/** # Flow in a Sandglass / Hourglass with a lateral orifice in a silo with height H = 3.9L We propose an implementation of the Jop Pouliquen Forterre µ(I) rheology for the flow in a hourglass. Like the case with an orifice at the bottom. We find that the flow through the lateral orifice alse follows the Beverloo-Hagen discharge law in a pure 2D case. And finally we add the influence of a moderate friction of the front and back wall to simulate a 3D case # 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; /** 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; scalar f[]; scalar * interfaces = {f}; face vector alphav[]; face vector muv[]; /** Boundary conditions for granular flow, pressure must be zero at the surface. The pressure is zero in the hole $x=1$ and $4\deltax < y< W+4\deltax$, but the lithostatic gradient is given elswhere on the right wall. No slip boundary conditions on the other walls. */ p[top] = dirichlet(0); u.n[top] = neumann(0); u.t[bottom] = dirichlet(0); u.n[bottom] = dirichlet(0); u.n[left] = dirichlet(0); u.t[left] = dirichlet(0); f[left]= neumann(0); u.t[right] = (fabs(y)>= 0.1 && fabs(y)<= (W+0.1)) ? neumann(0): dirichlet(0); u.n[right] = (fabs(y)>= 0.1&& fabs(y)<= (W+0.1)) ? neumann(0): dirichlet(0); p[right] = (fabs(y)>= 0.1 && fabs(y)<= (W+0.1)) ? dirichlet(0): neumann(0); /** the three cases in the main */ int main(int argc, char ** argv) { L0 = LDOMAIN; // number of grid points N = 1 << LEVEL; // maximum timestep DT = 0.001; // coefficient of friction of wall muwall=0.1; TOLERANCE = 1e-3; //silo height H0 = 3.9; R0=20.000; // Grain size D=1./90.; // size of the hole W=0.5; fwq = fopen ("outWQ", "w"); fclose(fwq); //film size Lz = LDOMAIN; //Silo thickness WDOMAIN = 0.5; const face vector g[] = {0.,-1.,0}; a = g; alpha = alphav; mu = muv; Q = 0; tmax = 1.; fpf = fopen ("interface.txt", "w"); run(); fclose (fpf); fprintf (stdout,"\n"); fwq = fopen ("outWQ", "a"); fprintf(fwq," %lf %lf \n", W, Q); fclose (fwq); } /** initial heap, a rectangle */ // note the v event init (t = 0) { mask (x < 3.*L0/4. ? left : none); scalar phi[]; foreach_vertex() phi[] = min(H0 - y, R0 - x); fractions (phi, f); /** lithostatic pressure, with a zero pressure near the hole to help */ foreach() p[] = (fabs(y-(W/2.+ 0.1 ))<= W/2. && fabs(x-LDOMAIN)<= .1) ? 0 : max(H0 - y,0) ; } /** 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++) { trash ({alphav}); scalar eta[]; foreach() { eta[] = mug; if (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.; D2 += sq(dxx) + sq(dxy); } if (D2 > 0.) { D2 = sqrt(2.*D2)/(2.*Delta); double In = D2*D/sqrt(p[]); double muI = .4 + .28*In/(.4 + In); double etamin = sqrt(D*D*D); eta[] = max((muI*p[])/D2, etamin); eta[] = min(eta[],100); } } } boundary ({eta}); 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])/2.; muv.x[] = (fm*(eta[] + eta[-1])/2. + (1. - fm)*mug); // mu.x[] = 1./(2.*fm/(eta[] + eta[-1,0]) + (1. - fm)/mug); alphav.x[] = 1./rho(fm); } // boundary_normal ({mu,alpha}); boundary ((scalar *){muv,alphav}); } /** 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)); } /** 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: $$\frac{u^{n+1}-u^n}{\delta t} = \frac{-2 \mu_w p u^{n+1}}{W ||u^n||} $$ */ #if dimension == 2 event friction (i++) { foreach() { double a = norm(u)==0 ? HUGE : 1. + 2.*muwall*dt*p[]/(norm(u)*WDOMAIN); foreach_dimension() u.x[] /= a; } } #endif /** 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; V=0; foreach() V = V + f[]* Delta * Delta; if(t>=0.) fprintf (stdout,"%lf %lf \n",t,V); fflush (stdout); } /** film output */ #if 1 event movie (t += 0.05) { static FILE * fp1 = popen ("ppm2mpeg > level.mpg", "w"); scalar l[]; foreach() l[] = level; output_ppm (l, fp1, min = 0, max = LEVEL, n = 256, box = {{0,0},{Lz,Lz}}); foreach() l[] = f[]*(1+sqrt(sq(u.x[]) + sq(u.y[]))); boundary ({l}); static FILE * fp2 = popen ("ppm2mpeg > velo.mpg", "w"); output_ppm (l, fp2, min = 0, max = 2., linear = true, n = 256, box = {{0,0},{Lz,Lz}}); static FILE * fp3 = popen ("ppm2mpeg > f.mpg", "w"); foreach() l[] = f[]*p[]; output_ppm (l, fp3, min = 0, linear = true, n = 256, box = {{0,0},{Lz,Lz}}); } event pictures (t==3) { output_ppm (f, file = "f.png", min=0, max = 2, spread = 2, n = 256, linear = true, box = {{0,0},{2,2}}); } #endif /** # Run to run ~~~bash qcc -g -O2 -Wall -o LateralUnsym LateralSandglassUnsym.c -lm ./LateralUnsym > out ~~~ Plot of pressure at time 1 ~~~gnuplot pressure 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 xrange[-3:3];set yrange[-3:3]; set view 0,0 sp'./field-1.txt' u 1:2:($1>3. ? $4*$3 :NaN) t 'p' ~~~ Plot of velocity at time 1 ~~~gnuplot velocity 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 xrange[-3:3];set yrange[-3:3]; set view 0,0 sp'field-1.txt' u 1:2:($1>3. ? sqrt($7*$7+$6*$6)*$3 :NaN) t '|v|' ~~~ */ /** #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 */