shear3D_z_yRotation.c

Rotation test y-direction: sphere freely rotating in a simple shear (Stokes) flow with DLMFD
Physical parameters
Output and numerical parameters
Results

Rotation test y-direction: sphere freely rotating in a simple shear (Stokes) flow with DLMFD

We measure the sphere’s rotation rate in the y-direction by imposition a simple shear along the z direction. The velocity field reads \vec{u} = \left(u,v,w\right) = \left(0,0,\dot{\gamma}x \right), with \dot{\gamma} the shear rate.

The surface velocity on the particle satisfies \bm{\Omega}\cdot \vec{x} = \vec{\omega^p}\times \vec{x}, with \bm{\Omega} being the rotation rate tensor, \vec{\omega^p} the (pseudo vector) rotation vector and \vec{x} the position vector of a point on the sphere. Note that \vec{\omega^p} is related to the vorticity \vec{\hat{\omega}} as \vec{\omega^p} = \frac{1}{2}\vec{\hat{\omega}}.

Given \vec{x} = \left(x,y,z\right), a point on the surface of the sphere and \vec{\omega^p} = \left(\omega^p_x,\omega^p_y,\omega^p_z \right), it comes

\displaystyle -\frac{1}{2}\dot\gamma z = \omega^p_y z - \omega^p_z y

\displaystyle \omega^p_z x = \omega^p_x z

\displaystyle \frac{1}{2}\dot\gamma x = \omega^p_x y - \omega^p_y x

for \vec{x} = \left(D/2,0,0\right), it comes \displaystyle -\frac{1}{2}\dot{\gamma} = \omega^p_y and \omega^p_z = 0

for \vec{x} = \left(0,D/2,0\right), it comes \omega^p_z = \omega^p_x = 0

for \vec{x} = \left(0,0,D/2\right), it comes \omega^p_x = 0. which is redundent.

# define LEVEL 4
# include "grid/octree.h"
# define DLM_Moving_particle 1
# define TRANSLATION 0
# define ROTATION 1
# define DLM_alpha_coupling 1

# define NPARTICLES 1

# define adaptive 1
# define MAXLEVEL (LEVEL + 5)

Physical parameters

# define Uc 1. //caracteristic velocity
# define rhoval 1. // fluid density
# define diam (1.) // particle diameter
# define ReD  (0.001) // Reynolds number based on the particle's diameter to setupd the viscosity
# define Ldomain 20.
# define rhosolid 2. //particle density
# define tval (rhoval*Uc*diam/ReD) // fluid dynamical viscosity

Output and numerical parameters

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

We include the ficitious-domain implementation

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

double deltau;
scalar un[];

int main() {
  L0 = Ldomain;

  stokes = true;
  
  /* set time step */
  DT = mydt;
     
  /* initialize grid */
  init_grid(1 << (LEVEL));

  /* boundary conditions */

The shear rate is \dot{\gamma} = 2Uc/L0

  periodic(front);
  periodic(top);
  
  /* right boundary */
  uf.n[right] = dirichlet(0.);
  uf.r[right] = dirichlet(Uc);
  uf.t[right] = dirichlet(0.);

  u.n[right] = dirichlet(0.);
  u.r[right] = dirichlet(Uc);
  u.t[right] = dirichlet(0.);
  p[right] = neumann(0.);
  pf[right] = neumann(0.);
    
  /* left boundary */
  uf.n[left] = dirichlet(0.);
  uf.r[left] = dirichlet(-Uc);
  uf.t[left] = dirichlet(0.);
  u.n[left] = dirichlet(0.);
  u.r[left] = dirichlet(-Uc);
  u.t[left] = dirichlet(0.);
  p[left] = neumann(0.);
  pf[left] = neumann(0.);
  
  
  /* Convergence criteria */
  TOLERANCE = 1e-4;

  run();
}

We initialize the fluid and particle variables.

event init (i = 0) {

  /* set origin */
  origin (0., 0., 0.);
  
  if (!restore (file = "dump")) {
    /* fluid initial condition: */
    foreach() {
      u.z[] = -Uc + 2*x*Uc/L0;
      un[] = u.x[];
    }
    /* initial condition: particles position */
    particle * p = particles;
    
    for (int k = 0; k < NPARTICLES; k++) {
      GeomParameter gp = {0};
    
      p[k].iscube = 0;
      p[k].iswall = 0;
    
      gp.center.x = L0/2.;
      gp.center.y = L0/2.;
      gp.center.z = L0/2.;
      gp.radius   = diam/2.;
   
      p[k].g = gp;

      /* initial condition: particle's velocity */
      coord c = {0., 0., 0.};
      p[k].w = c;
 
    }

  }
  else { // restart the run, the default init event will take care of
	 // it
  }
}

We log the number of iterations of the multigrid solver for pressure and viscosity.

event logfile (i++) {
  deltau = change (u.x, un);
  fprintf (stderr, "log output %d %g %d %d %g %g %g %ld\n", i, t, mgp.i, mgu.i, mgp.resa, mgu.resa, deltau, grid->tn);

}

event output_data (t += 0.01; t < maxtime) {
  stats statsvelox;
  /* scalar omega[]; */
  view (fov = 22.3366, quat = {0,1,0,1}, tx = -0.465283, ty = -0.439056, bg = {1,1,1}, width = 890, height = 862, samples = 1);
  /* vorticity(u, omega); */
  statsvelox =  statsf (u.z);
  clear();
  squares ("u.z", n = {0,1,0}, alpha = L0/2, map = cool_warm, min = statsvelox.min, max = statsvelox.max); 
  cells(n = {0,1,0}, alpha = L0/2);
  save ("movie.mp4");
}

event last_output (t=end){
  p.nodump = false;
  dump("dump_bc");
}

Results

set grid
show grid
plot "particle-data-0" u 1:10 w l title "\omega^p_z", "particle-data-0" u 1:8 w l title "\omega^p_x", 0 title "analytical omega^p_z",0 title "analytical omega^p_x"

$particle’s angular velocities \omega^p_z, \omega^p_x (script)$

particle’s angular velocities \omega^p_z, \omega^p_x (script)

set grid
show grid
Uc = 1.;
L = 20.;
shear = 2.*Uc/L
set yrange [-0.055:-0.025]
plot  "particle-data-0" u 1:9 w l title "\omega^p_y",-shear*0.5 title "analytical omega^p_y"

$particle’s angular velocity \omega^p_y (script)$

particle’s angular velocity \omega^p_y (script)

reset
set grid
show grid
plot "sum_lambda-0" u 1:5 w l title "T_x", "sum_lambda-0" u 1:6 w l title "T_y","sum_lambda-0" u 1:7 w l title "T_z"

Torque on the particle T_x, T_y, T_z (script)