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 incompressible;water being an excellent example. In contrast compressible fluids do undergo density changes. Gases are generally compressible;air 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 1.1: Classification of Fluids

Though gases are compressible, the density changes they undergo at low speeds may not be considerable. Take air for instance. Fig. 1.2 shows the density changes plotted as a function of Mach

Number. Density change is represented as where is the air density at zero speed (i.e., Zero

Mach Number).

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Compressible Flow

Figure 1.2: Density change as a function of Mach Number

We observe that for Mach numbers up to 0.3, 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 0.3, changes do become appreciable and at a Mach Number of 1, it is 36.5% . 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 0.3.

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 2.3 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. 1.3.

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Compressible Flow

Figure 1.3: 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. 1.4.

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 1.4: Definition of a System

Control Volume should now be familiar to you. Most of Integral Approach to Fluid Dynamics exploits

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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 also a characterise a system. To define the state of

a system (Fig. 1.5) uniquely we need to specify two properties say (p,T), (p, ), (T,s) etc., where p, T,

, s are pressure, temperature, density and specific entropy respectively.

Figure 1.5: 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,

(1.1)

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

This laws helps define Temperature. It states - "Two systems which are in thermal equilibrium with a

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third system are themselves in thermal equilibrium."

Figure 1.6: Zeroth Law of Thermodynamics

When in thermal equilibrium, we say that the two systems are at the same temperature. In the figure 1.6, 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 (1.2)

where u is Internal Energy. Bringing in Specific Enthalpy defined as

(1.3)

the statement for the first law can also be written as

dh = dq + v.dp (1.4)

While writing Eqn.1.4 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

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Compressible Flow

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 1.7: 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'.

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

(1.5)

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

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(1.6)

Generalising the equation 1.6, we have

(1.7)

where an '=' sign is used for reversible processes and > 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 rem