2D-3multidomain_overlapping_test-2.c

Three overlapping cylinders freely rotating in a sheared-flow with DLMFD
Physical parameters
Output and numerical parameters
Results
Annimation

Three overlapping cylinders freely rotating in a sheared-flow with DLMFD

Three cylinders, overlapping each other and twice as heavy as the surrounding fluid, are freely rotating in a linear shear-flow.

# define LEVEL 6 
# include "grid/quadtree.h"
# define DLM_Moving_particle 1
# define TRANSLATION 0
# define ROTATION 1
# define DLM_alpha_coupling 0
# define adaptive 1
# define NPARTICLES 3

# if adaptive
# define MAXLEVEL (LEVEL + 1)
# endif

Physical parameters

# define Uc 1. //caracteristic velocity
# define rhoval 1. // fluid density
# define diam (0.5) // particle diameter
# define ReD  (10) // Reynolds number based on the particle's diameter
# define Ldomain 1.0
# define rhosolid 2.0 //particle density
# define tval (rhoval*Uc*Ldomain/ReD) //fluid's dynamic viscosity

Output and numerical parameters

# define Tc (diam/Uc) // caracteristic time scale
# define mydt (Tc/200) // maximum time-step
# define maxtime (1.0)
# define tsave (Tc/1.)

We include the ficitious-domain implementation

# include "dlmfd.h"
# include "view.h"

double deltau;
scalar un[];

int main() {
  L0 = Ldomain;
  
  /* set time step */
  DT = mydt;
    
  /* initialize grid */
  init_grid(1 << (LEVEL));
  
  /* boundary conditions */
  u.n[left] = dirichlet(-Uc + 2*y*Uc/L0);
  u.t[left] = dirichlet(0);

  u.n[right] = dirichlet(-Uc + 2*y*Uc/L0);
  u.t[right] = dirichlet(0);
	
  u.n[top]   = dirichlet(0);
  u.t[top]   = dirichlet(Uc);

  u.n[bottom] = dirichlet(0);
  u.t[bottom] = dirichlet(-Uc);

   
  /* Convergence criteria */
  TOLERANCE = 1e-3;

  run();
}

We initialize the fluid and particle variables.

event init (i = 0) {

  /* set origin */
  origin (0, 0);
  
  if (!restore (file = "dump")) {
    /* fluid initial condition: */
    foreach() {
      u.x[] = -Uc + 2*y*Uc/L0;
      un[] = u.x[];
    }
    /* particles initial position */
    particle * p = particles;

    for (int k = 0; k < NPARTICLES; k++) {
      GeomParameter gp = {0};

We place the cylinders close to each others to test how the stencils are constructed when they are close to colision.

      if (k == 0) {
	gp.center.x = L0/4;
	gp.center.y = L0/4;
	gp.radius   = diam/2;
      }

      if (k == 1) { 
	gp.center.x = 3*L0/4;
	gp.center.y = L0/4;
	gp.radius   = diam/2;

      }

      if (k == 2) { 
	gp.center.x = L0/2;
	gp.center.y = L0/4;
	gp.radius   = diam/2;

      }
    
      p[k].g = gp;
       
      /* particle's initial angular velocity */
      coord c = {0., 0., 0.};
      p[k].w = c;
    }
  }
  else // Restart of a simulation
    {
     /* the default init=0 event will take of it */
  }
}



event output_data (t += maxtime/50; t < maxtime) {
  stats statsvelox;
  view (fov = 22.3366, quat = {0, 0, 0, 1}, tx = -0.465283, ty = -0.439056, bg = {1,1,1}, width = 890, height = 862, samples = 1);

  scalar constrained_cells[];
  int aa;
  foreach() {
    aa = 0;
    if ((int)index_lambda.y[] > -1 && flagfield[] < 1)
      aa = 1;
    constrained_cells[] = aa;
    constrained_cells[] += 3*flagfield[];
  }
  statsvelox =  statsf (constrained_cells);
  clear();
  squares ("constrained_cells", map = cool_warm, min = statsvelox.min, max = statsvelox.max); 
  cells();
  save ("movie.mp4");
}


event output_data_2 (t += maxtime/50; t < maxtime) {
  stats statsvelox;
  view (fov = 22.3366, quat = {0,0,0,1}, tx = -0.465283, ty = -0.439056, bg = {1,1,1}, width = 890, height = 862, samples = 1);
  statsvelox =  statsf (u.x);
  clear();
  squares ("u.x", map = cool_warm, min = statsvelox.min, max = statsvelox.max); 
  cells();
  save ("movie1.mp4");
}

event output_data_3 (t += maxtime/50; t < maxtime) {
  stats statsvelox;
  view (fov = 22.3366, quat = {0,0,0,1}, tx = -0.465283, ty = -0.439056, bg = {1,1,1}, width = 890, height = 862, samples = 1);
  statsvelox =  statsf (index_lambda.y);
  clear();
  squares ("index_lambda.y", map = cool_warm, min = statsvelox.min, max = statsvelox.max); 
  cells();
  save ("movie2.mp4");
}

Results


set grid
show grid
set xlabel "time"
set ylabel "Omegaz"
plot "particle-data-0" u 1:6 w l, "particle-data-1" u 1:6 w l, "particle-data-2" u 1:6 w l

particles angular velocities (script)

set grid
show grid
set xlabel "time"
set ylabel "Tz"
plot "sum_lambda-0" u 1:3 w l, "sum_lambda-1" u 1:3 w l, "sum_lambda-2" u 1:3 w l

Torque of the particles (script)


set grid
show grid
set xlabel "Time iteration"
set ylabel "Iterations to solve the saddle problem"
plot "converge-uzawa" u 1:2 w l

Number of iterations to solve the saddle problem with the Uzawa method (script)

Annimation

Constrained cells by the Lagrange multipliers

Red cells: constrained by the set of Lagrange multipliers which lie on the surface of the particle.

Light blue cells: constrained by Lagrange multipliers that are inside the particle.