differential equations of fluid dynamics

7/31/2019 Differential Equations of Fluid Dynamics

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Differential Equations of Fluid Dynamics

A short note on their derivation

D. Olivari

July 2012


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Differential Equations of Fluid Dynamics

A summary for the VKI DEFM Lectures 1999/2000

D. Olivari

1. The Continuum.

We will call a continuum any medium into which all physical, chemical and other variables1 A(x,t)

which define its properties are described by functions which have no discontinuities and which have

continuous derivatives up to any degree with respect to space or time.

If the medium allows for internal motion, i.e. a displacement of one point with respect to the others

in a continuous way, we call it a Fluid, and the corresponding motion a Flow.

Inside such a medium the quantities which define its properties can be transported, depending onthe nature of the medium, in essentially three ways:

Advection, transport by movement,

Diffusion, transport by molecular effects,

Radiation.

It is possible that two or more such media of different properties share a common boundary. At the

frontier between the two there may be discontinuities in some or all of the properties defining them: ifsuch is the case we define the ensemble of the media as multiphase.

We will now try to write the relevant equations for the transportof properties in a fluid.

2. The generalised Transport Equation.

The motion of a fluid can be described from two standpoints, or reference frames, respectively

called:

Eulerian,

Lagrangian.

In the first case the motion is seen from an external fixed frame 2, and the properties of all quantities

are described with respect to it. In the second one the observer is sitting on a particle in the flow and

moves with it: the description is then made with reference to this changing position. The differences

are relevant it is not always easy to pass from one description to the other. Each one has special

advantages for specific applications and the choice is made in function of this. In this introduction wewill use only theEulerian approach.

The generalised equation for the variation in time and the transport and of the quantity A in a

medium in motion with a vectorial velocity of components Ui (i =1..3), can be then written as:

DAi

Dt=^

ij

^xj

ASi

(1

1 Vectorial quantities will be indicated in the text by bold characters.2 We will see later that all the equation are invariant for a galileian transformation, and the term fixed frame is used in this context.

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where the meaning of each term is respectively:

1. The substantial derivative of the i-component ofA which, for flow in motion with velocity U,

can be expanded as DADt

= A^ tA A

xj

xj^ t

= A^ tA A

xj

Uj

(2,

2.A term representing the diffusion, or the molecular interaction in the fluid,

3.A source term, representing the action of the environment on the fluid and the flow fields.

It is important to remark that the term:

^ij

^xj

(3

is the divergence of the i-compnent of ij

. If the flow is limited in space, far away from its

limits it is equal to zero. This means that it is not sensitive to effects from outside, and only represents

a redistribution ofA inside the flow by diffusive action across and along the surfaces and its effects

are limited to the i-direction. . We will define these terms as short range effects or surface actions, as

opposed to the source terms S whichare longrange effects or volume actions.

Substituting forA , and S the relevant quantities, we can now obtain all the equations required,

namely the equations for:

O Conservation ofMass,

O Conservation ofMomentum,O Conservation ofEnergy,

OConservation ofSpecies, or of Concentration,

O Conservation of any other relevant quantity.

3. The transport equation for a fluid.

We will now briefly derive the equations relevant for a fluid flow field.

3.1.Mass Conservation.

For the Mass Flow the quantity transported is just the density times the elementary reference

volume V, there is no diffusion, and if there is no source term, we have:

Ai=V =0 S=0 (4

and the mass-transport, or continuity, equation becomes:

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DV

Dt=0

VD

DtA

DV

Dt=V

D

DtA

Ûi

^xi

dxdydz=0

^

^ t AUj^

^xjA

Ûi

^xj=0

(5

If a fluid is incompressible (we will see later under what conditions a fluid can be considered as

incompressible), then = const. , and the continuity equation simplifies to a purely kinematic

relation:

Ûi

^xj

=0 (6

i.e. the divergence of the flow is equal to zero. This can be seen as an external constraint imposed

on the dynamic of the flow.

3.2.Momentum Conservation.

For the Momentum Equation we have:

Ai=U

i

ij=

ij=normalA shear stresses

Si= g

i

(7

and it results:

DUi

Dt=^

ij

^xj

A gi

Ûi

^ tAU

j

Ûi

^xj

=^

ij

^xj

A gi

(8

The stress tensorji must now be determined taking into account the fact that our medium is a

fluid.

The determination of the structure of this term, derived from the mechanics of solid media, is due,

in its present form, to Navier and Stokes.

3.2.1.A formulation for ij .

For a continuum material it is generally assumed that the stresses are in some way function of thedeformations.

For a solid it is assumed that there is a direct relations between the stresses and the amount ofdeformation imposed:

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ij

Z^X

i

xj

(9

whereXi is the deformation of the material at point xi .

In a fluid, gas or liquid in normal conditions, the assumption, confirmed by experience, is that the

stresses are instead related to the velocity at which deformation takes place, so:

ij

Z^

xj

^Xi

^ t (10

In other worlds, the stress tensor is related to the strain rate in the flow.

ij

ZÛ

i

xj

(11

The strain rate itself can now be divided into a symmetric and an antisymmetric part as follows:

Ûi

xj

=1

2

Ûi

xj

A

Ûj

xi

A1

2

Ûi

xj

B

Ûj

xi

(12

The represents the pure deformation of the fluid, while the antisymmetric partrepresents the solid

body rotation, to which there is no associated deformation, of the fluid. It follows that the next stepis to assume that only the symmetric part (pure deformation) contributes to the generation ofstresses

in the fluid.

ij

Z1

2

Ûi

xj

AÛ

j

xi

Z1

2e

ij(13

Now a distinction should be made between the normal and the shear stresses.

3.2.2.The pressure term.

Thepressure or the normal stresses contribution is assumed to be identically equal to the sum of the

diagonal terms of the tensor ji , or to its contraction:

p=ij

ij

(14

The pressure term, or its contribution, is then, by definition, a scalarquantity.

3.2.3.The Shear Term.

The shear term can now be rewritten as:

ij

=dijB p

ij(15

The term d ,which by definition should have a zero sum diagonal is called the deviatoric stress

tensor.

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In a Newtonian fluidwe assume that dis linearly related to the deformation tensor e and that the

fluid properties are isotropic. So we can write:

dij=

ijmne

mn(16

where ijmn is the tensor representing the isotropic properties of the fluid. The expression for this

tensor is the contribution ofNavier3

and Stokes4

to the description of fluid properties.Using the properties of tensors it can be shown that the only possible isotropic relation between d

and e is:

dij=2Ae

ijAB

ije

kk(17

whereA andB are bulkscalar fluid properties.

The physical meaning ofA is the well known fluid viscosity , whileB, which plays a role only incompressible fluids, is variously defined as second viscosity, bulk viscosity,... and is associated to

isotropic volumetric variation of a fluid element. So:

dij=2e

ijAB

ije

kk(18

From the considerations made in 3.2.2, we mustassume that the contraction ofd, or the sum of itsdiagonal terms, should be equal to zero.

dii

=2eii

A3Beii

=0 (19

It is then immediately derived that:

B =B2

3 (20

And, finally we obtain for the stresses in a Newtonian fluid the relation:

ij

=Û

i

xj

AÛ

j

xi

B2

3

Ûi

xi

B pij

(21

3.2.4.The Final Formulation.

Substituting all the terms derived in the original equation we obtain the usual form of theMomentum equation as:

Û

i

^ tA U

j

Ûi

^ xj

= B^ p

^ xi

A^

^ xj

Û

i

^ xj

AÛ

j

^ xi

B2

3

Ûi

^ xi

A g i (22

It should be noted that for an incompressible flow the terms associated withB disappears becausethe divergence of the velocity is equal to zero.

3 Claude Luis Marie Henri Navier (1785-1836) was a french engineer known for the works on mechanics.4 Sir George Stokes (1819-1903) was a irish physicist who worked on hydrodynamics and fluorescence.

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3.3.Energy Conservation.

For the Energy Equation we have:

Ai=

1

2U

2

Ae

ij

=ij

UiAq

i= work doneby mechanical stressesA heat transfer

Si= g

iU

i

(23

where:

1

2U

2

= kinetic mechanical , energy

e=internal thermal energy.

(24

So we can write:

DE

Dt=

Û

2

2Ae

^ tAU

j

Û

2

2Ae

^xj

=^

^xj

ij

UiB

^ qi

^xi

A giU

i

(25

3.3.1.The Fourier law.

The relation used for the heat transferbetween adjacent layers of fluid is the classical Fourier5 law,

as for the solids:

qi=Bk

^T

^xi

(26

k being a scalar bulk property of the fluid, the thermal conductivity.

3.3.2.The mechanical work.

This work is the scalar product of the total surface stresses times the local fluid velocity and can be

expressed, per unit surface, as:

work= ij

j

Ui=

Ûi

^xj

AÛ

j

^xi

B2

3

Ûi

^xi

B p ij j Ui (27

[For simplicity from now on we will neglect the bulk viscosity term].

We can now write the complete expression.

5 Jean Baptiste Joseph de Fourier (1768-1830) is the originator of the Fourier series.

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3.3.3.The final formulation.

The usual formulation for the Total Energy equation is then:

^

U2

2A e

^ tA U

j

Û

2

2A e

^ xj

=

Ûi

Û

i

^ xj

AÛ

j

^ xi

B p ij

^xj

A k^

2

T

^xi^x

i

A giU

i

(28

However the total energy energy equation is of relatively little interest since the mechanical

component is practically known from the momentum equation. Of more interest, for its practical

aspects and for a physical insight is the Internal Energy equation.

3.3.4.The Internal Energy Equation.

The Internal Energy equation can be easily derived from the equation above by subtracting the

equation for the mechanical energy obtained from the momentum conservation.

Û

2

2

^ tA U

j

Û

2

2

xj

=Ui

^ Û

i

xj

AÛ

j

xi

B pij

xj

A giU

i

(29

Subtracting (22) from (21) we obtain the required result.

^ e

^ tA U

j

^ e

^ xj

= Û

i

^xj

AÛ

j

^ xi

B p ij Û

i

^ xj

A k^

2

T

^ xi^ x

i

e= CvT

(30

This is the equation for the Internal Energy Transport. It should be noted that it contains a

mechanical term (albeit in a different form than in the Total Energy equation) which represents the

dissipation of Mechanical Energy into Thermal Energy.

We will show now that this terms leads to irreversibility of the viscous flow motion.

3.3.5.The Role of the Stress Term.

The viscosity dependent component of the stress contribution can be rewritten using the

permutation of indexes allowed in tensorial notation, as:

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Û

i

xj

AÛ

j

xi

Û

i

xj

=1

2

Ûi

xj

AÛ

j

xi

Û

i

xj

A1

2

Ûj

xi

AÛ

i

xj

Û

j

xi

}=1

2

Û

i

xjA

Ûj

xi

2

(31

Since the last term is the square of the deformation tensor, it is always positive, and it shows that

the role of viscosity in the energy equation is always to dissipate kinetic energy into heat in airreversible way.

The pressure dependent term, on the other hand, can be rewritten, using the continuity equation,as:

Bp ij

Ûi

xi

=B pi

Ûix

i

=1

p

^ ^ t

AUi

^ x

i

(32

This is a reversible contribution to the internal energy due to the elastic work produced bycompression in compressible flows.

3.3.6.The equation for Entropy.

An important thermodynamic quantity to define the state of a system is theEntropy.

The energy equation can be rewritten in terms of Entropy S considering that:

S =dQ

dt =Cv ln T

1

R

Cv

Aconst. (33

and:

^ S

^ t=

Cv

T

^T

^ tB

R

^

^ t(34

After substitutions in (27) and (23) and a few manipulation, we obtain for the entropy:

^ S

^ tA

^ S Ui

^ xi

A^

^ xi

k

T

^T

^ xi

=

ij

T

Ûi

^xj

Ak

T2

^T

^xi

2

(35

For the same reasons seen before, the last two terms are always positive, so they always lead to anincrease in entropy. This is always the case for any closed system, so the above statement is just theapplication of the second principle of thermodynamics to the fluid motion.

For a perfect fluid:

=0 k=0

and:

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DS

Dt=0 (36

Inviscid, non-conductive, incompressible flows are isoentropic, S=const. This is the case, in

particular, for the flows described by theEulerequation.

3.4.The Scalar Transport Equation.

The equation for the Conservation of Passive Species, or the equations for the Transport of a

Passive Scalar, i.e. a scalar that do not react or interact with the surroundings can be derived by

replacing the scalar with its Concentration C. In the case where the mixing fluids are of the samenature, liquids or gases without free interfaces, we have:

Ai=C

ij=K

i=molecular Diffusion

S=Q=Sourcesof thevariousComponents

(37

In analogy with the Fourier law, we can write for the diffusion the Fick Law, as:

Ki=k

^C

xi

k=molecular Diffusivity, Fluid dependent. (38

The relevant transport equation can then be written as:

DC

Dt=^K

i

^xi

AQ

^C^ t

AUj^C^x

j

= k ^

2

C^x

jx

j

AQ

(39

This formulation is adequate for many practical application in incompressible flows.

3.5.The Initial and Boundary Condition.

No time-dependent partial derivative set of differential equations can be solved without specifying

adequate initial and boundary conditions. The number and type of these conditions depends on theorder and nature of the equation: these conditions are not obvious for non-linear equations and cannot

be specified with absolute mathematical rigour. Our Fluid Dynamic system of equations is a rather

high order partial differential system of equation of the evolutive6 type. Its ordercannot be clearly

defined, but is assumed to be 9. However there is no way of defining them from completely justified

physical arguments and there is no proof, at the moment, that the solution obtained is unique andcorrect, but all the experience we have at the moment tend to justify it.

Assuming that the flow conditions are such that the dimensions of the smallest solid body in the

flow field are much largerthan the man free path of the fluid molecules, we may say that:

6 This means that the time derivatives of all the unknown are expressed explicitly. This is true for the compressible flow case.

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O The initial conditions require that at initial time the values of all the variable should be

known in the entire flow field.

O The conditions at solid boundaries are more complex, and can be defined at the surface of an

impermeable fixed walls as:

The normal component of the velocity should be equal to zero.

The tangential component of the velocity is assumed to be equal to zero, the so-called

non-slip condition.

O At the surface the normal component of the heat flux into the fluid is equal to the heat flux

from the wall, orthat the wall and fluid temperatures are equal.

U t= 0

= known for all x Tt=0

= known for all x

Un wall =0

Ut wall = 0

k^T

^ n wall

= qwall

else U fluid wall =Twall

. (40

4. The non-dimensional Formulation of a Problem.

In the set of equations derived before we can identifyfour basic physical dimension:

a LengthL,

a Time t,

a MassM,

a Temperature T.

Each other quantity which appears in the equations has dimensions which are products of these

fundamental one, such as:

the Velocity, U=L/t,

the Pressure,p=M/Ltt,

...Once the basic physical quantities are selected and an appropriate, consistent, system of units

chosen, each variable and coefficient in the equation can be quantified by a dimensionless number,

which can always be transformed back into a physically measurable value. These dimensionless

numbers and the solution of the equations containing them are independent of the system of units

chosen and so, in a certain sense, universal.

Rewriting the basic equations in non-dimensional form has then not only a number of advantages,

but it is also a way the identify the dimensionless parameter, or similarity numbers, that play an

important role in any specific problem to be analysed and solved.

Then, considering the key points about the structure of the solution of the equations in

dimensionless form, we can make a few important statements.

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Non-dimensionalizing and scaling an equation and specifying the:

Initial and Boundary conditions in dimensionless form,

The Values of the dimensionless parameters and coefficients,

and solving them we have that:

For any setof the physical reference quantities (length, time,...) and fluid specific parameter(density, viscosity,...) that are consistent with the chosen values of the dimensionlesscoefficients, the flow fields obtained by reversing the scaling, i.e. making the result

dimensional, are valid solutions of the original unscaled equations.

Then: each set of dimensionless parameters defines a family ofequivalent solutions, which onlydiffers by a scale factor. This is the basis for similarity analysis, for numerical and experimental

modelling and order of magnitude evaluations.

Furthermore, close inspection of the dimensionless equations reveals that:

The contribution of some terms can be neglected as function of the relative value of the

dimensionless parameters,

Singularities may arise when one or more of these parameters approach zero or infinity,

making the eventual solution impossible or meaningless.

4.1.The Dimensionless Formulation of the Fluid Mechanic Equations.

Following the concept of the previous section, indicating for simplicity with a ' the dimensional

quantities (x', t', ...) and introducing some reference values as:

lref

, ref

, Uref,

pref,

tref,

Tref,

...

wecanwritex=

x'

lref

, '

ref

,U'

Uref

, t=t'

tref

, T=T'

Tref,

...

(41

and substitute to transform the equations as:

for Continuity:

lref

tref

Uref

^

^ tAÛ

i

^xi

= 0 }1

St

^ '

^ t'A^ 'U'

i

^x'i

=0 (42

For Momentum:

lref

tref

Uref

Û'i

^ t'AU'

j

Û'i

x'j

=Bp

ref

'ref

Uref

2

^ p

xi

A

ref

Uref

lref

^ 'ij

x'j

(43

And finally for the Momentum Equation:

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1

St '

^U'i

^ t'A 'U'

j

^ U'i

^ x'j

= BBe^ p'^ x'

i

A1

Rey

^ 'ij

^ x'j

(44

For the Internal Energy equation, with the equation of state and using a similarapproach :

p

=RT with M=

U

aa = RT (45

1

St '

^ e'^ t'

A 'U'j

^ e'A p' ^x'

j

=1

Rey

^ U'i

^ x'j

M2

'ij A

1

Pr

^2

T'

^ x'i^x'

i (46

4.1.1.The Dimensionless Parameters.

The coefficients which we have introduced are the Dimensionless Parameter that characterise the

universal solutions of our equations. They are:

Sr Strouhal Number tU/L,

M Mach Number U/c,

Rey Reynolds Number UL/ =UL/,

Pr Prandtl Number Cp/k.

Be Bernoulli Number p/U^2

Add we also considered mass diffusivity (D), chemical reaction rate () , heat transfer, ..., we willhave found, for the relevant equations other parameters, such as:

Sc Schmidt Number /D,

Da Damkohler Number L/U,

Pe Peclet Number ....,

Nu Nusselt Number ....,

...

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4.1.2.The Incompressible Formulation.

The effect ofMach Numberin the momentum equations is worth some consideration: if the fluid isconsidered as incompressible, then M=0, and a certain formulation of the Momentum equationcontains a singularity7. This is no a real problem since all fluids are compressible, but, written in this

way the equation is difficult to solve, especially numerically, because of the large difference in valuebetween the terms. So we may consider a simplification for the incompressible flow as a limiting casein the following manner.

If the flow is incompressible, then = cnst., and

the Continuity equation simplify to:

^ Ui

^xi

= 0 } divergence U = 0 (47

even if the flow is unsteady.

Thus in the whole set of equation there is a divergence constraintwhich destroys the velocity-density interaction, and it is no more possible to write the full set of equation in the Conservativeform.

For theMomentum equation, the equation of state remains valid, but we must write:

p

=RT

^ p^

T

=' c= speed of sound=' (48

in the formulation8:

^ U

i

^ tA

^

xj

UiU

jA

1

M2

^ p

xi A

1

Rey

^ ij

xj

(49

a singularity appears for M = 0 and unless we want to derive a complex limit formulation we mustcome back to the less sophisticated formulation of eq. 43:

1

St

^Ui

^ t AUj^ U

i

^x j = B

1

Be

1

^ p

^x i A

1

Rey

^ ij

^x j (50

The Internal Energy equation is modified even more drastically, considering eq. 25, (here M=0 donot lead to singularities) and reduces to:

7 See Klein R., Numerics in Combustion, VKI LS "Introduction to Turbulent Combustion", 1999.

8 Ibidem.

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1

St

^ e^ t

A Uj

^ e^x

j

=1

Pe

^2

T

^xi^x

i

(51

i.e., a purely thermal relation.

No fluid, and especially no gas, is perfectly incompressible, so we may look, using the previousassumptions, to the limits under which it can be considered as incompressible.

Using again the equation of state:

p

=RT }

^^p = 1RT but pZU

2

} ZU

2

RT

=B

ref

ref

ZU

2

c2 =

1

M2 Z 1BM

2

if MV1

(52

The usual limit to treat the flow as incompressible is considered to be M


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^

^ tAU

j

^

^ xj

A Û

i

xj

= 0 Continuity.

Û

i

^ tA U

j

Ûi

^ xj

= B^ p^ x

i

A gi

Momentum.

ê^ t

A Uj

^ e^ x

j

= B pÛ

i

^ xj

Internal Energy. e=CvT

^ S^ t

A^ S U

i

^ xi

=0 Entropy Equation.

^ h

^ xi

= BÛ

i

^ tBU

j

Ûi

^ xj

Enthalpy Equation. ^ h=Cp

^T

p=RT Equation of Status.

(55

It should be noted that the Euler Equation, and their solutions, are not at all equivalent to theequation which can be obtained by letting and k go to zero in the Navier Stokes Equations: they donot contain second order derivatives and have a totally different mathematical nature. Also, due to thelower order, one boundary condition has to be relaxed: the no-slip condition at the wall. The value of

the tangential component of the wall velocity is also part of the solution.

The Euler equation can be considered as a reasonable approximation for the flow field far awayfrom solid walls along which the action of the vorticity is concentrated because of the large localvelocity gradients existing there.

5.1.The Vorticity Equation.

From the Euler equations can be easily derived an equation for the Vorticity in an ideal fluid.

Recalling that:

= rot U

grad U2

=2 U Drot U AUj

Ûi

xj

(56

substituting in the Enthalpy equation and taking the rot( ) of the resultwe have the VorticityEquation:

rot^ U

^ t B rot U D rotU = 0

^ i

^ tAU

j

^ i

^ xj

= j

Ûi

^ xj

(57

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The interest of this equation is that thepressure term has disappeared, but a new term appears, the

velocity-vorticity interaction, or vortex stretching term10, on the right side of eq. 56.

5.2.The Bernoulli Equation.

From the Euler equation it is easy to derive the famousBernoulli Equation in its generalised form11.

If the fluid flow is irrotational that is rot(U) = 0 everywhere, then it exits a scalar function , called

the velocity potential such that:

Uix,y,z =

^ x,y,z

xi

(58

Then, from the Momentum equation in (54) we can derive:

^

xi

^

^ t A 12 ^xi

^

xk

2

=B1

^p

xi

A g (59

If the body forces are also conservative , that can be expressed by a potential which for the gravity

forces is = gz, then, expressing the pressure term as a function of the internal energy e, we have:

p

=

p

Ae and:

^ ^ t A12

^ ^xk

2

A p Ae A gz = const. (60

which is the generalised Bernoulli equation, that broadly states that the total energy of the flowremains constantfor an ideal, irrotational fluid.

The first term accounts for the time dependence of the flow and implies that the situation for:

an accelerating body in fluid at rest a body at rest in accelerating flow.

For a Steady, Incompressible Flow we have the well known relation:

1

2U2 A

p

A gz = const. (61

5.2.1.The equation for the Potential.

Continuing along the previous considerations it is possible to derive the generalised equation forthe potential in an ideal, irrotational flow as:

10 For more details see: Olivari, VKI CN 155.11 Daniel Bernoulli (1700-1782) is a swiss mathematician and physicist and first derived this principle.

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^2

^ t2 A 2

^ ^ x

i

^2

^ t x

i

A^ ^ x

i

^ ^ x

k

B a2

ik ^

2

^x

i^ x

k

= 0

a = RT= speed of sound.

(62

This equation is interesting because, in spite of the flow being compressible, it does not contain

explicitly the pressure nor the density. They can be derived, as function of velocity an temperature,from the Bernoulli equation.

For a steady flow they simplify to:

^ ^ x

i

^ ^ x

k

B a2

ik ^

2

^ x

i^ x

k

= 0 (63

The solution is relatively simple and many analytical solutions have been found. However a word ofcaution should be said, because depending on the value of the velocity they change of nature: they areelliptical for U < a and hyperbolic for U >a, and require a different approach for the solution in thetwo cases. The main problem is that there are flows which are elliptic in some part of the field andhyperbolic in others without well defined boundaries, such as the accelerating-decelerating flow on awing at transonic velocity.

It should also be noted that the steady potential flow is the object of the d'Alambert12 paradox,which states that bodies immersed in such a flow do not experience any aerodynamic force.

6. The Boundary Layer Simplification.

As well as the Euler simplification applies far from solid walls, another simplification can beapplied for the N.S. Equations close to them, because there the flow field is essentially aligned alonga dominant direction.

If in a 2-dimensional incompressible flow, say, Ux is the dominant velocity because the flow isalong a flat plate, then, at least near the plate, Uy >V , then the region y over which V velocity gradients exist is much smaller than theregionx for similar Ugradients. How much smaller it is can be derived by the momentum equation.

Considering thex-component:

UÛ

xAV

Û^y

=B^ p

xA

^y

Û

yA

^Vx

(65

and neglecting the pressure gradient, which is correct for a flat plate:

12 The French mathematician d'Alembert (1717-1783) discovered this around 1750.

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Û

^yW

^V

xU

Û

xZV

Û

^y

UÛ

xZ

^2

U

^y2 }

U

xZ

y 2 }

y

x=

xU}

y

x=

1

Reynolds

(66

So if the Reynolds' number is sufficiently high the transversal region interested by the velocitygradients is much smaller than the region interested by the longitudinal ones. The flow is advective

in the longitudinal direction and diffusive in the transversal one. This region is what Prandtl13 calledBoundary Layerover a solid surface.

Under these conditions the momentum equations take a considerably simpler form. In fact:

Û

^yW

^V

xV

Û

^yZU

Û

xU

^V

xZV

^V

^yVU

Û

x... (67

and thex-componentof momentum equation can be written as:

UÛ

xAV

Û

y=B

^p

xA

^

y

Û

y (68

while the y-component contribution to the overall vectorial momentum becomes negligible andreduces to:

^p

x=0 p=const. across the layer=p

out(69

pout , the pressure outside the boundary layer, can then be approximated by the pressure on the body

computed with theEuler formulation.

Similar simplification can be obtained for compressible flows and the Energy and Entropy14

equations, but this is outside the scope of the present summary.

This approach remains valid and acceptable until there is no flow separation from the bodycontour; if this happen then the only valid approach is given by the full N.S. Equations.

13 Ludwig Prandtl (1875-1953) defined it so in a now famous paper in 1904.14 See: G. Degrez, VKI Lecture Notes on Boundary Layers.

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Table of Contents

1.The Continuum.........................................................................................................2

2.The generalised Transport Equation.........................................................................2

3.The transport equation for a fluid.............................................................................33.1.Mass Conservation............................................................................................3

3.2.Momentum Conservation..................................................................................4

3.2.2.The pressure term..........................................................................................53.2.3.The Shear Term.............................................................................................5

3.2.4.The Final Formulation....................................................................................63.3.Energy Conservation.........................................................................................7

3.3.1.The Fourier law.............................................................................................7

3.3.2.The mechanical work.....................................................................................7

3.3.3.The final formulation......................................................................................8

3.3.4.The Internal Energy Equation........................................................................83.3.5.The Role of the Stress Term...........................................................................8

3.3.6.The equation for Entropy...............................................................................93.4.The Scalar Transport Equation........................................................................10

3.5.The Initial and Boundary Condition.................................................................10

4.The non-dimensional Formulation of a Problem......................................................114.1.The Dimensionless Formulation of the Fluid Mechanic Equations......... .... ......12

4.1.1.The Dimensionless Parameters.....................................................................13

4.1.2.The Incompressible Formulation..................................................................135.The Euler Equations...............................................................................................15

5.1.The Vorticity Equation....................................................................................16

5.2.The Bernoulli Equation...................................................................................175.2.1.The equation for the Potential......................................................................17

6.The Boundary Layer Simplification.........................................................................18

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differential equations of fluid dynamics

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