Continuity Equation

Continuity Equation

dxdydz x

)u( dz dy u - dy dz dx

x

)u(u

Net outflow in x direction

Continuity Equation

net out flow in y direction, 

dxdydz y

)v( dz dx v - dx dz dy

y

)v(v

Continuity Equation

Net out flow in z direction   dxdydz

z

wdydxw dx dydz

z

ww

)( -

)(

Net mass flow out of the element 

dxdydz z

)w(

y

)v(

x

)u(

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 z

w

y

v

x

u

)(

)(

)(

Continuity Equation

sec3m

kgm 0

z

)w(

y

)v(

x

)u(

t

0 .

Vt

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 )(

)(

)(

0

sec 0

)(

)(

)(

3

z

w

y

v

x

u

t

m

kgm

z

w

y

v

x

u

t

If the density is constant

0 )(

)(

)(

0 )(

)(

)(

z

w

y

v

x

u

z

w

y

v

x

u

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.  

Dt

Du

Dt

Du

u ).V( t

u

Dt

Du

z

u w

y

u v

x

u .u

t

u

v

ma

z

uw

y

uv

x

uu

t

u

Dt

Dua

v

m

Dt

Du

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  

Dt

Dv

z

v w

y

v v

x

v u

t

v

Inertial force / volume in z direction    Dt

Dw

z

w w

y

w v

x

w u

t

w

Dt

DuInertial force / volume in x direction

z

uw

y

uv

x

uu

t

u

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

xx g

dxdydz

dxdydz

v

mg

g

yg

zg

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

dydzP

dydzdxx

pP

dxdydzx

pdydzdx

x

pPP

x

p

dxdydz

dxdydz

x

p

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  

x

p

y

p

z

p

z

p

y

p

x

p

z

p

y

p

x

p

P

Viscous forces

Resolving in the X direction Net viscous forces 

dxdy dz z

dxdz dy y

dydz dx

dx

zxzx

zx

yxyx

yxxxxx

xx

  

  

 

 

 

 

dxdydz z

y

x

F zxyxxxv

a z

y

x

zxyxxx

b z

y

x

zyyyxy

c

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   

x

uexx

y

veyy

z

wezz

 

 

zzyyxxe e e

z

w

y

v

x

u

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 divx

u2xx

V divy

v2yy

V divz

w2zz

 

[ Effect of viscosity ‘ ’ is small in practice.

For gases a good working approximation can be obtained taking

Liquids are incompressible. div V = 0]

3/2

In this the second component is negligible

SHEAR STRESSES = ELASTIC CONSTANT X STRAIN RATE

n.deformatioangular rate average x 2 x yxxy

x

v

y

uyxxy

x

w

z

uzxxz

y

w

z

vzyyz

  

z

y

x

F xzxyxxvx

z

y

x

F xzyyyxvy

z

y

x

F zzzyzxvz

  

z

u

x

w

zy

u

x

v

xFvx

y .V

x

u 2

.V 2

z

v

y

w

zy

w

yy

u

x

v

xFvy

.V. 2

y

w

zz

v

y

w

yx

w

z

u

xFvz

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

  

  

 

z

y

x

g x

p

Dt

Du zxyxxx

x

z

y

x

g y

p

Dt

Dv

yzyyyxy

zyx

gz

p

Dt

Dw zzzyzxz

X direction:-

Y direction:-

Z direction:-

x

w

z

u

z

y

u

x

v

y

.V x

u 2

x g

x

p

z

u w

y

u v

x

u u

t

u x

X direction:-

z

v

y

w

z .V

y

v 2

y

y

u

x

v

x g

y

p

z

v w

y

v v

x

v u

t

v y

Y direction:-

.V z

w 2

z

z

v

y

w

y

x

w

z

u

x g

y

p

z

w w

y

w v

x

w u

t

w z

Z direction:-

z

y

x g

x

t

Du xzxyxx

x

+

. uV u V. Vu .

CONVERTING NON CONSERVATION FORM ONN-S EQUATION TO CONSERVATION FORM

  

  

 Navier-stokes equation in the X direction is given by 

zxz

y

xy

xxx xg

x

t

Du

uV. . . VuuV

VuuVu . . V.

Divergence of the product of scalar times a vector.

  

t

u t

u

t

u

t

u t

u

t

u

Taking RHS of N-S Equation we have

u.V

t

u u.V

t

u

z

u w

y

u v

x

u u

t

u

Dt

Du

V . u uV . t

u t

u

V . t

u uV . t

u

0 u uV . t

u

Dt

Du

CONTINUITY z

w

y

v

x

u

t

since

Is equal to zero

zxz

y

xy

xxx xg

x

p uV .

t

u

z

yz

y

yy

x

yx yg

y

p uV .

t

v

zzz

y

zy

xzx zg

z

p uV .

t

w

CONSERVATION FORM:-

zxz

y

xy

xxx xg

x

p

z

uw

y

uw

x

2u

t

u

z

yz

y

yy

x

yx yg

x

p

z

vw

y

2v

x

uv

t

v

zzz

y

zy

xzx zg

z

p

z

2w

y

vw

x

uw

t

w

xg x

w

z

u

z

y

u

x

v

y

x

u 2 .V

x

x

P

z

uw

y

uv

x

2u

t

u

SIMPLICATION OF NAVIER STOKES EQUATION

xg xz

w2

2z

u2

2y

u

xy

v2

x

u 2

z

w

y

v

x

u 3

2 x

x

P

z

uw

y

uw

x

2u

t

u

If is constant

xg xz

w2

2z

u2

2y

u

xy

v2

x

u 2

z

w

y

v

x

u 3

2 x

x

P

z

uw

y

uw

x

2u

t

u

xg xz

w2

2z

u2

2y

u2

xy

v2

2x

u2 2

zx

w2 3

2 yx

v2 3

2 2x

u2 3

2

x

P

z

uw

y

uw

x

2u

t

u

xz

w2 3

1 xy

v2 3

1 2z

u2

2y

u2

x

u2 3

11 x

P

z

uw

y

uv

x

2u

t

u

z

w

x 3

1 y

v

x 3

1 x

u

x 3

1

2y

u2

2x

u2

x

P

z

uw

y

uv

x

2u

t

u

z

w

y

v

x

u 3

1 2z

u2

2y

u2

2x

u2

x

P

z

uw

y

uv

x

2u

t

u

V. 31

2z

u2

2y

u2

2x

u2

x

P

z

uw

y

uv

x

2u

t

u

2z

u2

2y

u2

2x

u2

x

P

z

uw

y

uv

x

2u

t

u

For Incompressible flow

0 V .

Energy Equation

Energy 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

V.vF

dxdydz

zzxu

y

yx.u

xxxu

x

up

for Y direction

V.vF

dxdydz yzu y

yyv

x

yxv

y

vp

for Z direction

dxdydz xxw

y

zyw

xzxw

z

wp

V.vF

Body force is given by dxdydz V.g

wzg vyg uxg

Total work done 

dxdydz V.f

dxdydz

zzzw

y

yzw

xxzw

z

zyv

y

yyv

x

xyv

zzxu

y

yxu

xxxu

z

wp

y

vp

x

up

C

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

Volumetric heating dxdydz .q

Heat transfer in X direction = dydz dx

x

.xq

xq x q

dxdydz x

.q

dxdydz z

.zq

y

.yq

x

.xq

Heating of fluid element

dQ = B = dxdydz z

.zq

y

.yq

x

.xq

.q

dQ = B dxdydz z

Tk

z

y

Tk

y

x

Tk

x q

z

wp

y

vp

x

up

z

Tk

z

y

Tk

y

x

Tk

x q

2

2V e

Dt

D

f.V zzzw

y

yzw xzw

z

zyv

y

yyv

x

xyv

zzxu

y

yxu

xxxu

z

wp

y

vp

x

up

z

Tk

z

y

Tk

y

x

Tk

x q

2

2V e

Dt

D

Energy EquationNonconservation form

z

wp

y

vp

x

up

z

Tk

z

y

Tk

y

x

Tk

x q

2

2V e

Dt

D

f.V

zzzw

y

yzw

xxzw

z

zyv

y

yyv

x

xyv

zzxu

y

yxu

xxxu

Non conservation:-

z

wp

y

vp

x

up

z

Tk

z

y

Tk

y

x

Tk

x

q V 2

2V e .

2

2V e

Dt

D

f.V

zzzw

y

yzw

xxzw

z

zyv

y

yyv

x

xyv

zzxu

y

yxu

xxxu

Conservation:-

.V p 2z

T2k

2y

T2k

2x

T2k q

z

Tpc w

y

Tpcu

x

Tpcu

x

Tpc

.V p 2z

T2k

2y

T2k

2x

T2k q

z

wT

y

vT

x

uT pc

x

Tpc

xfzxz

yxy

xxx

x

p

tD

uD

fyz

yzyyy

xyx

y

p

tD

vD

fzz

zzy

zyx

zxz

p

tD

wD

Momentum Equation     Non conservation form 

X direction

Y direction

Z direction

Momentum Equation 

Conservation form 

X direction

Y direction

Z direction

xfzxz

yxy

xxx

x

pVu

tD

uD

)(.

fyz

yzyyy

xyx

y

pVv

tD

vD

).(

fzz

zzy

zyx

zxz

pVw

tD

wD

)(.

Vfz

zzw

yzyw

xzxw

zxz

zyzv

yyyv

xyxv

zxzu

yxyu

xxxu

z

wp

y

vp

x

up

z

Tk

zy

Tk

yx

Tk

xq

Ve

tD

D

.)()()(

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

)()()(

2

2)()()()(

Energy Equation

   Non conservation form

Vfz

zzw

yzyw

xzxw

zxz

zyzv

yyyv

xyxv

zxzu

yxyu

xxxu

z

wp

y

vp

x

up

z

Tk

zy

Tk

yx

Tk

xq

Ve

Ve

tV

.)(

)()()()()(

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

2

2.

2

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

Energy equation

Conservation form

FORMS OF THE GOVERNING EQUATIONS PARTICULARLY SUITED FOR CFD

energytotalofFluxVV

e

energyInternalofFluxVe

momentumofcomponentzofFluxVw

momentumofcomponentyofFluxVv

momentumofcomponentxofFluxVu

fluxMassV

)(2

2

Solution vectar

)(2

2Ve

w

v

u

U

Variation in x direction

xzwxyvxxux

Tkupu

Ve

xzuwxyuv

xxpu

u

F

)(2

2

2

Variation in y direction

zywyyvxyuy

Tkvpv

Ve

zyvwyypv

yxvu

v

G

)(2

2

2

Variation in z direction

zzwyzvxxzuz

Tkwpw

Ve

xzzpwxyzwv

xzwu

w

H

)(2

2

2

Source vectar

qzfwyfvxfu

zfyfxf

J

)(

0

Time marching

Jz

H

y

G

x

F

t

U

Types of time marching

1. Implicite time marching

2. Explicite time marching

Explicit FDM

Implicit FDM

Crank-Nicolson FDM

Space marching

Jz

H

y

G

x

F