Numerical Solution of Plane
Irrotational Flow
APPH 4200 Physics of Fluids
Columbia University
Solve Laplace's equation for the streamfunction, y[x,y], for two relatively simple examples. Since Mathematica
can solve thousands of simultaneous equations quickly, the only challenge to this problem is defining the
computational grid and the boundary conditions on the flow. Two dimensional solutions to elliptical PDE's can
be easily and accurately solved using today's computers.
Define Laplace's Equation on a Grid
In[1]:=
Off@General::spell1D;
Define Laplace's Equation…
In[2]:=
eqn@i_, j_D := H y@i + 1, jD - 2 y@i, jD + y@i - 1, jDL ê dx ^ 2 +
Hy@i, j + 1D - 2 y@i, jD + y@i, j - 1DL ê dy ^ 2 ã 0
In[3]:=
eqn@i, jD
y@i, -1 + jD - 2 y@i, jD + y@i, 1 + jD
Out[3]=
dy2
+
y@-1 + i, jD - 2 y@i, jD + y@1 + i, jD
dx2
ã0
2
Numerical_Potential_Flow.nb
Define a Grid
In[4]:=
height = 1.0 H* meter *L;
length = 5.0 H* meter *L;
wallHeight = 0.5 H* meter *L;
wallLength = 0.5 H* meter *L;
Grid points go from 1 to numX and from 1 to numY.
In[8]:=
numX = 80;
numY = 60;
Print@"Number of Grid Points = ", numX numYD;
Number of Grid Points = 4800
In[11]:=
In[13]:=
Out[13]=
In[14]:=
Out[14]=
In[15]:=
In[16]:=
In[17]:=
Out[17]=
In[18]:=
Out[18]=
In[19]:=
Out[19]=
In[20]:=
Out[20]=
dx = length ê HnumX - 1L;
dy = height ê HnumY - 1L;
wallLength ê dx
7.9
wallHeight ê dy
29.5
flow = 5.0 H* mês *L;
grid = Table@
If@length ê 2 - wallLength ê 2 § Hix - 1L dx § length ê 2 + wallLength ê 2 &&
Hiy - 1L dy § wallHeight »» ix ã 1 »» ix == numX »»
iy ã 1 »» iy == numY, 1, 0D, 8ix, 1, numX<, 8iy, 1, numY<D;
880, 60<
Dimensions@gridD
Count@Flatten@gridD, x_ ê; x ã 1D H* number of known grid values *L
508
Count@Flatten@gridD, x_ ê; x ã 0D H* number of unknown grid values *L
4292
Length@Flatten@gridDD
4800
Numerical_Potential_Flow.nb
In[21]:=
ArrayPlot@Transpose@gridD, Mesh Ø TrueD
Out[21]=
In[22]:=
wallBound@ix_D :=
If@length ê 2 - wallLength ê 2 § Hix - 1L dx § length ê 2 + wallLength ê 2,
Floor@wallHeight ê dyD + 1, 1D
3
4
Numerical_Potential_Flow.nb
In[23]:=
ListPlotBTableBwallBound@ixD, :ix, FloorB
numX
F, FloorB
3 numX
4
4
Frame Ø True, PlotStyle Ø [email protected]`D, Axes Ø False,
F>F,
FrameLabel Ø 8"x", "h", "Wall Boundary Index", ""<F
Wall Boundary Index
30
25
h
20
Out[23]=
15
10
5
0
0
10
20
30
40
x
Define Boundary Conditions
In[24]:=
Clear@yD; H* y is the variable used to store the streamfunction. *L
Top and bottom…
In[25]:=
y@ix_, numYD = flow height;
y@ix_, 1D = 0;
Left and right sides…
In[27]:=
y@1, iy_D = flow Hiy - 1L dy;
y@numX, iy_D = flow Hiy - 1L dy;
The wall…
In[29]:=
Table@Table@y@ix, iyD = 0, 8iy, 1, wallBound@ixD<D, 8ix, 1, numX<D;
Define the Equations
Numerical_Potential_Flow.nb
Define the Equations
In[30]:=
eqns = Flatten@Table@Table@eqn@ix, iyD, 8iy, wallBound@ixD + 1, numY - 1<D ,
8ix, 2, numX - 1<DD ê. dx Ø dx ê. dy Ø dy;
In[31]:=
eqns = Simplify@eqnsD;
In[32]:=
Dimensions@eqnsD
Out[32]=
In[33]:=
Out[33]=
In[34]:=
In[35]:=
Out[35]=
In[36]:=
Out[36]=
84292<
81. y@2, 2D ã 0.00283543 + 0.466542 y@2, 3D + 0.0334581 y@3, 2D,
0.0121551 + 1. y@2, 2D + 1. y@2, 4D + 0.071715 y@3, 3D ã 2.14343 y@2, 3D,
0.0182326 + 1. y@2, 3D + 1. y@2, 5D + 0.071715 y@3, 4D ã 2.14343 y@2, 4D<
eqnsP81, 2, 3<T
unknowns = Flatten@Table@
Table@y@ix, iyD, 8iy, wallBound@ixD + 1, numY - 1<D , 8ix, 2, numX - 1<DD;
8y@2, 2D, y@2, 3D, y@2, 4D<
unknownsP81, 2, 3<T
84292<
Dimensions@unknownsD
Finding the Solution
In[37]:=
Out[37]=
In[38]:=
Timing@sol = Flatten@Solve@eqns, unknownsDD;D
80.144978, Null<
ySol = Interpolation@Flatten@Table@8Hix - 1L dx, Hiy - 1L dy, y@ix, iyD<,
8ix, 1, numX<, 8iy, 1, numY<D ê. sol, 1DD;
5
6
Numerical_Potential_Flow.nb
In[39]:=
Out[39]=
ContourPlot@ySol@x, yD, 8x, 0, length<, 8y, 0, height<, PlotPoints Ø 80,
ContourShading Ø False, Contours Ø 18, FrameLabel Ø 8"x", "y"<,
PerformanceGoal Ø "Speed", PlotLabel Ø "Streamfunction Around Wall"D
Numerical_Potential_Flow.nb
In[40]:=
7
>,
2
2
2
2
8y, 0, height<, PlotPoints Ø 80, ContourShading Ø False,
Contours Ø 18, AspectRatio Ø Automatic, FrameLabel Ø 8"x", "y"<,
ContourPlotBySol@x, yD, :x,
length
-
height
,
length
+
height
PerformanceGoal Ø "Speed", PlotLabel Ø "Streamfunction Around Wall"F
Out[40]=
Computing the Velocity
In[41]:=
uX@xx_, yy_D = D@ySol@x, yD, yD ê. x Ø xx ê. y Ø yy;
uY@xx_, yy_D = -D@ySol@x, yD, xD ê. x Ø xx ê. y Ø yy;
u2@x_, y_D = uX@x, yD ^ 2 + uY@x, yD ^ 2;
8
Numerical_Potential_Flow.nb
In[44]:=
>,
2
2
2
2
2
8y, 0, height<, PlotPoints Ø 80, ContourShading Ø False,
Contours Ø 18, PerformanceGoal Ø "Speed", AspectRatio Ø Automatic,
ContourPlotB
1
u2@x, yD, :x,
length
-
height
,
length
+
height
FrameLabel Ø 8"x", "y"<, PlotLabel Ø "U^2ê2 Around Wall"F
Out[44]=
In[45]:=
Cp@x_, y_D = 1 - u2@x, yD ê flow ^ 2;
Numerical_Potential_Flow.nb
In[46]:=
9
>,
2
2
2
2
8y, 0, height<, PlotPoints Ø 80, ContourShading Ø False,
Contours Ø 18, PerformanceGoal Ø "Speed", AspectRatio Ø Automatic,
ContourPlotBCp@x, yD, :x,
length
-
height
,
length
+
height
FrameLabel Ø 8"x", "y"<, PlotLabel Ø "Normalized Pressure"F
Out[46]=
10
Numerical_Potential_Flow.nb
In[47]:=
Plot3DBCp@x, yD, :x,
length
2
-
height
2
,
length
Out[48]=
height
2
>,
8y, 0, height<, PlotPoints Ø 80, PlotLabel Ø "Normalized Pressure"F
Out[47]=
In[48]:=
2
+
Show@%, ViewPoint Ø 81.2`, 1.2`, 1.2`<D
Numerical_Potential_Flow.nb
In[49]:=
PlotBCpB
11
length
2
-
height
2
, yF, 8y, 0, height<, AxesLabel Ø 8"y", "Cp"<,
PlotLabel Ø "Pressure Along Leading Wall Edge"F
Pressure Along Leading Wall Edge
Cp
0.5
Out[49]=
0.2
0.4
0.6
0.8
1.0
y
-0.5
-1.0
In[50]:=
Plot@Cp@x, wallHeightD, 8x, 0, length<,
AxesLabel Ø 8"x", "Cp"<, PlotLabel Ø "Pressure Along Wall Height"D
Pressure Along Wall Height
Cp
1
-0.5
Out[50]=
-1.0
-1.5
Flow Past a Cylinder
2
3
4
5
x
12
Numerical_Potential_Flow.nb
Flow Past a Cylinder
In[51]:=
In[52]:=
Clear@y, uX, uY, u2D;
radius = 0.2 H* meter *L;
centerY = height ê 2 H* meter *L;
‡ Define Boundary Conditions
Top and bottom…
In[54]:=
y@ix_, numYD = flow height;
y@ix_, 1D = 0;
Left and right sides…
In[56]:=
y@1, iy_D = flow Hiy - 1L dy;
y@numX, iy_D = flow Hiy - 1L dy;
The cylinder
In[58]:=
Table@If@HHix - 1L dx - length ê 2L ^ 2 + HHiy - 1L dy - centerYL ^ 2 § radius ^ 2,
y@ix, iyD = flow centerYD, 8iy, 1, numY<, 8ix, 1, numX<D;
‡ Define the Equations
In[59]:=
In[61]:=
Out[61]=
eqns =
Flatten@Table@If@HHix - 1L dx - length ê 2L ^ 2 + HHiy - 1L dy - centerYL ^ 2 §
radius ^ 2, -1, eqn@ix, iyDD,
8iy, 2, numY - 1<, 8ix, 2, numX - 1<DD ê. dx Ø dx ê. dy Ø dy;
eqns = DeleteCases@eqns, _ ? NegativeD;
84408<
Dimensions@eqnsD
In[62]:=
eqns = Simplify@eqnsD;
In[63]:=
eqnsP81, 2, 3<T
Out[63]=
81. y@2, 2D ã 0.00283543 + 0.466542 y@2, 3D + 0.0334581 y@3, 2D,
1. y@2, 2D + 13.9441 y@3, 3D + 1. y@4, 2D ã 29.8882 y@3, 2D,
1. y@3, 2D + 13.9441 y@4, 3D + 1. y@5, 2D ã 29.8882 y@4, 2D<
Numerical_Potential_Flow.nb
In[64]:=
In[66]:=
Out[66]=
In[67]:=
Out[67]=
unknowns =
Flatten@Table@If@HHix - 1L dx - length ê 2L ^ 2 + HHiy - 1L dy - centerYL ^ 2 §
radius ^ 2, -1, y@ix, iyDD, 8iy, 2, numY - 1<, 8ix, 2, numX - 1<DD;
unknowns = DeleteCases@unknowns, _ ? NegativeD;
8y@2, 2D, y@3, 2D, y@4, 2D<
unknownsP81, 2, 3<T
84408<
Dimensions@unknownsD
‡ Solution
The solution is found quickly using Mathematica's built-in Solve function.
In[68]:=
Out[68]=
In[69]:=
Timing@sol = Flatten@Solve@eqns, unknownsDD;D
80.137636, Null<
ySol = Interpolation@Flatten@Table@8Hix - 1L dx, Hiy - 1L dy, y@ix, iyD<,
8ix, 1, numX<, 8iy, 1, numY<D ê. sol, 1DD;
13
14
Numerical_Potential_Flow.nb
In[70]:=
Out[70]=
ContourPlot@ySol@x, yD, 8x, 0, length<, 8y, 0, height<, PlotPoints Ø 80,
ContourShading Ø False, Contours Ø 24, FrameLabel Ø 8"x", "y"<,
PerformanceGoal Ø "Speed", PlotLabel Ø "Streamfunction Around Cylinder"D
Numerical_Potential_Flow.nb
In[71]:=
15
>,
2
2
2
2
8y, 0, height<, PlotPoints Ø 80, ContourShading Ø False,
PerformanceGoal Ø "Speed", Contours Ø 24, FrameLabel Ø 8"x", "y"<,
plt = ContourPlotBySol@x, yD, :x,
length
-
height
PlotLabel Ø "Streamfunction Around Cylinder"F
Out[71]=
,
length
+
height
16
Numerical_Potential_Flow.nb
In[72]:=
Out[72]=
cylinder = GraphicsB:[email protected]`D, CircleB:
length
2
, centerY>, radiusF>F
Numerical_Potential_Flow.nb
In[73]:=
Show@plt, cylinderD
Out[73]=
‡ Computing the Velocity
In[74]:=
uX@xx_, yy_D = D@ySol@x, yD, yD ê. x Ø xx ê. y Ø yy;
uY@xx_, yy_D = -D@ySol@x, yD, xD ê. x Ø xx ê. y Ø yy;
u2@x_, y_D = uX@x, yD ^ 2 + uY@x, yD ^ 2;
17
18
Numerical_Potential_Flow.nb
In[77]:=
>,
2
2
2
2
2
8y, 0, height<, PlotPoints Ø 80, ContourShading Ø False,
Contours Ø 18, PerformanceGoal Ø "Speed", AspectRatio Ø Automatic,
ContourPlotB
1
u2@x, yD, :x,
length
-
height
,
length
+
height
FrameLabel Ø 8"x", "y"<, PlotLabel Ø "U^2ê2 Around Cylinder"F
Out[77]=
In[78]:=
Cp@x_, y_D = 1 - u2@x, yD ê flow ^ 2;
Numerical_Potential_Flow.nb
In[79]:=
19
>,
2
2
2
2
8y, 0, height<, PlotPoints Ø 80, ContourShading Ø False,
Contours Ø 18, PerformanceGoal Ø "Speed", AspectRatio Ø Automatic,
ContourPlotBCp@x, yD, :x,
length
-
height
,
length
+
height
FrameLabel Ø 8"x", "y"<, PlotLabel Ø "Normalized Pressure"F
Out[79]=
20
Numerical_Potential_Flow.nb
In[80]:=
Plot3DBCp@x, yD, :x,
length
2
-
height
2
,
length
Out[81]=
height
2
>,
8y, 0, height<, PlotPoints Ø 80, PlotLabel Ø "Normalized Pressure"F
Out[80]=
In[81]:=
2
+
Show@%, ViewPoint Ø 81.2`, 1.2`, 1.2`<D
Numerical_Potential_Flow.nb
In[82]:=
21
Show@%%, PlotRange Ø AllD
Out[82]=
In[83]:=
PlotBCpB
length
2
, yF, 8y, 0, height<, AxesLabel Ø 8"y", "Cp"<,
PlotLabel Ø "Pressure Profile through Cylinder"F
Pressure Profile through Cylinder
Cp
1
0.2
-1
Out[83]=
-2
-3
-4
-5
-6
0.4
0.6
0.8
1.0
y
22
Numerical_Potential_Flow.nb
In[84]:=
PlotBCpB
length
2
+ 1.1` radius Cos@qD, centerY + 1.1` radius Sin@qDF,
8q, 0, 2 p<, AxesLabel Ø 8"q", "Cp"<, Ticks Ø :RangeB0, 2 p,
p
2
F, Automatic>,
PlotLabel Ø "Pressure Profile around Cylinder"F
Pressure Profile around Cylinder
Cp
2
q
3p
p
p
2
2p
-1
Out[84]=
-2
-3
-4
Summary
Two dimensional potential flow requires the solution of an elliptic partial differential equation. These problems
can be solved very easily with today's computers.