# 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.

We use the centered Navier-Stokes solver and advect the passive tracer f.

``````#include "navier-stokes/centered.h"
#include "tracer.h"

scalar f[];
scalar * tracers = {f};``````

The domain is eight units long, centered vertically.

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

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

``````  const face vector muc[] = {0.00078125,0.00078125};
μ = muc;
run();
}``````

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.);``````

We add a new boundary condition for the cylinder. The tangential velocity on the cylinder is set to zero.

``````bid cylinder;
u.t[cylinder] = dirichlet(0.);

event init (t = 0) {``````

To make a long channel, we set the top boundary for $y>0.5$ and the bottom boundary for $y<-0.5$. The cylinder has a radius of 0.0625.

``````  mask (y >  0.5 ? top :
y < -0.5 ? bottom :
sq(x) + sq(y) < sq(0.0625) ? cylinder :
none);``````

We set the initial velocity field.

``````  foreach()
u.x[] = 1.;
}``````

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.) {
static FILE * fp = popen ("ppm2gif > vort.gif", "w");
scalar vorticity[];
foreach()
vorticity[] = (u.x[0,1] - u.x[0,-1] - u.y[1,0] + u.y[-1,0])/(2.*Δ);
boundary ({vorticity});
output_ppm (vorticity, fp, box = {{-0.5,-0.5},{7.5,0.5}},
min = -10, max = 10, linear = true);
static FILE * fp1 = popen ("ppm2gif > f.gif", "w");
output_ppm (f, fp1, box = {{-0.5,-0.5},{7.5,0.5}},
linear = true, min = 0, max = 1);
}``````

If gfsview is installed on your system you can use this to visualise the simulation as it runs.

``````#if 0
event gfsview (i += 10) {
static FILE * fp = popen ("gfsview2D -s ../karman.gfv", "w");
output_gfs (fp);
}
#endif``````

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

``````event adapt (i++) {