# Friction in saint-venant : Fluvial Test case with Manning

In this example, we reproduce the MacDonald benchmark[1]. We test the friction source term in a fluvial flow. The leading equations can be written as : \displaystyle \begin{array}{cc} \partial_t h + \partial_x q = 0 \\ \partial_t q + \partial_x \left(\frac{q^2}{h}+ \frac{1}{2} g h^2 \right) = -gh (S_0 + S_f) \end{array} S_0 is the slope and S_f is the friction term in its kinematic form.

## Declarations

We call the saint venant solver in 1D and we add the Manning’s friction term as a source term.

#include "grid/cartesian1D.h"
#include "saint-venant.h"
#include "saintvenant/manning.h"

int LEVEL;

scalar e[];
double e1 = 0., e2 = 0., emax = 0.;
int ne = 0;
double pause, tmax, q0 = 1.5, z0, zf, h0, tb = 2000;

// Analytical solution for h and dh/dx
double hex(double x) {
return pow(4/G,1/3.)*(1 + 0.5*exp(-16*pow(-0.5 + x/1000.,2)));
}

double dhex(double x) {
return -0.016*pow(4/G,1/3.)*exp(-16*pow(-0.5 + x/1000.,2))*(-0.5 + x/1000.);
}

// Manning friction term in kinematic formulation
double sf(double x) {
return  -n*n*q0*q0/pow(hex(x),10/3.);
}

// Z and dz/dx
double dzex(double x) {
return (q0*q0/(G*pow(hex(x),3))-1)*dhex(x)+sf(x);
}
double zex(double x, double z){
double dx = L0/N;
return z + dx/4.*(dzex(x-dx)+2*dzex(x-0.5*dx)+dzex(x));
}

## Parameters

Definition of parameters and calling of the saint venant subroutine run().

int main()
{
n=0.0333;
pause=0.2;
L0 = 1000.;
X0 = 0;
G = 9.81;
tmax = 2501;
for( LEVEL = 6; LEVEL <= 10; LEVEL++){
e1 = e2 = emax = 0.;
ne = 0;
N = 1 << LEVEL;
run();
fprintf (stderr, "%d %g %g %g\n", N, e1/ne, sqrt(e2)/ne, emax);
}
}

## Boundary condition

We fix h and u at both boundaries (fluvial case).

h[left] = dirichlet(max(hex(0),0));
eta[left] =  dirichlet(max(hex(0)+zb[],zb[]));
u.n[left] = dirichlet(max(q0/hex(0),0));

h[right] = dirichlet(max(hex(1000),0));
eta[right] =  dirichlet(max(hex(1000)+zb[],zb[]));
u.n[right] = dirichlet(max(q0/hex(1000),0));

## Initial conditions

event init (i = 0)
{
// Because the slope is initially dry, we fix an artificial time-step.
DT = 1e-2;
z0=0;
foreach(){
zb[] = zex(x,z0);
z0=zb[];
u.x[] = 0;
h[]=0;
zf=z0;
}
boundary(all);
}

## Computing error

Noticing tb the time when the flow is already stationary, we define the following relative error norms [2]:

\displaystyle |h|_1 = \frac{\int_{tb}^T \int_0^L |h() - hex()| dx dt}{(T-tb)*L} \displaystyle |h|_2 = \frac{\sqrt{\int_{tb}^T \int_0^L (h() - hex()^2 dx dt}}{(T-tb)*L} \displaystyle |h|_{max} = \frac{\int_{tb}^T max(h() - hex())}{T-tb}

event error (i++; t<=tmax) {
foreach()
e[] = (h[] - hex(x));
norm no = normf (e);
if(t > tb){
e1 += no.avg;
e2 += no.rms*no.rms;
ne++;
if (no.max > emax)
emax = no.max;
}
if( N == 128){
static FILE * fp1 = fopen("ErrorN128.dat","w");
fprintf(fp1,"%g \t %g \t %g \t %g \n",t,no.avg,no.rms,no.max);
}
}

## Gnuplot output

We can use gnuplot to produce an animation of the water surface.

/*
event plot ( t <= tmax; t += 10 ) {
if( N == 32 || N == 128 ){
printf("set title 'Manning friction fluvial ----- t= %.3g , N = %i '\n"
"p[%g:%g][-5:1]  '-' u 1:($2+$4) t'free surface' w p pt 1,"
"'' u 1:4 t'topo' w l lt 4,"
"'' u 1:5 t'Analytical' w l lt -1 \n",t,N,X0,X0+L0);
foreach()
printf (" %g %g %g %g %g %g\n", x, h[], u.x[], zb[],hex(x)+zb[], t);
printf ("e\n"
"pause %.4g \n\n",pause);
}
}
*/

Print the water profile along the channel at final time.

event printprofile ( t = 2500 ){
char name[100];
FILE * fp;
sprintf(name,"profil-%i.dat",N);
fp=fopen(name,"w");
foreach() {
fprintf(fp,"%g  \t %g  \t %g \t %g \t %g   \n"
,x,h[],zb[],hex(x),u.x[]);
}
fclose(fp);
}

## References

[1] I. MacDonald, M. Baines, N. Nichols, and P. G. Samuels, “Analytic Benchmark Solutions for Open-Channel Flows,” no. November, pp. 1041–1045, 1997.

[2] S. Popinet, “Quadtree-adaptive tsunami modelling,” Ocean Dyn., vol. 61, no. January, pp. 1261–1285, 2011.