2dwo.c

An axi-symmetric Womersley flow in a tube
Results N fixe
- Bibliography

An axi-symmetric Womersley flow in a tube

It is almost identical to the code 2dwoflow.c. Here, we study the effects of womersley numbers and compare the results with theoretical results. Mesh adaptation is used by default.

#include "axi.h"
#include "navier-stokes/centered.h"
#include "view.h"
//difine if we want to use mesh adaption
#define ADAPT 1
// some mesh level and maxtime control
#define MIN_LEVEL 3
#define LEVEL 5
#define MAX_LEVEL 7
#define tmax   10*2.*M_PI
FILE * fp1;
FILE * fp2;
FILE * fp3;
int alpha_w;

main fonction

We change the value of womersley number at each simulation, which is set in the array womun[]

int main(){
  periodic (right);
  N = 2 << LEVEL;
  int wonum[] = {3,10,20};
  fp1 = fopen ("testpoint", "w");
  fp2 = fopen ("u.csv", "w");
  fp3 = fopen ("position.csv", "w");
  for(int K= 0 ; K <= 2 ; K++){
    alpha_w = wonum[K];
    fprintf(stderr,"# now womersley number = %d \n", alpha_w );
    run();
  }
}

we set the no slip boundary conditions for lateral wall

u.n[top]  = dirichlet(0);
u.t[top]  = dirichlet(0);

initial event

here we calculate the \mu based on womersley number \alpha, we apply a cosinus force in x direction to manipulate the system periodic in time

event init (t = 0) {
  double viscosity = 1. / sq(alpha_w);
  const face vector muc[] = {viscosity , viscosity};
  mu = muc;
  fprintf(stderr,"Corresponding viscosity: %g\n", viscosity);  
}

Mesh adaptation

We adapt the mesh according to the error on the volume fraction field and the velocity.

#if ADAPT
event adapt (i++) {
  adapt_wavelet((scalar *){u}, (double[]){5e-4,1e-3}, MAX_LEVEL, MIN_LEVEL) ;
}
#endif

Time integration

We record the pressure value of a choosen point in order to check if the result is converged after the simulation (periodic form)

event midpressure(t <= tmax; i += 1) {
  const face vector  g[] = {cos(t) , 0.};
  a = g;
  double px = L0/2.;
  double py = 0.;
  fprintf(fp1,"%d %g %g %g \n" , alpha_w, t, cos(t), interpolate(u.x, px, py));
  if (t == tmax)
    fprintf(fp1,"\n \n"); //to avoid the line between each draw 
}

We output the values of velocity and pressure in the middle position at the last period with time step \delta t = 0.1 T, so we can see the change of velocity profil, which we can compare with analytic results to check The accurecy of simulation.

event tracer (t <= tmax; t += (0.1* 2.*M_PI)) {
  if(t>= tmax - 2.*M_PI){
    for (double y = 0.; y < 1.; y += L0/50. ){
      double vxf = L0/2.;
      fprintf(fp2,"%d %g %g %g \n", alpha_w, t, y, interpolate(u.x , vxf, y));
    }
  }
}

we output the mesh grid and velocity field at tmax

event final(t = tmax){	
  foreach()
    fprintf(fp3,"%d %g %g %g \n",alpha_w, x, y, u.x[] );
}

Results N fixe

plot 'testpoint' u 2:4 w l t'testpointvelocity'

convergence testpoint (script)

plot 'u.csv' u ($1==20?$3:NaN):4 t'Wo-20',\
'thwo20' us 1:2 t'theory20' w l lc 1

Wo 20 compare (script)

plot 'position.csv' u ($1==20?$2:NaN):3 t'Mesh_Wo20'

Wo 20 mesh grid (script)

plot 'u.csv' u ($1==10?$3:NaN):4 t'Wo-10',\
'thwo10' us 1:2 t'theory10' w l lc 1

Wo 10 compare (script)

plot 'position.csv' u ($1==10?$2:NaN):3 t'Mesh_Wo10'

Wo 10 mesh grid (script)

plot 'u.csv' u ($1==3?$3:NaN):4 t'Wo-3',\
'thwo3' us 1:2 t'theory3' w l lc 1

Wo 3 compare (script)

plot 'position.csv' u ($1==3?$2:NaN):3 t'Mesh_Wo3'

Wo 3 mesh grid (script)

Bibliography

Womersley 1955
R. Ghigo Reduced-Order Models for Blood Flow in Networks of Large Arteries