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 Navierâ€“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.Â 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;
run();
}

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

event properties (i++)
{
foreach_face()
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.

  solid (cs, fs, x*x + y*y + z*z - sq(0.5));

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

  foreach()
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[];
foreach()
vyz[] = ((u.y[0,0,1] - u.y[0,0,-1]) - (u.z[0,1] - u.z[0,-1]))/(2.*Delta);
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", s.nf, s.nc);
}