cfd refresher course

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Computational Fluid DynamicsComputational Fluid Dynamics2D Convection- Diffusion Problems2D Convection- Diffusion Problems

Pravin NakodFluent [email protected]

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Overview

• Generalized 2D N-S Equation for • 2D Convection-Diffusion Problem • Discretization Schemes• Example of 2D Convection-Diffusion • Checkerboard Pressure Field• Staggered Grid Approach• Pressure Correction Equation• SIMPLE (Semi Implicit Method for Pressure Linked

Equation)• Types of Mesh and Quality of Mesh

Session-1

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Generalized 2D N-S Equation for

S)

y(

y)

x(

x)v(

y)u(

x)(

t

1 2 3 4

1 => Transient Term

2 => Convective Term

3 => Diffusion Term

4 => Source Term

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Generalized 2D N-S Equation for

xBxp

yByp

Pr

ypv

xpuTQradQch

Cp v1

Sc

Sr No Equation S

1 Bulk Mass Conservation

1 0 0

2 X - Momentum

u

3 Y - Momentum

v

4 Energy T

5 Species Conservation

j Rj

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2D Convection-Diffusion Problem

Assumptions:

1) X = Y

2) Mesh is uniform

3) Flow is steady

S)

y(

y)

x(

x)v(

y)u(

x

Governing equation:

YP EW

S

N

X

V

ws

e

n

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S)

yv(

y)

xu(

x

Lets define

) Sqx ix

i

iix x

uqi

I = 1, 2

Integrating over control volume

from w to e and

from s to n

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) n

s

e

wPx

n

s

e

w i

dydxSdydxqx i

Assuming quantities are uniform at cell faces

) ) YXSXqqYqq syenwxex

X Y = volume of the cell = V

Lets define

ee uYCe ww uYCw

nn uXCn ss uXCs These quantities represent mass flow rate at the cell faces

A

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Lets also define

XYde e

XYdw w

YXdn n

YXds s

XxPE

e

Disretization gives

Equation A reduces to

) ) ) ) VSdsCsdnCndwCwdeCe sPsPNnWPwPEe

B

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For Bulk mass conservation

= 1; = 0; S= 0

de = dw = dn = ds = 0

Equation B gives

Ce – Cw + Cn – Cs = 0 Equation of Conservation of mass

Let

e= e P + (1- e ) E

w= w W + (1- w ) W

n= n P + (1- n ) N

s= s S + (1- s ) P

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Where 0 < < 1 based on interpolation scheme used. There are different schemes to calculate valve of , which will be covered later on

Substituting values of e , w , n and s in equation B, we get

Ap P = AE E + AW W + AN N + AS S + S V

Ap P = Ai i + S V Required discretized equationwhere,

i = E, W, N, S

AP = Ai

AE= de - (1- e ) Ce

AW= dw +w Cw

AN= dn - (1- n ) Cn

As= ds +s Cs

If velocities in x and y directions are zero then the problem will become a simple conduction or diffusion problem

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Peclet number (Pc) is defined as

dcPc

Peclet number (Pc)

If Pc is based on cell length scale, it is referred as Cell Peclet number. Peclet number (Pc) is measure of relative dominance of convection and diffusion in the transport of

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AssignmentDerive the discretized equation for 2D unsteady convection diffusion problem.Steps:1. Consider unsteady term in the governing equation2. Integrate the governing equation over

w to e s to n and

t to t+t

3.We will get an unsteady term

)(t

) dtdydxt p

n

s

e

w

tt

t

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) dydx

n

s

e

w

OP

OPPP VYXdydx

n

s

e

w

But,

) VOP

OPPP

As we have integrated over t to t+t, other terms will have t in their numerator. Divide the whole equation by t

)tVO

PO

PPP

Finally, discretization equation for 2D unsteady convection diffusion problem.

Ap P = Ai i + S V + )tVO

PO

P

AP = Ai + tVO

P

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Schemes for Discretization

1. Central Difference Scheme

2. Upwind Difference Scheme

3. Exponential Scheme

4. Hybrid Scheme

5. Power Law

6. Second Order UDS

7. QUICK

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Central Difference Scheme

For steady one- dimensional convection diffusion problem without source, the governing differential equation reduces to

For continuity equation

)()(xx

ux

= 1; = 0

0 )u(x

ttanConsu

I

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Integrate the governing equation I over w to e

P EW

Xe

w e

Xw

) )we

we dxd

dxduu

For the convection term, lets assume piecewise-linear profile of

Therefore, )

)WPw

PEe

2121

Assuming that the interfaces are in midway

II

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) ) ) ) ) )

w

WPw

e

PEeWPwPEe XX

uu21

21

Value of can be found out by Harmonic mean

Lets define

Xd

uC

As earlier, C indicated the mass flow rate at the face and d is the diffusion conductance

d always positive, whereas C can be positive or negative depending on the direction of flow

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Ap P = AE E + AW W

WEP

wwW

eeE

AAA

CdA

CdA

2

2

The discretization equation represents the implications of the piecewise-linear profile for . This form of the discretization is known as Central Difference Scheme (CDS)

Required equation

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Upwind Difference Scheme

In the CDS we have difficulty of not satisfying the Scarborough’s criterion. The remedy for such difficulties is the First order Upwind difference Scheme (UDS)

In UDS, the formulation of diffusion term is left unchanged, but the convection term is calculated from the following assumption

The value of at an interface is equal to the value of at the grid point on the upwind side of the face.

Thus,

The value of w can be defined similarly

00

CeifCeif

Ee

Pe

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Lets define a new operator to denote a greater of A and B

For example, if A = 10 and B = 8 then means A

Then upwind scheme implies,

When this is done, discretized equation becomes

where,

B,AB,A

00 ,Ce,CeCe EPe

Ap P = AE E + AW W

WEP

wW

eE

AAA

,CwdA

,CedA

0

0

Required equation

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The Exact Solution

Fortunately, the governing equation I has an exact solution, if we assume as constant.

Lets take a domain as shown in the figure with following boundary conditions

The solution of the equation becomes

where,

Pc is the peclet number

L,LxAt,xAt

00

xL

) 1

1

0

0

PcexpL

xPcexp

L

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Variation of with x can be plotted for different values of Pc as

follows

Pc= 4Pc= 2

Pc= 1Pc= 0

Pc= -1Pc= -2

Pc= -4

x

Pc

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The Power law

Ce,deCe.

,deAe.

0

1010

50

For the power law,

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Second order UDSFor first order UDS, we have assumed

We may assume higher order profiles, such as linear or quadratic to derive a set of upwind weighted higher order schemes

When , we can write a Taylor series expansion up to linear term for in the neighborhood of the upwind point P as

00

CeifCeif

Ee

Pe

0Ce

)

xX

XXX

xXX

Pe

Pe

PePe

2

2

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)

) )

)

)2

44

2

WPPe

Wp

PEWPPe

WE

Xxb

Xxa

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QUICK (Quadratic Upwind Interpolation for Convective Kinetics)

) )

)

)

) )8

24

22

2

!2

22

2

22

2

PWEWEPe

PWE

WE

Pe

PePePe

Xx

Xx

XXX

xXX

xXX

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Overview



Session-2

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Example of 2D Convection-Diffusion

Ts

To V

L

g

x

y

s

Aim: To obtain equations for velocity and temperature profiles for a liquid film flowing over a hot plate

Assumptions: (1) Fully developed

(2) Steady

(3) Incompressible

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Overview



Session-3

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Checkerboard Pressure Field

YP EW

S

N

X

V

ws

e

n

• Pressure gradient is really a source term, but let us consider it separately since it is not known a priori

• Consider 2D uniform Cartesian mesh

• Store (tentatively) u, v, P at cell centroids

• This procedure is called co-located grid approachseparatelyconsideredbeTo

xP

i

xBxp

yu

yxu

xvu

yuu

xu

t

)()()()()(

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ew

e

w

PPdxxP

Integrating pressure gradient term over the control volume in x-direction

• We need pressure on faces, but it is available only on cell centroids

• Interpolating to face:

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222EWPEWP

ew

e

w

PPPPPPPPdxxP

Thus,

1. The momentum equation contains the pressure difference between two alternate grid points and not between adjacent ones

2. Sometimes if the pressure field is zig - zag, still the momentum equation will feel it as uniform for any grid point P

3. Similar pressure field would be produce for two dimensional case

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100 300 100 300 100

5 27 5 27 5

10 300 100 300 100

5 27 5 27 5

This type of pattern will produce no pressure force in x or y direction, Thus highly non-uniform field would be treated as uniform pressure field by particular discretized form of momentum equation.

Such pressure distribution is referred as Checkerboard Pressure Field

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Continuity EquationSimilar kind of difficulty arises for the continuity equation

3210 ,,ixu

i

i

0222

WEPWEP

ew

e

w

uuuuuuuudxxu

0 WE uu

Thus, the discretized continuity equation demands the equal velocities at alternate grid points and not at adjacent ones

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Staggered Grid Approach - a Remedy These difficulties in the velocity components and pressure

field must be resolved before formulating a numerical method

We can employ a different grid for each dependent variable

Sometimes it is not beneficial to employ different grid for all the variables

But, in case of velocity component, significant benefit is obtained by using different grids for velocity component and other quantities

This approach is called Staggered grid approach

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Staggered Grid Approach - a Remedy• Store pressure at main cell centroids• Store velocities on staggered control volumes• The control volumes are staggered in such a way that the

velocity components are calculated at the faces of the main control volume

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• Mass flow rates across the control faces can be calculated without any interpolation for the relevant velocity components

• The discretized continuity equation contains the differences of the adjacent velocity components and the wavy velocity field would be prevented

• Pressure difference between two adjacent grid points now becomes the driving force for the velocity component located between these grid points

• Thus, non uniform pressure field will no longer be treated as uniform field in staggered grid approach

Advantages of Staggered Grid Approach

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Nomenclature of Staggered Grid

YP EW

S

N

X

V

w

s

e

n

nn nne

Ne

ne

se

Se

ssess

nnE

NE

nE

ee

nwnW CV for ue velocity

CV for vn velocity

CV for Pressure

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Discretized equation for u component of velocity

Ae ue = Aeeuee + Aw uw + ANe uNe + ASeuSe + (PP – PE )Areae

+SuV

Ae ue = Ai ui + (PP – PE )Areae +Su V

where,

i = ee, w, Ne, Se• Areae is the area on which pressure difference acts (Here it is

given by Yx1)• The source term Su does not contain pressure gradient term as

we have considered it separately

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Discretized equation for v component of velocity

An vn = Annvnn + As vs + AnE vnE + AnWvnW + (PP – PN )Arean

+SvV

An vn = Ai vi + (PP – PN )Arean +Sv V

where,

i = nn, s, nE, sW• Arean is the area on which pressure difference acts (Here it is

given by Xx1)• The source term Sv does not contain pressure gradient term as

we have considered it separately

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• For three dimensional case, similar equation for w velocity can be written

• Unless the correct pressure field is obtained, the resulting velocities from the solution of above equations do not satisfy continuity equation

• The pressure field is not known a priori. Thus, at the start of the solution, we have to guess a pressure field

• Lets called the guessed pressure as P* and the velocities, thus, obtained as u*, v*, w*

• These “starred” velocity field will result from solution of following discretized equations

Pressure and velocity correction

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Ae u*e = Ai u*i + (P*P – P*E )Areae +Su V

An v*n = Ai v*i + (P*P – P*N )Arean +Sv V


• Our aim is to find a way of improving the guessed pressure P* such that the resulting starred velocities will progressively get closer to satisfy the continuity equation

• Lets say that the correct pressure is

P = P* + P’where, P’ is the pressure correction

• Similar corrections can be introduced for the velocity components

u = u* + u’ v = v* + v’

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• If we subtract the discretized equation of “starred” quantities from the discretised equation of absolute quantities, the resulting equation for the “primed” quantities would be

• At the convergence, the u’ will approach to zero• Thus, for the time being we may drop the term Ai u’i

Therefore,


Ae u’e = Ai u’i + (P’P – P’E )Areae

An v’n = Ai v’i + (P’P – P’N )Arean

Ae u’e = (P’P – P’E )Areae

u’e = (P’P – P’E )Areae/ Ae

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ue = ue* + ue’

= ue* +


(P’P – P’E )Areae/ Ae

• Thus, the “starred” velocities are to be corrected by the correction in the pressure to produce the absolute velocities

• Now, we shall obtain the discretized equation for the pressure correction, P’

vn = vn* + vn’

= vn* + (P’P – P’N )Arean/ An

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The Pressure-correction equation

• The pressure - correction equation is obtained from the continuity equation

• Assumptions areDensity is not dependent on pressureDensity is uniform over the control volumeVelocities are uniform over the faces of the control volume

• The continuity equation in 2D form is

0)()(

yv

xu

t

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The Pressure-correction equation

P E

N

S

Wueuw

vn

vs

0)()(

dydxdtyvdydxdt

xudydxdt

t

tt

t

e

w

n

s

tt

t

e

w

n

s

tt

t

e

w

n

s

Integrate the governing equation over

w to e s to n andt to t+t

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) ) ) ) ) 0

XvvYuut

YXsnwe

Po

P

)''*EP

e

eee PP

AAreauu )''*

PWw

www PP

AAreauu

)''*NP

n

nnn PP

AAreavv )''*

PSs

sss PP

AAreavv

Substituting,

Confidential 48

) ) )

) ) 0''''

''''

**

**

PSs

sssNP

n

nnn

PWw

wwwEP

e

eee

Po

P

PPA

AreavXPPA

AreavX

PPA

AreauYPPA

AreauYt

V

) ****' ssnnwwee

PPo

P XvXvYuYut

VS

Let,

) )

) ) '''''

''''

PPSs

ssNP

n

nn

PWw

wwEP

e

ee

SPPA

AreaXPPA

AreaX

PPA

AreaYPPA

AreaY

Therefore,

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'''''

'

PSs

ssN

n

nnW

w

wwE

e

ee

Ps

ss

n

nn

w

ww

e

ee

SPA

AreaXPA

AreaXPA

AreaYPA

AreaY

PA

AreaXA

AreaXA

AreaYA

AreaY

Collecting coefficients,

SNWEP

s

ssS

n

nnN

w

wwW

e

eeE

AAAAAA

AreaXA

AAreaXA

AAreaYA

AAreaYA

Lets define,

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'''''' PSSNNWWEEPP SPAPAPAPAPA Required Equation for Pressure Correction

Since the density is normally available at main control volume, the interface densities may be calculated by any interpolation scheme

The term in the pressure correction equation represents the discretized continuity equation in terms for starred velocities

If is zero, it means that the starred velocities along with the available value of do satisfy the continuity equation and no pressure correction is needed

)snwe ,,,

'PS

'PS )PP

o

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Overview



Session-4

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Semi Implicit Method for Pressure Linked Equation(SIMPLE)

1. Guess the pressure field P*2. Solve the momentum equation to obtain u*, v*, w*3. Solve the equation for P’4. Calculate P by adding P’ in P*5. Calculate the u and v from their starred values and their

corrections6. Solve the discretized equations for the other variables like

temperature, turbulence, species concentrations, etc if these quantities affect the flow field (the equations for variables, which do not affect the flow field can be solved once the converged solution for the flow is obtained)

The SIMPLE Algorithm

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7. Treat the corrected pressure as guessed pressure P* and return to step 2 and repeat the whole procedure until the converged solution is obtained

The words Semi-Implicit in the name SIMPLE is used to acknowledge the omission of the term Ai u’i. This term represents an indirect or implicit influence of the pressure correction on velocity; pressure correction at the nearby locations can alter the neighboring velocities and thus cause the velocity correction at the point under consideration. We do not include this influence and thus work with scheme that is only partially and not totally implicit (Semi-Implicit)

The SIMPLE Algorithm

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Types of Mesh

Regular and body-fitted meshes (Orthogonal Mesh)

Stair-stepped representation of complex geometry

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Types of Mesh

Non-conformal mesh

Hybrid mesh

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Cell shapes

a: Triangle, b: tetrahedron, c: quadrilateral

d: Hexahedron, e: Prism and f: Pyramid

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Thank you!

cfd refresher course

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