sandbox/zhenhai/LateralSandglassAngle40.c
Discharge of a silo through a lateral orifice with a bottom inclination
We propose an implementation of the Jop Pouliquen Forterre µ(I) rheology for the flow in a hourglass. We propose to identify the relative role of the acceleration and the friction on the discharge flow by varying the silo geometry (bottom inclination angle), adding the influence of a moderate friction on the front and back wall of the silo, and non-slip boundary condition is chosen for the silo bottom.
Code
Includes and definitions
#include "grid/quadtree.h"
#include "navier-stokes/centered.h"
#include "vof.h"
// Domain extent
#define LDOMAIN 4.
// heap definition
double H0, R0, D, W, tmax, Q, Wmin, muwall, WDOMAIN, Hmask, Lz, angle;
passive fluid small density to preserve 0 pressure and small viscocity
#define RHOF 1e-4
#define mug 1e-5
// Maximum refinement
#define LEVEL 8
//#define MINLEVEL 7
char s[80];
FILE * fpf, *fwq, *fqout;
scalar f[];
scalar * interfaces = { f };
face vector alphav[];
face vector muv[];
scalar rhov[];
Boundary conditions for granular flow, pressure must be zero at the surface. The pressure is zero in the hole x=0 and 0 < y< W, but the lithostatic gradient is given elsewhere on the right wall. No slip boundary conditions on the other walls.
[top] = dirichlet(0);
p.n[top] = neumann(0);
u.t[bottom] = dirichlet(0);
u.n[bottom] = dirichlet(0);
u.n[right] = dirichlet(0);
u.t[right] = dirichlet(0);
u[right] = neumann(0);
f.t[left] = (fabs(y) >= 0. &&
u(y) <= (W + 0.)) ? neumann(0) : dirichlet(0);
fabs.n[left] = (fabs(y) >= 0.&&
u(y) <= (W + 0.)) ? neumann(0) : dirichlet(0);
fabs[left] = (fabs(y) >= 0. &&
p(y) <= (W + 0.)) ? dirichlet(0) : neumann(0);
fabs
int main(int argc, char ** argv) {
= LDOMAIN;
L0 // number of grid points
= 1 << LEVEL;
N // maximum timestep
= 0.005;
DT // coefficient of friction of wall
= 0.1;
muwall = 1e-3;
TOLERANCE //silo height
= 3.84;
H0 = 20.000;
R0 // Grain size
= 1. / 90.;
D // size of the hole
= 0.5;
W = fopen("outWQ", "w");
fwq fclose(fwq);
//film size
= LDOMAIN;
Lz //Silo thickness
= 2.0;
WDOMAIN const face vector g[] = { 0., -1., 0 };
= g;
a = alphav;
alpha = muv;
mu = rhov;
rho //inclinaison angle
= 40.;
angle = 0.;
Q //maximum simulation time
= 20.*0.625 / W;
tmax = fopen("interface.txt", "w");
fpf = fopen("qout.txt", "w");
fqout run();
fclose(fqout);
fclose(fpf);
fprintf(stdout, "\n");
= fopen("outWQ", "a");
fwq fprintf(fwq, " %lf %lf \n", W, Q);
fclose(fwq);
}
initial heap, a rectangle
// note the v
event init(t = 0) {
mask(x > 1.*L0 / 4. ? right : none);
scalar phi[];
foreach_vertex()
[] = min(H0 - y, R0 - x);
phifractions(phi, f);
lithostatic pressure, with a zero pressure near the hole
foreach()
[] = (fabs(y - (W / 2. + 0.)) <= W / 2. &&
p(x) <= .1) ? 0 : max(H0 - y, 0);
fabs}
total density
#define rho(f) ((f) + RHOF*(1. - (f)))
Viscosity computing D_2=D_{ij}D_{ji};
In the pure shear flow D_{11}=D_{22}=0 et D_{12}=D_{21}=(1/2)(\partial u/\partial y), so that D_2=\sqrt{D_{ij}D_{ij}} =\sqrt{ 2 D_{12}^2} = \frac{\partial u}{ \sqrt{2}\partial y}. In a pure shear flow, \partial u/\partial y= \sqrt{2} D_2. The inertial number I is D \sqrt{2} D_2/\sqrt(p) and \mu = \mu_s+ \frac{\Delta \mu}{1+I/I_0} the viscosity is \eta = \mu(I)p/D_2:
note that if \eta is too small an artificial small viscosity \rho D \sqrt{gD} is taken see Lagrée et al. 11 § 2.3.1
event properties(i++) {
({ alphav });
trashscalar eta[];
foreach() {
[] = mug;
etaif (p[] > 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.;
+= sq(dxx) + sq(dxy);
D2 }
if (D2 > 0.) {
= sqrt(2.*D2) / (2.*Delta);
D2 double In = D2*D / sqrt(p[]);
double muI = .4 + .28*In / (.4 + In);
double etamin = sqrt(D*D*D);
[] = max((muI*p[]) / D2, etamin);
eta[] = min(eta[], 100);
eta}
}
}
boundary({ eta });
scalar fa[];
foreach()
[] = (4.*f[] +
fa2.*(f[-1, 0] + f[1, 0] + f[0, -1] + f[0, 1]) +
[1, 1] + f[-1, 1] + f[1, -1] + f[-1, -1]) / 16.;
fboundary({ fa });
foreach_face() {
double fm = (fa[] + fa[-1]) / 2.;
.x[] = (fm*(eta[] + eta[-1]) / 2. + (1. - fm)*mug);
muv.x[] = 1. / rho(fm);
alphav}
foreach()
[] = rho(fa[]);
rhovboundary({ muv, alphav, rhov });
}
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,
(log(mg.resb / mg.resa) / mg.i));
exp}
convergence stats
event logfile(i += 1) {
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);
}
wall friction \frac{du}{dt} = \frac{-2 \mu_w p u}{W ||u||} equation with discretization: \frac{u^{n+1}-u^n}{\delta t} = \frac{-2 \mu_w p u^{n+1}}{W ||u^n||}
event friction(i++) {
vector uold[], um[];
int it = 0;
double errmuw;
foreach() {
foreach_dimension(){
.x[] = u.x[];
uold}
}
do{
++;
itforeach() {
double a = norm(u) == 0 ? HUGE : 1. + 2.*muwall*dt*p[] / (norm(u)*WDOMAIN);
foreach_dimension(){
.x[] = u.x[];
um.x[] = uold.x[] / a;
u}
}
boundary((scalar *){ u });
= 0;
errmuw foreach() {
+= sqrt(sq(u.x[] - um.x[]) + sq(u.y[] - um.y[]));
errmuw }
/= pow(2, 2 * LEVEL);
errmuw }
convergence if |u^{m+1}-u^{m}| < 1e-6 or more 25 iterations
non-slip boundary condition in the inclination bottom area
event unul(i++) {
foreach() {
foreach_dimension(){
.x[] = u.x[] * (y >tan(angle / 180.0*pi)*(x)+0.);
u}
}
}
save some interfaces
event interface (t = 0; t += 1.; t <= tmax) {
#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 }, fp, linear = true);
fclose(fp);
}
Rate of flowing materials across the hole
event debit(t += 0.1) {
static double V = 1;
= 0;
V foreach()
= V + f[] * Delta * Delta;
V if (t >= 0.) fprintf(stdout, "%lf %lf \n", t, V);
fflush(stdout);
fprintf(fqout, "%lf %lf \n", t, V);
}
film output
#if 1
event movie(t += 0.05) {
static FILE * fp1 = popen("ppm2mpeg > level.mpg", "w");
scalar l[];
foreach()
[] = level;
loutput_ppm(l, fp1, min = 0, max = LEVEL,
= 512, box = { { 0, 0 }, { Lz, Lz } });
n
foreach()
[] = f[] * (1 + sqrt(sq(u.x[]) + sq(u.y[])))*(y >tan(angle / 180.0*pi)*(x)+0.);
lboundary({ l });
static FILE * fp2 = popen("ppm2mpeg > velo.mpg", "w");
output_ppm(l, fp2, min = 0, max = 2., linear = true,
= 512, box = { { 0, 0 }, { Lz, Lz } });
n
static FILE * fp3 = popen("ppm2mpeg > f.mpg", "w");
foreach()
[] = f[] * p[];
loutput_ppm(l, fp3, min = 0, linear = true,
= 512, box = { { 0, 0 }, { Lz, Lz } });
n }
event pictures(t == 3) {
output_ppm(f, file = "f.png", min = 0, max = 2, spread = 2, n = 512, linear = true,
box = { { 0, 0 }, { 2, 2 } });
}
#endif
Run
to run
qcc -g -O2 -Wall -o Lateral LateralSandglassAngle40.c -lm
./Lateral > out
~~~
#Bibliography 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/i2014-14005-6
L. 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: 10.1063/1.4757390
Y. Zhou, P. Ruyer, and P. Aussillous, Discharge flow of a bidisperse granular media from a silo: Discrete particle simulations Phys. Rev. E 92, 062204 (2015).
Zhou, Y., Lagree, P. Y., Popinet, S., Ruyer, P., and Aussillous, P. (2017). Experiments on, and ´ discrete and continuum simulations of, the discharge of granular media from silos with a lateral orifice. Journal of Fluid Mechanics, 829:459–485.