Continuity Equation

dxdydz x

)u( dz dy u - dy dz dx x

Net outflow in x direction

Continuity Equation

net out flow in y direction,

dxdydz y

)v( dz dx v - dx dz dy y

Continuity Equation

Net out flow in z direction dxdydz

zwdydxw dx dydz

zww )( - )(

Net mass flow out of the element

dxdydz z

Time rate of mass decrease in the element

dxdydzt

Net mass flow out of the element =

Time rate of mass decrease in the control volume

dxdydzt

dxdydz zw

xu )( )( )(

Continuity Equation

kgm 0 z

The above equation is a partial differential equation form of the continuity equation. Since the element is fixed in space, this form of equation is called conservation form.

0 )( )( )(

sec 0 )( )( )( 3

If the density is constant

0 )( )( )(

This is the continuity equation for incompressible fluid

Momentum equation is derived from the fundamental physical principle of Newton second law

Fx = m a = Fg + Fp + Fv

Fg is the gravity force Fp is the pressure force Fv is the viscous force Since force is a vectar, all these forces will have three components.

First we will go one component by next component than we will assemble all the components to get full Navier – Stokes Equation.

MOMENTUM EQUATION

[NAVIER STOKES EQUATION]

Fx – Inertial Force

Inertial Force = Mass X Acceleration derivative. Inertial Force in x direction = m X

represents instantaneous time rate of change of velocity of the fluid element as it moves through point through space.

u ).V( tu

Inertial force per unit volume in x direction =

Is called Material derivative or

Substantial derivative or

Acceleration derivative

‘u’ is variable

Inertial force / volume in y direction

Inertial force / volume in z direction Dt

DtDuInertial force / volume in x direction

Body forces act directly on the volumetric mass of the fluid element. The examples for the body forces are

Eg: gravitationalElectricMagnetic forces.

Body force =

Body force in y direction

Body force in z direction

dxdydzdxdydz

Body force per unit volume

Pressure on left hand face of the element

Pressure on right hand face of the element

Net pressure force in X direction is

Net pressure force per unit volume in X direction

dydzdxxpP

dxdydzxpdydzdx

dxdydzdxdydz

Pressure forces per unit volume

Net pressure force per unit volume in X direction

Net pressure force per unit volume in Y direction

Net pressure force per unit volume in Z direction

Net pressure force in all direction

Net pressure force in 3 direction

Viscous forces

Resolving in the X direction Net viscous forces

dxdy dz z

dxdz dy y

dydz dx

yxxxxx

dxdydz z

F zxyxxxv

zxyxxx

zyyyxy

zyxzzyzxz

Net viscous force per unit volume in X direction

Net viscous force per unit volume in Y direction

Net viscous force per unit volume in Z direction

UNDERSTANDING VISCOUS STRESSES

LINEAR STRESSES = ELASTIC CONSTANT X STRAIN RATE

strainlinear of rate average local x 2 x xxxx

Linear strain in X direction

zzyyxxe e e

V divor V . Volumetric strain

Three dimensional form of Newton’s law of viscosity for compressible flows involves two constants of proportionality. 1. dynamic viscosity.

2. relate stresses to volumetric deformation.

V divxu2xx

V divyv2yy

V divzw2zz

[ Effect of viscosity ‘ ’ is small in practice.

For gases a good working approximation can be obtained taking

Liquids are incompressible. div V = 0]

In this the second component is negligible

SHEAR STRESSES = ELASTIC CONSTANT X STRAIN RATE

n.deformatioangular rate average x 2 x yxxy

F xzxyxxvx

F xzyyyxvy

F zzzyzxvz

Having derived equations for inertial force per unit volume, pressure force per unit volume body force per unit volume, and viscous force per unit volume now it is time to assemble together the subcomponents.

vgfx F F F F

Assembly of all the components

DtDu zxyxxx

DtDv yzyyyx

DtDw zzzyzx

X direction:-

Y direction:-

Z direction:-

.V xu 2

X direction:-

Y direction:-

.V zw 2

Z direction:-

tDu xzxyxx

. uV u V. Vu .

CONVERTING NON CONSERVATION FORM ONN-S EQUATION TO CONSERVATION FORM

Navier-stokes equation in the X direction is given by

xxx xg

uV. . . VuuV

VuuVu . . V.

Divergence of the product of scalar times a vector.

Taking RHS of N-S Equation we have

tu u.V

V . u uV . t

u uV . tu

0 u uV . tu

CONTINUITY zw

Is equal to zero

xxx xg

xp uV .

xyx yg

yp uV .

xzx zg

zp uV .

CONSERVATION FORM:-

xxx xg

xyx yg

2v xuv

xzx zg

2w yvw

xu 2 .V

SIMPLICATION OF NAVIER STOKES EQUATION

xg xzw2

u xyv2

If is constant

xg xzw2

2 yxv2

1 2zu2

For Incompressible flow

Energy EquationEnergy is not a vector

So we will be having only one energy equation which includes the energy in all the direction.

The rate of Energy = Force X velocity

Energy equation can be got by multiplying the momentum equation with the corresponding component of velocity

dQ = dE + dW dE = dQ - dW = dQ + dW [Work done is negative] because work is done on the system.

Work done is given by dot product of viscous force and velocity vector.

for Xdirection

dxdydz

for Y direction

dxdydz yzu yyyv

for Z direction

dxdydz xxw

Body force is given by dxdydz V.g

wzg vyg uxg

Total work done

dxdydz V.f

dxdydz

Net Heat flux into element = Volumetric Heating + Heat transfer across surface.

Volumetric heating dxdydz .q

Heat transfer in X direction = dydz dx

xq x q

dxdydz x

dxdydz z

Heating of fluid element

dQ = B = dxdydz z

dQ = B dxdydz zTk

wp yvp

2V e DtD

f.V zzzw

wp yvp

2V e DtD

Energy EquationNonconservation form

wp yvp

2V e DtD

Non conservation:-

wp yvp

2V e . 2

2V e DtD

Conservation:-

.V p 2zT2

k 2yT2

.V p 2zT2

k 2yT2

uT pc x

Momentum Equation Non conservation form

X direction

Y direction

Z direction

Momentum Equation

Conservation form

X direction

Y direction

Z direction

VwtDwD

xyuxxxu

.)()()(

)()()()()()(

)()()(2

2)()()()(

Energy Equation

Non conservation form

zxwzxz

qVeVet

)()()()()(

)()()()()()(2

2)()()(])([])([

Energy equation

Conservation form

FORMS OF THE GOVERNING EQUATIONS PARTICULARLY SUITED FOR CFD

energytotalofFluxVVe

energyInternalofFluxVemomentumofcomponentzofFluxVwmomentumofcomponentyofFluxVvmomentumofcomponentxofFluxVu

fluxMassV

Solution vectar

Variation in x direction

xzwxyvxxuxTkupuVe

xzuwxyuv

Variation in y direction

zywyyvxyuyTkvpvVe

zyvwyypv

Variation in z direction

zzwyzvxxzuzTkwpwVe

xzzpwxyzwv

Source vectar

qzfwyfvxfuzfyfxf

Time marching

Types of time marching

1. Implicite time marching

2. Explicite time marching

Explicit FDM

Implicit FDM

Crank-Nicolson FDM

Space marching

Continuity Equation

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