Vortex shedding behind a sphere at Reynolds = 300

    Animation of the \lambda_2 vortices coloured with the vorticity component aligned with the flow.

    We solve the Navier–Stokes equations on an adaptive octree and use embedded boundaries to define the sphere.

    #include "grid/octree.h"
    #include "embed.h"
    #include "navier-stokes/centered.h"
    #include "navier-stokes/perfs.h"
    #include "view.h"

    We will use the \lambda_2 criterion of Jeong and Hussain, 1995 for vortex detection.

    #include "lambda2.h"

    This is the maximum level of refinement i.e. an equivalent maximum resolution of 256^3.

    int maxlevel = 8;

    We need a new field to define the viscosity.

    face vector muv[];

    The domain size is 16^3. We move the origin so that the center of the unit sphere is not too close to boundaries.

    int main()
      init_grid (64);
      size (16.);
      origin (-3, -L0/2., -L0/2.);
      mu = muv;

    The viscosity is just 1/Re, because we chose a sphere of diameter unity and an unit inflow velocity.

    event properties (i++)
        muv.x[] = fm.x[]/300.;

    The boundary conditions are inflow with unit velocity on the left-hand-side and outflow on the right-hand-side.

    u.n[left]  = dirichlet(1.);
    p[left]    = neumann(0.);
    pf[left]   = neumann(0.);
    u.n[right] = neumann(0.);
    p[right]   = dirichlet(0.);
    pf[right]  = dirichlet(0.);

    The boundary condition is no slip on the embedded boundary.

    u.n[embed] = dirichlet(0.);
    u.t[embed] = dirichlet(0.);
    u.r[embed] = dirichlet(0.);
    event init (t = 0) {

    We initially refine only in a sphere, slightly larger than the solid sphere.

      refine (x*x + y*y + z*z < sq(0.6) && level < maxlevel);

    We define the unit sphere.

      vertex scalar phi[];
        phi[] = x*x + y*y + z*z - sq(0.5);
      boundary ({phi});
      fractions (phi, cs, fs);

    We set the initially horizontal velocity to unity everywhere (outside the sphere).

        u.x[] = cs[] ? 1. : 0.;

    We log the number of iterations of the multigrid solver for pressure and viscosity.

    event logfile (i++)
      fprintf (stderr, "%d %g %d %d\n", i, t, mgp.i, mgu.i);

    We use Basilisk view to create the animated isosurface of \lambda_2 for 30 <= t <= 60.

    event movies (t = 30; t += 0.25; t <= 60)

    Here we compute two new fields, \lambda_2 and the vorticity component in the y-z plane.

      scalar l2[], vyz[];
        vyz[] = ((u.y[0,0,1] - u.y[0,0,-1]) - (u.z[0,1] - u.z[0,-1]))/(2.*Delta);
      boundary ({vyz});
      lambda2 (u, l2);
      view (fov = 11.44, quat = {0.072072,0.245086,0.303106,0.918076},
    	tx = -0.307321, ty = 0.22653, bg = {1,1,1},
    	width = 802, height = 634);
      draw_vof ("cs", "fs");
      isosurface ("l2", -0.01, color = "vyz", min = -1, max = 1,
    	      linear = true, map = cool_warm);
      save ("movie.mp4");

    We set an adaptation criterion with an error threshold of 0.02 on all velocity components and 10^{-2} on the geometry.

    event adapt (i++) {
      astats s = adapt_wavelet ({cs,u}, (double[]){1e-2,0.02,0.02,0.02},
    			    maxlevel, 4);
      fprintf (stderr, "# refined %d cells, coarsened %d cells\n",,;