Vortex shedding behind a sphere at Reynolds = 300

We solve the Navier–Stokes equations on an adaptive octree.

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

We will use the λ2 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 2563.

int maxlevel = 8;

The domain size is 163. We move the origin so that the center of the unit sphere is not too close to boundaries.

int main() {
  L0 = 16.;
  origin (-2, -L0/2., -L0/2.);

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

  const face vector muc[] = {1./300,1./300,1./300};
  μ = muc;

  init_grid (64);

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.);

We define a new, no-slip boundary condition (i.e. Dirichlet condition for the tangential velocity u.t) for the sphere.

bid sphere;
u.t[sphere] = dirichlet(0.);
u.r[sphere] = 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 by masking.

  mask (x*x + y*y + z*z < sq(0.5) ? sphere : none);

We set the initially horizontal velocity to unity everywhere.

    u.x[] = 1.;

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 gfsview and ppm2gif to create the animated isosurface of λ2 for 30<=t<=60.

event movies (t = 30; t += 0.25; t <= 60) {
  static FILE * fp =
    popen ("gfsview-batch3D sphere.gfv | ppm2gif > sphere.gif", "w");

Here we compute two new fields, λ2 and the vorticity component in the yz plane.

  scalar l2[], vyz[];
    vyz[] = ((u.y[0,0,1] - u.y[0,0,-1]) - (u.z[0,1] - u.z[0,-1]))/(2.*Δ);
  boundary ({vyz});
  lambda2 (u, l2);
  output_gfs (fp);

We tell gfsview-batch3D to save the images.

  fprintf (fp, "Save stdout { format = PPM width = 640 height = 480}\n");
Animation of the \lambda_2 vortices coloured with the vorticity component aligned with the flow.

Animation of the λ2 vortices coloured with the vorticity component aligned with the flow.

We set an adaptation criterion with an error threshold of 0.02 on all velocity components.

event adapt (i++) {
  astats s = adapt_wavelet ((scalar *){u}, (double[]){0.02,0.02,0.02},
			    maxlevel, 4);
  fprintf (ferr, "# refined %d cells, coarsened %d cells\n",,;