differential analysis cfd

8/10/2019 DIFFERENTIAL ANALYSIS CFD

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In this chapter we develop the mathematical basis for a comprehensivegeneral-purpose model of fluid flow and heat transfer from the basic prin-ciples of conservation of mass, momentum and energy. This leads to thegoverning equations of fluid flow and a discussion of the necessary auxiliaryconditions initial and boundary conditions. The main issues covered in thiscontext are:

Derivation of the system of partial differential equations (PDEs) thatgovern flows in Cartesian (x,y, z) co-ordinates

Thermodynamic equations of state Newtonian model of viscous stresses leading to the NavierStokes

equations Commonalities between the governing PDEs and the definition of the

transport equation Integrated forms of the transport equation over a finite time interval and

a finite control volume Classification of physical behaviours into three categories: elliptic,

parabolic and hyperbolic Appropriate boundary conditions for each category

Classification of fluid flows Auxiliary conditions for viscous fluid flows Problems with boundary condition specification in high Reynolds

number and high Mach number flows

The governing equations of fluid flow represent mathematical statements ofthe conservation laws of physics:

The mass of a fluid is conserved The rate of change of momentum equals the sum of the forces on a fluid

particle (Newtons second law) The rate of change of energy is equal to the sum of the rate of heat

addition to and the rate of work done on a fluid particle (first law ofthermodynamics)

The fluid will be regarded as a continuum. For the analysis of fluid flowsat macroscopic length scales (say 1 m and larger) the molecular structureof matter and molecular motions may be ignored. We describe the behaviourof the fluid in terms of macroscopic properties, such as velocity, pressure,density and temperature, and their space and time derivatives. These may

Chapter two Conservation laws of fluid

motion and boundary conditions

Governingequations of fluid

flow and heat transfer

2.1

OTONBARA LIBRARY


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10 CHAPTER 2 CONSERVATION LAWS OF FLUID MOTION

Figure 2.1 Fluid element for

conservation laws

be thought of as averages over suitably large numbers of molecules. A fluidparticle or point in a fluid is then the smallest possible element of fluid whosemacroscopic properties are not influenced by individual molecules.

We consider such a small element of fluid with sides x, y and z(Figure 2.1).

The six faces are labelled N, S, E, W, T and B, which stands for North,South, East, West, Top and Bottom. The positive directions along the co-ordinate axes are also given. The centre of the element is located at position(x,y, z). A systematic account of changes in the mass, momentum and energyof the fluid element due to fluid flow across its boundaries and, where appro-priate, due to the action of sources inside the element, leads to the fluid flowequations.

All fluid properties are functions of space and time so we would strictlyneed to write (x,y, z, t),p(x,y, z, t), T(x,y, z, t) and u(x,y, z, t) for thedensity, pressure, temperature and the velocity vector respectively. To avoidunduly cumbersome notation we will not explicitly state the dependence onspace co-ordinates and time. For instance, the density at the centre (x,y, z)

of a fluid element at time tis denoted by and the x-derivative of, say, pres-surep at (x,y, z) and time tby p/x. This practice will also be followed forall other fluid properties.

The element under consideration is so small that fluid properties at thefaces can be expressed accurately enough by means of the first two termsof a Taylor series expansion. So, for example, the pressure at the Wand Efaces, which are both at a distance of 12 x from the element centre, can beexpressed as

p x and p+ x

2.1.1 Mass conservation in three dimensions

The first step in the derivation of the mass conservation equation is to writedown a mass balance for the fluid element:

Rate of increase Net rate of flowof mass in fluid = of mass intoelement fluid element

1

2

p

x

1

2

p

x


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2.1 GOVERNING EQUATIONS OF FLUID FLOW AND HEAT TRANSFER 11

Figure 2.2 Mass flows in and

out of fluid element

The rate of increase of mass in the fluid element is

(xyz) = xyz (2.1)

Next we need to account for the mass flow rate across a face of the element,which is given by the product of density, area and the velocity componentnormal to the face. From Figure 2.2 it can be seen that the net rate of flow ofmass into the element across its boundaries is given by

u x yz u+ x yz

+ v y xz v+ y xz

+ w z xy w+ z xy (2.2)

Flows which are directed into the element produce an increase of mass in theelement and get a positive sign and those flows that are leaving the elementare given a negative sign.

DEF

12

(w)z

ABC

DEF

12

(w)z

ABC

DEF

1

2

(v)

y

ABC

DEF

1

2

(v)

y

ABC

DEF

1

2

(u)

x

ABC

DEF

1

2

(u)

x

ABC

t

t

The rate of increase of mass inside the element (2.1) is now equated to thenet rate of flow of mass into the element across its faces (2.2). All terms of theresulting mass balance are arranged on the left hand side of the equals signand the expression is divided by the element volume xyz. This yields

+ + + = 0 (2.3)

or in more compact vector notation

+ div(u) = 0 (2.4)

Equation (2.4) is the unsteady, three-dimensional mass conservationor continuity equation at a point in a compressible fluid. The first term

t

(w)

z

(v)

y

(u)

x

t


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on the left hand side is the rate of change in time of the density (mass per unitvolume). The second term describes the net flow of mass out of the elementacross its boundaries and is called the convective term.

For an incompressible fluid (i.e. a liquid) the density is constant andequation (2.4) becomes

div u= 0 (2.5)

or in longhand notation

+ + = 0 (2.6)

2.1.2 Rates of change following a fluid particle and for a fluid

element

The momentum and energy conservation laws make statements regardingchanges of properties of a fluid particle. This is termed the Lagrangianapproach. Each property of such a particle is a function of the position(x,y, z) of the particle and time t. Let the value of a property per unit massbe denoted by . The total or substantive derivative of with respect to timefollowing a fluid particle, written as D/Dt, is

= + + +

A fluid particle follows the flow, so dx/dt=u, dy/dt=v and dz/dt=w.Hence the substantive derivative of is given by

= +u +v +w = +u . grad (2.7)

D/Dt defines rate of change of property per unit mass. It is possibleto develop numerical methods for fluid flow calculations based on theLagrangian approach, i.e. by tracking the motion and computing the rates ofchange of conserved properties for collections of fluid particles. However,it is far more common to develop equations for collections of fluid elementsmaking up a region fixed in space, for example a region defined by a duct, apump, a furnace or similar piece of engineering equipment. This is termedthe Eulerian approach.

As in the case of the mass conservation equation, we are interested indeveloping equations for rates of change per unit volume. The rate of changeof property per unit volume for a fluid particle is given by the product ofD/Dtand density , hence

= +u . grad (2.8)

The most useful forms of the conservation laws for fluid flow computationare concerned with changes of a flow property for a fluid element that isstationary in space. The relationship between the substantive derivative of ,which follows a fluid particle, and rate of change of for a fluid element isnow developed.

DEFtABC

DDt

t

z

y

x

t

D

Dt

dz

dt

z

dy

dt

y

dx

dt

x

t

D

Dt

w

z

v

y

u

x



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The mass conservation equation contains the mass per unit volume (i.e.the density ) as the conserved quantity. The sum of the rate of change ofdensity in time and the convective term in the mass conservation equation

(2.4) for a fluid element is

+ div(u)

The generalisation of these terms for an arbitrary conserved property is

+ div(u) (2.9)

Formula (2.9) expresses the rate of change in time of per unit volume plusthe net flow of out of the fluid element per unit volume. It is now rewrittento illustrate its relationship with the substantive derivative of :

+ div(u) = +u . grad + + div(u)

= (2.10)

The term [(/t) + div(u)] is equal to zero by virtue of mass conserva-tion (2.4). In words, relationship (2.10) states

Rate of increase Net rate of flow Rate of increaseof of fluid + of out of = of for aelement fluid element fluid particle

To construct the three components of the momentum equation and theenergy equation the relevant entries for and their rates of change per unitvolume as defined in (2.8) and (2.10) are given below:

x-momentum u + div(uu)

y-momentum v + div(vu)

z-momentum w + div(wu)

energy E + div(Eu)

Both the conservative (or divergence) form and non-conservative form of therate of change can be used as alternatives to express the conservation of aphysical quantity. The non-conservative forms are used in the derivations ofmomentum and energy equations for a fluid flow in sections 2.4 and 2.5 forbrevity of notation and to emphasise that the conservation laws are funda-mentally conceived as statements that apply to a particle of fluid. In the final

(E)

t

DE

Dt

(w)

t

Dw

Dt

(v)

t

Dv

Dt

(u)

t

Du

Dt

D

Dt

JKL

tGHIJKL

tGHI()

t

()

t

t


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Figure 2.3 Stress components

on three faces of fluid element

section 2.8 we will return to the conservative form that is used in finite vol-ume CFD calculations.

2.1.3 Momentum equation in three dimensions

Newtons second law states that the rate of change of momentum of a fluidparticle equals the sum of the forces on the particle:

Rate of increase of Sum of forcesmomentum of = onfluid particle fluid particle

The rates of increase ofx-, y- andz-momentum per unit volume of afluid particle are given by

(2.11)

We distinguish two types of forces on fluid particles:

surface forces pressure forces viscous forces gravity force

body forces centrifugal force Coriolis force electromagnetic force

It is common practice to highlight the contributions due to the surface forcesas separate terms in the momentum equation and to include the effects of

body forces as source terms.The state of stress of a fluid element is defined in terms of the pressure

and the nine viscous stress components shown in Figure 2.3. The pressure,a normal stress, is denoted byp. Viscous stresses are denoted by . The usualsuffix notation ij is applied to indicate the direction of the viscous stresses.The suffices iand j in ij indicate that the stress component acts in the j-direction on a surface normal to the i-direction.

Dw

Dt

Dv

Dt

Du

Dt

First we consider the x-components of the forces due to pressure p andstress components xx, yx and zx shown in Figure 2.4. The magnitude of a


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force resulting from a surface stress is the product of stress and area. Forcesaligned with the direction of a co-ordinate axis get a positive sign and thosein the opposite direction a negative sign. The net force in the x-direction is

the sum of the force components acting in that direction on the fluid element.

Figure 2.4 Stress components

in the x-direction

On the pair of faces (E, W) we have

p x xx x yz+ p+ x

+ xx+ x yz= + xyz (2.12a)

The net force in the x-direction on the pair of faces (N, S) is

yx y xz+ yx+ y xz = xyz

(2.12b)

Finally the net force in the x-direction on faces Tand B is given by

zx z xy+ zx+ z xy = xyz

(2.12c)

The total force per unit volume on the fluid due to these surface stresses isequal to the sum of (2.12a), (2.12b) and (2.12c) divided by the volumexyz:

+ + (2.13)

Without considering the body forces in further detail their overall effectcan be included by defining a source SMx of x-momentum per unit volumeper unit time.

The x-component of the momentum equation is found by settingthe rate of change of x-momentum of the fluid particle (2.11) equal to the

zxz

yxy

(p+xx)x

zx

z

DEF

1

2

zx

z

ABC

DEF

1

2

zx

z

ABC

yxy

DEF12

yxy

ABCDEF12

yxy

ABC

DEF

xx

x

p

x

ABC

JKL

DEF

1

2

xx

x

ABC

DEF

1

2

p

x

ABC

GHI

JKL

DEF

1

2

xx

x

ABC

DEF

1

2

p

x

ABC

GHI


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total force in the x-direction on the element due to surface stresses (2.13)plus the rate of increase of x-momentum due to sources:

= + + +SMx (2.14a)

It is not too difficult to verify that the y-component of the momentumequation is given by

= + + +SMy (2.14b)

and the z-component of the momentum equation by

= + + +SMz (2.14c)

The sign associated with the pressure is opposite to that associated with thenormal viscous stress, because the usual sign convention takes a tensile stressto be the positive normal stress so that the pressure, which is by definition acompressive normal stress, has a minus sign.

The effects of surface stresses are accounted for explicitly; the sourceterms SMx, SMy and SMz in (2.14ac) include contributions due to body forcesonly. For example, the body force due to gravity would be modelled bySMx= 0, SMy = 0 and SMz= g.

2.1.4 Energy equation in three dimensions

The energy equation is derived from the first law of thermodynamics,which states that the rate of change of energy of a fluid particle is equal to therate of heat addition to the fluid particle plus the rate of work done on theparticle:

Rate of increase Net rate of Net rate of workof energy of = heat added to + done onfluid particle fluid particle fluid particle

As before, we will be deriving an equation for the rate of increase ofenergy of a fluid particle per unit volume, which is given by

(2.15)

Work done by surface forces

The rate of work done on the fluid particle in the element by a surfaceforce is equal to the product of the force and velocity component in thedirection of the force. For example, the forces given by (2.12ac) all act inthe x-direction. The work done by these forces is given by

DE

Dt

(p+zz)

z

yz

y

xz

x

Dw

Dt

zy

z

(p+yy)

y

xy

x

Dv

Dt

zx

z

yx

y

(p+xx)

x

Du

Dt


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pu x xxu x

pu+ x + xxu+ x yz

+ yxu y + yxu+ y xz

+ zxu z + zxu+ z xy

The net rate of work done by these surface forces acting in the x-direction isgiven by

+ + xyz (2.16a)

Surface stress components in they- and z-direction also do work on the fluidparticle. A repetition of the above process gives the additional rates of workdone on the fluid particle due to the work done by these surface forces:

+ + xyz (2.16b)

and

+ + xyz (2.16c)

The total rate of work done per unit volume on the fluid particle by allthe surface forces is given by the sum of (2.16ac) divided by the volumexyz. The terms containing pressure can be collected together and writtenmore compactly in vector form

= div(pu)

This yields the following total rate of work done on the fluid particle bysurface stresses:

[div(pu)] + + + + +

+ + + + (2.17)

Energy flux due to heat conduction

The heat flux vector q has three components: qx, qy and qz (Figure 2.5).

JKL

(wzz)

z

(wyz)

y

(wxz)

x

(vzy)

z

(vyy)

y

(vxy)

x

(uzx)

z

(uyx)

y

(uxx)

x

GH

I

(wp)

z

(vp)

y

(up)

x

JKL

(w(p+zz))

z

(wyz)

y

(wxz)

x

GHI

JKL

(vzy)

z

(v(p+yy))

y

(vxy)

x

GHI

JKL

(uzx)

z

(uyx)

y

(u(p +xx))

x

GHI

JKL

DEF

1

2

(zxu)

z

ABC

DEF

1

2

(zxu)

z

ABC

GHI

JKL

DEF

1

2

(yxu)

y

ABC

DEF

1

2

(yxu)

y

ABC

GHI

JKL

DEF

1

2

(xxu)

x

ABC

DEF

1

2

(pu)

x

ABC

DEF

1

2

(xxu)

x

ABC

DEF

1

2

(pu)

x

ABC

GHI


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The net rate of heat transfer to the fluid particle due to heat flow inthe x-direction is given by the difference between the rate of heat input

across face Wand the rate of heat loss across face E:

qx x qx+ x yz= xyz (2.18a)

Similarly, the net rates of heat transfer to the fluid due to heat flows in they- and z-direction are

xyz and xyz (2.18bc)

The total rate of heat added to the fluid particle per unit volume due to heatflow across its boundaries is the sum of (2.18ac) divided by the volumexyz:

= div q (2.19)

Fouriers law of heat conduction relates the heat flux to the local temperaturegradient. So

qx= k qy= k qz= k

This can be written in vector form as follows:

q= k grad T (2.20)

Combining (2.19) and (2.20) yields the final form of the rate of heataddition to the fluid particle due to heat conduction across elementboundaries:

div q= div(k grad T) (2.21)

Energy equation

Thus far we have not defined the specific energy E of a fluid. Often theenergy of a fluid is defined as the sum of internal (thermal) energy i, kineticenergy 12 (u

2 +v2 +w2) and gravitational potential energy. This definition

T

z

T

y

T

x

qz

zqy

yqx

x

qz

z

qy

y

qx

x

JKL

DEF

1

2

qx

x

ABC

DEF

1

2

qx

x

ABC

GHI


Figure 2.5 Components of the

heat flux vector


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takes the view that the fluid element is storing gravitational potential energy.It is also possible to regard the gravitational force as a body force, which doeswork on the fluid element as it moves through the gravity field.

Here we will take the latter view and include the effects of potentialenergy changes as a source term. As before, we define a source of energy SEper unit volume per unit time. Conservation of energy of the fluid particle isensured by equating the rate of change of energy of the fluid particle (2.15)to the sum of the net rate of work done on the fluid particle (2.17), the netrate of heat addition to the fluid (2.21) and the rate of increase of energy dueto sources. The energy equation is

= div(pu) + + + +

+ + + + +

+ div(k grad T) +SE (2.22)

In equation (2.22) we have E=i + 12 (u2 +v2 +w2).

Although (2.22) is a perfectly adequate energy equation it is commonpractice to extract the changes of the (mechanical) kinetic energy to obtainan equation for internal energy ior temperature T. The part of the energyequation attributable to the kinetic energy can be found by multiplying thex-momentum equation (2.14a) by velocity component u, the y-momentumequation (2.14b) by v and the z-momentum equation (2.14c) by w andadding the results together. It can be shown that this yields the followingconservation equation for the kinetic energy:

= u . gradp+u + +

+v + +

+w + + + u . SM (2.23)

Subtracting (2.23) from (2.22) and defining a new source term asSi=SE u . SM yields the internal energy equation

= p div u+ div(k grad T) +xx +yx +zx

+xy +yy +zy

+xz +yz +zz +Si (2.24)w

z

w

y

w

x

v

z

v

y

v

x

u

z

u

y

u

x

Di

Dt

DEF

zz

z

yz

y

xz

x

ABC

DEF

zy

z

yy

y

xy

x

ABC

DEF

zx

z

yx

y

xx

x

ABC

D[ 12 (u2 +v2 +w2)]

Dt

JKL

(wzz)

z

(wyz)

y

(wxz)

x

(vzy)

z

(vyy)

y

(vxy)

x

(uzx)

z

(uyx)

y

(uxx)

x

GHI

DE

Dt


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For the special case of an incompressible fluid we have i=cT, where c is thespecific heat and div u= 0. This allows us to recast (2.24) into a temperatureequation

c = div(k grad T) +xx +yx +zx +xy

+yy +zy +xz +yz +zz +Si (2.25)

For compressible flows equation (2.22) is often rearranged to give an equa-tion for the enthalpy. The specific enthalpy h and the specific total enthalpyh0 of a fluid are defined as

h=i+p/ and h0=h+12 (u

2 +v2 +w2)

Combining these two definitions with the one for specific energy Ewe get

h0=i+p/+12 (u

2 +v2 +w2) =E+p/ (2.26)

Substitution of (2.26) into (2.22) and some rearrangement yields the (total)enthalpy equation

+ div(h0 u) = div(k grad T) +

+ + +

+ + +

+ + + +Sh (2.27)

It should be stressed that equations (2.24), (2.25) and (2.27) are notnew (extra)conservation laws but merely alternative forms of the energy equation (2.22).

The motion of a fluid in three dimensions is described by a system of fivepartial differential equations: mass conservation (2.4), x-,y- and z-momentumequations (2.14ac) and energy equation (2.22). Among the unknowns arefour thermodynamic variables: ,p, iand T. In this brief discussion we point

out the linkage between these four variables. Relationships between thethermodynamic variables can be obtained through the assumption of thermo-dynamic equilibrium. The fluid velocities may be large, but they areusually small enough that, even though properties of a fluid particle changerapidly from place to place, the fluid can thermodynamically adjust itself tonew conditions so quickly that the changes are effectively instantaneous. Thusthe fluid always remains in thermodynamic equilibrium. The only exceptionsare certain flows with strong shockwaves, but even some of those are oftenwell enough approximated by equilibrium assumptions.

JKL

(wzz)

z

(wyz)

y

(wxz)

x

(vzy)

z

(vyy)

y

(vxy)

x

(uzx)

z

(uyx)

y

(uxx)

x

GHI

p

t

(h0)

t

w

z

w

y

w

x

v

z

v

y

v

x

u

z

u

y

u

x

DT

Dt

Equations ofstate

2.2


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2.3 NAVIER---STOKES EQUATIONS FOR A NEWTONIAN FLUID 21

We can describe the state of a substance in thermodynamic equilibriumby means of just two state variables. Equations of state relate the othervariables to the two state variables. If we use and Tas state variables wehave state equations for pressurep and specific internal energy i:

p=p(, T) and i=i(, T) (2.28)

For a perfect gas the following, well-known, equations of state are useful:

p=RT and i=CvT (2.29)

The assumption of thermodynamic equilibrium eliminates all but the twothermodynamic state variables. In the flow of compressible fluids theequations of state provide the linkage between the energy equation on theone hand and mass conservation and momentum equations on the other.This linkage arises through the possibility of density variations as a result ofpressure and temperature variations in the flow field.

Liquids and gases flowing at low speeds behave as incompressiblefluids. Without density variations there is no linkage between the energyequation and the mass conservation and momentum equations. The flowfield can often be solved by considering mass conservation and momentumequations only. The energy equation only needs to be solved alongside theothers if the problem involves heat transfer.

The governing equations contain as further unknowns the viscous stresscomponents ij. The most useful forms of the conservation equations forfluid flows are obtained by introducing a suitable model for the viscousstresses ij. In many fluid flows the viscous stresses can be expressed as func-tions of the local deformation rate or strain rate. In three-dimensional flowsthe local rate of deformation is composed of the linear deformation rate andthe volumetric deformation rate.

All gases and many liquids are isotropic. Liquids that contain signific-ant quantities of polymer molecules may exhibit anisotropic or directionalviscous stress properties as a result of the alignment of the chain-like polymermolecules with the flow. Such fluids are beyond the scope of this intro-ductory course and we shall continue the development by assuming that thefluids are isotropic.

The rate of linear deformation of a fluid element has nine componentsin three dimensions, six of which are independent in isotropic fluids(Schlichting, 1979). They are denoted by the symbol sij. The suffix systemis identical to that for stress components (see section 2.1.3). There are threelinear elongating deformation components:

sxx= syy= szz= (2.30a)

There are also six shearing linear deformation components:

sxy=syx= + and sxz=szx= +

syz=szy= + (2.30b)DEF

w

y

v

z

ABC

1

2

DEF

w

x

u

z

ABC

1

2

DEF

v

x

u

y

ABC

1

2

w

z

v

y

u

x

Navier---Stokesequations for aNewtonian fluid

2.3


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The volumetric deformation is given by

+ + = div u (2.30c)

In a Newtonian fluid the viscous stresses are proportional to the ratesof deformation. The three-dimensional form of Newtons law of viscosityfor compressible flows involves two constants of proportionality: the first(dynamic) viscosity, , to relate stresses to linear deformations, and the sec-ond viscosity, , to relate stresses to the volumetric deformation. The nineviscous stress components, of which six are independent, are

xx= 2 + div u yy= 2 + div u zz= 2 + div u

xy=yx= + xz=zx= +

yz=zy= + (2.31)

Not much is known about the second viscosity , because its effect is smallin practice. For gases a good working approximation can be obtained bytaking the value = 23 (Schlichting, 1979). Liquids are incompressible sothe mass conservation equation is div u= 0 and the viscous stresses are justtwice the local rate of linear deformation times the dynamic viscosity.

Substitution of the above shear stresses (2.31) into (2.14ac) yields theso-called NavierStokes equations, named after the two nineteenth-centuryscientists who derived them independently:

= + 2 +div u + +

+ + +SMx (2.32a)

= + + + 2 + div u

+ + +SMy (2.32b)

= + + + +

+ 2 + div u +SMz (2.32c)JKL

w

z

GHI

z

JKL

DEF

w

y

v

z

ABC

GHI

y

JKL

DEF

w

x

u

z

ABC

GHI

x

p

z

Dw

Dt

JKL

DEF

w

y

v

z

ABC

GHI

z

JKL

v

y

GHI

y

JKL

DEF

v

x

u

y

ABC

GHI

x

p

y

Dv

Dt

JKL

DEF

w

x

u

z

ABC

GHI

z

JKLDEFv

x

u

yABCGHI

yJKLu

xGHI

x

p

x

Du

Dt

DEF

w

y

v

z

ABC

DEF

w

x

u

z

ABC

DEF

v

x

u

y

ABC

w

z

v

y

u

x

w

z

v

y

u

x


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2.3 NAVIER---STOKES EQUATIONS FOR A NEWTONIAN FLUID 23

Often it is useful to rearrange the viscous stress terms as follows:

2 + div u + + + +

= + +

+ + + + ( div u)

= div(grad u) + [sMx]

The viscous stresses in they- and z-component equations can be recast in asimilar manner. We clearly intend to simplify the momentum equations byhiding the bracketed smaller contributions to the viscous stress terms in the

momentum source. Defining a new source by

SM=SM+ [sM] (2.33)

the NavierStokes equations can be written in the most useful form forthe development of the finite volume method:

= + div(grad u) +SMx (2.34a)

= + div(grad v) +SMy (2.34b)

= + div(grad w) +SMz (2.34c)

If we use the Newtonian model for viscous stresses in the internal energyequation (2.24) we obtain after some rearrangement

= p div u+ div(k gradT) + +Si (2.35)

All the effects due to viscous stresses in this internal energy equation aredescribed by the dissipation function , which, after considerable algebra,can be shown to be equal to

= 2

2

+

2

+

2

+ +

2

+ +

2

+ +

2

+(div u)2 (2.36)

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The dissipation function is non-negative since it only contains squared termsand represents a source of internal energy due to deformation work on thefluid particle. This work is extracted from the mechanical agency which

causes the motion and converted into internal energy or heat.

To summarise the findings thus far, we quote in Table 2.1 the conservativeor divergence form of the system of equations which governs the time-dependent three-dimensional fluid flow and heat transfer of a compressibleNewtonian fluid.


Table 2.1 Governing equations of the flow of a compressible Newtonian fluid

Continuity + div(u) = 0 (2.4)

x-momentum + div(uu) = + div(grad u) +SMx (2.37a)

y-momentum + div(vu) = + div(grad v) +SMy (2.37b)

z-momentum + div(wu) = + div(grad w) +SMz (2.37c)

Energy + div(iu) = p div u+ div(k grad T) + +Si (2.38)

Equations p=p(, T) and i=i(, T) (2.28)of state

e.g. perfect gasp=RTand i=CvT (2.29)

(i)

t

p

z

(w)

t

p

y

(v)

t

p

x

(u)

t

t

Momentum source SM and dissipation function are defined by (2.33)and (2.36) respectively.

It is interesting to note that the thermodynamic equilibrium assumption ofsection 2.2 has supplemented the five flow equations (PDEs) with two furtheralgebraic equations. The further introduction of the Newtonian model, whichexpresses the viscous stresses in terms of gradients of velocity components,has resulted in a system of seven equations with seven unknowns. With anequal number of equations and unknown functions this system is mathemat-ically closed, i.e. it can be solved provided that suitable auxiliary conditions,namely initial and boundary conditions, are supplied.

It is clear from Table 2.1 that there are significant commonalities betweenthe various equations. If we introduce a general variable the conservativeform of all fluid flow equations, including equations for scalar quantities suchas temperature and pollutant concentration etc., can usefully be written inthe following form:

+ div(u) = div(grad ) +S (2.39)()

t

Conservative formof the governing

equations of fluid flow

2.4

Differential and

integral formsof the general

transport equations

2.5


17/17

2.5 FORMS OF THE GENERAL TRANSPORT EQUATIONS 25

In words,

Rate of increase Net rate of flow Rate of increase Rate of increase

of of fluid + of out of = of due to + of due toelement fluid element diffusion sources

Equation (2.39) is the so-called transport equation for property . Itclearly highlights the various transport processes: the rate of change termand the convective term on the left hand side and the diffusive term ( =diffusion coefficient) and the source term respectively on the right handside. In order to bring out the common features we have, of course, had tohide the terms that are not shared between the equations in the source terms.Note that equation (2.39) can be made to work for the internal energy equa-tion by changing iinto Tor vice versa by means of an equation of state.

Equation (2.39) is used as the starting point for computational proceduresin the finite volume method. By setting equal to 1, u, v, w and i(or Torh0) and selecting appropriate values for diffusion coefficient and sourceterms, we obtain special forms of Table 2.1 for each of the five PDEs formass, momentum and energy conservation. The key step of the finite volumemethod, which is to be to be developed from Chapter 4 onwards, is the integ-ration of (2.39) over a three-dimensional control volume (CV):

dV+ div(u)dV= div(grad )dV+ SdV (2.40)

The volume integrals in the second term on the left hand side, the convec-tive term, and in the first term on the right hand side, the diffusive term, arerewritten as integrals over the entire bounding surface of the control volumeby using Gausss divergence theorem. For a vector a this theorem states

div(a)dV= n . adA (2.41)

The physical interpretation of n.a is the component of vector a in thedirection of the vector n normal to surface element dA. Thus the integralof the divergence of a vector a over a volume is equal to the component of ain the direction normal to the surface which bounds the volume summed(integrated) over the entire bounding surface A. Applying Gausss diver-gence theorem, equation (2.40) can be written as follows:

dV + n . (u)dA= n . ( grad )dA+ SdV (2.42)

The order of integration and differentiation has been changed in the firstterm on the left hand side of (2.42) to illustrate its physical meaning. Thisterm signifies the rate of change of the total amount of fluid property in the control volume. The product n.u expresses the flux com-ponent of property due to fluid flow along the outward normal vector n,so the second term on the left hand side of (2.42), the convective term,therefore is the net rate of decrease of fluid property of the fluidelement due to convection.

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