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mech7220-numerical-mmcode - xxxxxxx xxxxxxx xxxxxxx inflow...

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starting length X s y x inflow outflow surface length L = 1 H i ,0 i ,1 i , N y Figure 1: External boundary layer control volume. 4 Mathematica Code for streamfunction–vorticity CFD Listed below is a Mathematica code which implements the solution procedure for the flat plate, external boundary layer problem. The domain consists of a rectangular region show in Fig. ( 1 ). The mathematical formulation of the boundary conditions for this flow configuration was discussed in the previous set of notes. The objective here is to show how it is coded into Mathematica . The code consists of the following basic modules (or elements): A module to numerically solve for stream function values, as a function of the vorticity values and the stream function boundary conditions. The system of equations that are solved have the form ψ i +1 ,j + ψ i - 1 ,j - 2 ψ i,j x ) 2 + ψ i,j +1 + ψ i,j - 1 - 2 ψ i,j y ) 2 = - ω i,j (1) A module to explicitly calculate the values of either ω i,j or T i,j at the k time step from the flow information at the k - 1 time step. This procedure utilizes the explicit time integration, upwind– differencing scheme. Associated modules and functions to interpolate the numerical results, compute derivatives of results, produce plots, and so on. The code is presented as follows The first lines set up global definitions and define assorted utility functions. Remove["Global‘*"] Off[General::"spell1",General::"spell"] sp[x_]:=Simplify[PowerExpand[x]] $TextStyle={FontFamily->"Times",FontSize->16}; zeroprint[n_]:=Module[{s},s="";Do[s=s<>"0",{i,1,n}];s]; index[l_]:=Which[l<10,"00",l<100,"0",1==1,""]<>ToString[l]; 1
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printtodigits[x_,nd_]:=Module[{t,lt,ndot,ndrop,t2,nzero,t3}, t=ToString[x]; lt=SyntaxLength[t]; ndot=StringPosition[t,"."][[1,1]]; ndrop=Max[0,lt-ndot-nd]; t2=StringDrop[t,-ndrop]; nzero=Max[0,nd+ndot-lt]; t3=t2<>zeroprint[nzero]; t3] The system consists of a rectangular region, with length 1 + X s , where X s is the starting region, and a height of H . The number of interior control volumes in the x and y directions is nx-1 and ny-1 . The sizes of the control volume, delx and dely , are Δ x = 1 + X s N x , Δ y = H N y The module (or subroutine) psisoln[omega]
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mech7220-numerical-mmcode - xxxxxxx xxxxxxx xxxxxxx inflow...

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