cylinder-rotating.c

Cylinder rotating in a free-stream flow for Re=200 and \alpha =1

Cylinder rotating in a free-stream flow for Re=200 and \alpha =1

We reproduce here the results of Mittal and Kumar, 2003. The authors studied the flow induced by the rotation of a cylinder in a free-stream flow. Two dimensionless parameters govern this test case:

the Reynolds number Re = u d/\nu;
the non-dimensional rotation rate \alpha = \omega R/u.

We solve here the Navier-Stokes equations and add the cylinder using an embedded boundary for Re=200 and \alpha = 1.

#include "grid/quadtree.h"
#include "../myembed.h"
#include "../mycentered.h"
#include "../myembed-moving.h"
#include "view.h"

Reference solution

#define d    (1.)
#define uref (1.) // Reference velocity, uref
#define tref ((d)/(uref)) // Reference time, tref=d/u
#define Re   (200.) // Particle Reynolds number
#define A    (1.) // Adimensional rotation rate

We also define the shape of the domain.

#define cylinder(x,y) (sq ((x)) + sq ((y)) - sq ((d)/2.))

void p_shape (scalar c, face vector f, coord p)
{
  vertex scalar phi[];
  foreach_vertex()
    phi[] = (cylinder ((x - p.x), (y - p.y)));
  boundary ({phi});
  fractions (phi, c, f);
  fractions_cleanup (c, f,
		     smin = 1.e-14, cmin = 1.e-14);
}

Setup

We need a field for viscosity so that the embedded boundary metric can be taken into account.

face vector muv[];

We define the mesh adaptation parameters.

#define lmin (6) // Min mesh refinement level (l=6 is 2pt/d)
#define lmax (10) // Max mesh refinement level (l=10 is 32pt/d)
#define cmax (1.e-2*(uref)) // Absolute refinement criteria for the velocity field

int main ()
{

The domain is 32\times 32.

  L0 = 32.;
  size (L0);
  origin (-L0/2., -L0/2.);

We set the maximum timestep.

  DT = 1.e-2*(tref);

We set the tolerance of the Poisson solver.

  TOLERANCE    = 1.e-4;
  TOLERANCE_MU = 1.e-4*(uref);

We initialize the grid.

  N = 1 << (lmin);
  init_grid (N);
  
  run();
}

Boundary conditions

We use inlet boundary conditions.

u.n[left] = dirichlet ((uref));
u.t[left] = dirichlet (0);
p[left]   = neumann (0);
pf[left]  = neumann (0);

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

We give boundary conditions for the face velocity to “potentially” improve the convergence of the multigrid Poisson solver.

uf.n[left]   = (uref);
uf.n[bottom] = 0;
uf.n[top]    = 0;

Properties

event properties (i++)
{
  foreach_face()
    muv.x[] = (uref)*(d)/(Re)*fm.x[];
  boundary ((scalar *) {muv});
}

Initial conditions

event init (i = 0)
{

We set the viscosity field in the event properties.

  mu = muv;

We use “third-order” face flux interpolation.

#if ORDER2
  for (scalar s in {u, p, pf})
    s.third = false;
#else
  for (scalar s in {u, p, pf})
    s.third = true;
#endif // ORDER2

We use a slope-limiter to reduce the errors made in small-cells.

#if SLOPELIMITER
  for (scalar s in {u}) {
    s.gradient = minmod2;
  }
#endif // SLOPELIMITER
  
#if TREE

When using TREE and in the presence of embedded boundaries, we should also define the gradient of u at the cell center of cut-cells.

#endif // TREE

We initialize the embedded boundary.

#if TREE

When using TREE, we refine the mesh around the embedded boundary.

  astats ss;
  int ic = 0;
  do {
    ic++;
    p_shape (cs, fs, p_p);
    ss = adapt_wavelet ({cs}, (double[]) {1.e-30},
			maxlevel = (lmax), minlevel = (1));
  } while ((ss.nf || ss.nc) && ic < 100);
#endif // TREE
  
  p_shape (cs, fs, p_p);

We initialize the particle’s rotation speed, in the counter-clockwise direction. In 2D, both component of the rotation represent w_z and should be identical.

  p_w.x = (A)*(uref)/((d)/2.);
  p_w.y = (A)*(uref)/((d)/2.);

We initialize the velocity.

  foreach()
    u.x[] = cs[]*(uref);    
  boundary ((scalar *) {u});  
}

Embedded boundaries

We verify here that the velocity and pressure gradient boundary conditions are correctly computed.

event check (i++)
{
  foreach() {
    if (cs[] > 0. && cs[] < 1.) {

      // Normal pointing from fluid to solid
      coord b, n;
      embed_geometry (point, &b, &n);

      // Velocity
      bool dirichlet;
      double ub;

      ub = u.x.boundary[embed] (point, point, u.x, &dirichlet);
      assert (dirichlet);
      assert (ub +
	      p_w.x*(y + b.y*Delta - p_p.y) == 0.);
      ub = u.y.boundary[embed] (point, point, u.y, &dirichlet);
      assert (dirichlet);
      assert (ub -
	      p_w.y*(x + b.x*Delta - p_p.x) == 0.);

      // Pressure
      bool neumann;
      double pb;

      pb = p.boundary[embed] (point, point, p, &neumann);
      assert (!neumann);
      assert (pb +
	      (rho[])/(cs[] + SEPS)*(-sq (p_w.x)*(x + b.x*Delta - p_p.x)*n.x -
				     sq (p_w.y)*(y + b.y*Delta - p_p.y)*n.y) == 0.);

      // Pressure gradient
      double gb;
      
      gb = g.x.boundary[embed] (point, point, g.x, &dirichlet);
      assert (dirichlet);
      assert (gb - (-sq (p_w.x)*(x + b.x*Delta - p_p.x)) == 0.);
      gb = g.y.boundary[embed] (point, point, g.y, &dirichlet);
      assert (dirichlet);
      assert (gb - (-sq (p_w.y)*(y + b.y*Delta - p_p.y)) == 0.);
    }
  }
}

#if TREE
event adapt (i++)
{
  adapt_wavelet ({cs,u}, (double[]) {1.e-2,(cmax),(cmax)},
  		 maxlevel = (lmax), minlevel = (1));

We do not need here to reset the embedded fractions to avoid interpolation errors on the geometry as the is already done when moving the embedded boundaries. It might be necessary to do this however if surface forces are computed around the embedded boundaries.

}
#endif // TREE

Outputs

event logfile (i++; t <= 24.*(tref))
{
  coord Fp, Fmu;
  embed_force (p, u, mu, &Fp, &Fmu);

  double CD = (Fp.x + Fmu.x)/(0.5*sq ((uref))*(d));
  double CL = (Fp.y + Fmu.y)/(0.5*sq ((uref))*(d));

  fprintf (stderr, "%d %g %g %d %d %d %d %d %d %g %g %g %g %g %g\n",
	   i, t/(tref), dt/(tref),
	   mgp.i, mgp.nrelax, mgp.minlevel,
	   mgu.i, mgu.nrelax, mgu.minlevel,
	   mgp.resb, mgp.resa,
	   mgu.resb, mgu.resa,
	   CD, CL);
  fflush (stderr);
}

Snapshots

event snapshot (t = end)
{
  scalar omega[];
  vorticity (u, omega);
  
  char name2[80];

We first plot the whole box.

  view (fov = 20,
	tx = 0., ty = 0.,
	bg = {1,1,1},
	width = 800, height = 800);

  draw_vof ("cs", "fs", lw = 5);
  cells ();
  sprintf (name2, "mesh-level-%d.png", lmax);
  save (name2);
    
  draw_vof ("cs", "fs", filled = -1, lw = 5);
  squares ("u.x", map = cool_warm);
  sprintf (name2, "ux-level-%d.png", lmax);
  save (name2);

  draw_vof ("cs", "fs", filled = -1, lw = 5);
  squares ("u.y", map = cool_warm);
  sprintf (name2, "uy-level-%d.png", lmax);
  save (name2);

  draw_vof ("cs", "fs", filled = -1, lw = 5);
  squares ("p", map = cool_warm);
  sprintf (name2, "p-level-%d.png", lmax);
  save (name2);

  draw_vof ("cs", "fs", filled = -1, lw = 5);
  squares ("omega", map = cool_warm);
  sprintf (name2, "omega-level-%d.png", lmax);
  save (name2);

We then zoom on the cylinder.

  view (fov = 6,
	tx = -(p_p.x + 6.)/(L0), ty = -(p_p.x + 2.)/(L0),
	bg = {1,1,1},
	width = 800, height = 400);

  draw_vof ("cs", "fs", lw = 5);
  cells ();
  sprintf (name2, "mesh-zoom-level-%d.png", lmax);
  save (name2);

  draw_vof ("cs", "fs", lw = 5);
  squares ("u.x", map = cool_warm);
  sprintf (name2, "ux-zoom-level-%d.png", lmax);
  save (name2);

  draw_vof ("cs", "fs", lw = 5);
  squares ("u.y", map = cool_warm);
  sprintf (name2, "uy-zoom-level-%d.png", lmax);
  save (name2);

  draw_vof ("cs", "fs", lw = 5);
  squares ("p", map = cool_warm);
  sprintf (name2, "p-zoom-level-%d.png", lmax);
  save (name2);

  draw_vof ("cs", "fs", lw = 5);
  squares ("omega", map = cool_warm);
  sprintf (name2, "omega-zoom-level-%d.png", lmax);
  save (name2);
}

Results

We plot the time evolution of the drag and lift coefficients C_D and C_L. We compare the values with those obtained by Chen et al., 1993 and Mittal and Kumar,2003.

set terminal svg font ",16"
set key top right spacing 1.1
set xtics 0,5,50
set ytics -10,1,10
set grid y
set xlabel 't*u/R'
set ylabel 'C_{D}'
set xrange[0:50]
set yrange[0:5]
plot '../data/Chen1993/Chen1993-fig22.csv' u 1:2 w p pt 6 ps 0.75 lc rgb "black" t "Chen et al., 1993, C_D", \
     'log' u ($2/0.5):14 w l lc rgb "blue" t "Basilisk, l=10"

Time evolution of the drag coeffficient C_D (script)

set ylabel 'C_{L}'
set yrange[-5:1]
plot 'log' u ($2/0.5):15 w l lc rgb "blue" t "Basilisk, l=10"

Time evolution of the lift coeffficient C_L (script)

set key bottom right
set xtics -50,0.5,50
set ytics -100,5,100
set xlabel 'C_{D}'
set ylabel 'C_{L}'
set xrange[-2:1.5]
set yrange[-28:2]
plot 'log' u 14:15 w l lc rgb "blue" t "Basilisk, l=10"

Phase diagram of the drag and lift coeffficients (script)

References

[mittal2003]	S. Mittal and B. Kumar. Flow past a rotating cylinder. Journal of Fluid Mechanics, 476:303–334, 2003.
[chen1993]	Y.-M Chen, Y.-R. Ou, and A. J. Pearlstein. Development of the wake behind a circular cylinder impulsively started into rotatory and rectilinear motion. Journal of Fluid Mechanics, 253:449–484, 1993.