1. Fundamental Equations for Newtonian Fluids

1.1 Introduction

In this chapter we recall the standard form of the classical fluid dynamics

equations in eulerian form. For the evolution of a fluid in 1N spatial dimensions, the description involves ( )2N + fields. These are the mass density, the velocity field and the energy. The fundamental equations of

fluid dynamics are derived following the next three universal conservation

laws:

1. Conservation of Mass.

2. Conservation of Momentum or the Newtons Second Law.

3. Conservation of Energy or the First Law of Thermodynamics.

The resulting equations are the continuity equation, the momentum

equations and the energy equation, respectively. The system of governing

equations is closed by:

1. Statements characterizing the thermodynamic behavior of the fluid.

These are called state equations. The state equations do not depend

on the fluid motion and represent relationships between the

thermodynamic variable: pressure (p), density ( ), temperature (T), specific internal energy per unit mass (e) and total enthalpy per unit

mass (h). The microscopic transport properties like the dynamic

viscosity ( ) and the thermal conductivity (k) are also related to the thermodynamic variables.

2. Constitutive relations, which are postulated relationships between

stress and rate of strain and heat flux and temperature gradient. The

previously mentioned microscopic transport properties appear in the

constitutive relations as proportionality factors (between stress and

rate of strain and heat flux and temperature gradient) and must be

determined experimentally.

2

The flow is considered as known if, at any instant in time, the velocity field

and a precise number of thermodynamic variables are known at any point.

This number equals two for a real compressible fluid in thermodynamic

equilibrium (for example, the pressure and the density).

There is some freedom in choosing a set of variables to describe the fluid

flow. A possible choice is the set of primitive or physical variables,

namely the density, pressure and velocity components along the three

directions of a reference frame, i.e. the set ( ), , , ,p u v w . Another choice is the set of the so-called conservative variables which are the density, the

three momentum components and the total energy per unit mass, i.e. the set

( ), , , ,u v w E . The set of conservative variables results naturally from the application of the above fundamental laws of conservation. From a

computational point of view, there are some advantages in expressing the

governing equations in terms of the conservative variables. We will call

conservative methods the class of numerical methods based on the

formulation of the governing equations in conservative variable.

The primitive and conservative variables are dependent variables. They are

depending on time and space coordinates, i.e. on the independent

variables. For example, the density is ( ), , ,t x z y = . Recall that the number of the spatial coordinates is N.

The reader is invited here to notice an important difference between the

governing equations and the closure conditions. The governing equations

are exact mathematical conditions that must be satisfied by the flow

variables. The state equations and the constitutive relations are only

practical models and they depend on the degree of knowledge the user has

about the fluid and flow under consideration. Thus, they represent sources

of uncertainties that are introduced in the mathematical model of the flow.

The derivation of the fundamental equations mentioned above will not be

presented here. Almost all of the books devoted to fluid mechanics or to

CFD have chapters devoted to this issue. For example, more details on the

equations can be found in (Hirsch, 1990), (Ferziger and Peric, 1999) and

(Warsi, 1999).

3

1.2 Preliminary Concepts

Before proceeding to the derivation of the equations in integral form review

some preliminary concepts that are needed. Consider a scalar field function

),,,( tzyx , then the time rate of change of as registered by an observer moving with the fluid velocity ( . , )u v w=V is given by the substantial derivative (or material derivative) defined as:

D

gradDt t

= +

V (1.1)

The first term t

denotes the partial derivative of with respect to time

and represents the local rate of change of while the second term is the convective rate of change. If the scalar field is replaced by a vector field, the

operator Dt

Dcan also be applied in a component-wise manner. In particular,

we can obtain the substantial derivatives of ( . , )u v w=V , that is:

D

gradDt t

= +

V V

V V (1.2)

which in full becomes:

( , , )

T TDu u u v w

Dt t x x x

Dv v u v wu v w

Dt t u y y

Dw w u v w

Dt t z z z

= +

(1.3)

The left hand side of (1.3) is the acceleration vector of an element of moving

fluid. Let us now consider the new scalar field

( ) ( , , , )V

t x y z t dV = (1.4) where the volume of integration V is enclosed by a piece-wise smooth

boundary surface A V= that moves with the material under consideration. It can be shown that the material derivative of is given by:

( )V AD

dV dADt t

= +

n V (1.5)

4

where 1 2 3( , , )n n n=n is the outward pointing unit vector normal to the

surface A. Expression (1.5) can be generalized to vectors ( , , , )x y z t as

follows:

( )V AD

dV dADt t

= +

n V (1.6)

The surface integrals in the last equalities may be transformed to a volume

integral by virtue of Gausss theorem. This states that for any differentiable

vector field 1 2 3( , , ) = and a volume V with smooth bounding surface A the following identity holds:

( )A V

dA div dV = n (1.7) The above identity also applies to differentiable scalar and tensor fields. If

we use it, for example, to transform the right hand side of equation (1.5) into

a volume integral, one obtains:

( )VD

div dVDt t

= + V (1.8)

Please notice that in the above relations the volume V is arbitrary and

moving with the fluid particles. Let J be the jacobian of the transform that

links two successive positions of the volume V. There is a theorem due to

Euler which states that the time derivative of the jacobian is (Lions, 1996):

D J

J divDt

= V (1.9)

From (1.9) we remark that mathematically, the incompressibility means:

0div =V (1.10)

1.3 Integral Form of the Governing Equations

1.3.1 Conservation of Mass

The law conservation of mass can now be stated in integral form by

identifying the scalar in (1.4) as the density . In this case )(t in (1.5) becomes the total mass in the volume V. By assuming that no mass is

generated or annihilated within the material control volume V, we have

0=Dt

D, or:

5

( ) 0V AdV dAt

+ =

n V (1.11) This is the integral form of the law of the conservation of mass. This

integral conservation law may be generalised to include sources of mass ,

which will then appear as additional integral terms. A useful reinterpretation

of the integral form result if we rewrite it as

( )V A

dV dAt

=

n V (1.12) If now V is a fixed control volume then the left hand side of (1.12) becomes:

V V

ddV dV

t dt

=

(1.13) and thus is the time-rate of change of the mass enclosed by the volume V.

The right hand side of (1.12) shows that the mass enclosed by the control

volume V, in the absence of sources or skins, can only change by virtue of

mass flow through the boundary of the control volume V. One obtains from

(1.8) and (1.12):

( ) 0V

div dVt

+ = V (1.14)

As V is arbitrary it follows that the integrand must vanish, that is:

( ) ( ) ( ) 0t x yu v w + + + = (1.15) which is the differential form of the continuity equation.

We notice here that the differential and integral forms (1.15) and (1.14) are

not valid in general. Under the assumption of inviscid fluids, the governing

equations admit discontinuities in the solution, such as shock waves and

contract surfaces. The integral form (1.11), however, remains valid.

1.3.2 Conservation of Momentum

As done for the mass equation, we now provide the foundations for the law

of conservation of momentum, derive its integral form in quite general

terms and show that under appropriate smoothness assumptions the

differential form is implied by the integral form. A control volume V with

bounding surface A is chosen and the total momentum in V is given by:

( )V

t dV= V (1.16) The law of conservation of momentum results from the direct application of

Newtons law: the time rate of change of the momentum in V is equal to the

6

total force acting on the volume V. The total force is divided into surface

forces fs and volume fv given by:

, vA VdA dV= = sf S f g (1.17)

Here g is the specific volume-force vector and may account for inertial

forces, gravitational forces, electromagnetic forces and so on. S is the stress

vector, which is given in terms of a stress tensor as = S n . The stress tensor can be split into a spherical symmetric part due to pressure p, and

a viscous part :

p= I (1.18) Application of Newtons Law gives:

( ) ( ) sV A

dV dAt

= + + vV V n V f f (1.19)

Regarding V as a fixed volume in space, independent of time, we write:

( ) ( )V A

ddV dA

dt = + + s vV V n V f f ,

We interpret the above as saying that the time rate of change of momentum

within the fixed control volume V is due to the net momentum inflow over

momentum outflow, given by the first term in (1.19), plus surface and

volume forces.

Substituting the stress tensor from (1.18) into (1.19) and writing all surface

terms into a single integral we have:

[ ]( ) ( ) .V A V

ddV p dA dV

dt = + + V V n V n n g (1.20)

This is a general statement and is valid even for the case of discontinuous

solutions.

The differential conservation law can now be derived from (1.20) under

assumption that the integrand in the surface integral is sufficiently smooth

so that Gausss theorem may be invoked. The first term of the integrand of

the surface integral can be rewritten as:

( ) , = V n V n V V where the dyadic (tensor) product VV is a tensor. Its components are easily determined from the next equality. The three columns of the left hand

side are:

7

2

2

2

( ) , , ,

( ) , , ,

( ) , , .

T

T

T

u u uv uw

v uv v vw

w uw vw w

=

=

=

n V n

n V n

n V n

Now one can apply the Gausss theorem to each of the surface terms:

[ ]( ) ( ) .V V V

div gradp div dV dVt

= + + V V V g (1.21)

As this is valid for any arbitrary volume V the intregrand must vanish, i.e.:

[ ]( ) div pt

+ + =

V V V I g (1.22)

This is the differential conservative form of the momentum equation,

including a source term due to volume forces. In order to use it for a given

fluid flow the components of the viscous stress tensor must be provided

through a constitutive relation.

When the viscous stresses are identically zero, and the volume forces are

neglected, we obtain the so-called compressible Euler equation. If the

viscous stresses are given by a constitutive relation under the Newtonian

assumption we obtain the compressible Navier-Stokes equation in

differential conservation law form.

1.3.3 Conservation of Energy

As done for mass and momentum we now consider the total energy in a

control volume V, that is:

( ) .V

t EdV = (1.23) where E is the total energy per unit volume:

( )2 2 2 21 12 2

E e e u v w = + = + + +

V (1.24)

Here e is the specific internal energy and the second term represents the

specific kinetic energy.

Before we discuss the equations in more detail, let us make a physical

comment: the derivation of the energy equation relies upon the assumption

that, in a fluid in motion, the fluctuation around thermodynamic equilibrium

are sufficiently weak so that the classical thermodynamics results hold at

8

every point and at all times. In particular, the thermodynamical state of the

fluid is determined by the same state variables as in classical

thermodynamics, and these variables are determined by the same state

equations (see the devoted paragraph in this chapter).

The energy conservation law states that the time rate of change of total

energy )(t is equal to the work done, per unit time, by all forces acting on the volume plus the influx of energy per unit time into the volume. The

surface and volume forces in (1.17) respectively give rise to the following

terms:

( ) ( )surfA A

E p dA dA= + V n V n , ( ) .vol

VE dV= V g

The first term corresponds to the work done by the pressure while the

second term corresponds to the work done by the viscous stresses. volE is

the work done by the volume force g. To account for the influx of energy

into the volume we denote the energy flux vector by 1 2 3( , , )q q q=Q . The flow of energy per unit time across a surface dA is given by the flux

( )dA n Q . This gives the total influx of energy to be included in the equation of balance of energy:

inf ( )A

E dA= n Q , Thus, one can write that:

infsurf volD

E E EDt

= + + (1.25)

The left hand side of (1.25) can be transformed into:

( )

( )V A

D tEdV E dA

Dt t

= +

n V (1.26) As done for the laws of conservation of mass momentum we now reinterpret

the volume V as fixed in space and independent of time. This leads to:

[ ]( ) ( ( )V A V

dEdV E p dA dV

dt= + + + n V V Q V n V g (1.27)

The differential form of the conservation of energy law can now be derived

by assuming sufficient smoothness and applying Gausss theorem to all

surface integrals of (1.27). Direct application of Gausss theorem gives:

( ) ( )A V

p dA div p dV = n V V

9

.A V

dA div dV = n Q Q The second term of (1.27) can be transformed via Gausss theorem by first

observing that ( ) ( ). = V n n V This follows from the symmetry of the viscous stress tensor . Hence:

( ) ( )A V

dA div dV = V n V Substitution of these volume integrals into the integral form of the law of

conservation of energy (1.84) gives:

( ){ }( ) 0tVE div E p dV+ + + = V V Q V g (1.28)

Since the volume V is arbitrary the integrand must vanish identically, that

is:

( ) ( ) 0tE div E p + + + = V V Q V g (1.29) This is the differential conservative form of the law of conservation of

energy with a source term accounting for effect of body forces; if these are

neglected we obtain the homogeneous energy equation corresponding to the

Navier-Stokes equation. The energy flux vector Q must be specified through

a constitutive relation.

Further, when viscous and heat conduction effects are neglected we obtain

the energy equation corresponding to the compressible Euler equations.

1.3.4 The Role of the Integral Form of the Equations

The previous derivation of the governing equations, such as the

compressible Euler and Navier-Stokes equations stated earlier, is based on

integral relations on control volume and their boundaries.

The differential form of the equations results from further assumptions on

the flow variables (smoothness). In the case of the Navier-Stokes equations,

the smoothness of the solution is naturally assured by the diffusion due to

the viscosity and heat conductivity. On the contrary, in the absence of

viscous diffusion and heat conduction one obtains the Euler equations.

These admit discontinuous solutions and the smoothness assumption that

leads to the differential form no longer holds true. Thus, one must return to

the more fundamental integral from involving integrals over control

volumes and their boundaries. However, for most of the theoretical

developments the differential form of the equations is preferred.

10

Form a computational point of view there is another good reason for

returning to the integral form of the equations. Discretised domains result

naturally in finite control volumes or computational cells. Local

enforcement of the fundamental equations in these volumes leads to Finite

Volume numerical methods.

1.4 Equations of State

1.4.1 Generalities

The governing equations expressed in integral or differential form are

describing the dynamics of a compressible material. However, these

equations are insufficient to completely describe the physical processes

involved during a flow. There are more unknowns than equations and thus

closure conditions are required. For example, if we take as unknowns the

five conservative variables ( ), ,E V , it is easy to see that there are terms in the integral or differential form of the governing equations which must be

expressed in terms of the unknowns. Such terms are the pressure and the

specific internal energy. One therefore requires another relation defining e

and p in terms of quantities already given, such as the conservative

variables. For a number of applications, due to the complexity of the

physical phenomena other variables, such as temperature T or entropy s for

instance, may need to be introduced.

The formulation of the closure conditions we have referred to previously is

provided by Thermodynamics under the form of state equations. The

specific internal energy e has an important role in the First Law of

Thermodynamics, while the entropy s is involved with the Second Law of

Thermodynamics. We mention here that the entropy plays an important not

just in establishing the governing equations but also at the level of their

mathematical properties and the designing of numerical methods to solve

them. Details can be found in (Lions, 1996) and (Toro, 1997).

11

1.4.2 Equations of State for Systems in Thermodynamic Equilibrium

A system in thermodynamic equilibrium can be completely described by the

basic thermodynamic variables pressure p and the specific volume . The specific volume is defined as

1

= (1.30)

A family of states in thermodynamic equilibrium may be described by a

curve in the p plane each characterized by a particular value of a variable temperature T. There are physical situations that require additional

variables. Here we are only interested in p T systems. In these, one can relate the variables via the thermal equation of state:

( ),T T p = (1.31) The p T relationship changes from substance to substance. For thermally ideal gases one has the simple expression:

p p

TR R

= = (1.32)

where R is a constant which depends on the particular gas under

consideration.

The First Law of Thermodynamics states that for a non-adiabatic system

the change e in internal energy e in a process will be given by e=W

+Q, where W is the work done on the system and Q is the heat

transmitted to the system. Taking the work done as dW = - pdv one may

write:

dQ de pd= + (1.33) The internal energy e can also be related to the pressure and specific volume

through a caloric equation of state:

( ),e e p = (1.34) The relations (1.31) and (1.34) are invertible.

For a calorically ideal gas one has the simple expression or equation of

state (EOS):

( 1)

pe

=

(1.35)

where is a constant that depend on the particular gas under consideration.

12

The thermal and caloric equation of state for a given material are closely

related. Both are necessary for a complete description of the

thermodynamics of a system. Choosing a thermal EOS does restrict the

choice of a caloric EOS but does not determinate it. Note that for the Euler

equations one requires a caloric EOS , e.g. ( ),p p e= unless temperature T is needed for some other purpose, in which case a thermal EOS needs to

be given explicitly.

1.4.3 Other Thermodynamic Variables and Relations

The entropy s results as follows. We first introduce an integrating factor 1/T

so that that expression

e e

de pd p d dpv p

+ = + +

(1.36)

becomes an exact differential. Then for the Second Law of

Thermodynamics one introduces the new variable s, called entropy, via the

relation:

Tds de pd= + (1.37)

For any physical process the change in entropy is due to the entropy carried

into the system through the boundaries of the system, 0s , and to the

entropy generated in the system during the process, is . Examples of entropy-generating mechanism are heat transfer, and viscosity such as may

operate within the internal structure of shock waves.

The importance of the entropy in fluid dynamics and not only resides from

the Second Law of Thermodynamics. This states that

0is (1.38) in any physical process. If the process is reversible then the entropy does not

change, i.e. 0is = . An equivalent but more useful form of the equation (1.38) is the next one:

( ) ( ) 0t

s div s + V (1.39) As long as shocks do not form during the flow, the entropy equation is valid

and if initially s is constant, it remains constant. This is a very strong quality

13

criterion for the evaluation of different numerical schemes developed for the

Euler equations and not only.

Another variable of interest is the specific enthalpy h. This is defined in

terms of other thermodynamic variables, namely:

h e p= + (1.40)

One can also establish other relationship amongst the basic thermodynamic

variable already defined. For instance one may choose to express the

internal energy e in terms of the variables appearing in the differentials, i.e.

).,( vsee =

Also, taking the differential of (1.40) we have ,vdppdvdedh ++= which becomes ,vdpTdsdh += and thus we can choose to define h in terms of s and p, i.e. ).,( pshh =

We call the last two relations canonical equations of state and, unlike the

thermal and caloric equations of states (1.31) and (1.34), each of these

provides a complete description of the Thermodynamics.

Two more quantities can be defined if a thermal EOS ( ),p T = . These are the volume expansivity (or expansion coefficient) and the isothermal

conpressibility , namely:

pT

v

v

=1

and

Tp

v

v

=1

.

The heat capacity at constant pressure cp and the heat capacity at

constant volume cv (specific heat capacities) are now introduced. In

general, when an addition of heat dQ changes the temperature by dT the

ratio dTdQc /= is called the heat capacity of the system. For a thermodynamic process at constant pressure one obtains:

,),( dhvpddedQ =+=

14

where the definition of the specific enthalpy has been used. Assuming

),( pThh = , the heat capacity cp at constant pressure becomes:

pp p

h sc T

T T

= = (1.41)

The heat capacity cv at constant volume may be written, following a similar

argument, as:

vv v

e sc T

T T

= = (1.42)

The speed of sound is another variable of fundamental interest. For flows

in which particles undergo unconstrained thermodynamic equilibrium one

defines a new state variable a, called the equilibrium speed of sound or

just speed of sound. Given a caloric equation of state such as ),,( spp = one defines the speed of sound a as:

s

pa

= (1.43)

This basic definition can be transformed in various ways using established

thermodynamic relations. For instance, given a caloric EOS in the form

),,( phh = we can also write:

+

+=

+

dss

pd

pTdsd

hdp

p

h

spp

1

Settings ds=0 and using definition (1.43) we obtain successively:

1

2

=

p

h

h

ap

and for a thermally ideal gas characterized by the EOS of the form h= h(T) :

1

2

=

p

pp

h

h

a

15

From pp

ch

=

and for a thermally ideal gas (i.e., if the thermal EOS is

acceptable), we obtain the widely used relation for the speed of sound:

( )P

a R

= = (1.44)

The reader should notice the dependence on the temperature of the

coefficient .

For a general material the caloric EOS is a functional relationship involving

the variable p--e. The derived expression for the speed of sound a depends

on the choice of dependent variables. Two possible choices and their

respective expression for a are:

2

2

( , ),

( , ),

e

p p

pp p e a p p

epe e p a

e e

= = +

= =

(1.45)

where subscripts denote partial derivates.

1.4.4 Ideal Gases

We consider in what follows gases obeying the ideal thermal EOS of the

form:

pV nRT= (1.46) where V is the volume, R=8.134 x 10

3 J kilomole

-1, called the Universal

Gas Constant, and T is the absolute temperature measured in degrees

Kelvin (K). Recall that a mole of a substance is numerically equal to gram

and contains 6.02x1023 particles of that substance, where is the relative

atomic mass or relative molecular mass; 1 kilomole = kg. One kilomole of

substance contains NA = 6.02 x 1026 particles of that substance. The constant

NA is called the Avogadro Number. Sometimes this number is given in

terms of one mole. The Boltzmann constant k is k = RNA. The number of

kilomoles in volume V is n = N/NA and N is the number of molecules. On

division by the mass m = n have:

,p RT R

= = (1.47)

16

or

p

RT

= (1.48)

where R is called the Gas constant. Equation (1.48) is known as Mariottes

law. Further, we obtain from (1.47) the ideal gas thermal equation of state

as:

.),(p

RTpTvv ==

The volume expansivity and the isothermal compressibility defined

previously become 1 1and

T p = = , respectively. After some

manipulations, one deduce that:

0T

e

v

= (1.49)

This means that if the ideal thermal EOS is assumed, then it follows that the

internal energy e is a function of temperature alone, that is e=e(T). A

remarkable particular case is Joules law, that is one in which:

ve c T= (1.50) where the specific heat capacity cv is a constant. In this case one speaks of

a calorically ideal gas, or a polytropic gas.

It is now possible to relate cp and cv via the general expression 2

.p vT

c c

= + For a thermally ideal gas equations this expression

simplifies to:

p vc c R = (1.51)

We mention here that, a necessary condition for thermal stability is cv>0 and

a condition for mechanical stability is >0, see (Toro, 1997). From (1.51) the

following inequalities result 0p vc c> > .

The ratio of specific heats , or adiabatic exponent, is defined as:

p

v

c

c = (1.52)

which if used in conjuntion with (1.51) gives:

17

, .1 1

p v

R Rc c

= =

(1.53)

In order to determinate the caloric EOS (1.50) we need to determine the

specific heat capacities, cv in particular. Molecular Theory and the principle

of equipartition of energy (Toro, 1997) can also provide an expression for

the specific internal energy of molecule. In general a molecule, however

complex, has M degrees of freedom, of which three are translational. Other

possible degrees of freedom are rotational and vibrational. From Molecular

Theory it can be shown that if the energy associated with any degree of

freedom is a quadratic function in the appropriate variable expressing that

degree of freedom, then the mean value of the energy is kT2

1where k is the

Boltzmann constant. Moreover, from the principle of equipartition of energy

this is the same for each degree of freedom. Therefore, the mean total

energy of a molecule is 1

2e MkT= and for N molecule we have

1

2Ne MNkT=

From which the specific internal energy is

.2

1MRTe =

From the above considerations we obtain successively:

,2

1MR

T

ec

v

v =

=

and

.2

2R

Mcp

+=

Thus, the ratio of specific heat becomes can be expressed in terms of the

degrees of freedom M:

.2

M

M +=

And hence:

1

2

=

M

From the thermal EOS for ideal gases we have:

18

.2

1Mpve =

And

( ) ( )1 1pv p

ep

= =

(1.54)

Thus, we have found once again the expression for the specific internal

energy.

These theoretical expressions for the specific heats and their ratio in terms

of R and M are found to be very accurate for monatomic gases, when M = 3

(three translational degrees of freedom). For polyatomic gases rotational and

vibrational degrees of freedom contribute to M but now the expression

might be rather inaccurate when compared with experimental data. A strong

dependence on T is observed. However, the inequality ,3

51

19

The covolume EOS can be further corrected to account for the forces of

attraction between molecules, the so-called van der Waal forces. These are

neglected in both the ideal and covolume equations of state. Accounting for

such forces results in a reduction of the pressure by an amount of c/v2,

where c is a quantity that depend on the particular gas under consideration.

Thus from (1.49) the pressure is corrected as

.2v

c

bv

RTp

=

Then we can write:

2( )

cp v b RT

v

+ =

(1.56)

This is generally known as the van der Waals equation of state for real

gases.

So far, we have presented the governing equations for the dynamics of a

compressible medium along with some EOS so that a closed system is

obtained.

1.5 Viscous Stresses and Heat Conduction

The stresses in a fluid, given by a tensor , are due to the effects of the

thermodynamic pressure p and the viscous stresses. Thus the stress tensor

can be written as

p= + I (1.57) where pI is the spherically tensor due to p, I is the unit tensor and is the

viscous stress tensor. It is desirable to express in terms of flow variables

already defined. For the pressure contribution this has already been achieved

by defining p in terms of other thermodynamic variables via an equation of

state. Recall that equations of state are approximate statements about

the nature of a material. In defining the viscous stress contribution one

may resort to the Newtonian approximation whereby is related to the

derivatives of the velocity field ( ), ,u v w=V via the deformation tensor:

20

1 1( ) ( )

2 2

1 1( ) ( )

2 2

1 1( ) ( )

2 2

x x y x z

x y y y z

x z y z z

u v u w u

v u v w v

w u w v w

+ + = + + + +

D (1.58)

The Newtonian assumption is an idealisation in which the relationship

between and D is linear and homogeneous, that is will vanish only if

D vanishes, and the medium is isotropic with respect to this relation. An

isotropic medium is that in which there are no preferred directions. By

denoting the stress tensor by:

xx xy xz

yx yy yz

zx zy zz

=

(1.59)

the Newtonian approximation becomes:

2

2 ( )3

b div = +

D V I (1.60)

In full we have:

( )

( )

( )

4 2,

3 3

4 2,

3 3

4 2,

3 3

( ),

( ),

( ).

xx

x y x b

yy

y z x b

zz

z x y b

xy yx

y x

yz zy

z y

zx xz

x z

u v w divV

v w u divV

w u v divV

u v

v w

w u

= + + = + +

= + + = = += = +

= = +

(1.61)

In the Newtonian relationship (1.60) there are two scalar quantities that are

still undetermined. These are the coefficient of shear viscosity and the

coefficient of bulk viscosity b. Approximate expressions for there are

obtained mainly from experiments.

21

In particular, for monatomic gases Molecular Theory gives for the

coefficient of bulk viscosity ,0=b which is found to agree well with experiment. For polyatomic gases 0b and appropriate values for b are to be obtained experimentally.

Concerning the coefficient of shear viscosity , it is observed that, as long as

temperature are not too high, depends strongly on temperature and only

slightly on pressure. A relatively accurate relation between and T is the

Sutherland formula:

1

21 1

CC T

T

= +

(1.62)

where C1 and C2 are two experimentally adjustable constants. When T is

measured in Kelvin degrees, for case of air one has:

.112,1046.1 26

1 KCxC ==

Sutherlands formula describes the dependence of on T rather well for a

wide range of temperatures, provided no dissociation or ionisation take

place. These phenomena occur at very high temperatures where the

dependence of on pressure p, in addition to temperature T, cannot be

neglected.

In summary, the Navier-Stokes equations (momentum equation) can now be

written in differential conservation law form as:

( ) ( ) 0t

p + + =V V V I (1.63) where is given by (for monatomic gases):

2

2 ( )3

div = D V I (1.64)

The Euler equations written in differential conservation form are:

( ) ( ) 0t

p + + =V V V I (1.65)

Influx of energy contributes to the rate of change of total energy E. We

denoted by ( )1 2 3, ,T

q q q=Q the energy flux vector, which results from: i)

heat flow due to temperature gradients, ii) diffusion processes in gas

mixture and (iii) radiation.

In what follows we only consider the effect of the heat flux due to

temperature gradients. Thus, Q is identical to the heat flux vector caused by

22

temperature gradients. In a similar manner to that it which viscous stresses

were related to gradient of the velocity vector V, one can assume that the

fluid is isotropic and thus one can relate Q to gradients of temperature T via

Fouriers heat conduction law:

k T= Q (1.66) where k is a positive scalar quantity already called the coefficient of

thermal conductivity, and is yet to be determined.

Note the analogy between the two microscopic transport properties of the

fluid, and k. This analogy between and k goes further in that k, just as ,

depends on T but only slightly on pressure p.

In fact, Molecular Theory says that k is directly proportional to . Under the

assumption that the specific heat at constant pressure cp is constant, the

dimensionless quantity called the Prandtl number:

p

r

cP

k

= (1.67)

is a constant. For monatomic gases Pr is very nearly constant. For air in the

temperature range KTK 100200 Pr differs only slightly from its mean value of 0.7.

There is also a formula attributed to Eucken, see (Toro, 1997), that relates Pr

to the ratio of specific heats via

59

4

=

rP

to account for departures from calorically ideal gas behaviour.

1.6 Differential Form of the Governing Equations

We have determined previously the integral form of the governing

equations, as well as the conservative differential form of the equations.

Here we summarize the governing equations written in conservative

differential form. We notice that the name Euler and/or Navier-Stokes

originally given to the momentum equations (or equations of motion)

transfers to the entire system of equations.

We summarize below the general laws of conservation of mass, momentum

and total energy, written both in differential and integral form:

23

a) the continuity equation:

( ) 0t div + =V or

( ) 0V A

ddV dA

dt + = n V

b) the momentum eqution:

[ ]( ) ,t div p + + =V V V I g or

[ ]( ) ( ) .V V V

div gradp div dV dVt

+ + = V V V g

c) the energy equation

[ ]( ) ( ).tE div E p + + + = V V Q V g or

[ ]( ) ( ( )V A V

dEdV E p dA dV

dt+ + + = n V V Q V n V g

where ),,( 321 gggg = is a body force vector.

At this point, all the terms, such as the viscous stress tensor, the heat flux,

the pressure, the temperature and the specific internal energy are expressed

as functions of the conservative variables. For practical reasons, it is useful

to write the previous equations in a more compact form, which is called

conservation law form.

1.6.1 The Euler Equations

In this section we consider the time-dependent Euler equations. These are a

system of non-linear hyperbolic conservation laws that govern the dynamics

of a compressible material, such as gases or liquids at high pressure.

When body forces are included via a source term vector but viscous and

heat conduction effects are neglected we have the Euler equations:

( ) ( ) ( ) ( )t x y z+ + + =U F U G U H U S U (1.68)

where the vector of conservative variables is U and the convective fluxes

are F,G,H in the x, y and z directions. Their explicit forms are:

24

2

2

2

, ,

( )

,

( ) ( ).

u

u u p

v uv

w puw

E u E p

wv

uwuv

vwv p

vw w p

v E p w E p

+ = = +

= =+ + + +

U F

G H

(1.69)

First, it is important to note that the fluxes are nonlinear functions of the

conserved variable vector. Any set of partial differential equations written in

the form (1.68) is called a system of conservation law (in differential

formulation). Recall that the differential form assumes smooth solutions,

that is, partial derivatives are assumed to exist. It is also clear now that the

integral form of the equations is an alternative way of expressing the

conservation laws in which the smoothness assumption is relaxed so that to

include discontinuous solutions.

Here ( )=S S U is a source or forcing term. Due to the presence of the source term, the equations are said to be inhomogeneous. There are several

physical effects that can be included in the forcing term: body forces such as

gravity, injection of mass, momentum and/or energy. Usually, ( )S U is a

prescribed algebraic function of the flow variables and does not involve

derivatives of these, but there are exceptions. When ( ) 0=S U one speaks of homogeneous equations. We also mention here that there are situations in

which source terms arise as a consequence of approximating the

homogeneous equations to model situations with particular geometric

features (axy-symmetric flows, for instance). In this case the source term is

of geometric character, but we shall still call it a source term.

Sometimes it is convenient to express the equations in term of the primitive

or physical variables , u, v, w and p. By expanding derivatives in the

conservation law form and using the mass equation into the momentum

25

equations and in turn using these into the energy equation one can re-write

the thee-dimensional Euler equations for ideal gases with a body-force

source term as:

1

2

3

( ) 0

1,

1,

1,

( ) 0

t x y z x y z

t x y z x

t x y z y

t x y z z

t x y z x y z

u v w u v w

u uu vu wu p g

v uv vv wv p g

w uw vw ww p g

p up vp wp p u v w

+ + + + + + =

+ + + + =

+ + + + =

+ + + + =

+ + + + + + =

(1.70)

For computational purposes it is the conservation law form (1.68) that is

most useful. The formulation in primitive variables is more convenient to

theoretical developments.

1.6.2 The Navier-Stokes Equations

When the effects of viscosity and heat conduction are added to the basic

Euler equations one has the Navier-Stokes equations with heat conduction

in conservation law form:

c c c d d dt x y z x y z+ + + = + +U F G H F G H (1.71)

where U is once again the vector of conserved variables, the flux vectors Fc,

Gc and H

c are the inviscid fluxes (c stand for convection) for the Euler

equations as given previously and the respective flux vectors Fd, G

d and H

d

(d stands for diffusion) due to viscosity and heat conduction are:

26

1 2

3

0 0

, '

0

yxxx

yyd dxy

yzxz

xx xy xz yx yy yz

zx

zyd

zz

zx zy zz

u v w q u v w q

u v w q

= = + + + +

= + +

F G

H

(1.72)

The form of the equations given by (1.71) split the effect of convection on

the left-hand side from those of viscous diffusion and heat conduction on

the right-hand side. For numerical purpose, the particular form of the

equations adopted depends largely on the numerical technique to be used to

solve the equations. One possible approach is to split the convection effects

from those of viscous diffusion and heat conduction during a small time

interval t, in which case the above from is perfectly adequate. An

alternative form is obtained by combining the fluxes due to convection,

viscous diffusion and heat conduction into new fluxes so that the governing

equations look formally like a homogeneous system (zero right-hand side)

of conservation laws:

0,

with

, ,

t x y z

c d c d c d

+ + + =

= = =

U F G H

F F F G G G H H H

(1.73)

This form is only justified if the numerical method employed actually

exploits the coupling of convection, viscosity and heat conduction when

defining numerical approximations to the flux vectors F, G and H in (1.73).

27

1.7 Conclusion

We have presented in this chapter the time-dependent Euler and Navier-

Stokes equations of fluid dynamics. The equations are accompanied by

equations of state and by constitutive relations. Both models can be used for

homogeneous gases and/or liquids at high pressure. We appreciate that for

usual purposes the Euler and Navier-Stokes are the representative models.

We now conclude this chapter with a brief discussion of initial and

boundary conditions. Our goal in this book is to present numerical schemes

that can be used to solve the Cauchy problems for the models derived and

described above. In other words, we solve the systems of equations for 0t prescribing the values of all the unknowns at 0t = . The question of the boundary condition is much more delicate. Simply said, one impose

physical boundary condition at the solid boundaries for some of the

variables, and these can be of Dirichlet or Neumann type (depending on the

variable type). Since the boundary conditions are replacing the governing

equations on the boundaries, the rest of the variables are determined via

numerical techniques. These are strongly related to the numerical solvers

and thus we will discuss the imposition of the boundary conditions in the

context of numerical solvers.

A short comment on the Navier-Stokes model is necessary. The equations

presented in this chapter can be used for the numerical simulation of laminar

flows only. At practical Reynolds numbers (see the introduction), the effects

of the turbulence must be taken into account. The Reynolds Averaged

Navier-Stokes (RANS) equations and the Large Eddy Simulation (LES)

equations have formally the same mathematical aspect, with minor

differences. The huge difference comes from the necessity of modeling the

turbulence effects, mainly in the vicinity of solid boundaries. As we already

mentioned, it is beyond the purposes of this book to discuss about

turbulence modeling and turbulence models.

More details about the derivation of these and other equations can be found

in (Anderson et. all, 1984), (Danaila and Berbente, 2003), (Hirsch, 1990),

(Warsi, 1999). A very rigorous discussion of the models and their solutions

can be found in (Lions, 1996).

28

Two remarks we wish to make before at the end of this first chapter. The

first is that the Navier-Stokes and Euler equations are the cornerstones for

the development of practical CFD codes. From a mathematical point of

view, these models are systems of partial differential equations. Since with a

few exceptions we cannot find exact solutions of these equations for

practical problems, the only way is to solve them is to use numerical

techniques. It is generally accepted that the discretization techniques must

be based not only on the underlying physics bust also on the mathematical

properties of the partial differential equations. It is therefore useful to begin

with the analysis of the mathematical nature of the governing equations of

the flow before trying to solve them numerically.

The second remark is that the equations of compressible fluid flow reduce to

hyperbolic conservation laws (i.e. the Euler system) when the effects of

viscosity and heat conduction are neglected. Furthermore, the hyperbolic

part represents the convection and pressure gradient effects and it can be

identified in the equations even the above physical effects are not negligible.

This is the reason why we allocate a special chapter to study the hyperbolic

partial differential equations.

29

2. Approximated and Simplified Models

2.1 Generalities

In this chapter we wish to consider successively simplified version, or

submodels, of the governing equations and their closure conditions. There

are many reasons and many ways to use and derive simplified models. On

one hand, it is clear that any simplification leads to the reduction of the

generality of the equations. On the other hand, a rational simplification may

lead to a significant reduction of the computational effort without penalties

on the quality of the solution. There are some possibilities that can be

exploited to derive simplified models, and perhaps the most common of

them are the reduction of the dimensionality of the problem and the

incompressibility assumption.

One can also augment the basic equations by source terms to account for

additional physics. This is not exactly a simplification of the basic model;

on the contrary, adding some semi-empirical source terms may enlarge the

area of practical applications that can be solved with that model.

In the previous chapter we already done a significant simplification of the

Navier-Stokes equations. We have neglected the viscosity and the heat

conduction effects and we have arrived to the Euler equations. In this

chapter we discuss further simplifications of both the Euler and Navier-

Stokes equations.

Compressible submodels will include flows with variation; flows with axial

symmetry; flows with cylindrical and spherical symmetry; plane one-

dimensional flow and further simplifications of this to include linearised and

scalar submodels. Incompressible submodels will include various

formulations of the incompressible Navier-Stokes equations.

30

2.2 Compressible Submodels

2.2.1 Flow with Area Variation

Flows with area variation arise naturally in the study of fluid flow

phenomena in ducts, pipes, shock tubes and nozzles. One may start from the

two dimensional homogeneous version of Euler equations to produce, under

the assumption of smooth area variations, a quasi-two dimensional system

with a geometric source term ( )S U , namely:

( ) ( )t x+ =U F U S U (2.1) where

2 21

, ,

( ) ( )

x t

x

x

u uA A

u u p pu AA

E u E p u E p A

= = + = + +

U F S (2.2)

Here x denotes distance along the tube, nozzle, etc.; A is the cross-sectional

area and in general is a function of both space and time, that is ),( txAA = . The most common case is that in which A depends on x only, for which one

cam write governing equations in the more convenient form:

( ) ( )t x S+ =U F U U (2.3) where

2

0

, ( ) ,

( ) 0

x

A A u

A u A u p pA

AE Au E p

= = + = +

U F S

This is a convenient form in that by defining area weighted values for and

p, equation (2.3) may be interpreted as the usual one-dimensional equations

of Gas Dynamics plus a simple source term S.

2.2.2 Axi-Symmetric Flows

Here we consider domains that are symmetric around a coordinate direction.

We choose this coordinate to be the z-axis and is called the axial direction.

The second coordinate is r, which measures distance from the axis of

symmetry z and is called the radial direction. There are two component of

velocity, namely ),( zru and ),( zrv . These are respectively the radial (r)

and axial (z) components of velocity. Then the three dimensional

31

(inhomogeneous) conservation laws are approximated by a two dimensional

problem with geometric source terms )(US , namely:

( ) ( ) ( )t r z+ + =U F U G U S U (2.4) where

22

2

1, , ,

( )( ) ( )

u v u

uvu uu p

v uvruv u p

E u E pu E p v E p

+ = = = = +

++ +

U F G S (2.5)

2.2.3 Cylindrical and Spherical Symmetry

Cylindrical and spherical symmetric wave motion arises naturally in the

theory of explosion waves in water, air and other media. In these situations

the multidimensional equation may be reduced to essentially one-

dimensional equations with a geometrical source term vector S(U) to

account for the second and third spatial dimensions. We write:

( ) )t r+ =U F U S(U (2.6) where

2 2, ,

( ) ( )

u u

u u p ur

E u E p u E p

= = + = + +

U F S (2.7)

Here r is the radial distance from the origin and u is the radial velocity.

When 0= we have plane one-dimensional flow; when 1= we have cylindrically symmetric flow, an approximation to two-dimensional flow.

This is a special case of the axy-symmetric equations when no axial

variations are present )0( =v . For 2= we have spherically symmetric flow, an approximation to three-dimensional flow. Approximations of this

kind can easily be solved numerically to a high degree of accuracy by a

good one-dimensional numerical method. These accurate one-dimensional

solutions can then be very useful in partially validating two and three

dimensional numerical solutions of the full models.

32

2.2.4 Plain One-Dimensional Flow

We first consider the one-dimensional time depend case:

) 0t x+ =U F(U (2.8) where

2,

( )

u

u u p

E u E p

= = + +

U F

These equations also result from the previous equations, for example from

(2.3). They are useful for solving shock-tube type problems. Further, under

suitable physical assumptions they produce even simpler mathematical

models. In all the submodels studied so far we have assumed some

thermodynamic closure condition given by an Equation of State (EOS).

The isentropic equations result under the assumption that the entropy s is

constant everywhere, which is a simplification of the thermodynamics. Now

the EOS becomes ( )p p C = , where C = constant. This makes the energy equation redundant and we have 2x2 system:

2

) 0,

,

t x

u

u u p

+ =

= = +

U F(U

U F (2.9)

with the pressure p given by the above simple EOS. Notice please that even

in the isentropic assumption, the equation are still non-linear.

The isothermal equations are even a simpler model then the isentropic

equations, still non-linear. These may be viewed as resulting from the

isentropic equations (2.9) with a simpler EOS, that is 2)( app = , where a is a non-zero constant and represents the propagation speed of

sound. Thus, the isothermal equations are:

2 2

0,

,

t x

u

u u a

+ =

= = +

U F (U)

U F (2.10)

More submodels may be further obtained by writing the isentropic equations

as:

33

0

1

t x x

t x x

u u

u uu p

+ + = + =

(2.11)

The inviscid Burgers equation is a scalar (single equation) non linear

equation given by:

0t xu uu+ = (2.12) and can be obtained from the above momentum equation (2.11) by neglecting

density and thus pressure variations. In conservative form equation (1.115)

reads:

2

02

t

x

uu

+ =

(2.13)

The Linearised Equations of Gas Dynamics are obtain from (2.11) by

considering small disturbances ,u to a motionless gas. Set u u= and

0 = + , where 0 is a constant density value. Recall that )(pp = and neglecting products of small quantities we have:

0 0( ) ( )p

p p

= +

that is, 20p p a = + with )( 00 pp = , and constant.)( 02 =

= p

a

Substituting into (2.11) and neglecting squares of small quantities we obtain

the linear equations:

0

2

0

0

0

t x

t x

u

au

+ =

+ =

(2.14)

In matrix form system (1.118)-(1.119) reads:

02

0

0,

0

,0

t x

au

+ = = =

W AW

W A (2.15)

where bars have been dropped. The coefficient matrix A is now constant

and thus the system (2.15) is a linear system with constant coefficients, the

linearised equations of gas dynamics.

34

The linear advection equation, sometimes called linear convection,

equation is:

0t xu au+ = (2.16) where a is a constant speed of wave propagation. This is also known as the

one-way wave equation and plays a major role in the designing, analysing

and testing of numerical methods for wave propagation problems.

2.2.5 Steady Flow

The steady, or time-independent, homogeneous version of the three-

dimensional Euler equations are:

0x y z+ + =F G H (2.17)

In the steady regime it is important to identify subsonic and supersonic flow

regions. To this end we recall the definition of Mach number M:

( ) 21

2

222

++=

a

wvuM

where a is the speed of sound. Supersonic flow requires M>1, while for

subsonic flow we have M

35

Steady linearised models can be further obtained from the steady Euler

equations. An interesting submodel is the small perturbation, two-

dimensional steady supersonic system of equations:

2 0

0

x y

x x

u a v

v u

+ =

= (2.21)

with

1

12

2

=

Ma

and where M denotes the constant free-stream Mach number and

),(),,( yxvyxu are small perturbations of the x and y velocity components.

In matrix form these equations read

0x y+ =W AW (2.22)

with 20

,1 0

u a

v

= =

W A .

2.2.6 Viscous Scalar Equations

From the viscous compressible Navier-Stokes equations one can deduce,

after some manipulations, two representative scalar equations. The first is

the viscous Burgers equation, which is the viscous version of (2.13). It

reads:

2

2t xx

x

uu u

+ =

(2.23)

where is a coefficient of viscosity. A linearised form of this is the linear advection-diffusion equation:

t x xxu au u+ = (2.24) wich is the viscous version of (2.16).

36

2.3 Incompressible Submodels

2.3.1 About Incompressible Models and Low Mach Number Expansions

In the field of CFD, the incompressibility assumption is very important for

applications since many common fluids (liquids) are incompressible or only

very slightly compressible. Mathematically, the incompressibility

condition means:

0div =V (2.25) Therefore, the volume occupied by a group of fluid particles at the initial

time remains constant during the flow. The continuity equation written as:

( ) 0t x y z x y zu v w u v w + + + + + + = leads to

0tD

Dt

= + =V (2.26)

This means that if the density is constant initially and on the boundaries

from where the fluid comes inside the domain under consideration it

remains so. This is equivalent to say that the fluid is homogeneous.

Further, having in mind the previous derivation of the compressible models

and their EOS and constitutive models, it could be useful to make a

distinction between models obtained by:

1. Incompressibility hypothesis.

2. Low Mach number expansions.

We emphasize here that the incompressibility hypothesis does not impose

an explicit restriction on the magnitude of the velocity. Moreover, the

incompressible models aim to describe liquids where compression effects

are neglected and the density is taken as constant. For example, it is not

rational to expect from an incompressible model to describe accurately the

propagation of acoustic or pressure waves through liquids. This is due the

fact that the incompressibility condition (2.25) is normally associated with

other working hypothesis made on the EOS and on the fluid transport

properties. Further, the energy equation is firmly decoupled from the

continuity and momentum equations, by stating that the temperature field

can be calculated separately, after the velocity and pressure fields have been

determined.

37

On the other side, the compressible models presented in the previous

chapter are valid for gases. Starting from the compressible models it is

rational to discuss about the low Mach number expansions. The precise

definition of the Mach number is: ( )'M p = V . Therefore, letting M go to zero means that, keeping constant values for the density and the

temperature, the magnitude of the velocity is a small parameter. An

asymptotic analysis starting from the Navier-Stokes equation derived for a

compressible ideal gas shows that there is a lack of consistency between

compressible models and incompressible submodels, see (Lions, 1996), in

the presence of heat conduction. A possible physical explanation is the

following assertion: compressible models are valid for gases and the low

Mach number limit yields particular incompressible submodels. These

particular submodels are definitively determined by the EOS and transport

properties chosen for the gas. Further, such an asymptotic analysis reveals

that the incompressible submodel is very sensitive to the errors in the

pressure calculation.

In what follows, we assume the fluid to be incompressible, homogeneous,

non-heat conducting and viscous, with constant coefficient of viscosity . Body forces are also neglected, only for simplicity. We study three

mathematical formulations of the governing equations in Cartesian

coordinates and restrict our attention to the two-dimensional case.

2.3.2 The Incompressible Navier-Stokes Equations in Primitive

Variable Form

The primitive variable formulation of the incompressible two dimensional

Navier-Stokes equations is given by:

0

1

1

x y

t x y x xx yy

t x y y xx yy

u v

u uu vu p v u u

v uv vv p v v v

+ =

+ + + = +

+ + + = +

(2.27)

where the kinematic viscosity is:

v

= (2.28)

38

Recall that is the coefficient of shear viscosity. We have a set of three equations for the three unknowns u, v, p, the primitive variables. This is a

mixed elliptic-parabolic system. Due to the mixed nature of the

mathematical model, the solution cannot be obtained directly via time-

marching algorithms. In principal, given a domain along with initial and

boundary conditions for the equations one should be able to solve this

problem using the primitive variable formulation.

2.3.3 The Incompressible Navier-Stokes Equations in Stream-Function

Vorticity Form

The stream-function vorticity formulation is another way of expressing the

incompressible Navier-Stokes equations. This formulation is attractive for

the two-dimensional case but not so much in three dimensions, in which the

role of a stream function is replaced by that of a vector potential. The

magnitude of the vorticity vector can be written as:

x yv u = (2.29) Introducing a stream function we have for the velocity components:

,y xu v = = . By combining the momentum equations so as to eliminate the pressure p,

and using (2.29) we obtain the vorticity transport equation:

t x y xx yyu v v + + = + (2.30)

This is an advection-diffusion equation of parabolic type. In order to

solve it, one requires the solution for the stream function , which is in turn related to the vorticity via: xx yy + = (2.31) This is called the Poisson equation and is of elliptic type. There are

numerical schemes to solve (2.30)-(2.31) using the apparent decoupling of the

parabolic-elliptic problem to transform it into the parabolic equation vor the

vorticity and the elliptic equation for the stream function.. A relevant

observation, from the numerical point of view, is that the convection terms

of the left hand side of equation (2.30) can be written in conservative form

and hence we have:

( ) ( )t xx yyx yu v v + + = + (2.32) This follows from the fact that 0=+ yx vu , which was also used to obtain.

39

2.3.4 The Incompressible Navier-Stokes Equations in Artificial

Compressibility Form

The artificial compressibility formulation is yet another approach to

formulate the incompressible Navier-Stokes equations and was originally

put forward by Chorin, see (Chorin, 1968) for the steady case. Let us

consider the two-dimensional incompressible Navier-Stokes equations

written in non-dimensional form:

0x yu v+ = (2.33)

t x y x xx yy

t x y y xx yy

u uu vu p u u

v uv vv p v v

+ + + = +

+ + + = + (2.34)

where the following non-dimensionalisation has been used:

.,1

'',

,,,2

==

V

LVR

R

L

tVt

L

yy

L

xx

V

pp

V

vv

V

uu

eL

eL

Multiplying (2.33) by the non-zero parameter 2c and adding an artificial

compressibility term tp the first equations reads:

( ) ( )2 2 0t x yp uc vc+ + = (2.35) By using equation (2.33) the convective terms in the momentum equations

can be written in conservative form, so that the modified system becomes:

( ) ( )2 22

2

0

( ) ( )

( ) ( )

t x y

t x y xx yy

t x x xx yy

p uc vc

u u p uv u u

v uv v p v v

+ + =

+ + + = +

+ + + = +

(2.36)

The equations can be written in compact form as

t x y+ + =U F (U) G (U) S(U) (2.37)

where the vectors of unknowns, fluxes and source terms are:

40

2 2

2

2

0

, , , ( )

( )

xx yy

xx yy

p c u c v

u u p uv u u

v uv v p v v

= = + = = + + +

U F G S (2.38)

The above equations are called the artificial compressibility equations. Here

c2 is the artificial compressibility factor, usually taken as a constant

parameter. The source term vector in this case is a function of second

derivatives. Note that the modified equations are equivalent to the original

equations in the steady state limit only. The left-hand side of the artificial

compressibility equations form a non-linear hyperbolic system.

More recently, new formulations have been proposed for the solution of

steady and unsteady incompressible Navier-Stokes equations. Since time-

marching methods cannot be applied directly, the system (2.36) must be

transformed into a more convenient one. The dual time approach requires

the addition of derivatives of a fictitious pseudo-time to each of the three equations to give:

( ) ( )22

2

10

( ) ( )

( ) ( )

t x y

t x y xx yy

t x x xx yy

p p u v

u u u p uv u u

v v uv v p v v

+ + + =

+ + + + = +

+ + + + = +

(2.39)

where is a parameter and the term added to the continuity equation has the same form as the basic artificial compressibility method. A steady-state

solution in pseudo-time ( ), , 0p u v corresponds to an instantaneous unsteady solution in real time. A recommended value for the

parameter in the case the governing equations are written in

dimensionless form is ( )1 . The convective part of the system (2.39) is of hyperbolic type and therefore a time-marching solution procedure is

possible, (Gaitonde, 1998).