2011-Lecture-23

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Computational Fluid Dynamics

Solving the Navier-Stokes Equations-II

Grtar Tryggvason Spring 2011

http://www.nd.edu/~gtryggva/CFD-Course/ Computational Fluid Dynamics

Other Methods for the Navier-Stokes Equations:Articial Compressibility & SIMPLE

Higher Order Advection QUICK & ENO

Higher order in space

Compact schemes

Conservation of energy

Higher order in time for

the " -# formulation

Outline


Other Methods


The ArticialCompressibility Method(Chorin, 1967, JCP 2,12-26)

! u

! t = " A (u ) + # $ 2u " $ p

! p! t

=" 1%

$ & u

$ is a numerical parameter, in effect an articialvelocity of sound. At steady state ! " u = 0

! u

! t = " A (u ) + # $ 2u " $ p "

12

$ % u( )uSometimes we keep additional terms

! u! t

= " A (u ) + # $ 2u " $ p " 12

$ % u( )u " 1& $ $ % u( )


The SIMPLE (Semi-Implicit Method for Pressure LinkedEquations) family of methods for the steady state solution

At steady state we have

0 = ! A (u ) + " # 2u ! # p# $ u = 0

a pu p = a l u l neighbours

! " # ph# $ u = 0

After discretization the momentumequation can be written as:

Where the sum is overnonlinear terms sincethe coefcients dependon the velocity


In SIMPLE based methods we solve the discrete momentumequation

approximately using an estimation of the pressure and thencorrect it by adding pressure.

Since the momentum equation is an approximation, this isnot the nal velocity and we therefore continue iterating untilconvergence

a p u p = a l u l neighbours

! " # ph

p = p*

+ p'

u = u*

+ u '

! 2 p ' = " 1

# t ! $ u

p

*

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The SIMPLE Algorithm:

1. Guess the pressure eld p*

2. Solve the momentum equations toget u* and v* (using u to nd a p)

3. Solve for p &

4. Compute new pressure

5. Compute new velocity

6. Take the corrected pressure as thenew guess p* and go to step 2, usingthe new velocity for the coefcient

u p

= u p

*

! " ph

p = p*

+ p'

! 2 p ' = "

1

# t ! $ u *

a p u p*

= a l u l *

neighbours

! " # ph*

In the original

SIMPLE methodwe drop this term


There is no good reason for neglecting theneighboring points in step 2 when updating thevelocities (except that they are not known!) and in theSIMPLEC (SIMPLE Corrected) the neighboringvelocities are estimated using the values from the lastiteration.

In SIMPLER (SIMPLE Revisited) method pressure isfound from the new velocity.

The PISO (Pressure Implicit with Splitting ofOperators) the rst estimate for the velocities isfollowed by another one where the preliminaryvelocities are used to get a better estimate.


Advection byHigher Order

Methods


For the advection terms, the methodsdescribed for hyperbolic equations, includingENO, can all be applied, yielding stable androbust methods that can be forgiving forlow resolution.

QUICK, where a third order upstreamdifferencing is used is also popular.


s=1 f1

s=2 f2

s=3 f3

s=4 f4

s=5/2

At s = 5/2

! f 2

! x

" # $

i

% 1

h f i + 1/ 2

2 & f i&1/ 22{ }

=1

64 h3 f i + 1 + 6 f i & f i&1[ ]

2 & 3 f i + 6 f i&1 & f i&2[ ]2{ }

f 5/ 2 = 1/ 8( )

3 f 3 + 6 f 2 ! f 1[ ]

! f ! t

+1

2

! f 2

! x= D

! 2 f ! x2

Use to solve:


Centered

Upwind QUICK

! f ! t

+1

2

! f 2

! x= D

! 2 f ! x

2

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0 5 10 15 20 25 30 3 5 4 00

1

2

3

4

5

6

7

8

9

10

0 5 10 15 20 25 30 3 5 4 00

1

2

3

4

5

6

7

8

9

10

Re_cell=20 Re_cell=10

! f ! t

+1

2

! f 2

! x=

D

! 2 f ! x2

Centered

Upwind QUICK


! f ! t

+ u ! f ! x

= 0

f j *

= f j n !

" t h

u j n f j + 1/2

n ! f j ! 1/2n( )

f j n + 1 = f j

n ! " t h

1

2u j

n f j + 1/2n ! f j ! 1/2

n( )+ u j * f j + 1/2* ! f j ! 1/2*( )( )

f j + 1/2 = f j +

12 amin ! f j

+ , ! f j "( ), if 12 u j + u j + 1( )> 0

f j " 12 amin ! f j + 1

+ , ! f j + 1"( ), if 12 u j + u j + 1( )< 0

#$%

&%

! f j +

= f j + 1 " f j ! f j "

= f j " f j " 1

amin a , b( )=a , a < b

b , b ! a

"#$

%$

Second order ENO scheme for the linear advection equation


Second order ENO

Re_cell=20

! f ! t

+1

2

! f 2

! x= D

! 2 f ! x

2

0 5 1 0 1 5 2 0 2 5 3 0 35 400

1

2

3

4

5

6

7

8

9

10

Centered

Upwind QUICK

Upwind

ENO


! f ! t

+ u ! f ! x

= 0

Second order ENO scheme for the linear advection equation

Upwind

ENO


Higher order

in space


Higher order nite difference approximations

The simplest approach is to use more points:

h h f ( x-2h) f ( x-h) f ( x) f ( x+h ) f ( x+2h )

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! f ! x

# $

j

= f j %2 %8 f j %1 + 8 f j + 1 % f j + 212 h

+ O(h 4 )

! 2 f ! x2

# $

j

=% f j %2 + 16 f j %1 % 30 f j + 16 f j + 1 % f j + 2

12 h 2+ O(h 4 )

Centered

Skewed

! f ! x

# $

j

= f j %2 %6 f j %1 + 3 f j + 2 f j + 1

6h+ O(h 3 )


Compact schemes


The standard way to obtain higher orderapproximations to derivatives is to include morepoints. This can lead to very wide st encils and nearboundaries this requires a large number of ghost points outside the boundary. This can be overcome by

compact schemes, where we derive expressionsrelating the derivatives at neighboring points to eachother and the function values.

Compact Schemes Computational Fluid Dynamics

By a Taylor series expansion the following forth order relationsbetween the values of f and the derivatives of f can be derived

(1)

(2)

(3)

(4)

Adding

f i + 1 = f i +! f i! x

" x +! 2 f i! x2

" x2

2+

! 3 f i! x3

" x3

6+

! 4 f i! x4

" x4

24+ O(" x5 )

f i # 1 = f i #! f i! x

" x +! 2 f i! x2

" x2

2#

! 3 f i! x3

" x3

6+

! 4 f i! x4

" x4

24+ O(" x5 )

!! f i + 1 + !! f i " 1 = 2 !! f i + f i

iv# x2 + f ivi

# x4

12+ O(# x6 )

f i+ 1 + f i ! 1 = 2 f i + "" f i # x

2 + f iiv

# x4

12+ O(# x6 )

Taking the second derivative:

Eliminating the fourth derivative

!! f i + 1 + 10 !! f i + !! f i " 1 =12

# x 2 f i + 1 " 2 f i + f i " 1( )+ O # x4( )

Compact Schemes


By a Taylor series expansion the following forth order relationsbetween the values of f and the derivatives of f can be derived

(1)

(2)

(3)

(4)

f i + 1 = f i +! f i! x

" x +! 2 f i! x2

" x2

2+

! 3 f i! x3

" x3

6+

! 4 f i! x4

" x4

24+ O( " x5 )

f i # 1 = f i #! f i! x

" x +! 2 f i! x2

" x2

2#

! 3 f i! x3

" x3

6+

! 4 f i! x4

" x4

24+ O( " x5 )

! f i+ 1 + ! f i " 1 = 2 ! f i + !!! f i # x

2 + f iiv

# x4

12+ O(# x6 )

Adding and taking the rst derivative:

Subtracting 2 from 1

! f i+ 1 + 4 ! f i + ! f i" 1 =3

# x f i + 1 " f i " 1( )+ O # x4( )

f i + 1 ! f i ! 1 = 2 " f i # x + """ f i

# x3

3+ O(# x5 )

Eliminating the third derivative

Compact Schemes Computational Fluid Dynamics

To solve the nonlinear advection-diffusion equation

! f i + 1! x

+ 4 ! f i! x

+ ! f i " 1! x

= 3# x

f i+ 1 " f i " 1( )+ O # x 4( )

! 2 f i+ 1! x2

+ 10! 2 f i! x2

+! 2 f i" 1

! x2 =

12

# x 2 f i + 1 " 2 f i + f i " 1( )+ O # x4( )

! f i! t

= " f i! f i! x

+ D! 2 f i! x2

And use the values to compute the RHS. The time integration isthen done using a high order time integration method.

we rst solve nd the rst and second derivatives using theexpressions derived above:

Compact Schemes

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!" ! t

=!# ! x

!" ! y

$ !# ! y

!" ! x

+ % ! 2"

! x 2+

! 2" ! x 2

' (

* +

! 2" ! x 2

+! 2" ! x 2

= #$

Use compact schemes to nd O(h 4)approximations for the spatial derivatives

Solution of the vorticity-streamfunction equations


f x( )i + 1, j + 4 f x( )i, j + f x( )i! 1, j =3

h f i + 1, j ! f i! 1, j ( )

f y( )i, j + 1 + 4 f y( )i, j + f y( )i, j ! 1 =3

h f i, j + 1 ! f i, j ! 1( )

f xx( )i + 1, j + 10 f xx( )i, j + f xx( )i! 1, j =12

h 2 f i + 1, j ! 2 f i, j + f i! 1, j ( )

f yy( )i, j + 1 + 10 f yy( )i, j + f yy( )i, j ! 1 =12

h 2 f i, j + 1 ! 2 f i, j + f i, j ! 1( )

By a Taylor series expansion the following forth orderrelations between the values of f and the derivatives off can be derived

(1)

(2)

(3)

(4)


!" ! t

# $ %

& ' (

i, j

=!) ! x

# $ %

& ' (

i, j

!" ! y

$ %

' (

i, j

* !) ! y

$ %

' (

i, j

!" ! x

# $ %

& ' (

i, j

+ + ! 2"

! x2# $ %

& ' (

i, j

+! 2" ! x2

# $ %

& ' (

i, j

#

$ % %

&

' ( (

! 2" ! x2

$ %

' (

i, j

+! 2" ! y2

$ %

' (

i, j

= )* i, j

To nd the time derivative, we need

The vorticity-streamfunction equations at grid point i,j are

! x , ! y , ! xx , ! yy , " x , " y , " xx , " yy

(5)

(6)


1. Given the vorticity, # , we solve tridiagonalequations (1-4) for ! x , ! y , ! xx , ! yy

! xx , ! yy , ! 2. For " use (3) and (4) plus the elliptic equationfor the streamfunction (6) for

! x , ! y3. Then use 1 and 2 for (tridiagonal systems)

!" ! t

4. Everything on the right hand side of (5) is nowknown to O(h 4) accuracy and

can be found


The main advantage of compact schemesis that it is somewhat easier to implementboundary conditions than for high orderschemes that use a broad stencil


Conservation ofkinetic energy

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Conservation of kinetic energy on a centered

regular mesh ! f ! t

+ f ! f ! x

= 0

E kin =1

2 f 2 dx!

f ! f ! t

+ f 2 ! f ! x

" # $

% & '

( dx = 0

d dt

1

2 f 2! dx = " 1

3

# f 3

# x! dx = 0

Dene:

Multiply by f and integrate

Therefore: The kineticenergy isconserved

Consider:


! f ! t

= " f ! f ! x

# " f i f i + 1 " f i" 1( )/2 h

! f ! t

= "1

2

! f 2

! x# "

1

2 f i + 1

2" f i" 1

2( )/2 h

Consider two different discretizations of the

nonlinear term

and

Both conserve f

Is f2 conserved?


Check conservation:

= + f j ! 12 f j ! f j ! 2( )+ f j 2 f j + 1 ! f j ! 1( )+ " 0

d dt

1

2 f 2! dx = " f j h2

# f 2

# x

$ % &

' ( )

* j

= " 1

2 f j f j + 1

2 " f j "12( )+

, - . / 0 *

d dt

1

2 f 2! dx = " f j h f # f # x

$ % &

' ( )

* j

= " f j 2 f j + 1 " f j "1( ){ }*

= + 1

2 f j ! 1 f j

2! f j ! 2

2( )+1

2 f j f j + 1

2! f j ! 1

2( )+ " 0

Second scheme:

First scheme:


Combine both schemes: ! f ! t

" # 12 h

$ f i f i + 1 # f i#1( )# 1 # $ ( )

2 f i+ 1

2 # f i#12( )

% & '

( ) *

! f 2

! t dx" # $1

2% f j

2 f j + 1 $ f j $1( )$ 1 $ % ( )

2 f j f j + 1

2 $ f j $12( )

& ' (

) * +

,

= + ! f j "12 f j " f j " 2( )+

1 " ! ( )2

f j "1 f j 2 " f j " 2

2( )+

! f j 2 f j + 1 " f j "1( )" 1 " ! ( )2 f j f j + 1

2 " f j " 12( )+ = 0

Write out the terms for the kinetic energy


= + ! f j "12 f j " f j "1

2 f j " 2( )+1 " ! ( )

2 f j " 1 f j

2 " f j "1 f j " 22( )+

! f j 2 f j + 1 " f j

2 f j " 1( )" 1 " ! ( )

2 f j f j + 1

2 " f j f j " 12( )+ = 0

! " 1 " ! ( )2

= 0

3 ! " 12

= 0

! =1

3

! f ! t

" # 16 h

f i f i+ 1 # f i#1( )# f i + 12 # f i#1

2( ){ }

= # 12 h

f i + 1 + f i + f i#13

f i + 1 # f i#1( )$ % &

' ( )

Conserves both f and f 2


The same idea is used in Arakawa s scheme(JCP 119, 1966)

!"

! t + u # $ " = 0

u ! " # = $%& % y

%# % x

+%& % x

%# % y

= J & , # ( )

!" ! t

+ J # ," ( )= 0

Consider the vorticity advection equation

Rewrite the nonlinear terms to introduce the Jacobian

giving

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J ! , " ( ) = #! # x

#" # y

$#! # y

#" # x

=# # x

! #" # y

% & '

( ) *

$ # # y

! #" # x

% & '

( ) *

=# # y

" #! # x

% & '

( ) *

$ # # x

" #! # y

% & '

( ) *

! ! x

" !# ! y

% &

( )

=!" ! x

!# ! y

*" ! 2#

! x! y

since

and so on

The Jacobian can be written in several ways


J ! , " ( ) = # J " , ! ( )

we also have

J ! , " ( ) =#! # x

#" # y

$ #! # y

#" # x

= $ #" # x

#! # y

$ #" # y

#! # x

& '

) *

since

Discretization of the different forms ofthe Jacobian give schemes with slightlydifferent conservation properties


J i, j 1

=1

4 h 2 ! i + 1, j " ! i"1, j ( )# i, j + 1 " # i, j "1( ){" ! i, j + 1 " ! i, j "1( )# i+ 1, j " # i"1, j ( )}

Discretization

J i, j 2

=1

4 h 2 ! i + 1, j " i + 1, j + 1 # " i + 1, j #1( ){ # ! i#1, j " i#1, j + 1 # " i#1, j #1( )# ! i, j + 1 " i + 1, j + 1 # " i#1, j + 1( )# ! i, j #1 " i + 1, j #1 # " i#1, j #1( )}

J i, j 3

=1

4 h 2 !" i + 1, j # i + 1, j + 1 ! # i+ 1, j ! 1( ){ + " i! 1, j # i! 1, j + 1 ! # i! 1, j ! 1( )+ " i, j + 1 # i + 1, j + 1 ! # i! 1, j + 1( )! " i, j ! 1 # i+ 1, j ! 1 ! # i! 1, j ! 1( )}


Arakawa showed that

Conserves both the vorticity and the kinetic energy

J i, j 1

=1

3 J i, j

1+ J i, j

2+ J i, j

3( )

Arakawa also presented a fourth order scheme withthe same properties


Higher order

in time for the vorticity-streamfunctionformulation


Due to the relatively straightforward couplingbetween the elliptic equation for thestreamfunction and the vorticity advection-diffusion equation, the algorithms discussedalready can be used with ease.

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! i, j n+

1 " ! i, j n"

1

2# t = " J i, j

n + $ %h2

! i, j n " 1

Leapfrog

! i, j n + 1 " !

i, j n

# t = "

1

23 J i, j

n " J i, j n " 1( )+

$

2%

h2

! i, j n + 1 + %

h2

! i, j n( )

Adams-Bashford/Crank-Nicholson

Predictor-corrector

! i, j n + 1 " ! i, j

# t = "

1

2 J i, j

n + J i, j ( )+$

2%

h2

! i, j n + %

h2 ! i, j ( )

! i, j " ! i, j n

# t = " J i, j

n + $ %h2

! i, j n


2nd order Runge-Kutta

! i, j = ! i, j n +

" t 2

# J i, j n + $ %

h2

! i, j n( )

Runge Kutta methods: take intermidiate steps

!h2" i, j

n= # $ i, j

n

!h2 " i, j = # $ i, j

! i, j = ! i, j n + " t # J i, j + $ % h

2 ! i, j ( )

Half step

Final step


4th order Runge-Kutta method

! i, j n + 1/ 2 = !

i, j n +

" t 2

# J i, j n + $ %

h2

! i, j n( )

!h2" i, j

n= # $ i, j

n

!h2 " i, j

n + 1/ 2 = # $ i, j n + 1/ 2

! i, j n+ 1 = !

i, j n + " t # J i, j

n + 1/ 2 + $ %h2 ! i, j n

+ 1/ 2( )

First half step

Predicted

nal value

!h2 " i, j

n + 1/ 2 = # $ i, j n + 1/ 2

! i, j n + 1/ 2 = !

i, j n +

" t 2

# J i, j n + 1/ 2 + $ %

h2 ! i, j

n + 1/ 2( )Second half step


4th order Runge-Kutta method (continued)

! i, j n + 1 = !

i, j n +

" t 6

# J i, j n # 2 J i, j

n + 1/ 2 # 2 J i, j n + 1/ 2 # J i, j

n + 1(+ $ %

h2

! i, j n + 2$ % h

2 ! i, j n + 1/ 2 + 2$ % h

2 ! i, j n + 1/ 2 + $ %

h2 ! i, j

n + 1)

correctednal value

!h2 " i, j

n + 1 = # $ i, j n + 1

t

t + ! t

t + ! t /2


Generating higher order methods for theNavier-Stokes equations in the vorticity/ streamfunction form is relatively straightforward and any method developed for theadvection diffusion equation can be usedwithout much difculty


Other Methods for the Navier-Stokes Equations:Articial Compressibility & SIMPLE

Higher Order Advection QUICK & ENO

Higher order in space

Compact schemes

Conservation of energy

Higher order in time for

the " -# formulation

Outline

2011-Lecture-23

Documents

Transcript of 2011-Lecture-23