# Bénard–von Kármán Vortex Street for flow around a cylinder at Re=160

An example of 2D viscous flow around a simple solid boundary. Fluid is injected to the left of a channel bounded by solid walls with a slip boundary condition. A passive tracer is injected in the bottom half of the inlet.

Animation of the vorticity field.

Animation of the tracer field.

We use the centered Navier-Stokes solver, with embedded boundaries and advect the passive tracer f.

#include "embed.h"
#include "navier-stokes/centered.h"
// #include "navier-stokes/perfs.h"
#include "tracer.h"

scalar f[];
scalar * tracers = {f};
double Reynolds = 160.;
int maxlevel = 9;
face vector muv[];

The domain is eight units long, centered vertically.

int main() {
L0 = 8.;
origin (-0.5, -L0/2.);
N = 512;
mu = muv;

When using bview we can interactively control the Reynolds number and maximum level of refinement.

  display_control (Reynolds, 10, 1000);
display_control (maxlevel, 6, 12);

run();
}

We set a constant viscosity corresponding to a Reynolds number of 160, based on the cylinder diameter (0.125) and the inflow velocity (1).

event properties (i++)
{
foreach_face()
muv.x[] = fm.x[]*0.125/Reynolds;
}

The fluid is injected on the left boundary with a unit velocity. The tracer is injected in the lower-half of the left boundary. An outflow condition is used on the right boundary.

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

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

The top and bottom walls are free-slip and the cylinder is no-slip.

u.n[embed] = fabs(y) > 0.25 ? neumann(0.) : dirichlet(0.);
u.t[embed] = fabs(y) > 0.25 ? neumann(0.) : dirichlet(0.);

event init (t = 0)
{

The domain is the intersection of a channel of width unity and a circle of diameter 0.125.

  vertex scalar phi[];
foreach_vertex() {
phi[] = intersection (0.5 - y, 0.5 + y);
phi[] = intersection (phi[], sq(x) + sq(y) - sq(0.125/2.));
}
boundary ({phi});
fractions (phi, cs, fs);

We set the initial velocity field.

  foreach()
u.x[] = cs[] ? 1. : 0.;
}

We check the number of iterations of the Poisson and viscous problems.

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

We produce animations of the vorticity and tracer fields…

event movies (i += 4; t <= 15.)
{
scalar omega[], m[];
vorticity (u, omega);
foreach()
m[] = cs[] - 0.5;
boundary ({m});
output_ppm (omega, file = "vort.mp4", box = {{-0.5,-0.5},{7.5,0.5}},
min = -10, max = 10, linear = true, mask = m);
output_ppm (f, file = "f.mp4", box = {{-0.5,-0.5},{7.5,0.5}},
linear = false, min = 0, max = 1, mask = m);
}

We adapt according to the error on the embedded geometry, velocity and tracer fields.

event adapt (i++) {
}