embed-freeslip.c

Flow over an obstacle

Flow over an obstacle

On this page we embed a free slip mountain into a domain that is filled with an viscous fluid. The free-slip condition is not free of rotation along a curved boundary. Therefore, simple potential flow theory does not apply and we solve the equations for fluid motion.

#include "grid/multigrid.h"
#include "embed.h"
#include "navier-stokes/centered.h"
#include "view.h"
#define BVIEW 1
#include "particles.h"

We set a free-slip boundary for the x and y components.

u.n[embed] = neumann (0);
u.t[embed] = neumann (0);

u.n[left] = dirichlet(min(t, 1.)*fm.x[]);
u.n[right] = neumann(0);
p[right] = dirichlet(0);
pf[right] = dirichlet(0);

face vector muc[];
int main(){
  L0 = 8.;
  for (N = 64; N <= 256; N *= 2){
    init_grid(N);
    run();
  }
}

event properties(i++){
  foreach_face()
    muc.x[] = fm.x[] / 100.;
  boundary((scalar*){muc});
}

We use particles to check if the flow does not penetrate the embedded boundary.

event init (t = 0){
  init_particles_2D_square_grid(5, 1, 0.52, 1);
  mu = muc;
  scalar phi[];
  foreach_vertex()
    phi[] = y - 0.5*exp(-sq(x - L0/2));
  fractions(phi, cs, fs);
  boundary(all);
  output_ppm(cs, n = 512, file = "cs.png", min = 0, max = 1);
}

To help the accurate evaluation of the viscous term, we enforce the time step according to a limit cell-Peclet number.

double PECLET = 0.5;
event stability(i++){
  DT = 1.;
  foreach_face(){
    if (cs[] > 0.005 && fm.x[] > 0.005)
      DT = min(DT, PECLET*sq(Delta)*cs[]/muc.x[]);
  }
}

We generate a movie displaying the flow via u_x and the tracers, these do not penetrate the boundary.

The particles follow the flow along the boundary

event mover(t += 0.05){
  if (N == 128){
    clear();
    view(tx = -0.5, ty = -0.5);
    draw_vof("cs", filled = -1, fc = {1, 1, 1});
    squares("u.x", min = 0, max = 2);
    scatter(loc);
    draw_vof("cs", "fs");
    save("movie.mp4");
  }
}

We also check if Bernoulli’s law is statisfied.

event stop(t = 10){
  char fname[99];
  sprintf(fname, "data%d", N);
  FILE * fp = fopen(fname, "w");
  foreach(){
    if (y < L0/4 && fabs(x - L0/2.) < L0/4 && cm[] > 0.1)
      fprintf(fp, "%g\t%g\n", p[], sq(u.x[]) + sq(u.y[]));
  }

set output 'plot.png'
set term pngcairo
set xlabel 'Modified pressure'
set ylabel 'K'
set size square
set grid
set key outside
plot 'data256' t 'N = 256', 'data128' t 'N = 128', 'data64' t 'N = 64'

There is an anti colleration between K and p … (script)

De deviation from the perfect anti correlation becomes less pronounced when a smaller value for the viscosity is used (not shown). Vorticity may diffuse into the domain along a curved free-slip boundary. As such, we check for the presence of vorticity via the entrophy.

  scalar omega[];
  vorticity(u, omega);
  double Omega = 0;
  foreach()
    Omega += sq(omega[])*sq(Delta)*cm[];
  static FILE * fpo = fopen ("omega", "w");
  fprintf(fpo, "%d\t%g\n", N, Omega);

set output 'plot2.png'
set term pngcairo
set logscale x 2
set xr [32:512]
set xlabel 'N'
set ylabel 'Omega'
set size square
set grid
set key off
plot 'omega' u 1:2 pt 5

There is a small resolution dependence on the diagnosed enstrophy (script)

We check were in the domain the vorticity exists.

  if (N == 128){
    clear();
    view(tx = -0.5, ty = -0.5);
    draw_vof("cs", filled = -1, fc = {1, 1, 1});
    squares("omega");
    draw_vof("cs", "fs");
    save("vorticity.png");
  }
}

We can observe that indeed vorticty diffuses into the domain. Also note that the sign of the boundary curvature dictates the sign of the vorticity that diffuses into the domain.

A wake structure forms at the lee side of the free slip obstacle

A true eye-opener.