# src/examples/sphere.c

# Vortex shedding behind a sphere at Reynolds = 300

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.);
μ = 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.

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

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.*Δ);
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 (ferr, "# refined %d cells, coarsened %d cells\n", s.nf, s.nc);
}
```