sandbox/yixian/granular_bottom_NGF.c
Flow in a Sandglass / Hourglass / Silo with a bottom orifice
We propose an implementation of the non-local granular fluidity (NGF) model proposed by Kamrin and Koval (2012) for the flow in a silo with a bottom orifice.
Code
Includes and definitions
#include "grid/multigrid.h"
#include "navier-stokes/centered.h"
#include "vof.h"
#include "poisson.h"
#define LDOMAIN 1.
double H0,R0,D,W,tmax,zetaA;
#define RHOF 1e-4
#define mug 1e-5
#define LEVEL 6
char s[80];
FILE * fpf,*fwq;
scalar f[];
scalar * interfaces = {f};
face vector alphav[];
face vector muv[];
scalar rhov[];
scalar duxx[],duxy[],duyx[],duyy[],dpdx[],dpdy[];
/* Nonlocal fields and params */
scalar gloc[]; /* local fluidity */
scalar gnonloc[]; /* nonlocal fluidity */
scalar muIloc[]; /* local mu(I) */
scalar muI[];
scalar eta[];
scalar zeta_loc[];
scalar lambda[],rhs[];
face vector alphaf[];
double mus = 0.4; //
double mu2 = 0.68; //
double I0 = 0.4; //
double t_switch = 0.1;
// ==== Boundary conditions for g ====
gnonloc[top] = neumann(0);
gnonloc[left] = dirichlet(0.);
gnonloc[right] = dirichlet(0.);
gnonloc[bottom] = fabs(x-LDOMAIN/2)<= W/2 ? neumann(0): dirichlet(0);Boundary conditions for granular flow, pressure must be zero at the surface. The pressure is zero in the hole No slip boundary conditions on the other walls.
p[top] = dirichlet(0);
pf[top] = dirichlet(0);
u.n[top] = neumann(0);
u.t[bottom] = fabs(x-LDOMAIN/2)<= W/2 ? neumann(0): dirichlet(0);
u.n[bottom] = fabs(x-LDOMAIN/2)<= W/2 ? neumann(0): dirichlet(0);
p[bottom] = fabs(x-LDOMAIN/2)<= W/2 ? dirichlet(0): neumann(0);
pf[bottom] = fabs(x-LDOMAIN/2)<= W/2 ? dirichlet(0): neumann(0);
u.n[right] = dirichlet(0);
u.n[left] = dirichlet(0);
u.t[right] = dirichlet(0);
u.t[left] = dirichlet(0);
int main(int argc, char ** argv) {
disable_fpe (FE_DIVBYZERO|FE_INVALID);
L0 = LDOMAIN;
N = 1 << LEVEL;
TOLERANCE = 0.001;
H0 = 0.9;
R0 = 20.000;
const face vector g[] = {0.,-1.,0};
a = g;
alpha = alphav;
mu = muv;
rho = rhov;
DT = 0.001;
tmax = 4.;
W = 0.25;
zetaA = 0.5;
D = 0.04375;
fpf = fopen ("interface.txt", "w");
run();
fclose (fpf);
fprintf (stdout,"\n");
return 0;
}
/* initial heap, a rectangle */
event init (t = 0) {
scalar phi[];
foreach_vertex()
phi[] = min(H0 - y, R0 - x);
fractions (phi, f);
foreach() {
p[] = (fabs(x-LDOMAIN/2)<= W/2 && fabs(y)<= .1) ? 0 : max(H0 - y,0) ;
gloc[] = 0.;
gnonloc[] = 0.;
muIloc[] = 0.;
muI[] = 0.;
}
boundary({gloc, gnonloc, muIloc, muI});
}
#define rho(f) ((f) + RHOF*(1. - (f)))We solve for each time step the following additional set of equations using values obtained from the previous local formulation
g_{\mathrm{loc}} = \frac{|\dot{\gamma} |}{ \mu_{\mathrm{loc}}(I)} -[\xi(\mu_\mathrm{loc})]^2 \nabla^{2}g +g = g_{\mathrm{loc}} \mu =\frac{ |\dot{\gamma} | }{ g} \eta_{\mathrm{eff}} = \frac{\mu p}{ |\dot{\gamma} |}
where \xi(\mu) is the stress-dependent cooperativity length given by:
xi (\mu) = A\sqrt{\frac{\mu_2 - \mu}{\Delta\mu(\mu - \mu_s)}} d
with A a dimensionless non-local amplitude that quantifies the spatial extent of cooperativity in the flow. To avoid division by zero when computing the friction coefficient from the fluidity, we add a small regularisation term \varepsilon = 10^{-16} to the denominator, i.e., \mu = |\dot{\gamma}| / (g + \varepsilon).
event properties (i++) {
trash ({alphav});
trash ((scalar *){alphaf});
foreach() {
muIloc[] = 0.;
gloc[] = 0.;
rhs[] = 0.;
lambda[] = 1.0;
zeta_loc[] = 0.;
if (p[] > 0. && f[] > 0.) {
double D2 = 0.;
foreach_dimension() {
double dxx = u.x[1,0] - u.x[-1,0];
double dxy = (u.x[0,1] - u.x[0,-1] + u.y[1,0] - u.y[-1,0]) / 2.;
D2 += sq(dxx) + sq(dxy);
}
if (D2 > 0.) {
D2 = sqrt(D2) / (2.0 * Delta);
double sD2 = sqrt(2.) * D2;
double p_eff = p[];
double In_loc = sD2 * D / sqrt(p_eff);
double Delta_mu = mu2 - mus;
double mu_loc = mus + Delta_mu * In_loc / (I0 + In_loc);
muIloc[] = mu_loc;
gloc[] = sD2 / mu_loc;
double num = mu2 - mu_loc;
double den = Delta_mu * (mu_loc - mus);
double xi = zetaA * D * sqrt(num / den);
zeta_loc[] = xi;
}
}
rhs[] = gloc[];
}
boundary ({muIloc, gloc, rhs, lambda, zeta_loc});
if (t < t_switch) {
foreach() {
eta[] = mug;
muI[] = 0.;
if (p[] > 0. && f[] > 0.) {
double D2 = 0.;
foreach_dimension() {
double dxx = u.x[1,0] - u.x[-1,0];
double dxy = (u.x[0,1] - u.x[0,-1] + u.y[1,0] - u.y[-1,0]) / 2.;
D2 += sq(dxx) + sq(dxy);
}
if (D2 > 0.) {
D2 = sqrt(D2) / (2.0 * Delta);
double sD2 = sqrt(2.) * D2;
double muI_val = muIloc[];
muI[] = muI_val;
double etamin = sqrt(D*D*D);
eta[] = max((muI_val * p[]) / sD2, etamin);
eta[] = min(eta[], 100.);
}else {
eta[] = 100.;
muI[] = 0.;
}
}
else {
eta[] = mug;
muI[] = 0.;
}
}
boundary ({eta, muI});
}
else {
foreach_face(x) {
double zf = 0.5*(zeta_loc[] + zeta_loc[-1,0]);
alphaf.x[] = -sq(zf);
}
foreach_face(y) {
double zg = 0.5*(zeta_loc[] + zeta_loc[0,-1]);
alphaf.y[] = -sq(zg);
}
boundary ((scalar *){alphaf});
mgstats mg = poisson (gnonloc, rhs, alphaf, lambda);
if (mg.i > 0 && mg.resa > 0.)
fprintf(stderr, "poisson mg: iter=%d resb=%g resa=%g\n", mg.i, mg.resb, mg.resa);
foreach() {
if (gnonloc[] < 0.) {
fprintf(stderr, "Non-physical gnonloc < 0 at (x=%g,y=%g): %g\n", x, y, gnonloc[]);
gnonloc[] = 0.;
}
}
boundary ({gnonloc});
foreach() {
eta[] = mug;
muI[] = 0.;
if (p[] > 0. && f[] > 0.) {
double D2 = 0.;
foreach_dimension() {
double dxx = u.x[1,0] - u.x[-1,0];
double dxy = (u.x[0,1] - u.x[0,-1] + u.y[1,0] - u.y[-1,0]) / 2.;
D2 += sq(dxx) + sq(dxy);
}
if (D2 > 0.) {
D2 = sqrt(D2) / (2.0 * Delta);
double sD2 = sqrt(2.) * D2;
double g_eff = gnonloc[];
double muI_val = sD2 / (g_eff + 1e-16);
muI[] = muI_val;
double etamin = sqrt(D*D*D);
eta[] = max((muI_val * p[]) / sD2, etamin);
eta[] = min(eta[], 100.);
}
else {
eta[] = 100.;
gnonloc[] = 0.;
muI[] = 0.;
}
}
else {
eta[] = mug;
gnonloc[] = 0.;
muI[] = 0.;
}
}
boundary ({eta, gnonloc, muI});
}
scalar fa[];
foreach()
fa[] = (4.*f[] +
2.*(f[-1,0] + f[1,0] + f[0,-1] + f[0,1]) +
f[1,1] + f[-1,1] + f[1,-1] + f[-1,-1]) / 16.;
boundary ({fa});
foreach_face() {
double fm = (fa[] + fa[-1,0]) / 2.;
muv.x[] = (fm * (eta[] + eta[-1,0]) / 2. + (1. - fm) * mug);
alphav.x[] = 1.0 / rho(fm);
}
foreach()
rhov[] = rho(fa[]);
boundary ({muv, alphav, rhov});
foreach(){
duxy[] = (u.x[0,1] - u.x[0,-1])/2./Delta + (u.y[1,0] - u.y[-1,0])/2./Delta ;
duyx[] = (u.y[1,0] - u.y[-1,0])/2./Delta + (u.x[0,1] - u.x[0,-1])/2./Delta ;
duxx[] = 2 * (u.x[1,0] - u.x[-1,0])/2./Delta ;
duyy[] = 2 * (u.y[0,1] - u.y[0,-1])/2./Delta;
dpdx[] = (p[1,0] - p[-1,0])/2./Delta;
dpdy[] = (p[0,1] - p[0,-1])/2./Delta;
}
}
/* convergence outputs */
void mg_print (mgstats mg)
{
if (mg.i > 0 && mg.resa > 0.)
fprintf (stderr, "# %d %g %g %g\n", mg.i, mg.resb, mg.resa,
exp (log (mg.resb/mg.resa)/mg.i));
}
event logfile (i++) {
stats s = statsf (f);
fprintf (stderr, "%g %d %g %g %g %g\n", t, i, dt, s.sum, s.min, s.max - 1.);
mg_print (mgp);
mg_print (mgpf);
mg_print (mgu);
fflush (stderr);
}
/* output */
event interface (t = 2. ; t+=1. ; t<= 4.) {
#if dimension == 2
output_facets (f, fpf);
#endif
char s[80];
sprintf (s, "field-%g.txt", t);
FILE * fp = fopen (s, "w");
output_field ({f,p,u,uf,pf,muIloc,muI,gloc,gnonloc,eta,duxx,duxy,duyx,duyy,dpdx,dpdy}, fp, linear = true);
fclose (fp);
}Rate of flowing materials across the hole
event debit (t += 0.05; t <= tmax ) {
static double Vold,V=1,Qinst=0;
Vold=V;
V=0;
double dx = L0 / N;
foreach()
V = V + f[] * dx * dx;
Qinst = -(V-Vold)/0.05;
if(t>=.1) fprintf (stdout,"%lf %lf %lf \n",t,V,Qinst);
fflush (stdout);
}
/* End of file */to run
qcc -g -O2 -o granular_bottom_NGF granular_bottom_NGF.c -lm
./granular_bottom_NGF > volplots
reset
set pm3d map
set pm3d; set palette rgbformulae 22,13,-31;
splot 'field-2.txt' u 1:2:(abs($6)*$3) with pm3d notitle
set label "Vy" at graph 0.45, 0.85 font ",24" front 
set terminal postscript color enhanced set output “t2vy.eps” replot
Plot of pressure at time 2
reset
set pm3d map
set pm3d; set palette rgbformulae 22,13,-31;
splot 'field-2.txt' u 1:2:(abs($4)**$3) with pm3d notitle
set label "pressure" at graph 0.45, 0.85 font ",24" front
set terminal postscript color enhanced
set output “t2p.eps” replot
Plot of viscosity at time 2
reset
set cbrange [0:2]
set pm3d map
set pm3d; set palette rgbformulae 22,13,-31;
set label "Viscosity" at graph 0.45, 0.85 font ",24" front
splot'field-2.txt' u 1:2:(abs($14)**$3) with pm3d notitle
set terminal postscript color enhanced set output “t2eta.eps” replot
Links
https://basilisk.fr/sandbox/yixian/granular_bottom_NGF.c
https://basilisk.fr/sandbox/yixian/granular_bottom_constantNGF.c
https://basilisk.fr/sandbox/yixian/granular_bottom_dynamicNGF.c
https://basilisk.fr/sandbox/yixian/granular_bottom_linearisedconstantNGF.c
https://basilisk.fr/sandbox/yixian/granular_bottom_linearisedNGF.c
https://basilisk.fr/sandbox/yixian/granular_bottom_Local.c
https://basilisk.fr/sandbox/yixian/granular_bottom_muItheta.c
https://basilisk.fr/sandbox/yixian/granular_bottom_Igradient.c
Bibliography
Y. Zhou PhD https://theses.fr/2016AIXM4731
L. Staron, P.-Y. Lagrée, & S. Popinet (2014)
“Continuum simulation of the discharge of the granular silo, A validation test for the μ(I) visco-plastic flow law”
Eur. Phys. J. E (2014) 37: 5 DOI 10.1140/epje/i2014-14005-6L. Staron, P.-Y. Lagrée & S. Popinet (2012)
“The granular silo as a continuum plastic flow: the hour-glass vs the clepsydra”
Phys. Fluids 24, 103301 (2012); doi: 10.1063/1.4757390Kamrin, K. and Koval, G. (2012). Nonlocal constitutive relation for steady granular flow. Phys. Rev. Lett., 108:178301.
