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2011 Sajid

Chapter

Dr Muhammad Sajid

Assistant Professor

NUST, SMME.

Reference Text:Fundamental of

Computational Fluid

Dynamics, J. Anderson.

Email: [email protected]

Tel: 9085 6065

Computational Fluid

Dynamics

4-Nov-130

3 Governing Equations

Review

Conservation of mass

Conservation of LinearMomentum

Navier Stokes Equation

Computational Fluid Dynamics

Introduction

CFD is fundamentally based on the governing

equations of fluid dynamics.

These equations represent mathematical

statements of the laws of physicsregarding the

conservation of mass, momentum and energy.

The purpose of this chapter is to introduce thederivation and discussion of these equations.

All of CFD is based on these equations; we must

therefore begin our understanding at the most

basic description of the fluid flow processes.

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Introduction

After these equations are obtained, forms suited

for use in formulating CFD solutions will be

highlighted.

At the end of this chapter, some of the mysteries

surrounding CFD based predictions of fluid flow

problems will be replaced with an understanding

of the equations governing the fluid transport.

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2

Computational Fluid Dynamics 3

Review of basic concepts

Fluid properties:

Physical laws are stated in terms of parameters like v, a etc.

Let represent a fluid parameter/property and the amountof that parameter per unit mass, i.e. = m.

is extensive property and is an intensive property.

Fluid element is a volume stationary in space,

Fluid particle is a volume of fluid moving with the flow. A fluid particle in motion experiences two rates of

changes:

Due to changes in the fluid as a function of time.

Due to the fact that it moves to a different location in the fluidwith different conditions.

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The sum of these two rates of changes for aproperty per unit mass (extensive) is called

the totalor substantivederivative D/Dt:

With dx/dt=u, dy/dt=v, dz/dt=w, this results in:

dt

dz

zdt

dy

ydt

dx

xtDt

D

4-Nov-13

.u

tDt

D

w

vu

u zyx

zw

yv

xu

tDt

D



Fluid element and properties The behavior of the fluid is

described in terms of macroscopicproperties: Velocity u.

Pressure p.

Density r.

Temperature T. Energy E.

dy

dx

dz(x,y,z)

1 1

2 2W E

p pp p x p p x

x xd d

Properties at faces are expressed as first

two terms of a Taylor series expansion,

e.g. for p : and

Properties are averages of a sufficiently largenumber of molecules.

A fluid element can be thought of as the smallestvolume for which the continuum assumption isvalid.

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Field representation The flow field of a fluid can be thought of as being

comprised of a large number of finite sized fluid particleswhich have mass, momentum, internal energy, and otherproperties.

The distribution of fluid parameters (r, v, P& a) overspace and time is called field representation.

Velocity field Representation of fluid velocity as function of spatial

coordinates and time, V = f(x,y,z,t).

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k),,,(j),,,(i),,,( tzyxwtzyxvtzyxuV

222 wvuVV



System:A fixed quantity of matter, and no mass is allowed to

cross the system boundary.

Control Volume:A space of interest where mass can cross the boundary,

cv.

Control Surface: The boundary of the control volume is called the controlsurface, cs.

Reynolds transport theorem

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dAnVVdtDt

D

cscv

sys

rr

()()()

V

tDt

D

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Lagrangian approach: Fluid particles are tagged/identified

and their properties are determined

as they move in space.

Eulerian approach:

Fluid properties are determined at

fixed points in space as fluid flows

by.

Governing equations can bederived using each method and

converted to the other form.

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The governing equations include the followingconservation laws of physics: Conservation of mass.

Newtons second law: the change of momentumequals the sum of forces on a fluid particle.

First law of thermodynamics (conservation of energy):

rate of change of energy equals the sum of rate of heataddition to and work done on fluid particle.

The fluid is treated as a continuum. For lengthscales of, say, 1m and larger, the molecularstructure and motions may be ignored.

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Time rate of change of system mass = 0

For a fixed and nondeforming control volume:

t - dt, t (coincident), t + dt

From Reynolds transport theorem, we have.

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0sysmDt

D sys

sys Vdm r

dAnVVdt

mDt

D

cscv

sys

rr

dAnVVdtDt

D

cscv

sys

rr


Conservation of mass & Continuity

Or,

The continuity equation is an expression for the

conservation of mass in a control volume.

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rfacecontrol su

orughof mass th

f flownet rate o

lumecontrol vo

side theof mass in

of changetime rate

systemcoincident

s of theof the mas

of changetime rate

cv

Vdt

lumecontrol vo

side theof mass in

of changetime rate

r

cs

dAnV

rfacecontrol su

roughof mass th

f flownet rate o

r

0 dAnVVdt

cscv

rr

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Consider a differential fluid element.

Let density and velocities components at the

center of the element be , u, v and w.

The volume integral can be expressed as.

Next, we need to find the mass flow rate

through the surfaces

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0

dAnVVdtcscv

rr

zyxt

Vdt

cv

dddr

r



If ru is the horizontal mass flux at the center of

the element than at the faces the mass flux in

horizontal direction is.

Net rate of mass flowing through the surfaces is.

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zyx

x

uzy

x

x

uuzy

x

x

uu ddd

rdd

drrdd

drr

22

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The inflows (positive) and outflows (negative)in all directions are:

x

yz

( ) 1.2

ww z x y

z

rr d d d

( ) 1

.2

vv y x z

y

rr d d d

zyxx

uu ddd

rr

2

1.

)(

zxyy

vv ddd

rr

2

1.

)(

yxzz

ww ddd

rr

2

1.

)(

( ) 1.2

uu x y z

x

rr d d d

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Mass flow rate in the y and z directions is.

Combining these equations we get.

Thus,

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zyx

y

vddd

r

directiony''inrateflowmassnet

zyxz

w

y

v

x

udAnV

cs

dddrrr

r

rateflowmassnet

0

zyx

z

w

y

v

x

uzyx

tddd

rrrddd

r

zyx

y

wddd

r

directionz''inrateflowmassnet

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Continuity equation

Summing all terms in the previous slide anddividing by the volume dxdydzresults in:

In vector notation:

For incompressible fluids r/ t=0, and the

equation becomes: div u= 0. Alternative ways to write this:

0)()()(

zw

yv

xu

trrrr

0)(

urr

divt

Change in densityNet flow of mass across boundaries

Convective term

0

zw

yv

xu 0

i

i

x

u

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CONSERVATION OF MOMENTUM

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Momentum equation in three dimensions

Newtons second law:the sum of forces equals the rate of change ofmomentum.

Rate of increase of momentum along x, y, and zaxis's.

Forces on fluid particles are: Surface forces such as pressure and viscous forces.

Body forces, which act on a volume, such as gravity,centrifugal and electromagnetic forces.

Dt

Dw

Dt

Dv

Dt

Durrr

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()()()

V

tDt

D


Conservation of Momentum

The surface forces are due to the stresses,

exerted on the sides of the fluid element.

Stresses are forces per area. = N/m2or Pa.

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Two types of stresses:

normal stress (ij), often shown as

(ij) and

shear stress (ij).

i refers to the axis normal to the surface,

j represents the direction of the stress.

Forces in direction of an axis are

positive, else negative.

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All surface forces acting in the x-direction on thefluid element are:

Conservation of momentum4-Nov-13

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x

z

y

zyxx

pp ddd )

2

1.(

zyxx

pp ddd )

2

1.(

zyzz

zxzx

ddd

)2

1.(

yxzz

zxzx

ddd

)2

1.(

zxyy

yx

yx ddd

)

2

1.(

zxyy

yx

yx ddd

)

2

1.(

zyxx

xxxx

ddd

)2

1.(

zyxx

xxxx ddd

)

2

1.(



Set the rate of change of x-momentum for a

fluid particle Du/Dt equal to:

the sum of the forces due to surface stresses

shown in the previous slide, plus

the body forces. These are usually lumped

together into a source term SM:

p is a compressive stress and xxis a tensile

stress.

Mxzxyxxx S

zyx

p

Dt

Du

r

)(

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Similarly for y- and z-momentum:

Mxzxyxxx S

zyx

p

Dt

Du

r

)(

My

zyyyxyS

zy

p

xDt

Dv

r

)(

Mzzzyzxz

Sz

p

yxDt

Dw

)( r

4-Nov-13


Cauchys equation of motion

It is an expression for the acceleration, the

body forces, the pressure gradient forces, and

the viscous forces coupled with Newtons 2nd

Law:

Note that this is an incredibly succinct

equation, and is much more complicated than

it at first appears.

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ij

PkgDt

vDrr

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The full expanded version can be written foreach component,x, y, and zas:

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zyxzPkg

zvv

yvv

xvv

tv

zyxy

P

z

vv

y

vv

x

vv

t

v

zyxx

P

z

vv

y

vv

x

vv

t

v

zyyyxyzz

zy

zx

z

zyyyxyy

z

y

y

y

x

y

zxyxxxx

z

x

y

x

x

x

rr

r

r



Things to notice about these equations: only the zcomponent equation has a body force,

because gravity only works in the zdirection;

thex- and y-component equations are identical exceptfor subscripts;

the equations cannot be solved in their present form

because the stresses have not yet been recast interms of velocities.

In order to solve this equation for a specificsubstance, like a fluid, we need to substituteexpressions involving velocity for the viscousstress gradients.

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Constitutive Relationship for Viscous Fluids

The mechanical behavior of every substance canbe described in terms of stress and strain, and the

relationship between these variables is called a

constitutive relationship.

Generally, these must be determined through

experiments and differ for every type of substance

(i.e., rock, plastic, fluid, gas, etc.).

Mathematically, the relationship is between the

stress tensor and the strain tensor (for rigidsolids) or strain-rate tensor (for fluids).

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Constitutive Relationship for Viscous Fluids

Strain is a measure of distortion and the strain-rate tensor has nine components just like thestress tensor:

The diagonal entries , , and representnormal strain rates (elongation, contraction) andthe off-diagonal strains represent shear strainsrates. Remember

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zzzyzx

yzyyyx

xzxyxx

ij

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Strain and strain rate

The strain rates in the strain-rate tensor can be describedin terms of velocity gradients.

Consider a small elongate element of fluid moving in the

x-direction with a non-constant velocity.

The element is stretching as it is moving, resulting in a

normal strain-rate in thex-direction.

The element has a length ofxand undergoes strain

which stretches it tox+ xin the time t. xis the

small elongation ofxand is always smaller thanx.

The strain xxis:

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x

xxx

d

d



There are different ways to determine the

relationship between strain rate and velocity gradients.

First Method:

The only way the box can deform (i.e., stretch; i.e.,

strain) is if the right-hand side moves faster than the

left-hand slide, which happens when > 0.

The rateof strain will equal because this is the

amount by which the right-hand side is moving faster

than the left-hand side.

So we can deduce the answer as: =

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Second Method: Take the expression for strain = and

divide by the time increment, t;

The term in parentheses is the rate at which the

increment xgrows with time or differential velocity,

the difference in velocity between the left and right

sides of the original box.

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x

v

x

vx

xt

x

xt

xxxxxx

11

d

d

d

d

x

vxv

x

vxvvv

t

x xL

x

LLR

d

d

x

xxx

d

d

t

x

xtx

x

t

xx

d

d

d

d

d

d 1



Similar arguments show that the other diagonal

elements in the strain-rate tensor, are:

This is an example how just one of the 9

components of the strain-rate tensor is related to

the gradients of velocity in thex, y, and zdirections.

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z

v

y

vz

zz

y

yy

;

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The full strain-rate tensor can be expressed in terms ofvelocity gradients as follows:

This is a symmetric tensor across the diagonal elements. This strain-rate tensor is valid for all materials, including fluids.

It expresses the strain rates as a function of the velocity

gradients, and is constructed entirely from geometry.

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z

v

y

v

z

v

x

v

z

v

y

v

z

v

y

v

x

v

y

v

x

v

z

v

x

v

y

v

x

v

zzyzx

zyyyx

zxyxx

zzzyzx

yzyyyx

xzxyxx

ij

2

1

2

1

2

1

2

1

2

1

2

1

week 08 - gov eqs

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