Stretched Grids - University of Notre Damegtryggva/CFD-Course/2011-Lecture-25.pdf · 2011-03-22 ·...

Computational Fluid Dynamics

Elementary!Grid Generation!

Grétar Tryggvason !Spring 2011!

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

Stretched grids for rectangular geometries!!

Bilinear Interpolation!!

Elliptic grid generation!!

Unstructured hexahedron grids and block-structured grids!

!Imbedded boundaries!

!Adaptive Mesh Refinement!

Outline!


There are two main reasons for using grids that are not rectangular with uniform grid spacing!

!1.  Representing a domain with complex

boundaries!

2.  Put grid points in parts of the domain where high resolution is needed!

Frequently, it is necessary to deal with both issues!


Stretched Grids!


Fine grid needed

here!!

Gridlines are straight but unevenly spaced:!


Simple stretching!

!

x

x

!

Boundary layer! Internal layer!

x = x !( )


Stretching functions-examples!

“one-sided” boundary layer!

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

x

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

x

x = !n

x = !2

x = !

Computational Fluid Dynamics Stretching functions-examples!

Internal layer!

x = L! + A xC " L!( ) 1" !( )!Line!

Zero at endpoints!Changes signs at xc!

“strength” and gives concentrates points inside domain (A>0) or at the

ends (A<0)!

Computational Fluid Dynamics Stretching functions-examples!

0 0.2 0.4 0.6 0.8 10

0.5

1

1.5

2

x

A=2.5 x=1 % A program to test 1D grid refinement.!!L=2.0;xc=1.0; A=2.5!s=[0:0.05:1];n=size(s);!!for i=1:n(2), ! x(i)=L*s(i)+A*(xc-L*s(i))*s(i)*(1-s(i));!End!plot(s,x,'LineWidth',2);hold on!!for i=1:n(2),! plot([s(i),s(i)],[0.0, x(i)]);! plot([0.0,s(i)],[x(i), x(i)]);!end!xlabel('\xi','Fontsize',18)!ylabel('x','Fontsize',18)!set(gca,'Box','on'); !set(gca,'Fontsize',18, 'LineWidth',2)!plot([0,1],[0,L],'r')!text(0.05,L-0.1,['A=',num2str(A),...!' x=',num2str(xc)],'Fontsize',18)!!hold off; % print -depsc exampleplot!0 0.2 0.4 0.6 0.8 1

0

0.5

1

1.5

2

x

A= 1.5; x=1

x = x !( )

x = x !( )


The MAC method on Stretched

Grids!

Computational Fluid Dynamics Computational Fluid Dynamics Stretched grids!

Special case:!

x = x !( )y = y "( )

q1 = y!2

q2 = 0q3 = x"

2

J = x! y"

u = 1JUx!( ) = 1

x! y"Ux!( ) = U

y"

v = 1JVy!( ) = 1

y! x"Vy!( ) = V

x"

y!"u"#

+ x#"v"!

= 0

1x!

"u"!

+ 1y#

"v"#

= 00=!!+

!!

"#VU


Approximate the conservation equation!

x! = xi+1 " xi1

= #xi+1/ 2

y! =y j+1 " y j

1= #y j+1/ 2

ui+1/ 2, j ! ui!1/ 2, j"xi

+vi, j+1/ 2 ! vi, j!1/ 2

"y j

= 0

!" = !# =1

1x!

"u"!

+ 1y#

"v"#

= 0

!y j = 12!y j+1/ 2 + !y j"1/ 2( )

Define:!

!xi = 12

!xi+1/ 2 + !xi"1/ 2( )

Stretched grids!Computational Fluid Dynamics

!u!t

+1x"

!u2

!"+

1y#

!vu!#

= $1x"

!p!"

+ % 1x"

!!"

u"

x"

&

'(

)

*+ +

1y#

!!#

u#

y#

&

'(

)

*+

,

-..

/

011

u-Momentum Equation!

ui+1/ 2, jn+1 ! ui+1/ 2, j

n

"t= !

(u2 )i+1, jn ! (u2 )i, j

n

"xi+1/ 2

!(uv)

i+1/ 2, j+1/ 2

n ! (uv)i+1/ 2, j!1/ 2

n

"y j

!Pi+1, j ! Pi, j

"xi+1/ 2

+ # 1"xi+1/ 2

ui+3/ 2, j

n! u

i+1/ 2, j

n

"xi+1

!u

i+1/ 2, j

n! u

i!1/ 2, j

n

"xi

$

%&&

'

())

$

%

&&

+1"y j

ui+1/ 2, j+1

n! u

i+1/ 2, j

n

"y j+1/ 2

!u

i+1/ 2, j

n! u

i+1/ 2, j!1

n

"y j!1/ 2

$

%&&

'

())

'

(

))

Stretched grids!


!v!t

+1x"

!uv!"

+1y#

!v2

!#= $

1y#

!p!#

+ % 1x"

!!"

v"

x"

&

'(

)

*+ +

1y#

!!#

v#y#

&

'(

)

*+

,

-..

/

011

v-Momentum Equation!

vi, j+1/ 2n+1 ! vi, j+1/ 2

n

"t= !

(uv)i+1, j+1/ 2n ! (uv)i, j+1/ 2

n

"xi+1/ 2

!(v2 )

i, j+1

n ! (v2 )i, j

n

"y j

!Pi, j+1 ! Pi, j

"y j+1/ 2

+ # 1"xi

vi+1, j+1/ 2

n! v

i, j+1/ 2

n

"xi+1/ 2

!v

i, j+1/ 2

n! v

i!1, j+1/ 2

n

"xi!1/ 2

$

%&&

'

())

$

%

&&

+1

"y j+1/ 2

vi, j+3/ 2

n! v

i, j+1/ 2

n

"y j+1

!v

i, j+1/ 2

n! v

i, j!1/ 2

n

"y j

$

%&&

'

())

'

(

))


The pressure equation is:!

1!xi

Pi+1, j " Pi, j

!xi+1/ 2

"Pi, j " Pi"1, j

!xi"1/ 2

#

$%

&

'( +

1!y j

Pi, j+1 " Pi, j

!y j+1/ 2

"Pi, j " Pi, j"1

!y j"1/ 2

#

$%

&

'(

= !tu

i+1/ 2, j

*" u

i"1/ 2, j

*

!xi

"v

i, j+1/ 2

*" v

i, j"1/ 2

*

!y j

#

$%%

&

'((

Which can be solved by iteration!

Stretched grids!


Colocated stretched grids!

Re=10!


For one dimensional stretched grids, we simply replace the global Δx by the local Δx!

pi,j! ui+1/2,j!

Δxi+1/2!

Δxi!

ui-1/2,j! pi+1,j!

Stretched grids!


Bilinear Interpolation!


Simples grid generation is to break the domain into blocks and use bilinear interpolation within each block!

As an example,

we will write a simple

code to grid the domain to the right!

(x1,y1)!

(x2,y2)!(x3,y3)!

(x4,y4)!(x5,y5)!

(x6,y6)!(x7,y7)!

(x8,y8)!



Start by breaking the domain into blocks!

Then grid each block separately!

(x1,y1)!

(x2,y2)!(x3,y3)!

(x4,y4)!(x5,y5)!

(x6,y6)!(x7,y7)!

(x8,y8)!

Bilinear Interpolation!Computational Fluid Dynamics

P1,0!1!

2!

3!

P1,3!

P1,2=P2,3!

P1,1=P2,0! P2,1=P3,0!

P3,2! P3,1!P2,2=P3,3!





2!

P1,0!1!

P1,3!

P1,2!

P1,1!

3!

P3,0!

P3,2! P3,1!

P2,2!

P3,3!

P2,1!P2,0!

P2,3!




Consider an arbitrary shaped quadrilateral block!1. Select the ξ and ξ direction.!

(x0,y0)!

(x1,y1)!

(x2,y2)!(x3,y3)!

!

!

N!

M!2. Divide the

opposite sides evenly with N points in the ξ direction and !

M in the ξ direction and draw

straights line between the points

on the opposite sides!



x ! ,1( ) =N "!N "1

# $

% & x0 +

! "1N "1

# $

% & x1

Along the edge between points 0 and 1!

Along the edge between points 3 and 2!

x ! ,M( ) =N "!N "1

# $

% & x3 +

! "1N "1

# $

% & x2

Then interpolate again for points between the edges!

x ! ,"( ) =M #"M #1

$ %

& ' N #!N #1

$ %

& ' x0 +

! #1N #1

$ %

& ' x1

$ % ( &

' +

"#1M #1

$ %

& '

N #!N #1

$ %

& ' x3 +

! #1N #1

$ %

& ' x2

$ % ( &

'

The y-coordinate is found in the same way!


x ! ,"( ) =M #"M #1

$ %

& ' N #!N #1

$ %

& ' x0 +

! #1N #1

$ %

& ' x1

$ % ( &

' +

"#1M #1

$ %

& '

N #!N #1

$ %

& ' x3 +

! #1N #1$ %

& ' x2

$ % ( &

'

y ! ,"( ) =M #"M #1

$ %

& ' N #!N #1

$ %

& ' y0 +

! #1N #1

$ %

& ' y1

$ % ( &

' +

"#1M #1

$ %

& '

N #!N #1

$ %

& ' y3 +

! #1N #1$ %

& ' y2

$ % ( &

'

(x0,y0)!

(x1,y1)!

(x2,y2)!(x3,y3)!

!

!

N!

M!

For a single block, we therefore have:!



For many blocks:!1. we must interpolate for each block and !2. ensure that their boundaries are correct!

P1,0!1!

2!

3!

P1,3!

P1,2=P1,2!

P1,1=P2,0! P2,1=P3,0!

P3,2! P3,1!P2,2=P3,3!


Short MATLAB program to generate the grid!

xp(1,1:4)=[0,0,1,2];xp(2,1:4)=[0.5,0.5,1,1.5];!yp(1,1:4)=[1,0,0,0.5];yp(2,1:4)=[1,0.5,0.5,1];!

(0.0,1.0)!

(0.0,0.0)!

(0.5,1.0)!

(1.0,0.5)!

(1.0,0.0)!

(2.0,0.5)!(0.5,0.5)!

(1.5,1.0)!



xp(1,1:4)=[0,0,1,2];xp(2,1:4)=[0.5,0.5,1,1.5];!yp(1,1:4)=[1,0,0,0.5];yp(2,1:4)=[1,0.5,0.5,1];!

!NumBlocks=3;!

Nblock(1)=8;Nblock(2)=5;Nblock(3)=12; NTot=0;!Mblock(1)=6;Mblock(2)=6;Mblock(3)=6; MTot=6;!

!for l=1:NumBlocks!

N=Nblock(l);M=Mblock(l);! x0=xp(1,l);x1=xp(1,l+1);x3=xp(2,l);x2=xp(2,l+1);! y0=yp(1,l);y1=yp(1,l+1);y3=yp(2,l);y2=yp(2,l+1);!

Nf=2;if(l == 1),Nf=1;end! for i=Nf:N, for j=1:M !!

ii=i+NTot;! x(ii,j)=((M-j)/(M-1))*(((N-i)/(N-1))*x0+((i-1)/(N-1))*x1)+...!

((j-1)/(M-1))*(((N-i)/(N-1))*x3+((i-1)/(N-1))*x2);! y(ii,j)=((M-j)/(M-1))*(((N-i)/(N-1))*y0+((i-1)/(N-1))*y1)+...!

((j-1)/(M-1))*(((N-i)/(N-1))*y3+((i-1)/(N-1))*y2);! end, end;!

NTot=NTot+Nblock(l)-1;!end!!

axis(’equal'), hold on!for j=1:MTot;plot(x(1:NTot,j),y(1:NTot,j));end;!for i=1:NTot;plot(x(i,1:MTot),y(i,1:MTot));end;!


The grid!



Sometimes the grid can be improved by smoothing. The simples smoothing is to

replace the coordinate of each grid point by the average of the coordinates around it. This

process can be repeated several times to improve the smoothness.!


x i, j( ) = 0.25* x i +1, j( ) + x i !1, j( ) + x i, j +1( ) + x i, j !1( )( )y i, j( ) = 0.25* y i +1, j( ) + y i !1, j( ) + y i, j +1( ) + y i, j !1( )( )


The grid after two smoothing iterations!



Bilinear Interpolation can also be used for curved boundaries, if the points on the

boundaries are given.!

Higher order Interpolation functions can also be used to generate stretched grids for

complex boundaries.!


The grid generation results in an array of x and y coordinates for each grid point:!

!For simple problems we can include the grid generator in the fluid solver and generate the coordinates before

we solve for the fluid motion!!

For more complex problems the grid generation step is usually separated from the flow solver and the grid

points are read from a file!!

For commercial codes you can generally use many different grid generators—as long as the data format is

consistent !


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

Computational Fluid Dynamics Bilinear Interpolation!

Flow in an elbow, computed using a body fitted grid, using the streamfunction-vorticity formulation of the Navier-

Stokes equations!

Grid—bilinear interpolation with smoothing!

Streamfunction!

Computational Fluid Dynamics Bilinear Interpolation!

While bilinear interpolation is often the simplest approach for relatively simple domains, it usually

requires fairly large amount of human input.!!

Thus, there have been major attempts to make the grid generation more automatic. !


Elliptic Grid Generation!


Elliptic Scheme!

“Isotherms” of the conduction equation!

02 =! Thas nice properties of smoothness and concentrated !

contour spacing where solution has a large spatial gradient.!

Why not use the solution to the Laplace equation as the new coordinate?!

0,0 22 =!=! "#


Computational Fluid Dynamics Elliptic Grid Generation!

Elliptic Scheme! 0,0 22 =!=! "#

! = 0, "# /"! = 0

! = M, "# /"! = 0

! = 0"#"!

= 0

! = N"#"!

= 0


Few points in regions of interest!

Need more control over the point location!

0,0 22 =!=! "#



Further control can be applied by adding a source term:!

Proper choice of provides the shape of the mesh.!

In practice, instead of solving for in terms of! , we want to solve for in terms of .!

),(),,( 22 !"!!"" QP =#=#

QP,

!" ,yx, yx, !" ,

Transform the Poisson equation into space.!( )!" ,

Elliptic Grid Generation!Computational Fluid Dynamics

Adding the two equations:!

we obtain !

x! "2! + x#"

2# = x! P(! ,#) + x#Q(! ,#)

y! "2! + y#"

2# = y! P(! ,#) + y#Q(! ,#)

( )!"!!"!"" QxPxJxqxqxq +#=+# 2123 2

( )!"!!"!"" QyPyJyqyqyq +#=+# 2123 2

where !

223

2

221

!!

"!"!

""

yxq

yyxxq

yxq

+=

+=

+=



In discretized form (x-equation):!

! xi+1, j " 2xi, j + xi"1, j( ) " 0.5# xi+1, j+1 " xi+1, j"1 " xi"1, j+1 + xi"1, j"1( )

where !

! xi, j+1 " 2xi, j + xi, j"1( ) + 0.5# P xi+1, j " xi"1, j( ) + Q xi, j+1 " xi, j"1( )[ ] = 0

! = 0.25 xi, j+1 " xi, j"1( )2 + yi, j+1 " yi, j"1( )2[ ]

! = 0.25 xi+1, j " xi"1, j( ) xi, j+1 " xi, j"1( ) + yi+1, j " yi"1, j( ) yi, j+1 " yi, j"1( )[ ]

! = 0.25 xi+1, j " xi"1, j( )2 + yi+1, j " yi"1, j( )2[ ]

! = 116

xi+1, j " xi"1, j( ) yi, j+1 " yi, j"1( ) " yi+1, j " yi"1, j( ) xi, j+1 " xi, j"1( )[ ]2


suggested by Thompson (1977)!

P !,"( ) = # al sgn ! #! l( )exp #cl ! #! l( )l=1

L

$

QP,

! bm sgn " !"m( )exp !dm " !"m( )2 + # !#m( )2[ ]1/ 2$ % &

' ( )

m=1

M

*

Q !,"( ) = # al sgn " #"l( )exp #cl " #"l( )l=1

L

$

! bm sgn " !"m( )exp !dm # !#m( )2 + " !"m( )2[ ]1/ 2$ % &

' ( )

m=1

M

*where are chosen to generate appropriate !

grid clustering.!mlml dcba ,,,

( )!"

!#$

<%=>

=0if10if00if1

sgnxxx

x



Selection of!

P !,"( ) = #al sgn ! #! l( )exp #cl ! #! l( )QP,

Attracts grid lines to the line ξ = ξl.!

al determines how strongly and !

cl determines the “reach” of the attraction!

! l


Selection of! QP,

P(!,") = #bm sgn ! #!m( )exp #dm ! #!m( )2 + " #"m( )2[ ]1/ 2$ % &

' ( )

Attracts grid lines to the line ξ = ξm, near the point (ξ,η) = ξm, ηm).!bm determines how

strongly and !dm determines the

“reach” of the attraction!

!m

!m



Further Recommended Reading:!

Fletcher, C. A. J., Computational Techniques for Fluid!Dynamics, V. 2, Springer-Verlag (1991)!

!Thompson, J. F., Warsi, Z. U. A., and Wayne Mastin, C., !

Numerical Grid Generation, North-Holland (1985).!!

Hoffmann, K. A., Computational Fluid Dynamics for !Engineers (1991).!


Stretched Grids - University of Notre Damegtryggva/CFD-Course/2011-Lecture-25.pdf · 2011-03-22 ·...

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