CFD

Computational Fluid Dynamics

Natteri M. Sudharsan, PhD.

Natteri M Sudharsan, Ph.D., 2

CFD – A part of Knowledge Based Economy

Computational Fluid Dynamics (CFD) and Computational Structural Mechanics (CSM) is the study of fluid and structures subjected various external / internal flow and/or load conditions.

CFD is used in various fields such as: Aerospace AutomobileBio-medical

OceanOil and GasPower EngineeringTurbomachineryFluid – Structure InteractionEnvironmental Agencies


Aerospace Applications

External Aerodynamics (all speed regimes)

High Lift Internal Flows Thermal Management Aero acoustics Stability and Control Tank Sloshing

Plume Analysis Multi-Species Propeller Simulation Heat Exchangers Fluid Structure

Interaction (FSI) Combustion

KEY APPLICATIONS INCLUDE


CFD Simulations in Aerospace

Plume Analysis


HIGH LIFTROTORCRAFT


Environmental Control Systems


Automotive Applications

Fuel Injection – Break-up


Thermal Stress Analysis on Cylinder Head


Valve Simulation

After Treatment Devices for Pollution Control


Under the Hood


Comfort


Brake Cooling


Biomedical Applications

Grid Generation from MRI Scan


Streamline flow of Blood


Building Applications

Fire & Smoke Modeling

Heating Ventilation and Air conditioning

Pollutions Dispersion Modeling


Turbomachinery

Pelton Wheel Simulation


Wind Turbine

Power – conventional vs. Shrouded Turbine


Environmental Applications

ELECTROSTATIC PRECIPITATOR


Forest Fire & Control


Chemical Applications

Break-up and coalescence of bubbles


Spray Dryer


CFD Aided Design of HDDs

Objectives:•Identify the flow characteristics in HDDs at high rotating speed

•Seek solutions to reduce the flow-induced vibration of arms


Governing Equations&

Numerical Methods


Summary1. Derivation of Governing Equations for Energy.

2. Behavior of these Equations.

3. The three numerical schemes.

4. Discretization for transient problems.

5. Convection – Diffusion Equation and schemes.


Derivation of Energy Equations Consider a control volume of Area A and length ∆x. The energy balance is given as:

Heat in = Heat out qx + g’’’ = q x+∆x + qconv + qrad (1)


Figure: 1 Energy balance in a control volume

qx qx+∆x

∆x

g”’

qconv+ qrad


Steady State Heat Conduction

Neglecting convective and radiation loss and for a steady state heat conduction with no internal heat generation g’’’, the energy balance simplifies as

qx = q x+∆x (2)

Using Taylor series expansion to estimate q x+∆x we have,

...x)(qdxd

!x)(q

dxdqq xxxΔxx +∆+∆+=+

22

2

21


Neglecting higher order terms, we get,

(3)

Substituting the Taylor series expansion for qx+∆x In (2) yields

( ) xqdxdqq xxxx ∆+=∆+

( )

dxdTkAq

xqdxdqq

x

xxx

−=

∆+=

0=∆

− x

dxdTkA

dxd

(4)


Assuming k and A to be constants (4) becomes

02

2

=∆− xdx

TdkA

since k, A or ∆x cannot be zero, in coordinate invariant form yields:

02 =∇ T

This is a 2nd order Linear PDE. Classified as Laplace equation.


Steady State Heat Conduction with Internal Heat Generation

Assuming that there is internal heat generation within the control volume, equation (1) simplifies to qx + g’’’ = q x+∆x (5)

(6)

( )

02

2

=+

−=

∆+=∆+

kg

dxTd

dxdTkAq

xqdxdqxAgq

x

xxx

'''

'''


Steady State Heat Conduction with Convection

Heat in = Heat out by conduction + Convection

L

∅ d


02

2

=−−

−∆+∆

−+=

−∆+=

∞

∞

∞∆+

)(

)(

)(

TTkAhP

dxTd

TTxhPxdxdTkA

dxdqq

TTxhPqq

xx

xxx

Let,

222 dXLdxanddXLdxdTTdTLxX

TTTT

b

b

==−=

=−−=

∞

∞

∞

;)(

;

θ

θ

(7)


Substituting in (7) we get,

( ) 02

2

2 =−+−−−∞∞∞

∞ TTTTkAhP

dxd

LTT

bb )()( θθ

022

2

=− θθ )(mLdxd (8)

Fin tip boundary condition is convective then

)(,

)(

θθθθ Bik

hLkhL

dXd

TThAdxdTkA

=−=

−=− ∞


Steady state heat conduction with radiation

0442

2

=−− ∞ )( TTPdx

TdkA σ ε

Slug Flow

Tx Tx+∆x


xxpxxxpx AUTCqAUTCq ∆+∆+ +=+ |||| ρρ

αθθ

θ

α

ρ

ULPedXdPe

dXd

TTTT

LxX

dxdTU

dxTd

AUdTCxdx

TdkA

iL

i

p

==

−−==

=

=∆

,

,

2

2

2

2

2

2


Unsteady Heat Conduction

Heat in = Heat out + Heat stored Heat in – Heat out = Heat stored

Heat stored is τρ

∂∂∆=

• TxCAdTCm pp

.

τα

∂∂=

∂∂ T

xT2

2

Heat in – Heat out = 2

2

dxTdxkA∆


DISCRETIZATION METHODS

Finite Difference Method

211

211

3

33

32

2

2

33

32

2

2

02

31

2

322

31

21

31

21

hh

yyy

hyyh

yy

hyhyhxyhxy

hdx

ydhdx

ydhdxdyxyhxy

hdx

ydhdx

ydhdxdyxyhxy

ii

ii

)('

!'

!')()(

...!!

)()(

...!!

)()(

'''

'''

+−=

+=−

+=−−+

+−+−=−

++++=+

−+

−+

i-2 I -1 ∆x i i+1 i+2


The above is known as central difference. Theforward or backward difference for y’ can beobtained from yi and yi+1 or yi and yi-1 to yield

)(' xx

yyy ii ∆+∆

−= + 01 )(' xxyyy ii ∆+

∆−= − 01and

Similarly forward difference for y” is given as:-

212

212

32

32

22

2231

21

2312

2122

xyyyy

xyyyyhxyfromsubtractandbyhxyMultiply

hyhyhyyhxy

hyhyhyyhxy

iiii

iiii

iiii

iiii

∆+−=

∆+−=−++

+++=+

+++=+

++

++

''

''

''''''

''''''

)()()(

)(!

)(!

)()(

)(!

)(!

)()(


Backward difference

211 2

xyyyy iii

i ∆+−= +−''Central difference using

y(x+h) and y(x-h) yields

2212

xyyyy iii

i ∆+−= −−''

Non uniform grid size discretization

i-1 i i+1

2

2

211

2

2

2

11

2

2

2

1

22

2

2

1

21

1211

2

2

xT

axaaTTaT

axaxTTaaTT

xxTx

xTTT

xaxTxa

xTTT

iii

iii

ii

ii

∂∂=

+∆++−

+∆∂∂++=+

∆∂∂+∆

∂∂+=

∆∂∂+∆

∂∂−=

+−

+−

+

−

)()(

)()(

)(

)(

Let the distance from node i-1 to i be a∆x and i to i+1 be ∆x.


W P Ew e

∆xwP

∆xPE∆xWP

∆xPe

FVM Formulation


Γ

Γ+=

−+=

++=

=∆+∆−

∆

−Γ−

∆

−Γ

=+−

Γ

++

ofbehaviourtheondependingmeanharmonicormeanarithmeticbemay

SSSaslinearizedisSourceTheWhereSaaa

Saaa

xSxnxx

yieldsabovethegIntegratin

Sndxd

dxd

wore

uPP

PewP

uEeWwPp

PewPPewPPWP

WPw

PE

PEe

)()(

)()(

)(

φ

φφφ

φφφφφ

φφ

0

0


The 4 rules of Finite Volume

• Ensure that the flux across the face common to the two control volumes is represented by the same expression.

• Coefficients of ap and its neighbors should always be positive.

• SP should always be negative, at best should be less than 1.

• aP should be sum of all neighbors anb.


Discretization for Transient Problems

)()(

;

PewP

n

WP

WPw

PE

PEe

n

WP

WPw

PE

PEe

nP

nP

p

xSxx

xx

yieldsabovethegIntegratinck

xx

+

++

∆+

∆

−Γ−∆

−Γ−

+

∆

−Γ−∆

−Γ=

∆

−

=Γ∂∂=

∂∂Γ

∂∂

φφφφβ

φφφφβτ

φφρ

τφρφ

1

11

Where β = 0 is Explicit and β = 1 is Implicit


W P Ew e

∆xwP

∆xPE∆xWP

∆xPe

Convection – Diffusion Equation

Represents Upwind Formulation


)(/;/;

)()()()(

)()(

weewp

eeewwwEeWwPp

WPwPEeWPw

PEe

WP

WPw

PE

PEewwee

FFaaaFDaFDaaaa

DDFFSchemeCD

xDanduFLetxx

uu

yieldsgIntegratindxd

dxd

dxdu

−++=

−=+=+=

−−−=+−+

∆Γ==

∆

−Γ−

∆

−Γ=−

Γ=

2222

φφφ

φφφφφφφφ

ρ

φφφφφρφρ

φφρ

CD Scheme Limitation F/D i.e. Pe less than 2


Upwind Scheme

eeeww

eewww

PwEe

PeWw

FDaandDahaveWeFDaandFDahaveWeF

andhaveWeFandhaveWeF

−==<=+=>

==<==>

0000

φφφφφφφφ

Note: CD Scheme gives rise to artificial diffusion. Good for Pe < 2, hence ∆x needs to be small to ensure low Pe. Upwind gives reasonable results for higher Pe number. A hybrid scheme or power law can also be used.


Fluid Mechanics


TYPES OF FLUIDIdeal Fluid – Incompressible ‘zero viscosity’ fluid – Imaginary fluid.Real Fluid – Fluid with viscosityNewtonian – shear stress in proportional to ‘strain rate’ - grad V.Non – Newtonian – Ideal Plastic – shear stress in more than yield value and is proportional to strain rate.


Linear translation refers to bodily movement of fluid element with out deformation

Linear deformation refers to deformation in linear direction with axis remaining parallel


Angular deformation refers to average change contained by 2 adjacent sides

( )

∂∂+

∂∂=

∆+∆=

yu

xvratestrainshearOr

ndeformatioAngular

21

21

21 θθ


Pure rotation occurs when fluid particle rotates such that both axis moves with the same magnitude and direction

∂∂−

∂∂=

∂∂−

∂∂=

∂∂−

∂∂=

xw

zu

zv

yw

yu

xv

yxz 21

21

21 ωωω ;;

The rotational components are given as

ω2orVisVorticity ×∇


Reynold’s Transport TheoremConsider a fluid mass occupying an arbitrary volume. Let Φ be a transported quantity, and φ its intensive property, i.e. φ = Φ/m.

Let Sp be the space occupied by the fluid and cv the control volume overlapping fluid space Sp.


∫ ∫==Φ)(

)()(tSp cv

dVdV ρ φρ φ

∫∫

∫ ∫•∇=•

•+∂∂=Φ

cvcs

cv cs

dVudSnu

dSnudVtdt

d

)()(

)()(

ρ φρ φ

ρ φρ φ

By Gauss divergence theorem


u

Sp(∆ t) Sp(t+∆ t)

CV

Amount of fluid containedin time in time,t, is

∫)(

)(tsp

t dVρ φ

PROOF


Thus from time t to t+∆t we have

∫ ρ φ)t(sp

t dV)( ∫∆+

∆+ρ φ)tt(sp

tt dV)(

The increment ∆Φ = Φ(t+∆t) - Φ(t) =

∫−)(

)(tsp

t dVρ φ ∫∆+

∆++)(

)(ttsp

tt dVρ φ


this can be re-written as

∫∫

∫∫ρ φ−ρ φ

+ρ φ−ρ φ

∆+

∆+∆+

∆+

)t(spt

)t(sptt

)t(sptt

)tt(sptt

dV)(dV)(

dV)(dV)(

The first two terms deals with the deformation of the fluid and will occur at the rate of fluid velocity = u.n ds


∫ ρ φcs

u)( dSn.

The next two terms yields ∫ ∆ρ φ∂∂

)t(sp

tdV)(t

Thus proved.

∫∫

∫ ∫ρ φ•∇=•ρ φ

•ρ φ+ρ φ∂∂=Φ=

∆∆ Φ

cvcs

cv cs

dV)u(dSnu)(

dSnu)(dV)(tdt

dt

Lim


Derivation of Continuity Equation

Let Φ the transported quantity be mass, m, the intensive property φ = Φ/m =1. Law of conservation of Mass

0tt

m =∂Φ∂=

∂∂

0ndSudV)(t csCV

=•ρ+ρ∂∂

∫∫


Hence , using Gauss divergence theorem we have,

0dV)u(tcv

=

ρ•∇+

∂ρ∂

∫For an incompressible fluid we have

0u. =∇


Momentum Equation

By Newton’s II law, the rate of change of momentum is equal to the total force (both surface and body forces ) ∫

cv

dV.X


Normal and shear components are given in figure below. I subscript refers to the plane at which force acts x – YZ plane.II subscript refers to the direction.

σxy

σxx

σyx

σyy


ds dydx

Resolving the horizontal forces we have, dxdy yxxx ×σ−×σ

dxdy yyxy ×σ−×σAnd vertical forces =


nx=dy/ds and ny= - dx/ds or [dy -dx] = n.dsThus the external forces can be written as n.σ ds

[ ]

σσσσ

×−yyyx

xyxxdxdy


The surface + body forces =

Letting Φ to be momentum, mu, and φ=mu/m = u. Thus the rate of change of momentum

∫ ∫+σ•cs cv

dVXdsn

( ) dVXdtd

cv∫ +σ•∇=Φ


dV)uu()u(tdt

d

cv∫

ρ•∇+ρ

∂∂=Φ

from Reynolds transport theorem we have

and equating this rate of change of momentum to the forces =

( ) dVXcv∫ +σ•∇

X)uu()u(t

+σ•∇=ρ•∇+ρ∂∂


X

uuu.utu

tu

+σ•∇=

∇•ρ+ρ∇+∂∂ρ+

∂ρ∂

Taking the first and third term in LHS and second and fourth term of LHS we have,

Xuutuu

tu +σ•∇=

∇•+

∂∂ρ+

ρ•∇+

∂ρ∂


By continuity equation the first term of LHS is equal to zero. Thus,

Xuutu +σ•∇=

∇•+

∂∂ρ

Now connect the equation by substituting σ with known quantities viz. pressure and velocity.


( )

=

=

2

2

2

wwvwuvwvvuuwuvu

wvuwvu

uu

( ) Dz/wz/vz/uy/wy/vy/ux/wx/vx/u

wvuz/y/x/

u =

∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂

=

∂∂∂∂∂∂

=∇

zw

yv

xuu

∂∂+

∂∂+

∂∂=•∇


A B

D C

A’

C’D’

B’ δ2

δ1

It moves to a position A’B’C’D’.

Consider a fluid volume ABCD in space


Although the volume is conserved there is a strain in x and y direction as well as angular movement.

Strain, . xxx ∂ε∂=ε

Strain rate is given as,

xxx

xx

.u

txxtt=

∂ε∂

∂∂=

∂ε∂

∂∂=

∂ε∂

=ε•

yyy v=ε•

zzz w=ε•


The angular movement , ( )21

.

xy 21 δ+δ=ε

xand

yWhere

21

21

.

21xy∂

η∂=δ∂

ε∂=δ

δ+δ=ε

•••••

( )xyxy

x2y1

vu21

vxt

uyt

+=ε

=

∂η∂

∂∂=δ=

∂

ε∂∂∂=δ

•

••


If the fluid undergoes net rotation it is given by

( ) ( ) uuvt yx12 ×∇=−=δ−δ

∂∂

also known as vorticity ω.

Thus the full strain tensor is given as,

[ ]T

zzzyzx

yzyyyx

xzxyxx

DD +=

•••

•••

•••

21

εεε

εεε

εεε


=

∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂

=

zzz

yyy

xxx

wvuwvuwvu

z/wz/vz/uy/wy/vy/ux/wx/vx/u

D

Constitutive relation as per Hooke’s Law is ε=σ ]E[

For a Newtonian Fluid its given as:

( ) ijijkkdij 2p µ ε+δλ ε+−=σ

εkk=εxx+εyy+εzz=∇•u


Thus, ( ) ijijdij 2up µ ε+δ•∇λ+−=σ

The negative sign for p ensures positive flow along decreasing slope. pd stands for thermodynamic pressure, which includes the rotational and vibrational modes of energy as well as the mechanical pressure related to the translation energy (kinetic energy) of molecules.


p = static pressure = - (σxx+σyy+σzz)/3 which is the mechanical energy related to translation of kinetic energy of molecules.

The constitutive equation can be written as


)uw(212)uw(&

)vw(212)vw(&

)uv(212)uv(&

w2upv2upu2up

zxzxxzxz

zyzyyzzy

yxyxyxxy

zdzz

ydyy

xdxx

+µ×=+µ=σσ

+µ×=+µ=σσ

+µ×=+µ=σσ

µ+•∇λ+−=σ

µ+•∇λ+−=σµ+•∇λ+−=σ


Since σxx+σyy+σzz= -3p

Adding the constitutive equations yields -3p= - 3pd+3λ∇•u+2µ(∇•u) pd – p = ∇•u(λ + 2/3µ)

For incompressible flow ∇•u = 0, therefore pd = p. (3λ + 2µ) is called the Bulk viscosity.


By Stoke’s hypothesis setting bulk viscosity = 0 yields λ = - 2/3µThus,

δ•∇−εµ+δ−=σ ijijijij u

322p

Thus,

δ•∇−εµ•∇+∇−=

+σ•∇=

∇•+

∂∂ρ

ijij u312pX

Xuutu


For isothermal and incompressible flow ∇•u = 0, hence the III term in LHS equation simplifies as

( )

+•∇=•∇ jiijij εεµεµ212)(2


∂∂

∂∂+

∂∂

∂∂+

∂∂

∂∂+

∂∂

∂∂

∂∂+

∂∂

∂∂+

∂∂

∂∂+

∂∂

∂∂

•∇=

∂∂

∂∂+

∂∂

∂∂+

∂∂

∂∂+

∂∂

∂∂

∂∂+

∂∂

∂∂+

∂∂

∂∂+

∂∂

∂∂

•∇

zw

yw

zv

xw

zu

yw

zv

yv

xv

yu

xw

zu

xv

yu

xu

zw

yw

zv

xw

zu

yw

zv

yv

xv

yu

xw

zu

xv

yu

xu

µµµ

µµµ

µµµ

µ

2

2

2

21

21

21

21

21

21

2


∂∂

∂∂+

∂∂+

∂∂

∂∂+

∂∂+

∂∂

∂∂

∂∂+

∂∂

∂∂+

∂∂

∂∂+

∂∂+

∂∂

∂∂

∂∂+

∂∂

∂∂+

∂∂+

∂∂

∂∂+

∂∂

∂∂

=

zw

zyw

zv

yxw

zu

x

yw

zv

zyv

yxv

yu

x

xw

zu

zxv

yu

yxu

x

µµµ

µµµ

µµµ

2

2

2

∂∂∂+

∂∂∂+

∂∂+

∂∂+

∂∂

∂∂∂+

∂∂∂+

∂∂+

∂∂+

∂∂

∂∂∂+

∂∂∂+

∂∂+

∂∂+

∂∂

=

zyv

zxu

yw

xw

zw

zyw

yxu

zv

xv

yv

zxw

yxv

zu

yu

xu

22

2

2

2

2

2

2

22

2

2

2

2

2

2

22

2

2

2

2

2

2

2

2

2

µµµµµ

µµµµµ

µµµµµ


∂∂+

∂∂∂+

∂∂∂++

∂∂+

∂∂+

∂∂

∂∂∂+

∂∂+

∂∂∂+

∂∂+

∂∂+

∂∂

∂∂∂+

∂∂∂+

∂∂+

∂∂+

∂∂+

∂∂

=

2

222

2

2

2

2

2

2

2

2

22

2

2

2

2

2

2

22

2

2

2

2

2

2

2

2

zw

zyv

zxu

zw

yw

xw

zyw

yv

yxu

zv

yv

xv

zxw

yxv

xu

zu

yu

xu

µµµµµµ

µµµµµµ

µµµµµµ

∂∂+

∂∂+

∂∂

∂∂+

∂∂+

∂∂+

∂∂

∂∂+

∂∂+

∂∂

∂∂+

∂∂+

∂∂+

∂∂

∂∂+

∂∂+

∂∂

∂∂+

∂∂+

∂∂+

∂∂

=

zw

yv

xu

zzw

yw

xw

zu

yu

xu

yzv

yv

xv

zw

yv

xu

xzu

yu

xu

µµµµ

µµµµ

µµµµ

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

Re-arranging, we get:


But the sum in brackets at the end of each row is the divergence of the velocity, which is zero for Incompressible Flow.

zw

yv

xuv

∂∂+

∂∂+

∂∂=•∇

u

zw

yw

xw

zv

yv

xv

zu

yu

xu

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

∇=

∂∂+

∂∂+

∂∂

∂∂+

∂∂+

∂∂

∂∂+

∂∂+

∂∂

= µµ


upXuutu

0u

2∇µ+∇−=

∇•+

∂∂ρ

=•∇

( )u

jiijij

2

212)(2

∇=

+•∇=•∇

µ

εεµεµ

Thus,

We therefore have:


For a creeping flow, neglecting inertial terms yields the Stokes equation:-

Xuptu 2 +∇µ+− ∇=

∂∂ρ

For high fluids with high inertial terms the viscous terms can be neglected to yield the Euler equation:-

pXuutu ∇−=

∇•+

∂∂ρ


Integrating Euler equation over time yields the Bernoulli equation

Where, p* = p + X and

*puu − ∇=∇•ρ

( )

∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂∂

=∇•z/wz/vz/uy/wy/vy/ux/wx/vx/u

wvuuu


∂∂+∂∂+∂∂∂∂+∂∂+∂∂∂∂+∂∂+∂∂

=∇•∴z/wwy/wvx/wu

z/vwy/vvx/vuz/uwy/uvx/uu

uu

We have seen the derivation of the Navier-Stokes Equation and the simplification of the equation to yield the Stokes, Euler and Bernoulli equation.


Slug Flow using Reynold’s Transport Theorem

Γ==== UL

kULCULPe

Ck P

p

ρραρ

α ,PC

k=Γ

Continuity Equation

0=∂

∂+∂

∂ )( jj

uxt

ρρ

Generalized Transport Equation

( ) ( ) Sxx

uxt jj

jj

+

∂∂Γ

∂∂=

∂∂+

∂∂ φφρρ φ


Let density remain constant and let φ =CpT and Assuming 1-D and constant velocity, U,

2

2

xT

xTU

tT

∂∂Γ=

∂∂+

∂∂ ρρ

dividing by density and for steady state

2

2

xT

xTU

∂∂=

∂∂

α

αθθ

θ

ULPedXdPe

dXd

TTTT

LxX

iL

i

==

−−==

,

,

2

2


Boundary layer


Boundary Layer Flows

Flow around bodies immersed in a fluid streamFlow will have viscous effects near body surface, inviscid in far fieldFlow is unconfined unlike pipe flowField of study – Aerodynamics, Hydrodynamics, Transportation, Wind engineering (Tall structures) etc.,


Boundary layer analysis can be used to compute viscous effects near wall and patch the results to the outer inviscid motion. This patching is successful for large Re.

Consider figure with Re = 10. The fluid flow is retarded greatly with thick shear layer.


Re 1 to 1000


Flows with thick shear layers do not have simple theory for performing analysis.

Low range Re flows are studied experimentally or numerically (computational fluid dynamics).


Turbulent Flow – Re=10

Very Thin boundary layer – Amenable to boundary layer patching (Prandtl,1904)


The three regions in the boundary layer are:

1. The Wall Layer: Viscous shear dominates

2. Outer Layer: Turbulent shear dominates

3. Overlap Layer: Both shear types are important.


Laws relating to Velocity Profile

VelocityFrictionalu w ,ρ

ττ =


Linear Sub Layer y+<5 Law of the wall

wyuy τµτ ≈

∂∂=)(

Integrating with B.C’s u =0, y=0, gives a linear relationship and equated as,

++ ==

===

yuoryuuu

yuyyu ww

ν

νρ ντ

µτ

τ

τ

τ2


Far away form the wall one can expect that velocity at a point be influenced by the retarding effect of the wall through the value of wall shear stress, but not by viscosity itself. Thus,

)/(max δτ

ygu

uu =−

is known as the velocity defect law.


Log Layer – Turbulent region close to the smooth wall. The region outside viscous sub layer where both viscous and turbulent effects are important. 30 < y+ < 500. τ varies slowly with distance from the wall and within this inner region it is assumed to be constant and equal to wall shear stress.

]Ey[lnk1B,Byln

k1u +++ =+=


k and B are universal constants, E is the wall roughness, the constants are obtained by measurements. B=5.5, k=0.4 and E = 9.8. The log layer is shown to be valid from 0.02 < y/δ <0.2 by experiments.


Ayku

uu +

=−

δτ

lnmax 1

Outer Layer – Inertia dominated region. For larger values of y, the velocity defect law provides the correct form. For the overlap region between the log layer and the defect layer the values must be the same, i.e.

A is a constant and called the law of the wake


TWO – DIMENSIONAL FLOW

∂∂+

∂∂µ+

∂∂−=

∂∂+

∂∂ρ

∂∂+

∂∂µ+

∂∂−=

∂∂+

∂∂ρ

=∂∂+

∂∂

2

2

2

2

2

2

2

2

yv

xv

yp

yvv

xvu

yu

xu

xp

yuv

xuu

0yv

xu

Since shear layer is thin for large Re (Prandtl), following assumptions apply


only)x(ppor0yp

eqnmomentumytoapplyingyx

uv

==∂∂

−∂∂< <

∂∂

< <

For the outer inviscid flow applying the Bernoulli equation

2

2

2

2

yu

xu

dxdUU

dxdp

∂∂< <

∂∂

ρ−=


The three full equations are reduced to Prandtl’s two – boundary layer equation

yu

yu

ydxdUU

yuv

xuu

yv

xu

t

l

∂∂=

∂∂=

∂∂+≈

∂∂+

∂∂

=∂∂+

∂∂

τµτ

µτ

τρ1

0

LAMINAR

TURBULENT


Boundary Layers with Pressure Gradient

Flow separation is caused by excessive momentum loss near the wall in a boundary layer trying to move downstream against increasing pressure dp/dx>0 is called adverse pressure gradient. At the wall u = v = 0 thus

dxdp1

yu2

2

µ=

∂∂


Boundary Layers with Pressure Gradient




Turbulence


The uniform distribution of velocity tends to become irregular either due to a solid interaction or when neighboring streams of the same fluid flow past one another causing what is known as turbulence. It is to be noted that the irregularity associated with turbulence can be described by the laws of probability by being able to determine the fluctuation by statistical averaging.

TURBULENCE


N – S Equation for Turbulent Flow

The additional terms depend on the turbulent fluctuations of the stream. These terms can be interpreted as components of stress tensor. Therefore,

∂∂+

∂∂+

∂∂−∇+

∂∂−=

∂∂+

∂∂+

∂∂+

∂∂ ''''' wu

tvu

yu

xu

xP

zuw

yuv

xuu

tu 22 ρµρ

∂∂

∂∂+

∂∂+∇+

∂∂−=

∂∂+

∂∂+

∂∂+

∂∂

zxyxxx zyxu

xP

zuw

yuv

xuu

tu ''' ττσµρ 2


in 3-D comparing the above

−=

=

2

2

2

'''''''''''''''

'''''''''

wvwuwwvvuvwuvuu

zzyzxz

zyyyxy

zxyxxx

T ρστττστττσ

σ

The total laminar and turbulent stresses in the fluid are

'':'';'' wuxw

zuvu

xv

yuu

xuP xzxyxx ρµτρµτρµσ −

∂∂+

∂∂=−

∂∂+

∂∂=−

∂∂+−=

22


TURBULENCE MODLES

Mixing length – Prandtl’s mixing length hypothesis κ - ε Model Reynold’s Shear Stress (aka) 7 equation model


Models for evaluating ρui’uj

’ J Boussinesq introduced a mixing co-efficient for the Reynold’s stress term as µτ

yu

yu

yu

yuvu

l

t

∂∂=

∂∂=

∂∂=

∂∂=−=

ρ νµτ

ρ νµρτ ττ''


Thus the equation can be re-written as

( )

∂∂+

∂∂+

∂∂−=

∂∂+

∂∂

−

∂∂

∂∂+

∂∂−=

∂∂+

∂∂

yu

yxP

yuv

xuu

vuyu

yxP

yuv

xuu

tννρ

νρ

1

1 ''

term is known as eddie viscosity. The kinematic viscosity is a property and not influenced by flow, but the eddie viscosity is attributed to the random fluctuation and it is not a property of the fluid.

τν


There are various models that relate the apparent (or) eddie viscosity to the mean velocity gradient.Prandtl mixing length hypothesis

yul

yul

∂∂=

∂∂=

2

2

τ

τ

ν

ρµ

l is the Prandtl mixing length, For wall – boundary layers


.,.,,.,,.

)/()/(

wallfromdistyandrealisedisUwherelocationconstKarmanVonconstlengthmixing

yforlandyforyl

∞

==>=≤=

990410090

δχλ

δχλλ δδχλχ


−=

−+ ν

ρτ

χ Ay w

eyl21

1

/)/(

, is the wall shear stress, and A+ is the Von Driest damping constant = 26 for smooth surface without suction or blowing on low pressure gradient surface.

wτ

−=

−+ ν

ρτ

χ Ay w

eyl21

1

/)/(

−=

−+ ν

ρτ

χ Ay w

eyl21

1

/)/(

wτ

−=

−+ ν

ρτ

χ Ay w

eyl21

1

/)/(

, is the wall shear stress, and A+ is the Von Driest damping constant = 26 for smooth surface without suction or blowing on low pressure gradient surface.

wτ

Very close to the wall, viscous effects play a vital role. The mixing length should gradually go to zero

−=

−+ ν

ρτ

χ Ay w

eyl21

1

/)/(


For free shear flows, the mixing length scales with the shear layer thickness (δs). The eddie viscosity can be written as

sdCU δν τ =

where Ud, is the characteristic velocity defect across shear layer and C is the proportionality constant to be determined from experiments.


κ - ε Model for Turbulence

The instantaneous velocity of the fluid is decomposed into mean and fluctuating components. The effect of the fluctuating components on the mean motion is modeled after obtaining empirical relations for specific cases from experiments.


In κ - ε model, the local turbulent viscosity is determined from the solution of the transport equation for the turbulent kinetic energy, κ, and the rate of dissipation of kinetic energy, ε.

The x-momentum

∂∂+

∂∂

∂∂+

∂∂

∂∂+

∂∂−=

∂∂+

∂∂+

∂∂

xv

yu

yxu

xxPvu

yuu

xtu

effeff µµρρρ 2)()(


The y-momentum

∂∂+

∂∂

∂∂+

∂∂

∂∂+

∂∂−=

∂∂+

∂∂+

∂∂

xv

yu

xyv

yyPvv

yvu

xtv

effeff µµρρρ 2)()(

0902 .;/; ==+= µµττ ερ κµµµµ CCeff

The transport of passive scalar is written as Advection = Diffusion + Generation – dissipation


κερ

κεε

σµε

σµ

ερερ

ρ εκσµκ

σµ

κρκρ

εε

κκ

2

21 CGCyyxx

vy

ux

Gyyxx

vy

ux

effeff

effeff

−+

∂∂

∂∂+

∂∂

∂∂=

∂∂+

∂∂

−+

∂∂

∂∂+

∂∂

∂∂=

∂∂+

∂∂

)()(

)()(

G, is the generation of turbulent kinetic energy and is given as

∂∂+

∂∂+

∂∂+

∂∂=

222

2xv

yu

yv

xuG τµ

Empirical Values: C1 = 1.44, C2 = 1.92, σκ = 1.0, σε = 1.3 Cµ = 0.09


For near wall region the equation for wall shear stress is given as

µρ κχκρ

τ µτ

µ pp

pw

yCyanduuuwhere

EyCu 41212141 ////

,/,)ln(

=== ++

+

+

uτ, is the frictional velocity and yp+ >11 non-

dimensional distance from the wall to the point outside viscous layer, E an empirical constant, wall roughness = 9.8.


Since ,

ts is the characteristic time scale, vs=κ1/2 and ls=(κ3/2)/ε,

( )stt

andwvu κκεκ ~,'''21 222

∂∂=++=

ερ κµ µτ /2C=

B.C Inlet κ - ε distribution must be given, outlet or symmetry

00 =∂∂=∂∂ nandn // εκ

Free stream κ = ε = 0. Solid walls depends on Re.


kyu

CuEy

uuu

321 τ

µ

τ

τ

εκχ

==== ++ ;);ln(

High Reynold’s number 30<y+<500.

Low Reynold’s number

0902 .;/; ==+= µµµττ ερ κµµµµ CfCeff

κερ

κεε

σµε

σµ

ερερ

ρ εκσµκ

σµ

κρκρ

εε

κκ

2

2211 fCGfCyyxx

vy

ux

Gyyxx

vy

ux

effeff

effeff

−+

∂∂

∂∂+

∂∂

∂∂=

∂∂+

∂∂

−+

∂∂

∂∂+

∂∂

∂∂=

∂∂+

∂∂

)()(

)()(


ε νκµ

µ

/Re);Reexp(;.

Re.)]Re.exp([

222

3

1

2

10501

5201016501

=−−=

+=

+−−=

tt

ty

ff

f

f

Ret Turbulence Reynold’s number

Reynolds shear Stress Model 7 equation model, 6 PDE for the stresses

2

2

2

'''''''''

wwvvwuvuu


Disadvantages

Completely incapable of describing flows with separation and recalculation.Only calculates mean flow properties and turbulent shear stress.

AdvantagesMixing length model: Easy to implement and cheap in terms of computing resources.Good Predictions for thin shear layers i.e. jets, mixing layers, wakes and boundary layers.Well established


More expensive to implement than mixing length model (2 extra PDE’s)Poor performance in a variety of important cases such as Some unconfined flowsFlows with large extra strains (eg curved boundary layers, swirling flows)Rotating flowsFully developed flows in non-circular ducts.

k- ε Model:

Simplest turbulence models for which initial and/or boundary conditions need to be supplied.Excellent performance for many industrially relevant flows.Well established: The most widely validated turbulence model.


Very large computing cost (7 extra PDE’s)Not as widely validated as mixing length and κ - ε models.Performs just as poorly as the κ - ε model in some flows owing to identical problem with the ε equation modeling (eg, axi-symmetric jets and unconfined re-circulating flows.

Reynold’s Stress model:

Potentially the most general of all classical turbulence models.Only initial and / or boundary conditions need to be suppliedVery accurate calculation of mean flow properties and all Reynold’s stresses for many simple and more complex flows including wall jets, asymmetric channels and non circular duct flows and curved flows.


Unstructured Finite Volume Method


Summary

1. Discretization of equations in unstructured finite volume method.

2. Simple Algorithm.


0=−∫∂ S

s dSnvv ).(

∫ ∫ ∫∫∂ ∂

+=−+∂∂

S S Vbs

V

dVfdSnTdSnvvvvdVt

.).(ρρ

])([ TvvpIT ∇+∇+= µ

∫ ∫∫∫∫∂∂

+∇+∇−=−+∂∂

S Vb

VSs

V

dVfdSnvpdVdSnvvvvdVt

).().( µρρ

Continuity Equation

Momentum Equation

Governing Equations


Linear system (momentum)

0 0 0 0 01

kC j jP p P P p

ja a bφ φ

== +∑

1 10

1 ( ) ( )k k

j j j j jCj jP j

p A Aa= =

′∇ ⋅ = ⋅

∑ ∑n v n

∫∫∂

=−∂∂

SV

dSnvdVt

0.Space Conservation Law

Pressure correction


C

Njj

rc

rj

Arbitrary cell C with neighbor Nj across jth face


Spatial Representation

∑∫∫ ∆≈⇒=

−+=

jjjC

SV

CjC

SV

graddSdVgrad

rrgradj

φφφφ

φφφ

1)(

)).(()(

The gradient is obtained using the Gauss Theorem (2nd

order approximation). Since the value of j would be different when applied from either side of face a symmetric expression is used and presented as follows:

( ) [ ]).()().()(jjj NjNCjCNC

SOj rrgradrrgrad −+−++= φφφφφ

21

21


( )jNC φφ +

21 Term gives the value midway between cell

centers connected by a straight line.

[ ]).()().()(jj NjNCjC rrgradrrgrad −+− φφ

21

The above expression provides a correction which takes into account that the cell-face center may not lie on the line connecting the cell centers and/or that the distances to the two CV’s may not be equal. This would vanish otherwise.

( )FOj

SOj

FOjj

jN

jCFOj CtoNfromisflowWhen

NtoCfromisflowWhen

j

φφγφφ

φφ

φ

−+=

=

*


Time Integration

ττφφφ ∆+= +++ ),( 1

11n

nnn F

A first order approximation can be used where temporal accuracy is not important

Implicit Schemes that can be used are the Crank-Nicolson Method or a second order approximation assuming a quadratic profile for φ

[ ][ ] profileQuadraticF

NicolsonCrankFF

nnnnn

nn

nnnn

ττφφφφ

τφτφτφφ

∆+−=

−+∆+=

++−+

+++

),(

),(),(

1111

111

32

31

34

2


Moving Grid

tV

dSnvV

dSnvdVt

j

S

j

SV

∆==

=−∂∂

∫

∫∫

∂

•

∂

δ.

. 0

In second order the the swept volume by the cell face during the preceding time step is included to yield:

∫ ∆−

==•

jS

njj

sj tVV

dSvV2

3 δδ


Convective Fluxes

)(

..)(

*

.)(

*

**

*

FOj

SOj

FOjj

jjjjS

s

jj

Ssj

VSvdSvvm

valuemeanfacecellindicates

mdSvvC

j

j

φφγφφ

ρρ

φρ φ

−+=

−≈−=

−

≈−=

••

•

∫

∫


Diffusive Fluxes

( ) jS

jj SgraddSgradDj

j.. *∫ Γ≈Γ= φφ φφ

The approximation of the gradient based on the SO (second order) interpolation

jφΓ is diffusivity at face-center and obtained using:

( ) [ ]).()().()(jjjj NjNCjCNC

SO rrgradrrgrad −Γ+−Γ+Γ+Γ=Γ21

21

φ

∑∫∫ ∆≈⇒=

jjjC

SV

SV

graddSdVgrad φφφφ 1)(


This II order space-centered expression cannot sense oscillations that are twice the characteristic length of the numerical mesh. Unphysical oscillatory profiles if induced, remain superimposed.

RECTIFICATION: A III order dissipation term is added as:

( ) ( )

j

j

NCj

j

j

j

j

j

CNjj

rrdIIIIIITERMS

SS

ddgrad

dgradgrad

−=

−

−+=

.* φφφφφ


The over-bar in term III represents the arithmetic average of values calculated at node C and Nj.

The II & III term is the difference between the central difference approximation of the derivative in the direction of vector dj (II) and the value obtained by interpolating cell-center gradients (III). This term in brackets (II-III) vanishes if the spatial variation for phi is linear or quadratic. Its magnitude is proportional to the II order truncation error and reduces with grid refinement.

The I term is the contribution from the nearest neighbors and treated implicitly, the II & III term represent the cross-diffusion component and vanish with orthogonal grid. This is treated explicitly which is a deferred correction approach.


The resulting algebraic equation is of the form

φφφ φφ Cj

NjCC raaj

=− ∑

SIMPLE Algorithm is used to solve the equation where, the momentum equations are solved assuming that the pressure field is known (predictor step).The mass fluxes are corrected to satisfy the continuity requirement in the corrector step by correcting velocity and density.


SIMPLE Algorithm

Semi Implicit Pressure Linked Equation'' ; vvvpppLet oo +=+=

Substituting the values of p and v as estimated plus corrections in the momentum equations and neglecting other terms yields:

ρ

ρρ

tAypAvand

xpAu

yp

tvand

xp

tu

∆=∂∂−=

∂∂−=

∂∂−=

∂∂

∂∂−=

∂∂

;''''

''''

Note: Corrections are zero at the first iteration


ypAvvand

xpAuu

∂∂−=

∂∂−= ''

00

Substituting this in the continuity equation gives

( )02

2

2

2

2

1

0

0

vA

p

yp

xpA

yv

xu

ypAv

yxpAu

x

oo

oo

.'

''

''

∇=∇

=

∂

∂∂

∂−

∂

∂+∂

∂

=

∂∂−

∂∂+

∂∂−

∂∂

This is poisson’s equation for pressure correction


As seen the velocity obtained in the predictor step does not necessarily satisfy the continuity equation and there exist a mass imbalance in each CV.The velocity and pressure corrections are obtained from

'','' pp

pgradaVv vC ∂

∂=∆= ρρ

vCaV∆ This coefficient comes from the discretized

momentum equation.

p∂∂ ρ is calculated from the equation of

state.


The velocities used to calculated the mass fluxes are obtained by interpolation as given below:

( ) [ ]).()().()(jjj NjNCjCNC

SOj rrvgradrrvgradvvv −+−++=

21

21

Cell-center pressure gradients are not sensitive to oscillations between immediate neighbors. Hence a dissipation term is added to the interpolated velocity which can sense such oscillations and smoothen it.

This method is for co-located grids. A staggered grid may also be used. However, requires more memory storage for storing geometric information along with tedious interpolation.


j

j

j

CNvC

j

j

j

j

j

j

CNvC

jj

SS

d

ppaVv

SS

ddpgrad

d

ppaVvv

j

j

'''

.*

−

∆−=

−

−

∆−=

Thus the procedure is repeated till all corrections are within the limits prescribed.

Thank you

Date post:	29-Oct-2014
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