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;

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 :

We set the initial velocity field.

    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…

Animation of the vorticity field.

Animation of the vorticity field.

Animation of the tracer field.

Animation of the tracer field.

event movies (i += 4; t <= 15.) {
  static FILE * fp = popen ("ppm2gif > vort.gif", "w");
  scalar vorticity[];
    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);

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

event adapt (i++) {
  adapt_wavelet ({u,f}, (double[]){3e-2,3e-2,3e-2}, 9, 4);

See also