Numerical Solution of Plane Irrotational Flow

Transcription

1 Numerical Solution of Plane Irrotational Flow APPH 400 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[]:= eqn@i_, j_d := H y@i + 1, jd - y@i, jd + y@i - 1, jdl ê dx^ + Hy@i, j + 1D - y@i, jd + y@i, j - 1DL ê dy^ ã 0 In[3]:= eqn@i, jd Out[3]= y@i, -1 + jd - y@i, jd + y@i, 1 + jd dy + y@-1 + i, jd - y@i, jd + y@1 + i, jd dx ã 0

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]:= dx = length ê HnumX - 1L; dy = height ê HnumY - 1L; In[13]:= Out[13]= 7.9 In[14]:= Out[14]= 9.5 walllength ê dx wallheight ê dy In[15]:= In[16]:= flow = 5.0 H* mês *L; grid = Table@ If@length ê - walllength ê Hix - 1L dx length ê + walllength ê && Hiy - 1L dy wallheight»» ix ã 1»» ix == numx»» iy ã 1»» iy == numy, 1, 0D, 8ix, 1, numx<, 8iy, 1, numy<d; In[17]:= Dimensions@gridD Out[17]= 880, 60< In[18]:= Count@Flatten@gridD, x_ ê; x ã 1D H* number of known grid values *L Out[18]= 508 In[19]:= Count@Flatten@gridD, x_ ê; x ã 0D H* number of unknown grid values *L Out[19]= 49 In[0]:= Out[0]= 4800 Length@Flatten@gridDD

3 Numerical_Potential_Flow.nb 3 In[1]:= ArrayPlot@Transpose@gridD, Mesh Ø TrueD Out[1]= In[]:= wallbound@ix_d := If@length ê - walllength ê Hix - 1L dx length ê + walllength ê, Floor@wallHeight ê dyd + 1, 1D

4 4 Numerical_Potential_Flow.nb In[3]:= :ix, FloorB numx 3 numx F, FloorB F>F, 4 4 Frame Ø True, PlotStyle Ø PointSize@0.0`D, Axes Ø False, FrameLabel Ø 8"x", "h", "Wall Boundary Index", ""<F Wall Boundary Index Out[3]= h x Define Boundary Conditions In[4]:= Clear@yD; H* y is the variable used to store the streamfunction. *L Top and bottom In[5]:= y@ix_, numyd = flow height; y@ix_, 1D = 0; Left and right sides In[7]:= y@1, iy_d = flow Hiy - 1L dy; y@numx, iy_d = flow Hiy - 1L dy; The wall In[9]:= Table@Table@y@ix, iyd = 0, 8iy, 1, wallbound@ixd<d, 8ix, 1, numx<d;

5 Numerical_Potential_Flow.nb 5 Define the Equations In[30]:= eqns = Flatten@Table@Table@eqn@ix, iyd, 8iy, wallbound@ixd + 1, numy - 1<D, 8ix,, numx - 1<DD ê. dx Ø dx ê. dy Ø dy; In[31]:= eqns = Simplify@eqnsD; In[3]:= Dimensions@eqnsD Out[3]= 849< In[33]:= eqnsp81,, 3<T Out[33]= 81. y@, D ã y@, 3D y@3, D, y@, D + 1. y@, 4D y@3, 3D ã y@, 3D, y@, 3D + 1. y@, 5D y@3, 4D ã y@, 4D< In[34]:= unknowns = Flatten@Table@ Table@y@ix, iyd, 8iy, wallbound@ixd + 1, numy - 1<D, 8ix,, numx - 1<DD; In[35]:= unknownsp81,, 3<T Out[35]= 8y@, D, y@, 3D, y@, 4D< In[36]:= Dimensions@unknownsD Out[36]= 849< Finding the Solution In[37]:= Out[37]= In[38]:= Timing@sol = Flatten@Solve@eqns, unknownsdd;d , Null< ysol = Interpolation@Flatten@Table@8Hix - 1L dx, Hiy - 1L dy, y@ix, iyd<, 8ix, 1, numx<, 8iy, 1, numy<d ê. sol, 1DD;

6 6 Numerical_Potential_Flow.nb In[39]:= 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 Out[39]=

7 Numerical_Potential_Flow.nb 7 In[40]:= ContourPlotBySol@x, yd, :x, length - height, length + height >, 8y, 0, height<, PlotPoints Ø 80, ContourShading Ø False, Contours Ø 18, AspectRatio Ø Automatic, FrameLabel Ø 8"x", "y"<, 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; u@x_, y_d = ux@x, yd^ + uy@x, yd^;

8 8 Numerical_Potential_Flow.nb In[44]:= ContourPlotB 1 length u@x, yd, :x, - height, length + height >, 8y, 0, height<, PlotPoints Ø 80, ContourShading Ø False, Contours Ø 18, PerformanceGoal Ø "Speed", AspectRatio Ø Automatic, FrameLabel Ø 8"x", "y"<, PlotLabel Ø "U^ê Around Wall"F Out[44]= In[45]:= Cp@x_, y_d = 1 - u@x, yd ê flow ^ ;

9 Numerical_Potential_Flow.nb 9 In[46]:= ContourPlotBCp@x, yd, :x, length - height, length + height >, 8y, 0, height<, PlotPoints Ø 80, ContourShading Ø False, Contours Ø 18, PerformanceGoal Ø "Speed", AspectRatio Ø Automatic, FrameLabel Ø 8"x", "y"<, PlotLabel Ø "Normalized Pressure"F Out[46]=

10 10 Numerical_Potential_Flow.nb In[47]:= yd, :x, length - height, length + height >, 8y, 0, height<, PlotPoints Ø 80, PlotLabel Ø "Normalized Pressure"F Out[47]= In[48]:= Show@%, ViewPoint Ø 81.`, 1.`, 1.`<D Out[48]=

11 Numerical_Potential_Flow.nb 11 In[49]:= PlotBCpB length - height, 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]= y In[50]:= Plot@Cp@x, wallheightd, 8x, 0, length<, AxesLabel Ø 8"x", "Cp"<, PlotLabel Ø "Pressure Along Wall Height"D Cp Pressure Along Wall Height x -0.5 Out[50]=

12 1 Numerical_Potential_Flow.nb Flow Past a Cylinder In[51]:= In[5]:= Clear@y, ux, uy, ud; radius = 0. H* meter *L; centery = height ê 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 L^ + HHiy - 1L dy - centeryl^ radius^, y@ix, iyd = flow centeryd, 8iy, 1, numy<, 8ix, 1, numx<d; Define the Equations In[59]:= eqns = Flatten@Table@If@HHix - 1L dx - length ê L^ + HHiy - 1L dy - centeryl^ radius^, -1, eqn@ix, iydd, 8iy,, numy - 1<, 8ix,, numx - 1<DD ê. dx Ø dx ê. dy Ø dy; eqns = DeleteCases@eqns, _?NegativeD; In[61]:= Dimensions@eqnsD Out[61]= 84408< In[6]:= eqns = Simplify@eqnsD; In[63]:= eqnsp81,, 3<T Out[63]= 81. y@, D ã y@, 3D y@3, D, 1. y@, D y@3, 3D + 1. y@4, D ã y@3, D, 1. y@3, D y@4, 3D + 1. y@5, D ã y@4, D<

13 Numerical_Potential_Flow.nb 13 In[64]:= unknowns = Flatten@Table@If@HHix - 1L dx - length ê L^ + HHiy - 1L dy - centeryl^ radius^, -1, y@ix, iydd, 8iy,, numy - 1<, 8ix,, numx - 1<DD; unknowns = DeleteCases@unknowns, _?NegativeD; In[66]:= unknownsp81,, 3<T Out[66]= 8y@, D, y@3, D, y@4, D< In[67]:= Dimensions@unknownsD Out[67]= 84408< 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 , Null< ysol = Interpolation@Flatten@Table@8Hix - 1L dx, Hiy - 1L dy, y@ix, iyd<, 8ix, 1, numx<, 8iy, 1, numy<d ê. sol, 1DD;

14 14 Numerical_Potential_Flow.nb In[70]:= yd, 8x, 0, length<, 8y, 0, height<, PlotPoints Ø 80, ContourShading Ø False, Contours Ø 4, FrameLabel Ø 8"x", "y"<, PerformanceGoal Ø "Speed", PlotLabel Ø "Streamfunction Around Cylinder"D Out[70]=

15 Numerical_Potential_Flow.nb 15 In[71]:= plt = ContourPlotBySol@x, yd, :x, length - height, length + height >, 8y, 0, height<, PlotPoints Ø 80, ContourShading Ø False, PerformanceGoal Ø "Speed", Contours Ø 4, FrameLabel Ø 8"x", "y"<, PlotLabel Ø "Streamfunction Around Cylinder"F Out[71]=

16 16 Numerical_Potential_Flow.nb In[7]:= cylinder = GraphicsB:Thickness@0.0`D, CircleB: length, centery>, radiusf>f Out[7]=

17 Numerical_Potential_Flow.nb 17 In[73]:= cylinderd Out[73]= Computing the Velocity In[74]:= 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; u@x_, y_d = ux@x, yd^ + uy@x, yd^;

18 18 Numerical_Potential_Flow.nb In[77]:= ContourPlotB 1 length u@x, yd, :x, - height, length + height >, 8y, 0, height<, PlotPoints Ø 80, ContourShading Ø False, Contours Ø 18, PerformanceGoal Ø "Speed", AspectRatio Ø Automatic, FrameLabel Ø 8"x", "y"<, PlotLabel Ø "U^ê Around Cylinder"F Out[77]= In[78]:= Cp@x_, y_d = 1 - u@x, yd ê flow ^ ;

19 Numerical_Potential_Flow.nb 19 In[79]:= yd, :x, length - height, length + height >, 8y, 0, height<, PlotPoints Ø 80, ContourShading Ø False, Contours Ø 18, PerformanceGoal Ø "Speed", AspectRatio Ø Automatic, FrameLabel Ø 8"x", "y"<, PlotLabel Ø "Normalized Pressure"F Out[79]=

20 0 Numerical_Potential_Flow.nb In[80]:= yd, :x, length - height, length + height >, 8y, 0, height<, PlotPoints Ø 80, PlotLabel Ø "Normalized Pressure"F Out[80]= In[81]:= Show@%, ViewPoint Ø 81.`, 1.`, 1.`<D Out[81]=

21 Numerical_Potential_Flow.nb 1 In[8]:= Show@%%, PlotRange Ø AllD Out[8]= In[83]:= PlotBCpB length, yf, 8y, 0, height<, AxesLabel Ø 8"y", "Cp"<, PlotLabel Ø "Pressure Profile through Cylinder"F 1 Pressure Profile through Cylinder Cp y -1 Out[83]=

22 Numerical_Potential_Flow.nb In[84]:= PlotBCpB length + 1.1` radius Cos@qD, centery + 1.1` radius Sin@qDF, 8q, 0, p<, AxesLabel Ø 8"q", "Cp"<, Ticks Ø :RangeB0, p, p F, Automatic>, PlotLabel Ø "Pressure Profile around Cylinder"F Pressure Profile around Cylinder Cp p p 3 p p q Out[84]= 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.