siloaxi.c

Bagnold flow in cylindrical silo

Bagnold flow in cylindrical silo

#include "axi.h"
#include "navier-stokes/centered.h"
#define LEVEL 8
double mumax,dg,P0,R0,Q;
scalar mu_eq[];

Bagnold solution for comparison,

double Uba( double r){
    double dmu=.26,mu0=.38,I0=.3;
    //double rgs2=1;//
    double rgs2=1./2;
    double rs=mu0*P0/rgs2;
    double uu = 30*2*I0*((1 - r)*rgs2 + dmu*P0*log(rgs2*(rs-1) + dmu*P0 ) - 
       dmu*P0*log(dmu*P0 + rgs2*(rs-r) ));

    double uup = 30*2*I0*((1 - rs)*rgs2 + dmu*P0*log(rgs2*(rs-1) + dmu*P0 ) - 
       dmu*P0*log(dmu*P0 + rgs2*(rs-rs) ));
     return (r < rs? uup:uu ); }

Main with parameters

 int main() {
  L0 = 1.;

‘Jansen’ pressure, but in fact any pressure gives almost the same ‘Q’

  P0 = 1./2/.38;
  DT = 0.01/2;
  R0 = 0.1 ; 

  // P0 =2;2D

the regularisation value of viscosity

  mumax=1000;

Boundary conditions are periodic

no slip at the top, confinment pressure P_0

    u.t[top] = dirichlet(0);
    u.n[top] = dirichlet(0);
    p[left] =  dirichlet( (t < 1 )? P0 : P0);
    p[right] = (y <= R0 ? dirichlet(0) : neumann(0) );
    u.n[right] =  (y<= R0 ? neumann(0) :dirichlet(0));
    u.t[right] =  (y <= R0 ? neumann(0) :dirichlet(0));
    u.n[left] =  neumann(0);
    u.t[left] = neumann(0);

symmetry at the bottom

    u.n[bottom] = dirichlet(0);
    u.t[bottom] = neumann(0); 
 
  run(); 
}


face vector muv[];

event init (t = 0) {

prepare viscosity

  mu = muv;

equivalent gravity acceleration

    const face vector grav[] = {1,0};

note that in “accceleration” in “navier-stokes/centered.h” there is the fmmetric term in front.

event acceleration (i++,last)

uf.x[] = fm.x[]*(face_value (u.x, 0) + dt*a.x[]);

will be the same for alphav.x[] = fm.x[]/rho(ff); next…

  a = grav;

Initialy at rest

  foreach() {
    u.x[] = 0;
    u.y[] = 0;
    p[]=P0;
  }
}

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);

old value of the velocity is saved

scalar un[];
event init_un (i = 0) {
    foreach()
    un[] = u.x[];
}

so that when it does not more change we are converged

event conv (t += 1; t < 150) {
    double du = change (u.x, un);
    fprintf(stdout,"t= %g %g %g %g \n",t,interpolate (u.x, L0/2, 0),interpolate (p, L0/2, 0),du);
    if (i > 0 && du < 1.0e-6)
        return 1; /* stop */
}

Implementation of the \mu(I) viscosity

event nonnewviscosity(i++) {
    scalar eta_eq[];

computation of the second invariant as defined by Darby -II_2 = 2 D:D and D_2=\sqrt{D:D} \displaystyle 2 D:D = (2 [(\frac{\partial v}{\partial y})^2 + (\frac{ v}{ y})^2) +(\frac{\partial u}{\partial x})^2] + [\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y}]^2) Note that y is r

so viscosity is \displaystyle \eta_{eq} = \mu(I)P/(\sqrt(2.)D2) with regularisation

    scalar shear[];
    foreach()
    shear[] = fabs((u.x[0,1] - u.x[0,-1])/(2.*Delta));
    boundary ({shear});

 foreach() { 
      double mI2 = 0.,D2 = 0,In = 0, muI = 0;
      dg = 1./30;
      double duxx = (u.x[1,0] - u.x[-1,0])/(2 * Delta);
      double duxy = (u.x[0,1] - u.x[0,-1])/(2 * Delta);
      double duyx = (u.y[1,0] - u.y[-1,0])/(2 * Delta);
      double duyy = (u.y[0,1] - u.y[0,-1])/(2 * Delta); 
	    mI2 =  sq(duyx+duxy) + 2*(sq(duyy) + sq(duxx) + sq(u.y[]/ max(y, 1e-20)));
      D2 = sqrt(mI2/2.);
      In = sqrt(2.)*dg*D2/sqrt(fabs(p[])+1e-10);
      //In =  dg*shear[]/sqrt(fabs(P0));
      muI = .38 + (.26)*In/(.3 + In);
      if(D2>0){
        eta_eq[] = min(muI*fabs(p[])/(sqrt(2.)*D2) , mumax );}
       else {
        eta_eq[]=mumax;	
       }
    }   
      boundary ({eta_eq});

 
    boundary ({mu_eq});
    foreach_face() {
        muv.x[] = fm.x[]*(eta_eq[] + eta_eq[-1,0])/2.;
    }
    boundary ((scalar *){muv});
}

Save profiles computed, shear and exact

event profiles (t += 1)
{
    FILE * fp = fopen("xprof", "w");
    scalar shear[];
    foreach()
    shear[] = (u.x[0,1] - u.x[0,-1])/(2.*Delta);
    boundary ({shear});
    for (double y = 0.; y < 1.0; y += 1./pow(2.,LEVEL))
        fprintf (fp, "%g %g %g %g %g  %g \n", y, interpolate (u.x, L0/2, y), interpolate (u.y, L0/2, y),
        	   interpolate (shear, L0/2, y),interpolate (p, L0/2, y),
                  Uba(y));
    fclose (fp);



}

 event interface (  t += .25  ) {
  char s[80];
  //sprintf (s, "field-%g.txt", t);
  sprintf (s, "field.txt");
  FILE * fp = fopen (s, "w");
  output_field ({p,u,uf,pf}, fp, linear = true);
  fclose (fp);
}

We adapt according to the error on the velocity field.

event adapt (i++) {
      Q=0;
    double dy=1./pow(2.,LEVEL);
    for (double y = 0.; y < 1.0; y += dy)
        Q+=interpolate (u.x, L0-dy, y);
    Q=Q*dy;
	 fprintf (stderr," %6.4g  %g \n",t,Q);
	  // adapt_wavelet ({u}, (double[]){3e-3,3e-3}, 8, 6);
}

event profile (t = end) {
  foreach()
    printf ("%g %g %g %g %g\n", x, y, u.x[], u.y[], p[]);
}

Compilation

ln -s siloaxi.c.page siloaxi.c              
make siloaxi.tst;
make siloaxi/plots;
make siloaxi.c.html;

Results and plots

Plot of the exact and computed velocities.

set ylabel "u(y)";set xlabel "y" 
p'field.txt' u ($4+$1*3):2 w l

profiles (script)

reset
set pm3d; set palette rgbformulae 22,13,-31;unset surface;
set ticslevel 0;
unset border;
unset xtics;
unset ytics;
unset ztics;
unset colorbox; 
set view 0,0
sp'field.txt' u 1:2:3 not

(script)

Bibliography

mu(I) and silo and Co