cylinderflow.py

In this script we solve the Navier-Stokes equations around a cylinder, using the same Raviart-Thomas discretization as in drivencavity-compatible.py but in curvilinear coordinates. The mesh is constructed such that all elements are shape similar, growing exponentially with radius such that the artificial exterior boundary is placed at large (configurable) distance.

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import nutils, numpy

The main function defines the parameter space for the script. Configurable parameters are the mesh density (in number of elements along the cylinder wall), polynomial degree, Reynolds number, rotational velocity of the cylinder, time step, exterior radius, and the end time of the simulation (by default it will run forever).

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def main(nelems: 'number of elements' = 24,
         degree: 'polynomial degree' = 3,
         reynolds: 'reynolds number' = 1000.,
         rotation: 'cylinder rotation speed' = 0.,
         timestep: 'time step' = 1/24,
         maxradius: 'approximate domain size' = 25.,
         seed: 'random seed' = 0,
         endtime: 'end time' = numpy.inf):

  elemangle = 2 * numpy.pi / nelems
  melems = int(numpy.log(2*maxradius) / elemangle + .5)
  nutils.log.info('creating {}x{} mesh, outer radius {:.2f}'.format(melems, nelems, .5*numpy.exp(elemangle*melems)))
  domain, geom = nutils.mesh.rectilinear([melems, nelems], periodic=(1,))
  domain = domain.withboundary(inner='left', outer='right')

  ns = nutils.function.Namespace()
  ns.uinf = 1, 0
  ns.r = .5 * nutils.function.exp(elemangle * geom[0])
  ns.Re = reynolds
  ns.phi = geom[1] * elemangle # add small angle to break element symmetry
  ns.x_i = 'r <cos(phi), sin(phi)>_i'
  ns.J = ns.x.grad(geom)
  ns.unbasis, ns.utbasis, ns.pbasis = nutils.function.chain([ # compatible spaces
    domain.basis('spline', degree=(degree,degree-1), removedofs=((0,),None)),
    domain.basis('spline', degree=(degree-1,degree)),
    domain.basis('spline', degree=degree-1),
  ]) / nutils.function.determinant(ns.J)
  ns.ubasis_ni = 'unbasis_n J_i0 + utbasis_n J_i1' # piola transformation
  ns.u_i = 'ubasis_ni ?lhs_n'
  ns.p = 'pbasis_n ?lhs_n'
  ns.sigma_ij = '(u_i,j + u_j,i) / Re - p δ_ij'
  ns.h = .5 * elemangle
  ns.N = 5 * degree / ns.h
  ns.rotation = rotation
  ns.uwall_i = '0.5 rotation <-sin(phi), cos(phi)>_i'

  inflow = domain.boundary['outer'].select(-ns.uinf.dotnorm(ns.x), ischeme='gauss1') # upstream half of the exterior boundary
  sqr = inflow.integral('(u_i - uinf_i) (u_i - uinf_i)' @ ns, degree=degree*2)
  cons = nutils.solver.optimize('lhs', sqr, droptol=1e-15) # constrain inflow semicircle to uinf

  sqr = domain.integral('(u_i - uinf_i) (u_i - uinf_i) + p^2' @ ns, degree=degree*2)
  lhs0 = nutils.solver.optimize('lhs', sqr) # set initial condition to u=uinf, p=0

  numpy.random.seed(seed)
  lhs0 *= numpy.random.normal(1, .1, lhs0.shape) # add small velocity noise

  res = domain.integral('(ubasis_ni u_i,j u_j + ubasis_ni,j sigma_ij + pbasis_n u_k,k) d:x' @ ns, degree=9)
  res += domain.boundary['inner'].integral('(N ubasis_ni - (ubasis_ni,j + ubasis_nj,i) n_j) (u_i - uwall_i) d:x / Re' @ ns, degree=9)
  inertia = domain.integral('ubasis_ni u_i d:x' @ ns, degree=9)

  bbox = numpy.array([[-2,46/9],[-2,2]]) # bounding box for figure based on 16x9 aspect ratio
  bezier0 = domain.sample('bezier', 5)
  bezier = bezier0.subset((bezier0.eval((ns.x-bbox[:,0]) * (bbox[:,1]-ns.x)) > 0).all(axis=1))
  interpolate = nutils.util.tri_interpolator(bezier.tri, bezier.eval(ns.x), mergetol=1e-5) # interpolator for quivers
  spacing = .05 # initial quiver spacing
  xgrd = nutils.util.regularize(bbox, spacing)

  with nutils.log.iter.plain('timestep', nutils.solver.impliciteuler('lhs', residual=res, inertia=inertia, lhs0=lhs0, timestep=timestep, constrain=cons, newtontol=1e-10)) as steps:
    for istep, lhs in enumerate(steps):

      t = istep * timestep
      x, u, normu, p = bezier.eval(['x_i', 'u_i', 'sqrt(u_k u_k)', 'p'] @ ns, lhs=lhs)
      ugrd = interpolate[xgrd](u)

      with nutils.export.mplfigure('flow.png', figsize=(12.8,7.2)) as fig:
        ax = fig.add_axes([0,0,1,1], yticks=[], xticks=[], frame_on=False, xlim=bbox[0], ylim=bbox[1])
        im = ax.tripcolor(x[:,0], x[:,1], bezier.tri, p, shading='gouraud', cmap='jet')
        import matplotlib.collections
        ax.add_collection(matplotlib.collections.LineCollection(x[bezier.hull], colors='k', linewidths=.1, alpha=.5))
        ax.quiver(xgrd[:,0], xgrd[:,1], ugrd[:,0], ugrd[:,1], angles='xy', width=1e-3, headwidth=3e3, headlength=5e3, headaxislength=2e3, zorder=9, alpha=.5)
        ax.plot(0, 0, 'k', marker=(3,2,t*rotation*180/numpy.pi-90), markersize=20)
        cax = fig.add_axes([0.9, 0.1, 0.01, 0.8])
        cax.tick_params(labelsize='large')
        fig.colorbar(im, cax=cax)

      if t >= endtime:
        break

      xgrd = nutils.util.regularize(bbox, spacing, xgrd + ugrd * timestep)

  return lhs0, lhs

If the script is executed (as opposed to imported), nutils.cli.run() calls the main function with arguments provided from the command line.

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if __name__ == '__main__':
  nutils.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.

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class test(nutils.testing.TestCase):

  @nutils.testing.requires('matplotlib', 'scipy')
  def test_rot0(self):
    lhs0, lhs = main(nelems=6, reynolds=100, timestep=.1, endtime=.05, rotation=0)
    with self.subTest('initial condition'): self.assertAlmostEqual64(lhs0, '''
      eNqtjD8OwWAcQJ/JNSQ20Tbf135RkUjEZO8RJA7gChYXsDgEkZjN+k/zbQYDCU06Y2Co3yG86S3vtb27
      C8fiXMDM0Q7s7MHCRHUUpPkqh42eaxhlvQzKQAewTMIEQjM2MEyuMUylrOxDykt3Id63Rrzprj0YFJ8T
      7L2vHMlfcqlU6UMrjVJ4+8+gwS3exnUdye8//AB+zDQQ''', atol=2e-13)
    with self.subTest('left-hand side'): self.assertAlmostEqual64(lhs, '''
      eNoB2AAn/5A0jjV/MIDKj8rFMoE4Rjcwz4LI7sery545+Dm5MwTGEsa8NVY8pjtWNSzE18OpyXI9VD02
      M5zCnsJazE0+Hj76NsPByMH/yhA3izOMyGPIyC+gN5Y4JcofyEbI+csJOGk4OzXZxrTGLzIKOXo7Acj2
      xOjENMk3O8Y85DcZwyTDAzjaPFY+sMfJwavBhDNPPvbFV8cxOKk3ADtFOFI86zqjN9o8D8hcNFjCfsXV
      Pd47Vj/qPdZBa0F5QUZD7UEJQYi527zjROVETUeVRfZIfrfvRKZKs7s6SVXLZ9k=''')

  @nutils.testing.requires('matplotlib', 'scipy')
  def test_rot1(self):
    lhs0, lhs = main(nelems=6, reynolds=100, timestep=.1, endtime=.05, rotation=1)
    with self.subTest('initial condition'): self.assertAlmostEqual64(lhs0, '''
      eNqtjD8OwWAcQJ/JNSQ20Tbf135RkUjEZO8RJA7gChYXsDgEkZjN+k/zbQYDCU06Y2Co3yG86S3vtb27
      C8fiXMDM0Q7s7MHCRHUUpPkqh42eaxhlvQzKQAewTMIEQjM2MEyuMUylrOxDykt3Id63Rrzprj0YFJ8T
      7L2vHMlfcqlU6UMrjVJ4+8+gwS3exnUdye8//AB+zDQQ''', atol=2e-13)
    with self.subTest('left-hand side'): self.assertAlmostEqual64(lhs, '''
      eNoB2AAn/4s0kDW8MIHKjcq1MoE4RzdQz4PI7sely545+Dm6MwTGEsa8NVY8pjtWNSzE18OpyXI9VD02
      M5zCnsJazE0+Hj76NsPByMH/yi03ODSmyHbI0jGyN5M4FcoayEHI2MsEOGs4PjXZxrXGXTILOXo7AMj2
      xOfEMsk3O8Y85DcZwyTDAzjaPFY+sMfJwavBhDNPPvTFXMc6OK43/zo7OFI87DqpN9o8Dcg2NFfCgcXX
      Pd87Vj/pPdZBbEF5QUZD7UEIQYe527zjROVETUeVRfZIfrfvRKZKsrs6ScqLaQk=''')