-Meyer Expansion Waves Shock Interactions and Detached Shocks Shock-Expansion Technique Thin Aerofoil Theory Method of Characteristics

Oblique Shock Waves

We have discussed Normal Shocks which occur in one-dimensional flows Although one can come across Normal Shocks in ducts and pipes most of the times we encounter only Oblique Shocks the ones that are not normal to the flow - shocks formed at the nose of wedge when a supersonic flow flows past the shock in front of a body in a supersonic flow These have been sketched in Fig41 How do we analyse such these shocks


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Compressible Flow

Figure 41 Examples of Oblique Shock

Figure 42 Velocity Components for an Oblique Shock

Consider an oblique shock as shown in Fig42 Looking at the velocity components and comparing with that for a normal shock it is clear that we now have an additional one ie a tangential componentv Accounting for this is not a major problem The shock mechanism is such that this component is unchanged across the oblique shock However the normal component u1 does undergoe a change as with a normal shock and comes out with a value u2

on passing through the shock We have seen with normal shocks that u2 lt u1 A close look at Fig42 reveals that the flow undergoes a turn as it passes

through an oblique shock and the turn is towards the shock

Relations across an Oblique Shock

The angle between the shock and the incoming flow is called the Shock Angle The angle through which the flow turns is termed Deflection Angle If M1 is the incoming mach number we have


Compressible Flow

Define

(41)

Oblique shock in Fig 42 can be viewed as a normal shock with an incoming Mach Number equal to but with a tangential (to the shock)

velocity componentv superposed everywhere Then it is a simple matter to calculate conditions across an oblique shock In the normal shock relations

Eqns 222 to 230 just replace Mach Number with Accordingly we have

(42)

It is to be noted that is the normal component of Mach Number downstream of the shock and is equal to

(43)

The other relations across the shock are given by

(44)

(45)


Compressible Flow

(46)

(47)

(48)

We have shown before that the upstream flow for a normal shock must be supersonic ie

(410)

For a given Mach Number M1 we have a minimum shock angle which is given by the maximum inclination is Accordingly the

condition we have for an oblique shock is that

(411)

The lower limit gives a Mach Wave which gives a zero flow turn The higher limit gives a normal shock which also gives a

zero flow turn but perturbs the flow strongly This gives the highest pressure jump across the shock

It may also be mentioned that the normal component of Mach Number downstream of the shock must be less than 1 ie must be

subsonic

Relation between and

A question that naturally arises is - for a given Mach Number M1 and a shock angle what is the flow deflection angle An expression for this can


Compressible Flow

derived as follows-

which gives

leading to

(412)

It is easily seen that has two zeros one at and the other at These correspond to the two limits we have

on the shock angle ( Eqn 411 Having two zeros it is evident that the expression should have a maximum somewhere in between Fig 43 shows

the relationship between and plotted for various Mach Numbers For a given value of we see that there are two values of indicating that two

shock angles are possible for a given flow turning and an upstream Mach Number For a given Mach Number there is a maximum flow turning


Compressible Flow

Figure 43 Relationship between and

Clearly for there are two solutions ie two values of The smaller value gives what is called a Weak Solution The other solution with

a higher value of is called a Strong Solution

Figure also shows the locus of solutions for which M2 = 1 It is clearly seen that a strong solution gives rise to a subsonic flow downstream of it

ie M2 The weak solution gives a supersonic flow downstream of it except in a narrow band with slightly smaller than

Conditions across an oblique shock can be found in a table in the Downloadable Information and Data Sheets The aerodynamics calculator at httpwwwaoevteduaoe3114calchtml is very useful in this regard

Supersonic Flow past Concave Corners and Wedges


Compressible Flow

We consider a few weak solutions for the oblique shocks now Strong solutions are considered later The examples we discuss are the flow past a concave corner and the flow past a wedge

Figure 44 Supersonic Flow past Concave Corners and Wedges

Let a supersonic flow at Mach Number M1 flow past a concave corner inclined at an angle to the incoming flow (Fig44) At the corner the

flow has to turn though an angle because of the requirement that the normal component of velocity at any solid surface has to be zero for an inviscid

flow To facilitate this turn we require an oblique shock at an angle to form at the corner We can calculate the shock angle for a given Mach Number

and angle The same theory can be applied in case of a symmetric or asymmetric wedge of half angle as shown in Fig44

In these cases the shocks are formed at the corner or the nose of the body They are called Attached Shocks Recalling that with supersonic flows we have limited upstream influence we can see that flow on the lower surface of the wedge is independent of the flow on the upper surface


Compressible Flow

Weak Oblique Shocks

For small deflection angles it is possible to reduce the oblique shock relations to simple expressions Writing out only the end results we have

(413)

The above expression indicates that the strength of the shock denoted by the term is proportional to the deflection angle It can be shown

that the change of entropy across the shock is proportional to the cube of the deflection angle

(414)

Other useful expression for a weak oblique shock is for the change of speed across it


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Compressible Flow

(415)

Supersonic Compression by Turning

Indirectly we have come across a method to compress a supersonic flow If the flow is turned around a concave corner an oblique shock is produced As the flow passes through the shock its pressure increases ie the flow is compressed This lends itself to a simple method of compression But the question is how to perform it efficiently

Consider first a compression through a single oblique shock (Fig45) The flow turns through an angle We have seen that

(416)

There is a price to be paid for this compression Entropy increases across the shock (which is the same as saying that total pressure decreases)

proportional to


Compressible Flow

Figure 45 Compression of a Supersonic Flow by Turning

Second consider compressing the flow through a number of shocks say n Let each shock be such that the flow downstream is supersonic The flow

turning through each of the shocks is Consequently through n shock we have

Total turning

Change in pressure

Entropy rise (417)


Compressible Flow

Thus we see that this arrangement gives us the same compression as before but the entropy rise is reduced enormously implying that losses are controlled

Now the third possibility suggests itself Why not make n tend to That is compression is effected by an infinite number of waves ie Mach Waves Instead of a concave corner we now have a smooth concave surface (Case c in Fig 45) Now the compression is effected as before but the losses are zero because

(418)

The entire process is thus isentropic and the most efficient

Convergence of Mach Waves

Figure 46 Coalescence of Mach Waves to form an Oblique Shock

Figure 47 Isentropic compression and expansion of a flow

Consider the compression around a concave surface which takes place through a series of Mach Waves As the flow passes through each Mach Wave


Compressible Flow

pressure rises and the Mach Number decreases Consequently the wave angle increases This leads to a convergence of Mach Waves far from the

concave surface as shown in Fig 46 This phenomenon is called Coalescence The waves merge and become an oblique shock When this happens the flow is no longer isentropic severe non-linearities build up If one wishes to have an isentropic compression it is necessary to see that waves do not converge This can be brought about by placing a wall forming a duct as shown in Fig47 It is interesting that the flow compresses itself as it moves from left to right and expands when it is driven in the opposite direction

Prandtl-Meyer Expansion

Now we consider expansion of a flow Obviously a supersonic flow negotiating a convex corner undergoes expansion First question that arises is Can we expand a gas through a shock This means we will send a supersonic flow through a shock and expect an increase in Mach Number and a decrease in pressure as sketched in Fig48 This demands that u2 gt u1 That is we are expecting the flow velocity normal to the shock to increase as it passes

through the shock This we have seen before violates the second law of thermodynamics Expansion shocks are a physical impossibility

Figure 48 Flow expansion through a shock

Actually expansion of a flow takes place isentropically through a series of Mach Waves The waves may be centered as happens at a convex corner or spread out as in the case of a convex surface The Mach Waves are divergent in both the cases A centered wave is called a Prandtl-Meyer Expansion fan


Compressible Flow

Figure 49 Prandtl-Meyer Expansion

Consider a Prandtl-Meyer fan (Fig 49) through which a flow expands from a Mach Number M1 to M2 The leading wave is inclined to the flow at an

angle and the expansion terminates in a wave inclined at an angle We can derive an expression

connecting the flow turning angle and the change in Mach Number as follows

Consider a differential element within the fan (Fig 410) through which the Mach Number changes from M to M + dM as the flow turns through an angle


Compressible Flow

Figure 410 Prandtl-Meyer Expansion continued

For a Mach Wave we have from Eqn 415

(419)

On integrating it gives

(420)

where we have introduced a function called the Prandtl-Meyer function

Starting from

(421)


Compressible Flow

one can deduce that

(422)

Prandtl-Meyer function is very significant in calculating supersonic flows Note that for For every Mach Number greater than one

there is a unique Prandtl-Meyer function Tables for supersonic flow in Appendix do list as a function of Mach Number In fact is the angle through

which a sonic flow should be turned in order to reach a Mach Number of M In addition consider a flow turning through an angle We have

(423)

With the knowledge of one can calculate M2 (or read it off the Table) One can also calculate the Mach Number following an isentropic compression

using Prandtl-Meyer function -

(424)

Figure 411 Using Prandtl-Meyer Function

It is to be noted that (Mach Number) decreases in compression and it increases in expansion See Fig411

Shock Interactions and Detached Shocks


Compressible Flow

Many interesting situations arise concerning oblique shocks These include reflection of shocks from solid walls intersection of shocks etc

Shock Reflection from a Wall

Consider an oblique shock impinging on wall at an angle as shown in Fig412 This oblique shock could be the result of a wedge being placed in a

supersonic flow at a Mach Number M1 The flow is deflected through an angle But the presence of the wall below pulls the flow back and renders it to

be parallel to itself This requires another shock inclined to the wall at an angle One other way of explaining the phenomenon is that the shock is

incident on the wall at an angle and gets reflected at an angle A question to ask is whether the angle of incidence be equal to angle of reflection

The answer is a NO It is true that both the shocks turn the flow by the same amount in different directions But the Mach Numbers are different Incident shock is generated in a flow where the Mach number is M1 while the reflected shock is generated where the Mach Number is M2

Figure 412 Shock reflection from a wall

The resulting pressure distribution is also given in the figure and can be calculated easily from the tables

Intersection of two Shocks


Compressible Flow

Intersection of two shocks occurs when they hit each other at an angle as shown in Fig413 Let us first consider shocks of equal strength The interaction takes place as if each of the shocks is reflected of the centreline of the flow which is in fact a streamline We can always treat a streamline as a wall so that the calculation procedure is identical to that for a single shock reflection

Figure 413 Intersection of two symmetric Shocks


Compressible Flow

Figure 414 Intersection of two asymmetric Shocks

When the shocks are of unequal strengths interact the flow field loses symmetry A new feature appears downstream of the interaction Fig 414 This is what is called a Slip Stream This divides the flow into two parts - (1) flow which has been processed by shocks on top and (2) flow processed by the

shocks at the bottom But the slip stream is such that the pressure p3 and the flow angle ( ) are continuous across it Density temperature and other

properties are different It requires an iteration to solve for the pressure distribution and other features in this case

Strong Solutions - Detached Shocks

Consider the case of a flow which is flowing past a body whose nose is such that (with reference to Fig43 Question is how is this geometry

dealt with by the flow What happens in such a case is the shock is not attached to the nose but stands away from it - ie is Detached as shown in Fig 415 The shock is no longer a straight line but is curved whose shape and strength depend upon M1 and the geometry of the body On the centreline the

shock is a normal shock at a As we move away from the centreline the shock weakens and approaches a Mach Wave at d From a to d one sees the entire range of solutions given by Fig43


Compressible Flow

Figure 415 Detached Shock in front of a Wedge

The flow field downstream of the shock is somewhat complicated Recall that the flow downstream of a normal shock is subsonic Accordingly we have a subsonic patch of flow near the centreline downstream of the shock The extent of this region depends upon the body geometry and the freestream Mach Number As we move away from the centreline the shock corresponds to a weak solution with a supersonic flow behind it A sonic line separates the supersonic flow from the subsonic patch

If the body is blunt as shown in Fig416 the shock wave is detached at all Mach Numbers

Figure 416 Detached Shock in front of a Blunt Body

The distance between the body and the shock is called Shock Stand Off Distance and it decreases with Mach number It is possible to compute this distance using CFD techniques or measure it experimentally


Compressible Flow

Mach Reflection

Sometimes the reflection of shock at a solid surface is not as simple as indicated before It may so happen that the Mach Number downstream M2 is

such that a simple reflection is not possible In these circumstances reflection of the shock does not take place at the solid wall but a distance away from it As shown in Fig417 We now have a triple point in the flow followed by a slip stream This phenomenon is called Mach Reflection

Figure 417 Mach Reflection

Shock-Expansion Technique

One could think of a general two-dimensional supersonic flow to be a combination of uniform flow shocks and expansion waves We have developed tools to handle each one of these in the preceding sections A technique to calculate such a flow reveals itself and is called Shock-Expansion Technique In plain words it can be described as follows Employ shock relations where there is a shock and Prandtl-Meyer expansion relations where there is an expansion

We now consider typical examples are flow past corners and aerofoils

Flat Plate Aerofoil

Figure 418 Flat Plate Aerofoil at zero angle of attack


Compressible Flow

Consider a thin flat plate placed in a supersonic stream as shown in Fig418 For a zero angle of attack there is no flow turning anywhere on the flat plate Consequently the pressure is uniform on the suction and the pressure surfaces Drag and lift are both zero Now consider the flow about the flat

plate at an angle of attack equal to

Figure 419 Flat Plate Aerofoil at an angle of attack

Interesting features are produced on the plate as shown in Fig 419 The flows sees the leading edge on the suction surface as a convex corner A Prandtl-Meyer expansion results At the trailing edge the flow compresses itself through a shock At the leading edge on the pressure surface is a shock since it forms a concave cornerThe flow leaves the trailing edge through an expansion fan A close look at the flow past the trailing edge shows that there are two streams of gas - one processed by expansion and shock on the suction surface and two gas processed by similar features on the pressure side The two shocks are not of the same strength Consequently the gas streams are of different densities and temperatures However the pressures and flow angles are equalised at the trailing edge This gives rise to a slip stream at the trailing edge From the pressure distribution shown lift and drag can be calculated as

(425)


Compressible Flow

where c is the chord Note that this drag is not produced by viscosity as with incompressible or subsonic flows It is brought about by the waves (shock and expansion) which are unique to supersonic flows This is an example of Supersonic Wave Drag

It is desirable to express drag and lift as drag and lift coefficients CL and CD These are obtained by non-dimensionalising the corresponding forces with

the term where A is the area over which lift or drag force acts This can be shown to be equal to

Consequently it can be shown that

(426)

where Cpl and Cpu are the pressure coefficients on lower and upper surfaces

Diamond Aerofoil

Figure 420 Flow about a Diamond Aerofoil


Compressible Flow

Consider a typical aerofoil for a supersonic flow ie a Diamond Aerofoil as shown in Fig420 A flow at zero angle of attack produces the features as shown At the leading edge we have a shock each on the pressure and suction sides Then at maximum thickness we have expansion waves The flow leaves the trailing edge through another shock system

The flow is symmetrical in the flow normal direction and lift is zero But there is a drag which is given by

(427)

It is possible to generalise the aerofoil and develop a formula for drag and lift Consider an aerofoil with a half wedge angle of Let be the

orientation of any side of the aerofoil The pressures on each of the sides can now be summed to determine lift and drag coefficients as follows -

(428)

In terms of Cpfor each side we have

(429)

Similarly for drag we have

(430)


(431)

Interaction between shocks and expansion waves

In the case of diamond aerofoil considered above interactions can take place between shocks and expansion In general these have insignificant effect on the flow Still for an accurate analysis the interactions should be considered But this is beyond the scope of an introductory textbook as the present one The effect of interaction is in general to attenuate the shock The flow configuration is given in Fig421


Compressible Flow

Figure 421 Interaction between Expansion Waves and Shock

Thin Aerofoil Theory

The shock-Expansion technique we developed is accurate and also simple However it is a numerical device requiring considerable book keeping This is no problem today that we have computers that can handle this efficiently But in the past people were looking for methods which gave a closed form solution One such is the Thin Aerofoil Theory Now we consider aerofoils that are thin and angles of attack small such that the flow is deflected only slightly from the freestream direction Consequently the shocks belong to the weak shock category ( See Weak Oblique Shocks ) Now the pressure change anywhere in the flow is given by

(432)


Compressible Flow

As per our assumption pressure p is not far from and the local Mach Number on the aerofoil is not far from making the above equation

reduce to

(433)

Referring all pressures to and flow direction to that of the freestream we have

(434)

This gives

(435)

(436)

(437)

Thus we have a simple expression for calculating Cp on any surface in the flow say an aerofoil The interesting feature is that Cp depends upon the local

flow inclination alone What feature caused that flow turning is of no consequence We can now re-look at the examples we considered before

Flow about a Flat Plate Aerofoil at an Angle of Attack

Consider the Flat Plate Aerofoil previously treated in Section The flow is inclined at an angle on both the surfaces Accordingly

(438)

The lift and drag coefficients are given by


Compressible Flow

(439)

Substituting for Cp and noting that for small we have

(440)

Diamond Aerofoil

For the aerofoil we have for the flow behind the shock

(441)

For the flow behind the expansion waves

(442)

While using Eqn 437 a positive sign is used for compression and a negative one for expansion The drag coefficient is given by

(443)

which can be written as

(444)

An Arbitrary Aerofoil


Compressible Flow

Consider a general aerofoil placed in a supersonic flow as in Fig422 The aerofoil can be thought of having a thickness h(x) an angle of attack and

a camber One can show that for this aerofoil

Figure 422 Flow about an Arbitrary Aerofoil

(445)

Second Order Theory

The approximate theory we have developed is of first order in that it retains only the first significant term involving in an expansion for Cp

Busemann has provided a second order theory which includes terms as well As per this theory

(446)

which is also written as

(447)

even while using this equation a positive sign for compression and a negative one for expansion is used

Note that the coefficients C1 and C2 are functions of Mach Number and only These are also listed in Tables in appendix


Compressible Flow

Thus we have three methods to calculate pressure in a turning supersonic flow Of these Shock-Expansion technique is the most accurate The remaining are for small flow turnings only The Busemanns method may provide better answers for small flow turnings

Reduction of Drag by cancelling the Waves

It is clear that in supersonic flows waves are the main sources of drag An idea suggests itself that we can reduce drag by removing the waves from the system What do we mean by this Let us take the example of a shock impinging on a solid wall We have seen that this produces an incident shock and a reflected shock The latter one comes about in order to turn the flow to be parallel to the wall Suppose we turn the wall itself at the point o through

an angle in the other direction as shown in Fig423 Then the flow follows the wall and there is no need for a reflected shock This phenomenon can also be interpreted as saying that an expansion wave is produced at 0 that cancels the reflected shock Now the system is free of waves and so free of wave drag

Figure 423 Cancellation of Waves

A clever device built based on the idea of wave cancellation is Busemann Biplane (Fig 424) The geometry and incoming Mach Number are so arranged that a perfectly symmetrical system of shocks is produced and at the exit there are no waves whatever This gives a zero wave drag If the Busemann plane is run under off-design conditions as in Fig425 the exit flow is not wave-free There is a resulting wave drag


Compressible Flow

Figure 424 Busemann Biplane under design conditions

Figure 425 Busemann Biplane under off-design conditions

Method of Characteristics

Method of Characteristics is a very convenient tool to calculate isentropic portions of supersonic flows This is a numerical method but the merit is that the method itself determines the grid (or mesh) it requires Researchers and others today seem to prefer a Finite-Volume method to compute supersonic or any other flow But there are a few who still prefer the Method of characteristics notably the ones that design supersonic nozzles There have also been efforts to extend the method to accommodate shocks by patching solutions across them These have seen only a limited success It is significant that even those that employ Finite-Volume Methods depend on Method of Characteristics to provide the boundary conditions

There is an elaborate mathematical theory behind the Method of Characteristics But we restrict ourselves to the application However we do bring out the essential features of the theory

Theory of Method of Characteristics

Governing Equations for a two-dimensional compressible irrotational flow can be written as

(448)

(449)

It is easy to realise that Eqn449 is the irrotationality condition Equation 448 is a non-linear Partial Differential Equation It is classified as follows


Compressible Flow

Elliptic if (u2 + v2)a2 lt 1 Parabolic if (u2 + v2)a2 = 1 Hyperbolic if (u2 + v2)a2 gt 1

We see that supersonic flows with M gt 1 belong to the Hyperbolic class One of the properties of Hyperbolic Equations is that there exist what are called the characteristic lines or directions Recall that in a supersonic flow at every point there are what are called Mach Waves These are in fact the characteristic lines Direction of Mach lines is the characteristic direction Across a characteristic line velocity derivatives may be discontinuous but velocity itself will be discontinuous Along the characteristic lines what are called the Compatibility Relations hold good

Compatibility Relations

Figure 426 Riemann Invariants

Consider a stream line in a supersonic flow as in Fig 426 We can have one coordinate s axis aligned along the streamline and the other n normal to it

Now consider the Mach lines at a point P There are two of them- the one to the left of the streamline is a characteristic and the one to the right is

called a characteristic Note that each of these is inclined at an angle to the streamline It can be shown that along a characteristic

ie a constant (450)

Similarly along a characteristic we have


Compressible Flow

ie a constant (451)

Equations 450 and 451 are the Compatibility Relations Essentially they say that and are invariant in and directions respectively These

are known as Riemann Invariants and are in a simple form because of the simple situation we have considered In complex situations Riemann Invariants could even be differential equations

Computing with Method of Characteristics

Working with the Method of Characteristics is made easy if we formulate the problem in terms of and Once we determine these two at any point in the flow other quantities of interest such as Mach Number Flow Velocity and Pressure can be determined using isentropic relations the energy equation

etc Accordingly consider a curve AB in the flow along which and are known This curve is known as a Starting Curve The working for the method should be clear from Fig 427

Now we have

Solving for C and Cwe have

Or (452)


Compressible Flow

Figure 427 Computing using Method of Characteristics

Now that we know the flow at C it is possible to continue and calculate the flow downstream

In practice a number of points on the starting curve are considered Mach Lines are drawn from each of them Then we march downstream calculating the flow at every point we arrive at In this process we create a net or a grid of points as shown in Fig 428

Figure 428 The Method of Characteristics Procedure


Compressible Flow

One should be aware of the accuracy of the procedure Note that we approximate the characteristics by straight lines For example consider Fig 429 We have treated the characteristic at 4 or 2 to be a straight line This is true only if Mach Number is constant between 4 and 7 or between 2 and 7 If the Mach Number varies between these points then we have curves which intersect at 7 instead of 7 Therefore what we calculate as properties for point 7 are actually those at point 7 However it is easy to realise that error due to this approximation may be minimised by bringing 2 and 4 closer In other words we need to have a large number of points on the Starting Curve

Figure 429 Accuracy of the Procedure

The procedure needs to be modified near a solid boundary and a free boundary At a solid boundary one knows the flow inclination ie while at a

free boundary one knows These will be clear in the worked example given below

Flow through a Diverging Duct

Consider a flow at Mach 1605 entering a two-dimensional diverging duct whose sides make an angle of 60 with the centreline ie a 120 divergence It is required to calculate the flow in the duct See Fig 430


Compressible Flow

Figure 430 Flow through a Diverging Duct

Let us have four points on the starting line - abc and d At each of these points Mach Number is the same 1605or The flow will be

horizontal on the centreline and will be aligned with the wall at the two side boundaries Accordingly will be 6020-20 and -60 at abc and d respectively First let us tabulate the known values

Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow

From Tables this refers to a Mach Number of 1672

Boundary Point

Consider the boundary point h

Mach Number corresponding to =190 is 1741

Cancellation of Waves

In a manner to similar to Bausemanns biplane we need to cancel waves (as well as shock waves) This happens in situations such as a supersonic nozzle where we need a test section flow free of all waves But we should not forget that the very waves expand the flow to the desired conditions It becomes necessary to cancel all these waves The principle behind the cancellation is the same as we have seen before


Compressible Flow

Figure 431 Reflection and cancellation of Waves

Consider the wave reflection from a wall as shown in Fig431 By suitably turning the wall the wave can be cancelled This also applies to a series of waves

Design of a Supersonic Nozzle

Supersonic nozzle is he basic element of any supersonic wind tunnel and is the one that accelerates the flow at rest in the reservoir to the required Mach Number MT at the test section It is a converging-diverging nozzle as shown in Fig432 As discussed before the converging section of the nozzle is

provided to obtain sonic conditions at the throat The section of this section is somewhat arbitrary

Figure 432 Design of a Supersonic Nozzle only upper half shown


Compressible Flow

The design of a supersonic nozzle means the design of the diverging section The conditions at the throat are sonic ie while at the section we

need a Mach Number or a value of It is required that the flow be turned through an angle equal to In theory it is possible to effect

this in one go meaning turn the flow once through But this may not be an efficient design A sudden turn could result in separation of flow

Further it is required that the flow entering the test section be uniform implying the absence of waves This is where one needs to take care

The flow expansion in the diverging section is carried out in two stages - (a) Expansion Section and (b) Cancellation Section In the Expansion Section

flow is expanded up to a In the subsequent Straightening Section is reduced progressively to cancel the waves When complete on

the walls and at the centreline Each of these operations could be carried out in a number of steps uniform or otherwise

Fig432 is self explanatory

Depending upon the application it may be necessary to minimise the length of the nozzle Then it is usual to keep the Expansion Section of zero length and have a Prandtl-Meyer expansion at the corner as shown in Fig 433

Figure 433 Design of a Minimum Length Supersonic Nozzle only upper half shown

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Compressible Flow

Compressible Flow

Figure 12 Density change as a function of Mach Number

We observe that for Mach numbers up to 03 density changes are within about 5 of So for all practical purposes one can ignore density changes

in this region But as the Mach Number increases beyond 03 changes do become appreciable and at a Mach Number of 1 it is 365 it is interesting to note that at a Mach Number of 2 the density changes are as high as 77 It follows that air flow can be considered incompressible for Mach Numbers below 03

Another important difference between incompressible and compressible flows is due to temperature changes For an incompressible flow temperature is generally constant But in a compressible flow one will see a significant change in temperature and an exchange between the modes of energy Consider a flow at a Mach Number of 2 It has two important modes of energy-Kinetic and Internal At this Mach Number these are of magnitudes 23 x 105 Joules and 2 x 105 Joules You will recognise that these are of the same order of magnitude This is in sharp contrast to incompressible flows where only the kinetic energy is important In addition when the Mach 2 flow is brought to rest as happens at a stagnation point all the kinetic energy gets converted into internal energy according to the principle of conservation of energy Consequently the temperature increases at the stagnation point When the flow Mach number is 2 at a temperature of 200C the stagnation temperature is as high as 260 0C as indicated in Fig 13


Compressible Flow













Compressible Flow






(11)





Compressible Flow






du = dq - dw (12)


(13)


dh = dq + vdp (14)


Compressible Flow








Compressible Flow


(15)


(16)


(17)



Perfect Gas Law


(18)


(19)




Compressible Flow

Jkg K Jkg K cpcv

Air 2897 2870 1004 14

Ammonia 1703 4882 2092 13

Argon 3994 2081 519 167


Helium 4003 2077 5200 167

Hydrogen 2016 4124 14350 14

Oxygen 32 2598 916 14



u = u(T) h = h(T) (110)



(111)


(112)



Compressible Flow

(113)





(114)


(115)


(116)




Compressible Flow





Continuity Equation


(117)


(118)


Momentum Equation


Compressible Flow


(119)


(120)


Energy Equation




p1v1 - p2v2 (122)




(124)



Compressible Flow


(125)



= constant (126)


(127)


(128)


(129)





Compressible Flow




Rewrite Eqn126 as

(130)


(131)






(132)


(133)



Compressible Flow

(134)

Since we have

(135)







(136)



Compressible Flow

(137)


(138)



Compressible Flow








(139)


(140)


(141)


(142)


Compressible Flow



(143)

(144)




(145)


(146)



Compressible Flow

(147)


Mass Flow Rate





(148)



Compressible Flow






Compressible Flow

(149)

Wave Propagation


Press RED to start

Speed of Sound


>

Compressible Flow




respectively




(21)


(22)


Compressible Flow




(24)


(25)


(26)


(27)




Compressible Flow


Stationary Source






Compressible Flow







(28)



Compressible Flow





>

Compressible Flow


Subsonic Flow


Supersonic Flow



Compressible Flow



Shock Waves











Compressible Flow




Compressible Flow



Normal Shock Waves


>
>

Compressible Flow





(29)

(210)

(211)



Compressible Flow

(212)


(213)


h01 = h02


T01 = T 02 (214)


(215)


(216)


(217)



Compressible Flow

(218)


(219)



(220)

(221)




equations are

(222)

(223)


Compressible Flow

(224)

(225)


(226)

(227)


(228)

and

(229)


(230)



Compressible Flow












Compressible Flow















Compressible Flow

(31)





Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow













Compressible Flow






(11)





Compressible Flow






du = dq - dw (12)


(13)


dh = dq + vdp (14)


Compressible Flow








Compressible Flow


(15)


(16)


(17)



Perfect Gas Law


(18)


(19)




Compressible Flow

Jkg K Jkg K cpcv

Air 2897 2870 1004 14

Ammonia 1703 4882 2092 13

Argon 3994 2081 519 167


Helium 4003 2077 5200 167

Hydrogen 2016 4124 14350 14

Oxygen 32 2598 916 14



u = u(T) h = h(T) (110)



(111)


(112)



Compressible Flow

(113)





(114)


(115)


(116)




Compressible Flow





Continuity Equation


(117)


(118)


Momentum Equation


Compressible Flow


(119)


(120)


Energy Equation




p1v1 - p2v2 (122)




(124)



Compressible Flow


(125)



= constant (126)


(127)


(128)


(129)





Compressible Flow




Rewrite Eqn126 as

(130)


(131)






(132)


(133)



Compressible Flow

(134)

Since we have

(135)







(136)



Compressible Flow

(137)


(138)



Compressible Flow








(139)


(140)


(141)


(142)


Compressible Flow



(143)

(144)




(145)


(146)



Compressible Flow

(147)


Mass Flow Rate





(148)



Compressible Flow






Compressible Flow

(149)

Wave Propagation


Press RED to start

Speed of Sound


>

Compressible Flow




respectively




(21)


(22)


Compressible Flow




(24)


(25)


(26)


(27)




Compressible Flow


Stationary Source






Compressible Flow







(28)



Compressible Flow





>

Compressible Flow


Subsonic Flow


Supersonic Flow



Compressible Flow



Shock Waves











Compressible Flow




Compressible Flow



Normal Shock Waves


>
>

Compressible Flow





(29)

(210)

(211)



Compressible Flow

(212)


(213)


h01 = h02


T01 = T 02 (214)


(215)


(216)


(217)



Compressible Flow

(218)


(219)



(220)

(221)




equations are

(222)

(223)


Compressible Flow

(224)

(225)


(226)

(227)


(228)

and

(229)


(230)



Compressible Flow












Compressible Flow















Compressible Flow

(31)





Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow






(11)





Compressible Flow






du = dq - dw (12)


(13)


dh = dq + vdp (14)


Compressible Flow








Compressible Flow


(15)


(16)


(17)



Perfect Gas Law


(18)


(19)




Compressible Flow

Jkg K Jkg K cpcv

Air 2897 2870 1004 14

Ammonia 1703 4882 2092 13

Argon 3994 2081 519 167


Helium 4003 2077 5200 167

Hydrogen 2016 4124 14350 14

Oxygen 32 2598 916 14



u = u(T) h = h(T) (110)



(111)


(112)



Compressible Flow

(113)





(114)


(115)


(116)




Compressible Flow





Continuity Equation


(117)


(118)


Momentum Equation


Compressible Flow


(119)


(120)


Energy Equation




p1v1 - p2v2 (122)




(124)



Compressible Flow


(125)



= constant (126)


(127)


(128)


(129)





Compressible Flow




Rewrite Eqn126 as

(130)


(131)






(132)


(133)



Compressible Flow

(134)

Since we have

(135)







(136)



Compressible Flow

(137)


(138)



Compressible Flow








(139)


(140)


(141)


(142)


Compressible Flow



(143)

(144)




(145)


(146)



Compressible Flow

(147)


Mass Flow Rate





(148)



Compressible Flow






Compressible Flow

(149)

Wave Propagation


Press RED to start

Speed of Sound


>

Compressible Flow




respectively




(21)


(22)


Compressible Flow




(24)


(25)


(26)


(27)




Compressible Flow


Stationary Source






Compressible Flow







(28)



Compressible Flow





>

Compressible Flow


Subsonic Flow


Supersonic Flow



Compressible Flow



Shock Waves











Compressible Flow




Compressible Flow



Normal Shock Waves


>
>

Compressible Flow





(29)

(210)

(211)



Compressible Flow

(212)


(213)


h01 = h02


T01 = T 02 (214)


(215)


(216)


(217)



Compressible Flow

(218)


(219)



(220)

(221)




equations are

(222)

(223)


Compressible Flow

(224)

(225)


(226)

(227)


(228)

and

(229)


(230)



Compressible Flow












Compressible Flow















Compressible Flow

(31)





Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow






du = dq - dw (12)


(13)


dh = dq + vdp (14)


Compressible Flow








Compressible Flow


(15)


(16)


(17)



Perfect Gas Law


(18)


(19)




Compressible Flow

Jkg K Jkg K cpcv

Air 2897 2870 1004 14

Ammonia 1703 4882 2092 13

Argon 3994 2081 519 167


Helium 4003 2077 5200 167

Hydrogen 2016 4124 14350 14

Oxygen 32 2598 916 14



u = u(T) h = h(T) (110)



(111)


(112)



Compressible Flow

(113)





(114)


(115)


(116)




Compressible Flow





Continuity Equation


(117)


(118)


Momentum Equation


Compressible Flow


(119)


(120)


Energy Equation




p1v1 - p2v2 (122)




(124)



Compressible Flow


(125)



= constant (126)


(127)


(128)


(129)





Compressible Flow




Rewrite Eqn126 as

(130)


(131)






(132)


(133)



Compressible Flow

(134)

Since we have

(135)







(136)



Compressible Flow

(137)


(138)



Compressible Flow








(139)


(140)


(141)


(142)


Compressible Flow



(143)

(144)




(145)


(146)



Compressible Flow

(147)


Mass Flow Rate





(148)



Compressible Flow






Compressible Flow

(149)

Wave Propagation


Press RED to start

Speed of Sound


>

Compressible Flow




respectively




(21)


(22)


Compressible Flow




(24)


(25)


(26)


(27)




Compressible Flow


Stationary Source






Compressible Flow







(28)



Compressible Flow





>

Compressible Flow


Subsonic Flow


Supersonic Flow



Compressible Flow



Shock Waves











Compressible Flow




Compressible Flow



Normal Shock Waves


>
>

Compressible Flow





(29)

(210)

(211)



Compressible Flow

(212)


(213)


h01 = h02


T01 = T 02 (214)


(215)


(216)


(217)



Compressible Flow

(218)


(219)



(220)

(221)




equations are

(222)

(223)


Compressible Flow

(224)

(225)


(226)

(227)


(228)

and

(229)


(230)



Compressible Flow












Compressible Flow















Compressible Flow

(31)





Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow








Compressible Flow


(15)


(16)


(17)



Perfect Gas Law


(18)


(19)




Compressible Flow

Jkg K Jkg K cpcv

Air 2897 2870 1004 14

Ammonia 1703 4882 2092 13

Argon 3994 2081 519 167


Helium 4003 2077 5200 167

Hydrogen 2016 4124 14350 14

Oxygen 32 2598 916 14



u = u(T) h = h(T) (110)



(111)


(112)



Compressible Flow

(113)





(114)


(115)


(116)




Compressible Flow





Continuity Equation


(117)


(118)


Momentum Equation


Compressible Flow


(119)


(120)


Energy Equation




p1v1 - p2v2 (122)




(124)



Compressible Flow


(125)



= constant (126)


(127)


(128)


(129)





Compressible Flow




Rewrite Eqn126 as

(130)


(131)






(132)


(133)



Compressible Flow

(134)

Since we have

(135)







(136)



Compressible Flow

(137)


(138)



Compressible Flow








(139)


(140)


(141)


(142)


Compressible Flow



(143)

(144)




(145)


(146)



Compressible Flow

(147)


Mass Flow Rate





(148)



Compressible Flow






Compressible Flow

(149)

Wave Propagation


Press RED to start

Speed of Sound


>

Compressible Flow




respectively




(21)


(22)


Compressible Flow




(24)


(25)


(26)


(27)




Compressible Flow


Stationary Source






Compressible Flow







(28)



Compressible Flow





>

Compressible Flow


Subsonic Flow


Supersonic Flow



Compressible Flow



Shock Waves











Compressible Flow




Compressible Flow



Normal Shock Waves


>
>

Compressible Flow





(29)

(210)

(211)



Compressible Flow

(212)


(213)


h01 = h02


T01 = T 02 (214)


(215)


(216)


(217)



Compressible Flow

(218)


(219)



(220)

(221)




equations are

(222)

(223)


Compressible Flow

(224)

(225)


(226)

(227)


(228)

and

(229)


(230)



Compressible Flow












Compressible Flow















Compressible Flow

(31)





Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow


(15)


(16)


(17)



Perfect Gas Law


(18)


(19)




Compressible Flow

Jkg K Jkg K cpcv

Air 2897 2870 1004 14

Ammonia 1703 4882 2092 13

Argon 3994 2081 519 167


Helium 4003 2077 5200 167

Hydrogen 2016 4124 14350 14

Oxygen 32 2598 916 14



u = u(T) h = h(T) (110)



(111)


(112)



Compressible Flow

(113)





(114)


(115)


(116)




Compressible Flow





Continuity Equation


(117)


(118)


Momentum Equation


Compressible Flow


(119)


(120)


Energy Equation




p1v1 - p2v2 (122)




(124)



Compressible Flow


(125)



= constant (126)


(127)


(128)


(129)





Compressible Flow




Rewrite Eqn126 as

(130)


(131)






(132)


(133)



Compressible Flow

(134)

Since we have

(135)







(136)



Compressible Flow

(137)


(138)



Compressible Flow








(139)


(140)


(141)


(142)


Compressible Flow



(143)

(144)




(145)


(146)



Compressible Flow

(147)


Mass Flow Rate





(148)



Compressible Flow






Compressible Flow

(149)

Wave Propagation


Press RED to start

Speed of Sound


>

Compressible Flow




respectively




(21)


(22)


Compressible Flow




(24)


(25)


(26)


(27)




Compressible Flow


Stationary Source






Compressible Flow







(28)



Compressible Flow





>

Compressible Flow


Subsonic Flow


Supersonic Flow



Compressible Flow



Shock Waves











Compressible Flow




Compressible Flow



Normal Shock Waves


>
>

Compressible Flow





(29)

(210)

(211)



Compressible Flow

(212)


(213)


h01 = h02


T01 = T 02 (214)


(215)


(216)


(217)



Compressible Flow

(218)


(219)



(220)

(221)




equations are

(222)

(223)


Compressible Flow

(224)

(225)


(226)

(227)


(228)

and

(229)


(230)



Compressible Flow












Compressible Flow















Compressible Flow

(31)





Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow

Jkg K Jkg K cpcv

Air 2897 2870 1004 14

Ammonia 1703 4882 2092 13

Argon 3994 2081 519 167


Helium 4003 2077 5200 167

Hydrogen 2016 4124 14350 14

Oxygen 32 2598 916 14



u = u(T) h = h(T) (110)



(111)


(112)



Compressible Flow

(113)





(114)


(115)


(116)




Compressible Flow





Continuity Equation


(117)


(118)


Momentum Equation


Compressible Flow


(119)


(120)


Energy Equation




p1v1 - p2v2 (122)




(124)



Compressible Flow


(125)



= constant (126)


(127)


(128)


(129)





Compressible Flow




Rewrite Eqn126 as

(130)


(131)






(132)


(133)



Compressible Flow

(134)

Since we have

(135)







(136)



Compressible Flow

(137)


(138)



Compressible Flow








(139)


(140)


(141)


(142)


Compressible Flow



(143)

(144)




(145)


(146)



Compressible Flow

(147)


Mass Flow Rate





(148)



Compressible Flow






Compressible Flow

(149)

Wave Propagation


Press RED to start

Speed of Sound


>

Compressible Flow




respectively




(21)


(22)


Compressible Flow




(24)


(25)


(26)


(27)




Compressible Flow


Stationary Source






Compressible Flow







(28)



Compressible Flow





>

Compressible Flow


Subsonic Flow


Supersonic Flow



Compressible Flow



Shock Waves











Compressible Flow




Compressible Flow



Normal Shock Waves


>
>

Compressible Flow





(29)

(210)

(211)



Compressible Flow

(212)


(213)


h01 = h02


T01 = T 02 (214)


(215)


(216)


(217)



Compressible Flow

(218)


(219)



(220)

(221)




equations are

(222)

(223)


Compressible Flow

(224)

(225)


(226)

(227)


(228)

and

(229)


(230)



Compressible Flow












Compressible Flow















Compressible Flow

(31)





Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow

(113)





(114)


(115)


(116)




Compressible Flow





Continuity Equation


(117)


(118)


Momentum Equation


Compressible Flow


(119)


(120)


Energy Equation




p1v1 - p2v2 (122)




(124)



Compressible Flow


(125)



= constant (126)


(127)


(128)


(129)





Compressible Flow




Rewrite Eqn126 as

(130)


(131)






(132)


(133)



Compressible Flow

(134)

Since we have

(135)







(136)



Compressible Flow

(137)


(138)



Compressible Flow








(139)


(140)


(141)


(142)


Compressible Flow



(143)

(144)




(145)


(146)



Compressible Flow

(147)


Mass Flow Rate





(148)



Compressible Flow






Compressible Flow

(149)

Wave Propagation


Press RED to start

Speed of Sound


>

Compressible Flow




respectively




(21)


(22)


Compressible Flow




(24)


(25)


(26)


(27)




Compressible Flow


Stationary Source






Compressible Flow







(28)



Compressible Flow





>

Compressible Flow


Subsonic Flow


Supersonic Flow



Compressible Flow



Shock Waves











Compressible Flow




Compressible Flow



Normal Shock Waves


>
>

Compressible Flow





(29)

(210)

(211)



Compressible Flow

(212)


(213)


h01 = h02


T01 = T 02 (214)


(215)


(216)


(217)



Compressible Flow

(218)


(219)



(220)

(221)




equations are

(222)

(223)


Compressible Flow

(224)

(225)


(226)

(227)


(228)

and

(229)


(230)



Compressible Flow












Compressible Flow















Compressible Flow

(31)





Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow





Continuity Equation


(117)


(118)


Momentum Equation


Compressible Flow


(119)


(120)


Energy Equation




p1v1 - p2v2 (122)




(124)



Compressible Flow


(125)



= constant (126)


(127)


(128)


(129)





Compressible Flow




Rewrite Eqn126 as

(130)


(131)






(132)


(133)



Compressible Flow

(134)

Since we have

(135)







(136)



Compressible Flow

(137)


(138)



Compressible Flow








(139)


(140)


(141)


(142)


Compressible Flow



(143)

(144)




(145)


(146)



Compressible Flow

(147)


Mass Flow Rate





(148)



Compressible Flow






Compressible Flow

(149)

Wave Propagation


Press RED to start

Speed of Sound


>

Compressible Flow




respectively




(21)


(22)


Compressible Flow




(24)


(25)


(26)


(27)




Compressible Flow


Stationary Source






Compressible Flow







(28)



Compressible Flow





>

Compressible Flow


Subsonic Flow


Supersonic Flow



Compressible Flow



Shock Waves











Compressible Flow




Compressible Flow



Normal Shock Waves


>
>

Compressible Flow





(29)

(210)

(211)



Compressible Flow

(212)


(213)


h01 = h02


T01 = T 02 (214)


(215)


(216)


(217)



Compressible Flow

(218)


(219)



(220)

(221)




equations are

(222)

(223)


Compressible Flow

(224)

(225)


(226)

(227)


(228)

and

(229)


(230)



Compressible Flow












Compressible Flow















Compressible Flow

(31)





Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow


(119)


(120)


Energy Equation




p1v1 - p2v2 (122)




(124)



Compressible Flow


(125)



= constant (126)


(127)


(128)


(129)





Compressible Flow




Rewrite Eqn126 as

(130)


(131)






(132)


(133)



Compressible Flow

(134)

Since we have

(135)







(136)



Compressible Flow

(137)


(138)



Compressible Flow








(139)


(140)


(141)


(142)


Compressible Flow



(143)

(144)




(145)


(146)



Compressible Flow

(147)


Mass Flow Rate





(148)



Compressible Flow






Compressible Flow

(149)

Wave Propagation


Press RED to start

Speed of Sound


>

Compressible Flow




respectively




(21)


(22)


Compressible Flow




(24)


(25)


(26)


(27)




Compressible Flow


Stationary Source






Compressible Flow







(28)



Compressible Flow





>

Compressible Flow


Subsonic Flow


Supersonic Flow



Compressible Flow



Shock Waves











Compressible Flow




Compressible Flow



Normal Shock Waves


>
>

Compressible Flow





(29)

(210)

(211)



Compressible Flow

(212)


(213)


h01 = h02


T01 = T 02 (214)


(215)


(216)


(217)



Compressible Flow

(218)


(219)



(220)

(221)




equations are

(222)

(223)


Compressible Flow

(224)

(225)


(226)

(227)


(228)

and

(229)


(230)



Compressible Flow












Compressible Flow















Compressible Flow

(31)





Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow


(125)



= constant (126)


(127)


(128)


(129)





Compressible Flow




Rewrite Eqn126 as

(130)


(131)






(132)


(133)



Compressible Flow

(134)

Since we have

(135)







(136)



Compressible Flow

(137)


(138)



Compressible Flow








(139)


(140)


(141)


(142)


Compressible Flow



(143)

(144)




(145)


(146)



Compressible Flow

(147)


Mass Flow Rate





(148)



Compressible Flow






Compressible Flow

(149)

Wave Propagation


Press RED to start

Speed of Sound


>

Compressible Flow




respectively




(21)


(22)


Compressible Flow




(24)


(25)


(26)


(27)




Compressible Flow


Stationary Source






Compressible Flow







(28)



Compressible Flow





>

Compressible Flow


Subsonic Flow


Supersonic Flow



Compressible Flow



Shock Waves











Compressible Flow




Compressible Flow



Normal Shock Waves


>
>

Compressible Flow





(29)

(210)

(211)



Compressible Flow

(212)


(213)


h01 = h02


T01 = T 02 (214)


(215)


(216)


(217)



Compressible Flow

(218)


(219)



(220)

(221)




equations are

(222)

(223)


Compressible Flow

(224)

(225)


(226)

(227)


(228)

and

(229)


(230)



Compressible Flow












Compressible Flow















Compressible Flow

(31)





Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow




Rewrite Eqn126 as

(130)


(131)






(132)


(133)



Compressible Flow

(134)

Since we have

(135)







(136)



Compressible Flow

(137)


(138)



Compressible Flow








(139)


(140)


(141)


(142)


Compressible Flow



(143)

(144)




(145)


(146)



Compressible Flow

(147)


Mass Flow Rate





(148)



Compressible Flow






Compressible Flow

(149)

Wave Propagation


Press RED to start

Speed of Sound


>

Compressible Flow




respectively




(21)


(22)


Compressible Flow




(24)


(25)


(26)


(27)




Compressible Flow


Stationary Source






Compressible Flow







(28)



Compressible Flow





>

Compressible Flow


Subsonic Flow


Supersonic Flow



Compressible Flow



Shock Waves











Compressible Flow




Compressible Flow



Normal Shock Waves


>
>

Compressible Flow





(29)

(210)

(211)



Compressible Flow

(212)


(213)


h01 = h02


T01 = T 02 (214)


(215)


(216)


(217)



Compressible Flow

(218)


(219)



(220)

(221)




equations are

(222)

(223)


Compressible Flow

(224)

(225)


(226)

(227)


(228)

and

(229)


(230)



Compressible Flow












Compressible Flow















Compressible Flow

(31)





Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow

(134)

Since we have

(135)







(136)



Compressible Flow

(137)


(138)



Compressible Flow








(139)


(140)


(141)


(142)


Compressible Flow



(143)

(144)




(145)


(146)



Compressible Flow

(147)


Mass Flow Rate





(148)



Compressible Flow






Compressible Flow

(149)

Wave Propagation


Press RED to start

Speed of Sound


>

Compressible Flow




respectively




(21)


(22)


Compressible Flow




(24)


(25)


(26)


(27)




Compressible Flow


Stationary Source






Compressible Flow







(28)



Compressible Flow





>

Compressible Flow


Subsonic Flow


Supersonic Flow



Compressible Flow



Shock Waves











Compressible Flow




Compressible Flow



Normal Shock Waves


>
>

Compressible Flow





(29)

(210)

(211)



Compressible Flow

(212)


(213)


h01 = h02


T01 = T 02 (214)


(215)


(216)


(217)



Compressible Flow

(218)


(219)



(220)

(221)




equations are

(222)

(223)


Compressible Flow

(224)

(225)


(226)

(227)


(228)

and

(229)


(230)



Compressible Flow












Compressible Flow















Compressible Flow

(31)





Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow

(137)


(138)



Compressible Flow








(139)


(140)


(141)


(142)


Compressible Flow



(143)

(144)




(145)


(146)



Compressible Flow

(147)


Mass Flow Rate





(148)



Compressible Flow






Compressible Flow

(149)

Wave Propagation


Press RED to start

Speed of Sound


>

Compressible Flow




respectively




(21)


(22)


Compressible Flow




(24)


(25)


(26)


(27)




Compressible Flow


Stationary Source






Compressible Flow







(28)



Compressible Flow





>

Compressible Flow


Subsonic Flow


Supersonic Flow



Compressible Flow



Shock Waves











Compressible Flow




Compressible Flow



Normal Shock Waves


>
>

Compressible Flow





(29)

(210)

(211)



Compressible Flow

(212)


(213)


h01 = h02


T01 = T 02 (214)


(215)


(216)


(217)



Compressible Flow

(218)


(219)



(220)

(221)




equations are

(222)

(223)


Compressible Flow

(224)

(225)


(226)

(227)


(228)

and

(229)


(230)



Compressible Flow












Compressible Flow















Compressible Flow

(31)





Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow








(139)


(140)


(141)


(142)


Compressible Flow



(143)

(144)




(145)


(146)



Compressible Flow

(147)


Mass Flow Rate





(148)



Compressible Flow






Compressible Flow

(149)

Wave Propagation


Press RED to start

Speed of Sound


>

Compressible Flow




respectively




(21)


(22)


Compressible Flow




(24)


(25)


(26)


(27)




Compressible Flow


Stationary Source






Compressible Flow







(28)



Compressible Flow





>

Compressible Flow


Subsonic Flow


Supersonic Flow



Compressible Flow



Shock Waves











Compressible Flow




Compressible Flow



Normal Shock Waves


>
>

Compressible Flow





(29)

(210)

(211)



Compressible Flow

(212)


(213)


h01 = h02


T01 = T 02 (214)


(215)


(216)


(217)



Compressible Flow

(218)


(219)



(220)

(221)




equations are

(222)

(223)


Compressible Flow

(224)

(225)


(226)

(227)


(228)

and

(229)


(230)



Compressible Flow












Compressible Flow















Compressible Flow

(31)





Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow



(143)

(144)




(145)


(146)



Compressible Flow

(147)


Mass Flow Rate





(148)



Compressible Flow






Compressible Flow

(149)

Wave Propagation


Press RED to start

Speed of Sound


>

Compressible Flow




respectively




(21)


(22)


Compressible Flow




(24)


(25)


(26)


(27)




Compressible Flow


Stationary Source






Compressible Flow







(28)



Compressible Flow





>

Compressible Flow


Subsonic Flow


Supersonic Flow



Compressible Flow



Shock Waves











Compressible Flow




Compressible Flow



Normal Shock Waves


>
>

Compressible Flow





(29)

(210)

(211)



Compressible Flow

(212)


(213)


h01 = h02


T01 = T 02 (214)


(215)


(216)


(217)



Compressible Flow

(218)


(219)



(220)

(221)




equations are

(222)

(223)


Compressible Flow

(224)

(225)


(226)

(227)


(228)

and

(229)


(230)



Compressible Flow












Compressible Flow















Compressible Flow

(31)





Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow

(147)


Mass Flow Rate





(148)



Compressible Flow






Compressible Flow

(149)

Wave Propagation


Press RED to start

Speed of Sound


>

Compressible Flow




respectively




(21)


(22)


Compressible Flow




(24)


(25)


(26)


(27)




Compressible Flow


Stationary Source






Compressible Flow







(28)



Compressible Flow





>

Compressible Flow


Subsonic Flow


Supersonic Flow



Compressible Flow



Shock Waves











Compressible Flow




Compressible Flow



Normal Shock Waves


>
>

Compressible Flow





(29)

(210)

(211)



Compressible Flow

(212)


(213)


h01 = h02


T01 = T 02 (214)


(215)


(216)


(217)



Compressible Flow

(218)


(219)



(220)

(221)




equations are

(222)

(223)


Compressible Flow

(224)

(225)


(226)

(227)


(228)

and

(229)


(230)



Compressible Flow












Compressible Flow















Compressible Flow

(31)





Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow






Compressible Flow

(149)

Wave Propagation


Press RED to start

Speed of Sound


>

Compressible Flow




respectively




(21)


(22)


Compressible Flow




(24)


(25)


(26)


(27)




Compressible Flow


Stationary Source






Compressible Flow







(28)



Compressible Flow





>

Compressible Flow


Subsonic Flow


Supersonic Flow



Compressible Flow



Shock Waves











Compressible Flow




Compressible Flow



Normal Shock Waves


>
>

Compressible Flow





(29)

(210)

(211)



Compressible Flow

(212)


(213)


h01 = h02


T01 = T 02 (214)


(215)


(216)


(217)



Compressible Flow

(218)


(219)



(220)

(221)




equations are

(222)

(223)


Compressible Flow

(224)

(225)


(226)

(227)


(228)

and

(229)


(230)



Compressible Flow












Compressible Flow















Compressible Flow

(31)





Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow

(149)

Wave Propagation


Press RED to start

Speed of Sound


>

Compressible Flow




respectively




(21)


(22)


Compressible Flow




(24)


(25)


(26)


(27)




Compressible Flow


Stationary Source






Compressible Flow







(28)



Compressible Flow





>

Compressible Flow


Subsonic Flow


Supersonic Flow



Compressible Flow



Shock Waves











Compressible Flow




Compressible Flow



Normal Shock Waves


>
>

Compressible Flow





(29)

(210)

(211)



Compressible Flow

(212)


(213)


h01 = h02


T01 = T 02 (214)


(215)


(216)


(217)



Compressible Flow

(218)


(219)



(220)

(221)




equations are

(222)

(223)


Compressible Flow

(224)

(225)


(226)

(227)


(228)

and

(229)


(230)



Compressible Flow












Compressible Flow















Compressible Flow

(31)





Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow




respectively




(21)


(22)


Compressible Flow




(24)


(25)


(26)


(27)




Compressible Flow


Stationary Source






Compressible Flow







(28)



Compressible Flow





>

Compressible Flow


Subsonic Flow


Supersonic Flow



Compressible Flow



Shock Waves











Compressible Flow




Compressible Flow



Normal Shock Waves


>
>

Compressible Flow





(29)

(210)

(211)



Compressible Flow

(212)


(213)


h01 = h02


T01 = T 02 (214)


(215)


(216)


(217)



Compressible Flow

(218)


(219)



(220)

(221)




equations are

(222)

(223)


Compressible Flow

(224)

(225)


(226)

(227)


(228)

and

(229)


(230)



Compressible Flow












Compressible Flow















Compressible Flow

(31)





Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow




(24)


(25)


(26)


(27)




Compressible Flow


Stationary Source






Compressible Flow







(28)



Compressible Flow





>

Compressible Flow


Subsonic Flow


Supersonic Flow



Compressible Flow



Shock Waves











Compressible Flow




Compressible Flow



Normal Shock Waves


>
>

Compressible Flow





(29)

(210)

(211)



Compressible Flow

(212)


(213)


h01 = h02


T01 = T 02 (214)


(215)


(216)


(217)



Compressible Flow

(218)


(219)



(220)

(221)




equations are

(222)

(223)


Compressible Flow

(224)

(225)


(226)

(227)


(228)

and

(229)


(230)



Compressible Flow












Compressible Flow















Compressible Flow

(31)





Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow


Stationary Source






Compressible Flow







(28)



Compressible Flow





>

Compressible Flow


Subsonic Flow


Supersonic Flow



Compressible Flow



Shock Waves











Compressible Flow




Compressible Flow



Normal Shock Waves


>
>

Compressible Flow





(29)

(210)

(211)



Compressible Flow

(212)


(213)


h01 = h02


T01 = T 02 (214)


(215)


(216)


(217)



Compressible Flow

(218)


(219)



(220)

(221)




equations are

(222)

(223)


Compressible Flow

(224)

(225)


(226)

(227)


(228)

and

(229)


(230)



Compressible Flow












Compressible Flow















Compressible Flow

(31)





Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow







(28)



Compressible Flow





>

Compressible Flow


Subsonic Flow


Supersonic Flow



Compressible Flow



Shock Waves











Compressible Flow




Compressible Flow



Normal Shock Waves


>
>

Compressible Flow





(29)

(210)

(211)



Compressible Flow

(212)


(213)


h01 = h02


T01 = T 02 (214)


(215)


(216)


(217)



Compressible Flow

(218)


(219)



(220)

(221)




equations are

(222)

(223)


Compressible Flow

(224)

(225)


(226)

(227)


(228)

and

(229)


(230)



Compressible Flow












Compressible Flow















Compressible Flow

(31)





Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow





>

Compressible Flow


Subsonic Flow


Supersonic Flow



Compressible Flow



Shock Waves











Compressible Flow




Compressible Flow



Normal Shock Waves


>
>

Compressible Flow





(29)

(210)

(211)



Compressible Flow

(212)


(213)


h01 = h02


T01 = T 02 (214)


(215)


(216)


(217)



Compressible Flow

(218)


(219)



(220)

(221)




equations are

(222)

(223)


Compressible Flow

(224)

(225)


(226)

(227)


(228)

and

(229)


(230)



Compressible Flow












Compressible Flow















Compressible Flow

(31)





Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow


Subsonic Flow


Supersonic Flow



Compressible Flow



Shock Waves











Compressible Flow




Compressible Flow



Normal Shock Waves


>
>

Compressible Flow





(29)

(210)

(211)



Compressible Flow

(212)


(213)


h01 = h02


T01 = T 02 (214)


(215)


(216)


(217)



Compressible Flow

(218)


(219)



(220)

(221)




equations are

(222)

(223)


Compressible Flow

(224)

(225)


(226)

(227)


(228)

and

(229)


(230)



Compressible Flow












Compressible Flow















Compressible Flow

(31)





Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow



Shock Waves











Compressible Flow




Compressible Flow



Normal Shock Waves


>
>

Compressible Flow





(29)

(210)

(211)



Compressible Flow

(212)


(213)


h01 = h02


T01 = T 02 (214)


(215)


(216)


(217)



Compressible Flow

(218)


(219)



(220)

(221)




equations are

(222)

(223)


Compressible Flow

(224)

(225)


(226)

(227)


(228)

and

(229)


(230)



Compressible Flow












Compressible Flow















Compressible Flow

(31)





Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow




Compressible Flow



Normal Shock Waves


>
>

Compressible Flow





(29)

(210)

(211)



Compressible Flow

(212)


(213)


h01 = h02


T01 = T 02 (214)


(215)


(216)


(217)



Compressible Flow

(218)


(219)



(220)

(221)




equations are

(222)

(223)


Compressible Flow

(224)

(225)


(226)

(227)


(228)

and

(229)


(230)



Compressible Flow












Compressible Flow















Compressible Flow

(31)





Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow



Normal Shock Waves


>
>

Compressible Flow





(29)

(210)

(211)



Compressible Flow

(212)


(213)


h01 = h02


T01 = T 02 (214)


(215)


(216)


(217)



Compressible Flow

(218)


(219)



(220)

(221)




equations are

(222)

(223)


Compressible Flow

(224)

(225)


(226)

(227)


(228)

and

(229)


(230)



Compressible Flow












Compressible Flow















Compressible Flow

(31)





Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow





(29)

(210)

(211)



Compressible Flow

(212)


(213)


h01 = h02


T01 = T 02 (214)


(215)


(216)


(217)



Compressible Flow

(218)


(219)



(220)

(221)




equations are

(222)

(223)


Compressible Flow

(224)

(225)


(226)

(227)


(228)

and

(229)


(230)



Compressible Flow












Compressible Flow















Compressible Flow

(31)





Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow

(212)


(213)


h01 = h02


T01 = T 02 (214)


(215)


(216)


(217)



Compressible Flow

(218)


(219)



(220)

(221)




equations are

(222)

(223)


Compressible Flow

(224)

(225)


(226)

(227)


(228)

and

(229)


(230)



Compressible Flow












Compressible Flow















Compressible Flow

(31)





Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow

(218)


(219)



(220)

(221)




equations are

(222)

(223)


Compressible Flow

(224)

(225)


(226)

(227)


(228)

and

(229)


(230)



Compressible Flow












Compressible Flow















Compressible Flow

(31)





Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow

(224)

(225)


(226)

(227)


(228)

and

(229)


(230)



Compressible Flow












Compressible Flow















Compressible Flow

(31)





Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow












Compressible Flow















Compressible Flow

(31)





Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow















Compressible Flow

(31)





Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow

(31)





Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow















Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow








Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow





Oblique Shock Waves



>

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow









Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow

Define

(41)




(42)


(43)


(44)

(45)


Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow

(46)

(47)

(48)


(410)



(411)




subsonic




Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow

derived as follows-

which gives

leading to

(412)






Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow









Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow









Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow

Weak Oblique Shocks


(413)



(414)



>

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow

(415)




(416)


proportional to


Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow




Total turning

Change in pressure

Entropy rise (417)


Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow



(418)







Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow









Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow







Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow



(419)


(420)


Starting from

(421)


Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow

one can deduce that

(422)




(423)



(424)





Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow












Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow




Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow










Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow







Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow

Mach Reflection







Flat Plate Aerofoil



Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow





(425)


Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow





(426)


Diamond Aerofoil



Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow



(427)



(428)


(429)


(430)


(431)




Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow




(432)


Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow


reduce to

(433)


(434)

This gives

(435)

(436)

(437)





(438)



Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow

(439)


(440)

Diamond Aerofoil


(441)


(442)


(443)


(444)



Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow




(445)

Second Order Theory



(446)


(447)




Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow








Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow








(448)

(449)



Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow








ie a constant (450)



Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow

ie a constant (451)






Now we have


Or (452)


Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow






Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow








Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow




Point M Q R

a 1605 15 6 21 9

b 1605 15 2 17 13

c 1605 15 -2 13 17

d 1605 15 -6 9 21

Interior Point

Consider e We have


Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow


Boundary Point






Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow








Compressible Flow














wwwaesuozau

Compressible Flow

Compressible Flow














wwwaesuozau

Compressible Flow

compressible flow

Documents

flow mach number

flow of compressible

density changes

temperature changes

air flow

pm compressible flow

function of mach number

incompressible fluids