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Computational Fluid Dynamics CFD Basic Discretisation

Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

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Page 1: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Computational Fluid Dynamics

CFD

Basic Discretisation

Page 2: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Governing equations

System of equations:

ii

i

i

i

ijj

iii

i fux

pu

x

u

x

Tk

xt

q

x

Eu

t

E

j

ij

j

i

j

jii

xx

pf

x

uu

t

u

0

i

i

x

u

t

Mass

Momentum

Energy

Page 3: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Governing equations

ii

i

i

i

ijj

iii

i fux

pu

x

u

x

Tk

xt

q

x

Eu

t

E

j

ij

j

i

j

ij

i

xx

pf

x

uu

t

u

0

i

i

i

ix

u

xu

t

Mass

Momentum

Energy

Non-conserved forms

Page 4: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Classification of PDEs

21221122112212

1221 dxdbdbdxdycbcbdadadycacaA

0122112211221

2

1221

dbdb

dx

dycbcbdada

dx

dycaca

0

2

c

dx

dyb

dx

dya

a

acbb

dx

dy

2

42

Three situations:

04

04

04

2

2

2

acb

acb

acb hyperbolic

parabolic

elliptic

Page 5: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Classification of PDEs

Hyperbolic

PDomain of

dependence Region of influence

Charateristic lines

y

x

Page 6: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Classification of PDEs

parabolic

PDomain of

dependence Region of influence

y

x

Known boundary conditions

Known boundary conditions

Page 7: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Classification of PDEs

Elliptic

P

y

x

Every point influences all

other points

Page 8: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Governing equations

and boundary

conditions

Discretisation,

choice of grid

System of

algebraic

equations

Equation

system solver

Approximate

solution

Mathematical

description of

physical ”reality”

FV, FD, FE?

Page 9: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Finite differences

02

2

ixt

Consider the equation

To solve this numerically we create a discrete

approximation in time and space. Hence, we get a

system of algebraic equations and obtain the solution

only at certain points.

xx is the grid spacing and

t is the time step

Page 10: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Finite differences

02

2

xt

For simplicity we use 1D-equation

First derivative.

Use Taylor expansion :

H.O.T.!3!2 3

33

2

221

n

j

n

j

n

j

nj

nj

t

t

t

t

tt

H.O.T.!3!2 3

32

2

21

n

j

n

j

nj

nj

n

j t

t

t

t

tt

Rearrange and divide by t:

Page 11: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Finite differences

H.O.T.!3!2 3

32

2

21

n

j

n

j

nj

nj

n

j t

t

t

t

tt

tOtt

nj

nj

n

j

1

Truncation

error

First order forward difference, the truncation error is directly

proportional to the time step.

Note that we can not from this say anything about the exact size of

the TE, only how it behaves as t goes to zero.

Page 12: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Finite differencesFirst derivative.

Use Taylor expansion :

H.O.T.!3!2 3

33

2

221

n

j

n

j

n

j

nj

nj

t

t

t

t

tt

H.O.T.32 3

3211

n

j

nj

nj

n

j t

t

tt

H.O.T.!3!2 3

33

2

221

n

j

n

j

n

j

nj

nj

t

t

t

t

tt

Second order central difference, the truncation error is proportional

to the time step squared.

Subtract these two expressions, rearrange and divide by t:

Page 13: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Finite differences

H.O.T.32 3

3211

n

j

nj

nj

n

j t

t

tt

Second order central difference, the truncation error is proportional

to the time step squared.

211

2tO

tt

nj

nj

n

j

Page 14: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Finite differencesSecond derivative.

Use Taylor expansion:

H.O.T.!4!3!2 4

44

3

33

2

22

1

n

j

n

j

n

j

n

j

nj

nj

x

x

x

x

x

x

xx

H.O.T.4

24

42

2

11

2

2

n

j

n

j

n

j

n

j

n

jx

x

xx

H.O.T.!4!3!2 4

44

3

33

2

22

1

n

j

n

j

n

j

n

j

nj

nj

x

x

x

x

x

x

xx

Second order central difference, the truncation error is proportional

to the node distance squared.

Page 15: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Finite differences

ttn-1 tn tn+1

First

order

Second order

1st and 2nd order approximations to the

time derivative at point tn

Page 16: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Finite differences

02

2

xt

In total, the discrete approximation to

can be written as

02

2

111

xt

nj

nj

nj

nj

nj

Called the FTCS scheme (Forward in Time, Central in Space)

Page 17: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Finite differencesDissipation error

02

2

xxc

Convection-diffusion equation

1st order FD appoximation of the

first derivative and 2nd order for

the second derivative:

2

2

21

2xO

x

x

xx

jj

2

2

111

2

2

2

2 2

2xO

xxc

xx

xc

xc

jjjjj

Numerical dissipation

4

4

42

2

11

2

2

O4

2x

x

x

xxj

jjj

Page 18: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Finite differencesDispersion error

02

2

xxc

Convection-diffusion equation

2nd order FD appoximation of the

first derivative and 2nd order for

the second derivative:

022

4

3

3211

xO

x

x

xx

jj

4

4

42

2

11

2

2

O4

2x

x

x

xxj

jjj

Even derivatives are dissipative

Odd derivatives are dispersive

Peclet number

(Cell Reynolds number)

xcPe

Dispersive schemes are unstable if

2Pe

Page 19: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Finite Volumes

0

y

G

x

F

t

q

vG

uF

q

0

dxdyy

G

x

F

t

q

ABCD

Green’s theorem

GdxFdyds

GF

dsqdVdt

d

ABCD

nH

H

nH

,

0

Page 20: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Finite Volumes

Discrete approximation, second order

0, DA

AB

ABCDkj xGyFAqdt

d

ABAB

ABAB

xxx

yyy

2

2

,1,

,1,

kjkjAB

kjkjAB

GGG

FFF

etc.

etc.

etc.

Page 21: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Finite Volumes

022

22

22

22

,,1,,1

1,,1,,

,1,,1,

,1,,1,

DAkjkj

DAkjkj

CDkjkj

CDkjkj

BCkjkj

BCkjkj

ABkjkj

ABkjkj

ABCD

xGG

yFF

xGG

yFF

xGG

yFF

xGG

yFF

dt

dqA

Discrete approximation, second order

Page 22: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Finite Volumes

Discrete approximation, first order

0, DA

AB

ABCDkj xGyFAqdt

d

ABAB

ABAB

xxx

yyy

1,

1,

kjAB

kjAB

GG

FF

etc.

etc.

etc.

Page 23: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Finite Volumes

0,1,1

,,

,,

1,1,

DAkjDAkj

CDkjCDkj

BCkjBCkj

ABkjABkjABCD

xGyF

xGyF

xGyF

xGyFdt

dqA

Discrete approximation, first order

Page 24: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Finite Volumes

02

2

2

2

yx

02

2

2

2

dsdxdyyx

ABCD

nH

dxy

dyx

ds

nH

dy

Adxdy

xAx DCBADCBAkj

11

2/1,

'''',''

''''

''1, ADADCkjCBB

DCBA

BAkj yyyydy

Page 25: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Finite Volumes

'''',''

''''

''1, ADADCkjCBB

DCBA

BAkj yyyydy

If the mesh is not too distorted:

kkABkkABDCBAAB

kkADCB

ABDCBA

xyyxAA

yyy

yyy

,1,1''''

,1''''

''''

AB

ABkkkjkjAB

kj A

yy

x

,1,1,

2/1,

AB

ABkkkjkjAB

kjA

xx

y

,1,1,

2/1,

Page 26: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Example

Backward facing step

Questions:

• Discretisation?

• Type of mesh?

• Boundary conditions?

Page 27: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Example

Backward facing step

Page 28: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Example

Backward facing step

residuals

Page 29: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Example

Backward facing step

Velocity

1st order upwind 2nd order upwind

Page 30: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Example

Backward facing step

Page 31: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

• Question:

• Given a discrete approximation to the

governing equations can we ensure that

we get a solution and that the solution is

an approximation of reality?

Page 32: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Lax equivalence theorem:

Given a properly posed linear initial value

problem and a finite difference

approximation to it that satisfies the

consistency condition, stability is a

necessary and sufficient condition for

convergence.

Note! For an non-linear problem this is a

necessary but NOT sufficient condition.

Page 33: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Governing

Partial

Differential

Equations

System of algebraic

equations

Approximate

solution

Exact solution

discretisation

consistency

convergence

stability

CONSISTENCY+STABILITY=CONVERGENCE

Page 34: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Solution error: The difference between the

exact solution of the governing PDEs and

the exact solution to the system of

algebraic equations

njnj

nj tx ,

Convergence: The exact solution

to the system of algebraic

equations will approach the

exact solution of the governing

PDEs when grid spacing and

time step go to zero

0lim0,

nj

tx

Page 35: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Consistency: The system of algebraic

equations will be equivalent to the

governing PDEs at each grid point when

grid spacing and time step go to zero

Stability: If spontaneous perturbations

in the solution to the system of

algebraic equations decay, we have

stability

Page 36: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Consistency

02

2

11

111

1

xt

nj

nj

nj

nj

nj

Consider fully implicit form of the equation

Expand and around the j:th node1

1nj

11nj

022

!6!5!4!3!2

!6!5!4!3!2

7

2

1

2

1

6

661

5

551

4

441

3

331

2

2211

2

1

6

661

5

551

4

441

3

331

2

2211

2

1

xOxx

x

x

x

x

x

x

x

x

x

x

xx

x

x

x

x

x

x

x

x

x

x

x

xx

x

t

nj

n

j

n

j

n

j

n

j

n

j

n

j

nj

n

j

n

j

n

j

n

j

n

j

n

j

nj

nj

nj

Page 37: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Consistency

3

3

32

2

2

4

3

33

2

221

O!3!2

O!3!2

1

tt

t

t

t

t

tt

t

t

t

tt

tt

n

j

n

j

n

j

nj

n

j

n

j

n

j

nj

nj

nj

0...36012

1

6

641

4

421

2

21

n

j

n

j

n

j

nj

nj

x

x

x

x

xt

Now expand 1nj

1

2

2

n

jx

and

1

4

4

n

jx

,

Page 38: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Consistency

3

22

42

2

3

2

21

2

2

2tO

xt

t

xtt

xx

n

j

n

j

n

j

n

j

3

42

62

4

5

4

421

4

42

21212tO

xt

t

xtt

x

x

x

xn

j

n

j

n

j

n

j

Page 39: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Consistency

0....360

...12

...2

....62

6

64

4

5

4

42

22

42

2

3

2

2

3

32

2

2

n

j

n

j

n

j

n

j

n

j

n

j

n

j

n

j

n

j

x

x

xtt

x

x

xt

t

xtt

x

t

t

t

t

x

Page 40: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

ConsistencyFrom the governing equation:

6

63

3

3

4

42

2

2

2

2

2

2

2

2

2

2

xt

xxxxtttt

xt

02

2

nj

n

j

Ext

We can now write:

...720366122 3

3

2

422

2

22

n

j

n

j

nj

t

xtxt

t

xtE

2x

ts

Page 41: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Consistency

02

2

nj

n

j

Ext

...120

1

4

11

36

11

2

...1204

136

12

3

3

2

2

2

2

3

3

22

422

2

22

n

j

n

j

n

j

n

j

nj

tss

t

ts

t

tt

x

t

xt

tt

xtE

2x

ts

Consistent if 0lim2

0,

t

x

xt

Page 42: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Stability

Consider the numerical error *nj

nj

nj

Exact solution to system

of algebraic equations

Numerical solution to

system of algebraic

equations

Stable if decreases towards round off

Unstable if increases

nj

nj

Page 43: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Stability

02

2

11

1

xt

n

j

n

j

n

j

n

j

n

j

n

j

n

j

n

j

n

j sss 11

1 21

2x

ts

*1

1

**

1

*1 21

n

j

n

j

n

j

n

j sss

If the system is linear we can subract these equations

n

j

n

j

n

j

n

j sss 11

1 21

Page 44: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Stabilityvon Neumann stability analysis

The errors are expanded as finite Fourier series. Stability is

determined by considering whether a Fourier component decays

or amplifies when progressing to the next time level.

1,.....,3,2 ,2

1

0

JjeaJ

m

ji

mjmInitial error: xmm

n

j

n

j

n

j

n

j sss 11

1 21

Assume exponetial growth or decay in time:tn

m ea

Page 45: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Stabilityvon Neumann stability analysis

jitnn

j ee

Due to linearity it is sufficient to study one a single

term of the series.

Substitute into the error equation:

111 21 jitnjitnjitnjitn eseeeseseee

2/sin41 2 se t

t

n

j

n

jeG

1

21 iit eese 2

cos

ii ee

2

cos1

2sin 2

Amplification

factor

Page 46: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Stabilityvon Neumann stability analysis

Stable if 1G for all

hence for this scheme 12/sin411 2 s

true if

2

1s

2x

ts

What does this mean physically?

tx

xs

physical information speed

travelling speed

i.e. speed of physical information should be half of what can be resolved

on the grid.

Page 47: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Stability

02

11

1

xc

t

n

j

n

j

n

j

n

j

The convection equation.

01

1

xc

t

n

j

n

j

n

j

n

j

0

xc

t

Central difference for convective term

sin1 iCG

1C

x

tcC

Courant number

Unstable!

Upwind difference for convective term

sincos11 iCCG

Stable if CFL (Courant-Friedrich-Levi) condition

Physical interpretation:

tx

cC

Convection speed

travelling speed

Page 48: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Stabilityvon Neumann stability analysis

02

2 2

11111

xxc

t

nj

nj

nj

nj

nj

nj

nj

The convection-diffusion equation. 02

2

xxc

t

Central difference for convective term

sincos121 iCsG

120 2 sCx

tcC

Courant numberStable if

2x

ts

Page 49: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Stabilityvon Neumann stability analysis

The convection-diffusion equation.

02

2

1111

xxc

t

nj

nj

nj

nj

nj

nj

nj

02

2

xxc

t

Upwind difference for convective term

sincos121 iCsG

12 sCx

tcC

Courant number

Page 50: Computational Fluid Dynamics CFD€¦ · Computational Fluid Dynamics CFD Basic Discretisation . Governing equations System of equations: i i i i i j ij i i i i u f x u p x u x T

Stability

02

2

11

111

1

xt

nj

nj

nj

nj

nj

2x

ts

Explicit vs. implicit schemes

Explicit: 02

2

11

1

xt

n

j

n

j

n

j

n

j

n

j

Stable if2

1s

Implicit:

2/sin41

12 s

G

2/sin41 2 sG

nj

nj

nj

nj sss

11

111 21

Stable if 0s i.e. unconditionally stable