# drivencavity.py¶

In this script we solve the lid driven cavity problem for stationary Stokes and Navier-Stokes flow. That is, a unit square domain, with no-slip left, bottom and right boundaries and a top boundary that is moving at unit velocity in positive x-direction.

 8 import nutils, numpy 

The main function defines the parameter space for the script. Configurable parameters are the mesh density (in number of elements along an edge), element type (square, triangle, or mixed), polynomial degree, and Reynolds number.

 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 def main(nelems: 'number of elements' = 12, etype: 'type of elements (square/triangle/mixed)' = 'square', degree: 'polynomial degree for velocity' = 3, reynolds: 'reynolds number' = 1000.): domain, geom = nutils.mesh.unitsquare(nelems, etype) ns = nutils.function.Namespace() ns.Re = reynolds ns.x = geom ns.ubasis, ns.pbasis = nutils.function.chain([ domain.basis('std', degree=degree).vector(2), domain.basis('std', degree=degree-1), ]) ns.u_i = 'ubasis_ni ?lhs_n' ns.p = 'pbasis_n ?lhs_n' ns.stress_ij = '(u_i,j + u_j,i) / Re - p δ_ij' sqr = domain.boundary.integral('u_k u_k d:x' @ ns, degree=degree*2) wallcons = nutils.solver.optimize('lhs', sqr, droptol=1e-15) sqr = domain.boundary['top'].integral('(u_0 - 1)^2 d:x' @ ns, degree=degree*2) lidcons = nutils.solver.optimize('lhs', sqr, droptol=1e-15) cons = numpy.choose(numpy.isnan(lidcons), [lidcons, wallcons]) cons[-1] = 0 # pressure point constraint res = domain.integral('(ubasis_ni,j stress_ij + pbasis_n u_k,k) d:x' @ ns, degree=degree*2) with nutils.log.context('stokes'): lhs0 = nutils.solver.solve_linear('lhs', res, constrain=cons) postprocess(domain, ns, lhs=lhs0) res += domain.integral('ubasis_ni u_i,j u_j d:x' @ ns, degree=degree*3) with nutils.log.context('navierstokes'): lhs1 = nutils.solver.newton('lhs', res, lhs0=lhs0, constrain=cons).solve(tol=1e-10) postprocess(domain, ns, lhs=lhs1) return lhs0, lhs1 

Postprocessing in this script is separated so that it can be reused for the results of Stokes and Navier-Stokes, and because of the extra steps required for establishing streamlines.

 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 def postprocess(domain, ns, every=.05, spacing=.01, **arguments): ns = ns.copy_() # copy namespace so that we don't modify the calling argument ns.streambasis = domain.basis('std', degree=2)[1:] # remove first dof to obtain non-singular system ns.stream = 'streambasis_n ?streamdofs_n' # stream function sqr = domain.integral('((u_0 - stream_,1)^2 + (u_1 + stream_,0)^2) d:x' @ ns, degree=4) arguments['streamdofs'] = nutils.solver.optimize('streamdofs', sqr, arguments=arguments) # compute streamlines bezier = domain.sample('bezier', 9) x, u, p, stream = bezier.eval(['x_i', 'sqrt(u_k u_k)', 'p', 'stream'] @ ns, **arguments) with nutils.export.mplfigure('flow.png') as fig: # plot velocity as field, pressure as contours, streamlines as dashed ax = fig.add_axes([.1,.1,.8,.8], yticks=[], aspect='equal') import matplotlib.collections ax.add_collection(matplotlib.collections.LineCollection(x[bezier.hull], colors='w', linewidths=.5, alpha=.2)) ax.tricontour(x[:,0], x[:,1], bezier.tri, stream, 16, colors='k', linestyles='dotted', linewidths=.5, zorder=9) caxu = fig.add_axes([.1,.1,.03,.8], title='velocity') imu = ax.tripcolor(x[:,0], x[:,1], bezier.tri, u, shading='gouraud', cmap='jet') fig.colorbar(imu, cax=caxu) caxu.yaxis.set_ticks_position('left') caxp = fig.add_axes([.87,.1,.03,.8], title='pressure') imp = ax.tricontour(x[:,0], x[:,1], bezier.tri, p, 16, cmap='gray', linestyles='solid') fig.colorbar(imp, cax=caxp) 

If the script is executed (as opposed to imported), nutils.cli.run() calls the main function with arguments provided from the command line. To keep with the default arguments simply run python3 drivencavity.py (view log).

 85 86 if __name__ == '__main__': nutils.cli.run(main) 

Once a simulation is developed and tested, it is good practice to save a few strategicly chosen return values for routine regression testing. Here we use the standard unittest framework, with nutils.numeric.assert_allclose64() facilitating the embedding of desired results as compressed base64 data.

  94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 class test(nutils.testing.TestCase): @nutils.testing.requires('matplotlib') def test_square(self): lhs0, lhs1 = main(nelems=3, etype='square', reynolds=100, degree=3) nutils.numeric.assert_allclose64(lhs0, 'eNp1zj1IQlEUB/BrCJKEQxLRFNFQxvN1vTcpo' 'qWhzZaGElr7WKOGirApiIaipcEKoiXCpaKEiCKnhjznXX1PejaEJGGFRCCiCH153YrXOXCG3+F' 'w/oT8rZFeQpaVqDGVmjHNxEKSJmxM2rOIal1aDlsxKyK+gF/asZbHEA5gDmL6FduuWRnHsAQXc' 'ABEXeGP/5rVrdUPqyxWma1q2ih3u1g7/+JnPf3+BiYtr5ToBGvm33yNd/C3pLTrTi9d9Y2yCku' 'xU2Z6pa17CqpKMzTo+6AbdLJmc3eupC7axKFmF7NiR5c2aBpiUYugAxUcRk/Nmgyn2MVXsME83' 'INblRZW6hMFfIA6CMRvbotonTgL7/ACWQjBfjwcT8MT6HAJSxCEI8hAvroxIQZ7cA7FX+3ET3C' 'gG1Ucxz5sRDu2IMctTONQNVkFbNW5iScGIT8HbdXq') nutils.numeric.assert_allclose64(lhs1, 'eNptzktoU0EUBuC7KeLGguKioS4MBdPekNyZS' 'WIwEihowVVBxJW0pYuiFgpiXSh0F0ltELvoC2zAVuorRuiTJlRLC6Hof2cml0wwCxVqCl1XFOq' 'i4p27LPlXP985HI5hHM/1i4aRMzvVL7VqOs4j5VMhS9un8k2ZkEnZLL+271v3mLYb8oG4KuKiR' '0yGtkk6om1MODzLH/Ma/xZK0b+eXROveJzX7Vs8ZcXYUFTbkYiJp7yFb9i3VTO765m/fFL+5IM' '8ZBfFHJvybCD4WvVWi86BZPIsj3j3Gv3cKKXKUDhJovQ7TbBhdsrSdjl4xcqSbtrEZukM7VDa3' 'ge2wnHSRAt0lmboSFjbCfNMuGItkH7aSxdpi9Q2c+Gf80JFgpdIHxkgdaJtt3aufFq2iRXxUPq' 'chLfnV63yLT/Pd2CKLXqfadsL9DmGmLeruPPl42diN/44jyV8wBuMogvteIe827MYxwTWkMOiK' '1k8QxrTbl9xZQpPMIzn2EDR3cgjg5dYxzYKKIHjDzbx252sY9mdHuKHaRj/AYh1yFc=') @nutils.testing.requires('matplotlib') def test_mixed(self): lhs0, lhs1 = main(nelems=3, etype='mixed', reynolds=100, degree=2) nutils.numeric.assert_allclose64(lhs0, 'eNpjYICAiRePnWdg0D736SyIF3P2nK6VYSWQH' 'WS+1SjI3MAkyLz6rMbZI2BZhXMJZxyMNp/xMbwMFA8yLzNhYNh6YdUFiElzzykYgGg94yBzkH6' 'oBQwvLm80YmA4r6dkCOYZq5h4GZUYgdg8QHKbJpA2OHhp8zmQiM8Vp6tpV03PMp1TPQ/ipwPJc' 'IOtZyAmvT69Bcy6BOXHnM0+m3w28ezmM+ZnY88EnW0/O+vs2bO7zq48W352FdA8ABC3SoM=') nutils.numeric.assert_allclose64(lhs1, 'eNpjYICA1RezLjIwPD639hyIl31umX6vgQGQH' 'WTuaRhkLmYcZB54bvvZq2dBsofPqZ4tMoo4o22oaxJkHmReasLAsOrihAsQkxzOJl0B0TJAOZB' '+qAUMtZefGzIwxOjtNgDxfho9MbI1UjcCsV/pMTA802VgqDNYqrsEbL+I7nGD0/o655ouMIFN3' 'QLUqWSUcQZiEvMZbrA7npyG8IXPyJ2RPiN65ubpn6dPn+Y9I3XG4AwfUMzlDPuZ60A9AH73RT0' '=')