# 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 9 10 11 12 13 14 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 53 54 55 56 57 58 59 from nutils import mesh, function, solver, export, cli, testing import numpy, treelog def main(nelems:int, etype:str, degree:int, reynolds:float): ''' Driven cavity benchmark problem. .. arguments:: nelems [12] Number of elements along edge. etype [square] Element type (square/triangle/mixed). degree [2] Polynomial degree for velocity; the pressure space is one degree less. reynolds [1000] Reynolds number, taking the domain size as characteristic length. ''' domain, geom = mesh.unitsquare(nelems, etype) ns = function.Namespace() ns.Re = reynolds ns.x = geom ns.ubasis, ns.pbasis = 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 = solver.optimize('lhs', sqr, droptol=1e-15) sqr = domain.boundary['top'].integral('(u_0 - 1)^2 d:x' @ ns, degree=degree*2) lidcons = 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 treelog.context('stokes'): lhs0 = solver.solve_linear('lhs', res, constrain=cons) postprocess(domain, ns, lhs=lhs0) res += domain.integral('.5 (ubasis_ni u_i,j - ubasis_ni,j u_i) u_j d:x' @ ns, degree=degree*3) with treelog.context('navierstokes'): lhs1 = 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.

 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 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'] = 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 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).

 92 93 if __name__ == '__main__': cli.run(main) 

Once a simulation is developed and tested, it is good practice to save a few strategic return values for regression testing. The nutils.testing module, which builds on the standard unittest framework, facilitates this by providing nutils.testing.TestCase.assertAlmostEqual64() for the embedding of desired results as compressed base64 data.

 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 class test(testing.TestCase): @testing.requires('matplotlib') def test_square(self): lhs0, lhs1 = main(nelems=3, etype='square', reynolds=100, degree=3) with self.subTest('stokes'): self.assertAlmostEqual64(lhs0, ''' eNqNkE8og3EYx99JLZKDJTktOfjT9nr3+3kjuTi4zcUB5cpcxcGkOa3kQC4OGyUXaRfESiI7OdjzfX/v 3nd5OUiWhpbUWlsr/1Za48Tzrec5PJ/69v1K0t8z3PN9F11TZtQsdPmS9WzaauWW/sH9iaNuRe9TbayO lblHkXFFtbzSqU2wNJq4E588JZZ5xNPGvepLoszta+ftGbiVAJY8/RhhaSqymJFkZ+yQhVXLXeYKWOkY RVbOk6yc0J2yQ2MeSX5TgnxVuVcnf3CzV6OoT+TJECfUInZoV5PkahHkM+Je3TAqvgNWBqYIYF7rRwRp siNmuHDGhhBWO4xKjkYzqtWKTm0TaTyTEzZKiTmKeG7IqzrkSi8hV9Ss0X3JLKatW7L0KvInvHFFv7i0 sRzK3H96TtErPVGKArQdD8Wv6YEMOqUFGqM9uqNM6WNRjLbomHK/VIv3UgoHZIyjFw2oRjM41nGNQdhR JFtpr+HAlKQvrHfV6g==''') with self.subTest('navier-stokes'): self.assertAlmostEqual64(lhs1, ''' eNpjYCAMgswh9ErtA5c9ruTp/7xiZbhX28jo8MVbRn1XLIxVLhcbBxuIGsDUHbhgoxNx3tnA9vwcQ4fz PkbC59mNP2rONOIxnGiUaOx9GaYu5PxOjfJzB/Xdz5kbWp9jNYo9t81ont4so1OGXUbNJtX6MHVd515p XT4rqT//7EWDprPzDR+eTTY6pBdltNhoq9ELE+3zMHWe58rVl53lv2R1Vvbq+zOTdCLPmhpONkgwlDMO NawyvWAIU/f7nMRN03PyF3rOSp6XOqt3bv+ZA+cirnSclzf2vxhvGgo3T/pC5xXW80Ln751ddzb1rOpZ wzOXTp89l3iu1vic0WrTImOYugCd8uvrriy/aHf1w9nkizPPvDl/7rTe+cRTBmf3nVQx0T4DU0dMOK8/ vfX0xtOrT3ef9jitfXrN6Q1A9oLTk0/POL339LrTW4AiC05POt0F5G85vR0oMut0z+mK04tP7wfK7zi9 /nTf6aWnt50+enrP6ROnL57+cXrfacYze4G8rUD843SpLgMDAGbLx+o=''') @testing.requires('matplotlib') def test_mixed(self): lhs0, lhs1 = main(nelems=3, etype='mixed', reynolds=100, degree=2) with self.subTest('stokes'): self.assertAlmostEqual64(lhs0, ''' eNpjYEAFEy++uHzs/EYjEFv73Hm9T2eVDGFyMWdfGJ/TVTGxMvQyqjxbAlYTZM7AsNWIxwhEG5hs0wTR 1Wd5DDTOHrx05OzmczC9Cud8riSccbrqYJR2dfMZ07M+hkznLpuongepB+Eyk/TzIHVbL4QbrLqw9Qyy m+aee31awWALXEzP+NIZkB6Y/TFns88mn008u/mM+dnYM0Fn28/OOnv27K6zK8+Wn10FdAEAKXRKgw==''') with self.subTest('navier-stokes'): self.assertAlmostEqual64(lhs1, ''' eNpjYEAFvRcfXLa8eNsQxP5xLkBv17ktBjC5iHOfjbr1XxotNbAy0j4nbQQSCzJnYLA0lNIH0XLGi3VB dNK5HoNjZ2frXj+77BxM775zfbq6Z+cYxBpd1vc543jursGSC0omz86D1IPwBJOzYDszL5oaLbgQcQbZ TTZnec9svix4FsZXNbl9GqQHZj/rGb4zPGfYz5w5/fr03tOMZzjOKJz5d5rzjMmZL6cvAk0CAOBkR7A=''')