Posts Tagged ‘Fortran

15
Mar
07

1D Shock Tube


A shock tube is a tube containing high and low pressure gas separated by a thin diaphragm. A shock wave is produced when the diaphragm is quickly removed. The color in the upper plot shows the pressure. The lower plot shows the density. The following Mathematica code solves Euler’s equations using the finite volume method with the Jameson-Schmidt-Turkel (JST) scheme and Runge-Kutta time stepping.


(* runtime: 5 seconds *)
gamma = 1.4;
R[W_] := Module[{}, rho = W[[All, 1]]; u = W[[All, 2]]/rho; p = (gamma - 1)(W[[All, 3]] - rho u^2/2); F = u W + Transpose[{Table[0, {n}], p, u p}]; h = Table[(F[[Min[n, i + 1]]] + F[[i]])/2, {i,1, n}]; Q = Table[h[[Max[i, 2]]] - h[[Max[i, 2] - 1]], {i, 1, n}]; nu = Table[i = Max[2, Min[n - 1, i]]; Abs[(p[[i + 1]] - 2p[[i]] + p[[i - 1]])/(p[[i + 1]] + 2p[[i]] + p[[i - 1]])], {i, 1, n}]; S = Table[Max[nu[[Min[n, i + 1]]], nu[[i]]], {i, 1, n}]; alpha1 = 1/2; beta1 = 1/4;alpha2 = alpha1; beta2 = beta1; epsilon2 = Map[Min[alpha1, alpha2#] &, S];epsilon4 = Map[Max[0, beta1 - beta2#] &, epsilon2];dW = Table[W[[Min[n - 1, i] + 1]] - W[[Min[n - 1, i]]], {i, 1, n}];dW3 = Table[i = Max[2, Min[n - 2, i]]; -W[[i - 1]] + 3W[[i]] - 3W[[i + 1]] + W[[i + 2]], {i, 1, n}];d = (epsilon2 dW - epsilon4 dW3)(Abs[u] + a); Dflux = Table[d[[Max[2, i]]] - d[[Max[2, i] - 1]], {i, 1, n}]; (Q - Dflux)/dx];
n = 50; dx = 1.0/n; a = 1.0; dt = dx/(1.0 + a); W = Table[{If[i > n/2, 0.125, 1], 0, If[i > n/2, 0.1, 1]/(gamma - 1)}, {i, 1, n}];
Do[W -= dt R[W - dt R[W - dt R[W]/4]/3]/2;ListPlot[W[[All, 1]], PlotJoined -> True,PlotRange -> {0, 1}, AxesLabel -> {"i", "rho"}], {t, 0, 100dt, dt}];

16
Nov
06

Driven Cavity

Here is the flow inside a square box where the flow across the top of the box is moving to the right at Reynolds number 1000. This program uses the finite volume method to solve the Navier-Stokes equations assuming steady, incompressible, viscous, laminar flow. This was calculated on a 300×300 non-uniform grid and took 3 hours to run on my laptop. The animation shows the motion along the streamlines/pathlines.

Driven Cavity – calculated in Fortran 90, plotted in Mathematica 4.2, 11/2/05

The following Mathematica code is not completely accurate at the boundaries, but it gives the basic idea:

(* runtime: 20 minutes *)
n = 50; dx = 1.0/(n - 1); RE = 1000; relax = 0.4; residu = residv = residp = 1;
u = Table[If[j == 1 || i == 2 || i == n, 0, 1], {i, 1, n}, {j, 1, n}];
v = p = pstar = ap = an = as = aw = ae = sc = sp = du = dv = Table[0, {n}, {n}];
lisolv[i0_, j0_, phi0_] := Module[{phi = phi0, p = q =Table[0, {n}]}, Do[q[[j0 - 1]] = phi[[i, j0 - 1]]; Do[p[[j]] = an[[i, j]]/(ap[[i, j]] -as[[i, j]]p[[j - 1]]); q[[j]] = (as[[i, j]]q[[j - 1]] + ae[[i, j]] phi[[i + 1, j]] + aw[[i, j]]phi[[i - 1, j]] + sc[[i, j]])/(ap[[i, j]] - as[[i, j]]p[[j - 1]]), {j, j0, n - 1}]; Do[phi[[i,j]] = p[[j]]phi[[i, j + 1]] + q[[j]], {j, n - 1, j0, -1}], {i, i0, n - 1}]; phi];
calc[i0_, j0_] := Module[{}, {cn, cs, ce, cw} = 0.5 {v[[i,j + 1]] + v[[i0, j0 + 1]], v[[i, j]] + v[[i0, j0]], u[[i + 1, j]] + u[[i0 + 1, j0]], u[[i, j]] + u[[i0, j0]]}dx; cp = Max[0, cn - cs + ce - cw]; {an[[i, j]], as[[i, j]], ae[[i, j]], aw[[i, j]]} = Map[Max[0, 1 - 0.5 Abs[#]] &, RE{cn, cs, ce, cw}]/RE + Map[Max[0, #] &, {-cn, cs, -ce, cw}]; sp[[i, j]] = -cp; ap[[i, j]] = an[[i, j]] + as[[i, j]] + ae[[i, j]] + aw[[i, j]] - sp[[i, j]]];
While[Max[residu, residv, residp] > 0.001, residu = residv = residp = 0; Do[calc[i - 1, j]; sc[[i, j]] = cp u[[i, j]] + dx(p[[i - 1, j]] - p[[i, j]]); du[[i, j]] = relax dx/ap[[i, j]]; residu += Abs[an[[i, j]]u[[i, j + 1]] + as[[i, j]]u[[i, j - 1]] + ae[[i, j]]u[[i + 1, j]] + aw[[i, j]]u[[i - 1, j]] - ap[[i, j]]u[[i, j]] + sc[[i, j]]], {i, 3, n - 1}, {j, 2, n - 1}]; ap /= relax; sc += (1 - relax) ap u; u = lisolv[3, 2, u]; Do[calc[i, j - 1]; sc[[i, j]] = cp v[[i, j]] + dx(p[[i, j - 1]] - p[[i, j]]); dv[[i, j]] = relax dx/ap[[i, j]];residv += Abs[an[[i, j]]v[[i, j + 1]] + as[[i, j]]v[[i, j - 1]] + ae[[i, j]]v[[i + 1, j]] + aw[[i, j]]v[[i - 1, j]] - ap[[i, j]]v[[i, j]] + sc[[i, j]]], {i, 2, n - 1}, {j, 3, n - 1}]; ap /= relax; sc += (1 - relax)ap v; v = lisolv[2, 3, v];Do[an[[i, j]] = dx dv[[i, j + 1]]; as[[i, j]] = dx dv[[i, j]]; ae[[i, j]] = dx du[[i + 1, j]]; aw[[i, j]] = dx du[[i, j]]; sp[[i, j]] = 0; sc[[i, j]] = -((v[[i, j + 1]] - v[[i, j]])dx + (u[[i + 1, j]] - u[[i, j]]) dx); residp += Abs[sc[[i, j]]]; ap[[i, j]] = an[[i, j]] + as[[i, j]] + ae[[i, j]] + aw[[i, j]] - sp[[i, j]]; pstar[[i, j]] = 0, {i, 2, n - 1}, {j, 2, n - 1}]; Do[pstar = lisolv[2, 2, pstar], {2}]; Do[If[i != 2, u[[i, j]] += du[[i, j]](pstar[[i - 1, j]] - pstar[[i, j]])]; If[j != 2, v[[i, j]] += dv[[i, j]](pstar[[i, j - 1]] - pstar[[i, j]])]; p[[i, j]] += 0.3(pstar[[i, j]] - pstar[[n - 1, n - 1]]), {i, 2, n - 1}, {j, 2, n - 1}]];

Here is some Mathematica code to plot streamlines. The blue lines are rotating clockwise and the red lines are rotating counter-clockwise. The plot on the left agrees fairly well with Ercan Erturk’s results.
psi = Table[0, {n - 1}, {n - 1}]; Do[psi[[i, j]] = psi[[i, j - 1]] + dx(u[[i + 1, j - 1]] + u[[i + 1, j]])/2, {j, 2, n - 1}, {i, 1, n - 1}];
psi = ListInterpolation[psi, {{0, 1}, {0, 1}}];
ContourPlot[psi[x, y], {x, 0, 1}, {y, 0, 1}, PlotPoints -> 50, PlotRange -> All, ContourShading -> False,Contours -> {-0.08, -0.077, -0.07, -0.06, -0.045, -0.025, -0.01, -0.0025, 0, -8*^-6, 2*^-6, 3*^-5, 8*^-5, 1*^-4, 3*^-4, 6*^-4, 9*^-4}, ContourStyle -> Table[{Hue[2(1 - x)/3]}, {x, 0, 1, 1/16}]]


Here is some Mathematica code to plot vorticity contours. The plot on the left agrees fairly well with Ghia’s results.
omega = ListInterpolation[Table[(v[[i, j]] - v[[i - 1, j]])/dx - (u[[i, j]] - u[[i, j - 1]])/dx, {i, 2, n - 1}, {j, 2, n - 1}], {{0, 1}, {0, 1}}];
ContourPlot[omega[x, y], {x, 0, 1}, {y, 0, 1},PlotPoints -> 50, PlotRange -> All, ContourShading -> False, Contours -> Range[-14, 7],ContourStyle -> Table[{Hue[2(1 - x)/3]}, {x, 0, 1, 1/21}]]

Computational Fluid Dynamics (CFD) Links

Driven CavityMathematica 4.2, 11/16/06


Here is the same driven cavity using the finite difference method. This code uses the Alternating Direction Implicit (ADI) method to solve the vorticity transport equation assuming unsteady, incompressible, viscous flow.
(* runtime: 1 minute *)
<< LinearAlgebra`Tridiagonal`; n = 20; RE = 1000; dx = 1.0/(n - 1); dt = 0.25; cxy = dt/(4dx); sxy = dt/(2RE dx^2); lambda = 1.5; free = Range[2, n - 1];
omega = v = psi = Table[0, {n}, {n}]; u = Table[If[j == n, 1, 0], {n}, {j, 1, n}];
Do[omega[[All, 1]] = 2psi[[All, 2]]/dx^2; omega[[All, n]] = 2 (psi[[All, n - 1]] + dx)/dx^2; omega[[1, All]] = 2 psi[[2, All]]/dx^2; omega[[n, All]] = 2 psi[[n - 1, All]]/dx^2; omega[[free, free]] = Transpose[Partition[TridiagonalSolve[Drop[Flatten[Table[If[i == n - 1, 0, -(cxy u[[i, j]] + sxy)], {j, 2, n - 1}, {i, 2, n - 1}]], -1], Flatten[Table[1 + 2sxy, {j, 2, n - 1}, {i, 2, n - 1}]], Drop[Flatten[Table[If[i == n - 1,0, cxy u[[i + 1, j]] - sxy], {j, 2,n - 1}, {i, 2, n - 1}]], -1], Flatten[Table[(cxy v[[i, j - 1]] + sxy)omega[[i, j - 1]] + (1 - 2sxy)omega[[i, j]] + (-cxy v[[i, j + 1]] + sxy)omega[[i, j + 1]] + If[i == 2, (cxy u[[i - 1, j]] + sxy)omega[[i - 1, j]],0] - If[i == n - 1, (cxy u[[i + 1, j]] - sxy) omega[[i + 1, j]], 0], {j, 2, n - 1}, {i, 2, n - 1}]]], n - 2]]; omega[[free, free]] = Partition[TridiagonalSolve[Drop[Flatten[Table[If[j == n - 1, 0, -(cxy v[[i, j]] + sxy)], {i, 2, n - 1}, {j, 2, n - 1}]], -1], Flatten[Table[1 + 2sxy, {i, 2, n - 1}, {j, 2, n - 1}]], Drop[Flatten[Table[If[j == n - 1, 0, cxy v[[i, j + 1]] - sxy], {i, 2, n - 1}, {j, 2, n - 1}]], -1], Flatten[Table[(cxy u[[i - 1, j]] + sxy)omega[[i - 1, j]] + (1 - 2sxy)omega[[i, j]] + (-cxy u[[i + 1, j]] +sxy)omega[[i + 1, j]] + If[j == 2, (cxy v[[i, j - 1]] + sxy)omega[[i, j - 1]], 0] - If[j == n - 1, (cxy v[[i, j + 1]] - sxy) omega[[i, j + 1]], 0], {i, 2, n - 1}, {j, 2, n - 1}]]], n - 2]; resid = 1; While[resid > 0.001, resid = 0; Do[resid += Abs[psi[[i, j]] - (psi[[i,j]] = lambda(psi[[i, j - 1]] + psi[[i - 1, j]] + psi[[i + 1, j]] + psi[[i, j + 1]] - dx^2omega[[i, j]])/4 + (1 - lambda)psi[[i, j]])], {i, 2, n - 1}, {j, 2, n - 1}]]; Do[u[[i, j]] = (psi[[i, j + 1]] - psi[[i, j - 1]])/(2dx); v[[i, j]] = -(psi[[i + 1, j]] - psi[[i - 1, j]])/(2dx), {i, 2, n - 1}, {j, 2, n - 1}]; ListContourPlot[Transpose[psi], PlotRange -> All, ContourShading -> False], {100}];

Link: Driven Cavity – simple Fortran program by Zheming Zheng that uses this method

28
Apr
06

Flapping Wing

This flapping wing was calculated using the unsteady vortex panel methodAlan Lai’s Fortran code. It assumes inviscid incompressible potential flow (irrotational). I also have a working Mathematica version of this code, but it is a little lengthy to show here.

Flapping Wing – Fortran 90, rendered in POV-Ray 3.6.1, 4/28/06

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