Computational Methods for Domains with Complex …gtryggva/CFD-Course/2011-Lecture-24.pdfComputational Fluid Dynamics! x y Boundary-Fitted Coordinates! • Coordinate mapping: transform

Computational Fluid Dynamics

Computational Methods for Domains with !

Complex Boundaries-I!

Grétar Tryggvason!Spring 2011!

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

For most engineering problems it is necessary to deal with complex geometries, consisting of arbitrarily curved and oriented boundaries!


•  Overview of various strategies!•  Boundary-fitted coordinates!!- Navier-Stokes equations in vorticity form!!- Navier-Stokes equations in primitive form!

•  Grid generation for body-fitted coordinates!!- Algebraic methods!!- Differential methods!

Outline!

How to deal with irregular domains!


- Staircasing!- Boundary-fitted coordinates!- Immersed boundary method (no grid change)!- Unstructured grids: triangular or tetrahedral!- Adaptive mesh refinement (AMR)!

Various Strategies for Complex Geometries and to concentrate grid points in specific regions!

Overview!


Staircasing!

Approximate a curved boundary by a the nearest grid lines!


Boundary-Fitted Coordinates!


x

y


• Coordinate mapping: transform the domain into a simpler (usually rectangular) domain.!

• Boundaries are aligned with a constant coordinate line, thus simplifying the treatment of boundary conditions!

• The mathematical equations become more complicated!


!

!

x

y

0=!1=!0=!

1=!2=!3=!

),(),( !"#yx

),( yx!! =),( yx!! =

),(),( !"fyxf =



f (x) = f (x(!)) = f (!)

First consider the 1D case:!


dfdx

=dfd!

d!dx

=dfd!

dxd!

"#$

%&'

(1

For the first derivative the change of variables is straightforward using the chain rule:!

For the second derivative the derivation becomes considerably more complex:!


The second derivative is given by!

!"" /x

Where we have used the expression for the fist derivative for the final step. However, since the equations will be discretized in the new grid system, it is important to end up!with terms like , not . !x!! /"


dfdx

=dfd!

d!dx

=dfd!

dxd!

"#$

%&'

(1

d 2 fdx2 =

ddx

dfdx

!"#

$%&=

ddx

dfd'

d'dx

!"#

$%&

!

"#$

%&=

dd'

dfd'

d'dx

!"#

$%&

!

"#$

%&d'dx

!"#

$%&

=d 2 fd'2

d'dx

!"#

$%&

2

+dfd'

!"#

$%&

d'dx

!"#

$%&

dd'

d'dx

!"#

$%&=

d 2 fd'2

d'dx

!"#

$%&

2

+dfd'

!"#

$%&

ddx

d'dx

!"#

$%&

=d 2 fd'2

d'dx

!"#

$%&

2

+dfd'

!"#

$%&

d 2'dx2


To do so, we look at the second derivative in the new system!


dfd!

=dfdx

dxd!

d 2 fd!2 =

dd!

dfd!

"#$

%&'=

ddx

dfd!

"#$

%&'

dxd!

=ddx

dfdx

dxd!

"#$

%&'

dxd!

"#$

%&'

=d 2 fdx2

dxd!

"#$

%&'

2

+dfdx

"#$

%&'

dxd!

"#$

%&'

ddx

dxd!

=d 2 fdx2

dxd!

"#$

%&'

2

+dfdx

"#$

%&'

d 2xd!2

=d 2 fdx2

dxd!

"#$

%&'

2

+dfd!

"#$

%&'

dxd!

"#$

%&'

(1d 2xd!2

d 2 fdx2 =

d 2 fd!2

dxd!

"#$

%&'

(2

(dfd!

"#$

%&'

dxd!

"#$

%&'

(3d 2xd!2

Solving for the original derivative (which we need to transform) we get:!


dxdx

=dxd!

d!dx

= 1

dxdξ

=dξdx

⎛⎝⎜

⎞⎠⎟

−1

dd!

dxd!

d!dx

"#$

%&'=

d 2xd!2

d!dx

+dxd!

dd!

d!dx

"#$

%&'=

d 2xd!2

d!dx

+dxd!

"#$

%&'

2d 2!dx2 = 0

dxd!

"#$

%&'

2d 2!dx2 = (

d 2xd!2

d!dx

= (d 2xd!2

dxd!

"#$

%&'

(1

d 2!dx2 = "

d 2xd!2

dxd!

#$%

&'(

"3

By the chain rule we have!

For the second derivative we differentiate the above:!

Often we need the derivatives of the transformation itself: !

Giving:!



),()),(),,((),( !"!"!" fyxfyxf ==Change of variables!

2D: First Derivatives!

!"" /x

The equations will be discretized in the new!grid system . Therefore, it is important to end up!with terms like , not . !

),( !"x!! /"

Boundary-Fitted Coordinates!Computational Fluid Dynamics

!!!

"""

##

##+

##

##=

##

##

##+

##

##=

##

yyfx

xff

yyfx

xff

Using the chain rule, as we did for the 1D case:!

We want to derive expressions for !in the mapped coordinate system. !

yfxf !!!! /,/



!"!"!" ##

##

##+

##

##

##=

##

## xy

yfxx

xfxf

!"!"!" ##

##

##+

##

##

##=

##

## xy

yfxx

xfxf

Solving for the derivatives!

Subtracting!

!!"

#$$%

&''

''(

''

''

''=

''

''(

''

''

)**))**)xyxy

yfxfxf

!!!

"""

##

##+

##

##=

##

##

##+

##

##=

##

yyfx

xff

yyfx

xff

!""x

!""x


!!"

#$$%

&''

''(

''

''=

''

!!"

#$$%

&''

''(

''

''=

''

)**)

*))*

xfxfJy

f

yfyfJx

f

1

1

( )( )!""!!" ,,

##=

##

##$

##

##= yxyxyx

J

Solving for the original derivatives yields:!

where!

is the Jacobian.!



!"!"

"!

!"!"

"!

##=

##=

##=

##=

##=

##=

##=

##=

xx

xx

yy

yy

ff

ff

yf

fxf

f yx

;;;

;;;

A short-hand notation:!


)(1

)(1

!""!

"!!"

xfxfJ

f

yfyfJ

f

y

x

#=

#=

Rewriting in short-hand notation!

!""! yxyxJ #=where!

is the Jacobian.!



fx =1J

fy!( )" # fy"( )!

$%&

'()

fy =1J

fx"( )!# fx!( )"$

%&'()

These relations can also be written in conservative form:!


Since:!

fx =1J

fy!( )" # fy"( )!

$%&

'() =

1J

fy!" + f"y! # fy"! # f!y"$% '( =1J

f"y! # f!y"$% '(

And similarly for the other equation!


( ) ( )!!"

#

$$%

&'()*

+, --'

()*

+, -=

''(

)**+

,'()*

+,..-'

()*

+,..='

()*

+,..

..=

.

.

/0

/00/0/

/00/

/0

0/

yyfyfJ

yyfyfJJ

yxf

yxf

Jxf

xxf

111

12

2

The second derivatives is found by repeated application of the rules for the first derivative!

( ) ( )!!"

#

$$%

&'()*

+, --'

()*

+, -=

.

./

0/00/0

//00/ xxfxf

Jxxfxf

JJyf 1112

2

Similarly!

2D: Second Derivatives!



!2 f!x2 =

1J

1J

f" y# $ f# y"( )%&'

()* "

y# $1J

f" y# $ f# y"( )%&'

()*#

y"

+

,--

.

/00

Adding!

!2 f!y2 =

1J

1J

f"x# $ f#x"( )%&'

()*"

x# $1J

f"x# $ f#x"( )%&'

()* #

x"+

,--

.

/00

and!

yields an expression for the Laplacian:!

2

2

2

22

yf

xf

f!!+

!!="


!2 f!x2 +

!2 f!y2 =

1J

1J

f" y# $ f# y"( )%&'

()* "

y# $1J

f" y# $ f# y"( )%&'

()*#

y"

+

,--

.

/00+

1J

1J

f#x" $ f"x#( )%&'

()*#

x" $1J

f#x" $ f"x#( )%&'

()* "

x#+

,--

.

/00



q1 = x!2 + y!

2

q2 = x"x! + y"y!q3 = x"

2 + y"2

where!

!2 f ="2 f"x2 +

"2 f"y2 =

1J 2

""#

q1 f# $ q2 f%( ) + ""%

q3 f% $ q2 f#( )&

'(

)

*+

+1J 3 $q1J# f# + q2 J% f# + q2 J# f% $ q3J% f%&' )*


!2 f = 1J 2

q1 f"" # 2q2 f"$ + q3 f$$( ) + !2"( ) f" + !2$( ) f$

Expanding the derivatives yields!

where!

!2" = 1J 3

J x"# x# $ x## x" $ y## y" + y"# y#( ) $ q1J" + q2J#[ ]

!2" = 1J 3

J x#" x# $ x## x" $ y## y" + y#" y#( ) + q2J# $ q3J"[ ]!!"!"!"!!"!!! yxyxyxyxJ ##+=

!""!""""!"!"" yxyxyxyxJ ##+=



( )!""! yfyfJ

f x #= 1Derivation of!

if !

yyxx !!! +="2

!=f

!x = 1J!! y" #!" y!( ) =

y"J

hence!

!xx = !x( )x=

1J

y"J

#

$%

&

'(!

y" )y"J

#

$%

&

'("

y!

*

+

,,

-

.

//

similarly!

!yy = !y( )y=

1J

"x#J

$

%&

'

()#

x! " "x#J

$

%&

'

()!

x#*

+

,,

-

.

//


Putting them together, it can be shown that!(prove it!)!

!2" = 1J 3

q1 x# y"" $ y# x""( ) $ 2q2 x# y"# $ y# x"#( ) + q3 x# y## $ y# x##( )[ ]

!2" = 1J 3

q1 y# x## $ x# y##( ) $ 2q2 y# x#" $ x# y#"( ) + q3 y# x"" $ x# y""( )[ ]



gy fx ! gx fy = 1Jg" f# ! g# f"( )

We also have, for any function and!f g


!

!

x

y

0=!1=!0=!

1=!2=!3=! ),( yx!! = ),( yx!! =

),(),( !"fyxf =

A complex domain can be mapped into a rectangular domain where all grid lines are straight. The equations must, however, be rewritten in the new domain. !

)(1

)(1

!""!

"!!"

xfxfJ

f

yfyfJ

f

y

x

#=

#=Thus:!

And more complex expressions for the higher derivatives!



Vorticity-Stream Function

Formulation!


The Navier-Stokes equations in vorticity form are:!

Using the transformation relations obtained earlier,!

!"!t

+# y" x $# x" y = %&2"

!2" = #$

Vorticity-Stream Function Formulation!


The Navier-Stokes equations in vorticity form become:!

!"!t

+1J

(#$"% &#%"$ ) =

' 1J 2 q1"%% & 2q2"%$ + q3"$$( ) + (2%( )"% + (2$( )"$

)*+

,-.

223

2

221

!!

"!"!

""

yxq

yyxxq

yxq

+=

+=

+=


1J 2

q1!"" # q2!"$ + q3!$$( ) + %2"( )!" + %2$( )!$ = #&


0=!

Boundary Conditions!!

Q=!!

Inflow!Outflow!

12

2

1

2

121 )( !!! "=="= ## dvdxudyQ

slipNo: 0,Outflow:Inflow:0

!===

MN

"##

1=!=! "#



HOT21)0,(1)0,()0,()1,( ++!+== "#"#"#$"# $$$

0=!

)0,()0,( 2

22

!"!# $$!!

Jyx +

%=

!(",0) = #2x"2 + y"

2

J 2$(",0) #$(",1)[ ]

Lower wall!Stream function:!Vorticity:! 0= (no-slip)!

( )0=!


Using that!

We have:!


Q = (udy ! vdx) =0

L" d# =0

M" #M !#0

Q=!

!(",M) = #2x"2 + y"

2

J 2$(",M) #$(",M #1)[ ]

Upper wall!Stream function:!

Vorticity:!

η = M( )



Inlet flow!

)(),0( yLCyyu !=

( )0=!

Considering a fully-developed parabolic profile!

3

3

0

2 6;6

)(LQCLCdyyLyCQ

L==+!= "

!=""#=yuyLy

LQyu );(6),0( 3

!!"

#

$$%

&'()*

+,-'

()*

+,=-= .

32

0

23 23)(6)(

Ly

LyQdyyyL

LQyQ

y

Vorticity-Stream Function Formulation!Computational Fluid Dynamics

Inlet flow!

( ) !"#$

%& '==

MMLQu ((() 16,0

( ) !!"

#$$%

&!"#$

%&'!

"#$

%&=

32

23,0MM

Q ((()

( ) !"#$

%& '=

MLQ (() 216,0 2

( )0=!

Considering a fully-developed parabolic profile!and assume that!

MLy !=



Outflow!

!

( )N=!

Typically, assuming straight streamlines!

If is normal to the outflow boundary, this yields!

0=!!n"

0=!!"#

If not, then a proper transformation is needed for!n!

!"

Vorticity-Stream Function Formulation!Computational Fluid Dynamics

Velocity-Pressure Formulation!


!u!t

+!uu!x

+!vu!y

= "!p!x

+ # !2u!x2

+!2u!y2

$%&

'()

!v!t

+!uv!x

+!vv!y

= "!p!y

+ # !2v!x2 +

!2v!y2

$

%&'

()

The Navier-Stokes equations in primitive form!


and continuity equation!

0=!!+

!!

yv

xu

Notice that we have absorbed

the density into the pressure!


!uu!x

+ !vu!y

= 1J

uu( )" y# $ uu( )# y" + uv( )# x" $ uv( )" x#[ ]Advection Terms!

= 1J

y!uu( )" # uuy"! # y" uu( )! + uuy"![

+ x! uv( )" # uvx!" # x"uv( )! + uvx!" ]

= 1J

u uy! " vx!( ){ }#

+ u vx# " uy#( ){ }!

$ % &

' ( )

( ) ( )!"

#$%

&''+

''= uVuU

J ()1



Contravariant Velocity!

!!"" uyvxVvxuyU #=#= ;

xu,

yv,

C=!

UV

C=!

Unit tangent vector along! C=!

x! ,y!( )

y! ,"x!( ) = n! Unit normal vector!

!!!! vxuyxyvuU "="#= ),(),(

Therefore, is in the direction.!!U!V is in the direction.!

Velocity-Pressure Formulation!Computational Fluid Dynamics

!p!x

= 1Jp" y# $ p# y"( )

Pressure Term!

Diffusion Term!

!!"

#

$$%

&'()*

+,--.'

()*

+,--='

()*

+,--

--=

--

/0

0/

yxu

yxu

Jxu

xxu 12

2

= 1J1Ju! y" # u" y!( )$

% &

' ( ) !

y" #1Ju! y" # u" y!( )$

% &

' ( ) "

y!*

+ ,

-

. /

!2u!y 2

= 1J1Ju" x# $ u# x"( )%

& '

( ) * "

x# $1Ju" x# $ u# x"( )%

& '

( ) * #

x"+

, -

.

/ 0



!2u!x 2

+ !2u!y 2

= 1J

!!"

# $ %

1Ju" y&

2 ' u& y" y& ' u& x" x& + u" x&2( )(

) * + , -

= 1J

!!"

q1u" # q2u$( ) + !!$

q3u$ # q2u"( )%

& ' (

) *

+ !!"

1Ju" x#

2 $ u# x# x" $ u# y# y" + u" y#2( )%

& ' ( ) * + , -

223

2

221

!!

"!"!

""

yxq

yyxxq

yxq

+=

+=

+=


!u!t

+ 1J

!Uu!"

+ !Vu!#

$

% &

'

( ) = * 1

J+y#

!p!"

* y"!p!#

$

% &

'

( )

+!J 2

""#

q1u# $ q2u%( ) + ""%

q3u% $ q2u#( )&

'(

)

*+

u-Momentum Equation!

!v!t

+1J

!Uv!"

+!Vv!#

$%&

'()= *

1J+

x"

!p!#

* x#!p!"

$%&

'()

+!J 2

""#

q1v# $ q2v%( ) + ""%

q3v% $ q2v#( )&

'(

)

*+

v-Momentum Equation!

U = uy! " vx! ; V = vx# " uy#( )Velocity-Pressure Formulation!


0=!!+

!!

yv

xu

Continuity Equation!

Using!)(1),(1 !""!"!!" xfxf

Jfyfyf

Jf yx #=#=

1J(u! y" # u" y! ) + (v" x! # v! x" )[ ] = 0

Continuity equation becomes!

or!

uy! " vx!( )# + vx# " uy#( )! = 0

0=!!+

!!

"#VU


The momentum equations can be rearranged to!

!!"

#$$%

&''+

''(!!"

#$$%

&''+

''+

''

)*)* ))VvUv

xVuUu

ytU

J

= ! q1

"p"#

! q2

"p"$

%&'

()*++J

y$""#

q1u# ! q2u$( ) + ""$

q3u$ ! q2u#( ),

-.

/

01

234

54

!"#$

%& +

''(!

"#$

%& +

''

!!tv

Jxtu

Jy ))

U-Momentum Equation!

!x "##$

q1v$ ! q2v"( ) + ##"

q3v" ! q2v$( )%

& ' (

) * + , -



J !V!t

+ x"!Uu!"

+ !Vu!#

$

% &

'

( ) * y"

!Uv!"

+ !Vv!#

$

% &

'

( )

= ! q3

"p"#

! q2

"p"$

%&'

()*++J

x$

""$

q1v$ ! q2v#( ) + ""#

q3v# ! q2v$( ),

-.

/

01

234

54

V-Momentum Equation!

!y"##"

q1u" ! q2u$( ) + ##$

q3u$ ! q2u"( )%

& ' (

) * + , -

where!

u = 1JUx! +Vx"( )

v = 1JVy! +Uy"( )


!" #In the plane, a staggered grid system can be used.!

Same MAC grid and!projection method !can be used. !

p

C=!

UuVv

C=!



http://www.hpc.msstate.edu/publications/gridbook/!

The “classic”!

Here we have examined the elementary aspects of representing the governing equations in a mapped coordinate system. This is a complex topics with its own “language” that needs to be mastered for further studies. !

Documents

Computational Methods for Domains with Complex …gtryggva/CFD-Course/2011-Lecture-24.pdfComputational Fluid Dynamics! x y Boundary-Fitted Coordinates! • Coordinate mapping: transform