test_2vortex_rings1.c

Motion of two thin vortex rings (using the vortex filament framework)
References

Motion of two thin vortex rings (using the vortex filament framework)

In this example, we consider the motion of two vortex rings of radius R and core size a using the vortex filament framework.

The filament approach is perfectly justified as long as the core size remains small compared to other spatial scales (local curvature). We use the same framework as in Durán Venegas & Le Dizès (2019), Castillo-Castellanos, Le Dizès & Durán Venegas (2021), and Castillo-Castellanos & Le Dizès (2019). All the vorticity is considered as being concentrated along space-curves \vec{x}\in\mathcal{C} which move as material lines in the fluid according to: \begin{aligned} \frac{d\vec{x}_c}{dt} = \vec{U}_{ind} + \vec{U}_{\infty} \end{aligned} where \vec{x}_c is the position vector of vortex filament, \vec{U}_{ind} the velocity induced by the vortex filament and \vec{U}_{\infty} an external velocity field. The induced velocity \vec{U}_{ind} is given by the Biot-Savart law.

In this example, we’re going to create a simple solver for the equation above and compute the external field in the Eulerian grid. We’re also going to compare the motion of the vortex rings to the case of a single vortex ring. To this end, we’ll use the translational velocity of the circular vortex ring as \vec{U}_{\infty}. The two vortex rings are characterized by this leapfrogging motion.

Motion of the vortex rings

#include "grid/octree.h"
#include "run.h"
#include "view.h"
#include "acastillo/input_fields/filaments.h"
#include "acastillo/input_fields/draw_filaments.h"
#include "acastillo/filaments/biot-savart.h"

In this example, R_i=1.0, a_i=0.05 and the vortex rings are discretized into n_{seg}=64 filaments. Also, the separation distance d=2.0

int nseg = 64;
double R = 1.0;
double d = 2.0;
double a = 0.05;
double dtmax = 0.005;
struct vortex_filament ring1;
struct vortex_filament ring2;

coord Uinfty = {0., 0., 0.365405};

The main time loop is defined in run.h.

int minlevel = 4;
int maxlevel = 7;
vector u[];
int main(){
  L0 = 10.0;
  X0 = Y0 = Z0 = -L0 / 2;

  periodic(back);
  periodic(top);
  periodic(right);
  
  N = 1 << minlevel;
  init_grid(N);
  run();
}

Initial conditions

We consider two space-curves, \mathcal{C}_i(\xi,t) (i=1,2), parametrized as function of \phi(\xi,t). At time t=0, \begin{aligned} x_i &= R\cos(\phi), \quad \\ y_i &= R\sin(\phi), \quad \\ z_i &= z_0 \pm d/2 \end{aligned}

We also set the core size a_{seg} such that the total vorticity is that of a vortex ring of nominal core size a_{ring}, such that each segment has constant volume \begin{aligned} \pi a_{seg}^2 \ell_{seg} = a_{ring}^2 R \frac{2\pi^2}{n_{seg}} \end{aligned} This also means that vortex stretching with modify the local core size.

The curves will be stored as struct vortex_filament which must be released at the end of the simulation.

event init (t = 0) {
  double dphi = 2*pi/((double)nseg);
  double phi[nseg];
  double a1[nseg];
  double a2[nseg];
  double vol1[nseg];
  double vol2[nseg];
  coord C1[nseg];
  coord C2[nseg];

  // Define a curve 
  for (int i = 0; i < nseg; i++){
    phi[i] = dphi * (double)i;
    C1[i] = (coord) { R * cos(phi[i]), R * sin(phi[i]), -d};
    C2[i] = (coord) { R * cos(phi[i]), R * sin(phi[i]), -2*d};
    
    vol1[i] = pi * sq(a) * R * dphi;
    vol2[i] = pi * sq(a) * R * dphi;
    
    a1[i] = a;
    a2[i] = a;
  }

  // We store the space-curve in a structure 
  allocate_vortex_filament_members(&ring1, nseg);
  memcpy(ring1.phi, phi, nseg * sizeof(double));
  memcpy(ring1.C, C1, nseg * sizeof(coord));
  memcpy(ring1.a, a1, nseg * sizeof(double));  
  memcpy(ring1.vol, vol1, nseg * sizeof(double));

  local_induced_velocity(&ring1, Gamma=1.0);

  allocate_vortex_filament_members(&ring2, nseg);
  memcpy(ring2.phi, phi, nseg * sizeof(double));
  memcpy(ring2.C, C2, nseg * sizeof(coord));
  memcpy(ring2.a, a2, nseg * sizeof(double));  
  memcpy(ring2.vol, vol2, nseg * sizeof(double));
    
  local_induced_velocity(&ring2, Gamma=1.0);

  for (int j = 0; j < nseg; j++) {
    ring1.a[j] = sqrt(ring1.vol[j]/(pi*ring1.s[j]));
    ring2.a[j] = sqrt(ring2.vol[j]/(pi*ring2.s[j]));
  }

  view (camera="iso");  
  draw_tube_along_curve(ring1.nseg, ring1.C, ring1.a);
  draw_tube_along_curve(ring2.nseg, ring2.C, ring2.a);
  save ("prescribed_curve.png");

  FILE *fp = fopen("curve1.txt", "w"); 
  fclose(fp);  

  fp = fopen("curve2.txt", "w"); 
  fclose(fp);

We also initialize the Cartesian grid close to the vortex filaments and initialize the velocity field to zero.

  scalar dmin[];
  for (int i = (maxlevel-minlevel-1); i >= 0; i--){
    foreach(){      
      struct vortex_filament params1;
      params1 = ring1;
      params1.pcar = (coord){x,y,z};
      double dmin1 = get_min_distance(spatial_period=0, max_distance=4*L0, vortex=&params1);

      struct vortex_filament params2;
      params2 = ring2;
      params2.pcar = (coord){x,y,z};
      double dmin2 = get_min_distance(spatial_period=0, max_distance=4*L0, vortex=&params2);

      dmin[] = 0;
      dmin[] = ( max(dmin1, dmin2) < (i+1)*ring1.a[0])*noise();    
    }    
    adapt_wavelet ((scalar*){dmin}, (double[]){1e-12}, maxlevel-i, minlevel);    
  }  
  foreach(){
    foreach_dimension(){
      u.x[] = 0.;
    }
  }
}

We release the vortex filaments at the end of the simulation.

event finalize(t = end){
  free_vortex_filament_members(&ring1);
  free_vortex_filament_members(&ring2);
}

A Simple Time Advancing Scheme

Time integration is done in two steps. First, we evaluate the (self-)induced velocity at \vec{x}_c, then the mutually induced velocities

event evaluate_velocity (i++) {  
  
  memcpy(ring1.Uprev, ring1.U, nseg * sizeof(coord));
  memcpy(ring2.Uprev, ring2.U, nseg * sizeof(coord));

  local_induced_velocity(&ring1, Gamma=1.0);
  local_induced_velocity(&ring2, Gamma=1.0);  
  for (int j = 0; j < nseg; j++) {
    ring1.Uauto[j] = nonlocal_induced_velocity(ring1.C[j], &ring1, Gamma=1.0);
    ring2.Uauto[j] = nonlocal_induced_velocity(ring2.C[j], &ring2, Gamma=1.0);

    ring1.Umutual[j] = nonlocal_induced_velocity(ring1.C[j], &ring2, Gamma=1.0);
    ring2.Umutual[j] = nonlocal_induced_velocity(ring2.C[j], &ring1, Gamma=1.0);
    foreach_dimension(){
      ring1.U[j].x = ring1.Uauto[j].x + ring1.Umutual[j].x + ring1.Ulocal[j].x - Uinfty.x;
      ring2.U[j].x = ring2.Uauto[j].x + ring2.Umutual[j].x + ring2.Ulocal[j].x - Uinfty.x;      
    }
  }
}

Then, we advect the filament segments using an explicit Adams-Bashforth scheme

event advance_filaments (i++) {      
  for (int j = 0; j < nseg; j++) {
    foreach_dimension(){
      ring1.C[j].x += dt*(3.*ring1.U[j].x - ring1.Uprev[j].x)/2.;
      ring2.C[j].x += dt*(3.*ring2.U[j].x - ring2.Uprev[j].x)/2.;
    }
  } 
}

Finally, we compute the new arch-lenghts and update the core sizes to preserve the total vorticity

event advance_filaments (i++, last) {      
  local_induced_velocity(&ring1, Gamma=1.0);  
  local_induced_velocity(&ring2, Gamma=1.0);  
  for (int j = 0; j < nseg; j++) {
    ring1.a[j] = sqrt(ring1.vol[j]/(pi*ring1.s[j]));
    ring2.a[j] = sqrt(ring2.vol[j]/(pi*ring2.s[j]));
  } 
  dt = dtnext (dtmax);
}

Outputs

Displacement of the vortex filaments

event sequence (t += 0.20){
  view(camera="iso", fov=20);
  draw_tube_along_curve(ring1.nseg, ring1.C, ring1.a);  
  draw_tube_along_curve(ring2.nseg, ring2.C, ring2.a);  
  save("2vortex_rings.mp4");
}

The solution outside the vortex tube

Vortex filaments are treated as Lagrangian particles and may exist outside the mesh. We may also evaluate the induced velocity at any point \vec{x}\in\mathbb{R}^3.

Axial velocity induced at the mid-plane taken at regular intervals

event slice (t += 5.0){  
  foreach(){
    coord p = {x,y,z};
    coord u1_BS = nonlocal_induced_velocity(p, &ring1, Gamma=1.0);
    coord u2_BS = nonlocal_induced_velocity(p, &ring2, Gamma=1.0);
    foreach_dimension(){
      u.x[] = u1_BS.x + u2_BS.x - Uinfty.x;
    }
  }
  adapt_wavelet ((scalar*){u}, (double[]){1e-4, 1e-4, 1e-4}, maxlevel, minlevel);

  {
    char filename2[100]; 
    sprintf(filename2, "axial_velocity_%g.png", t);
    
    view(camera="left");  
    squares ("u.z", linear = true, spread = -1, n={1,0,0}, min=-1.00, max=1.00, map = cool_warm); 
    draw_tube_along_curve(ring1.nseg, ring1.C, ring1.a);  
    draw_tube_along_curve(ring2.nseg, ring2.C, ring2.a);  
    save(filename2);  
  }
}

Time evolution

We may also follow the position of \mathcal{C}_i and the induced velocities over time,

event final (t += 0.2, t <= 75.0){
  if (pid() == 0){
    FILE *fp = fopen("curve1.txt", "a"); 
    write_filament_state(fp, &ring1);
    fclose(fp);

    fp = fopen("curve2.txt", "a"); 
    write_filament_state(fp, &ring2);
    fclose(fp);
  }
}

Leapfrogging can be seen by following the axial coordinates. The outer ring becomes slower, while the inner ring shoots forward. Together, the two rings move faster than a single ring.

import numpy as np
import matplotlib.pyplot as plt
	
n = 64
ax = plt.figure()
data1 = np.loadtxt('curve1.txt', delimiter=' ', usecols=[0,16])
data2 = np.loadtxt('curve2.txt', delimiter=' ', usecols=[0,16])
plt.plot(data1[:,0], data1[:,1], label='Vortex 1')
plt.plot(data2[:,0], data2[:,1], label='Vortex 2')

plt.xlabel(r'Time $t$')
plt.ylabel(r'Translational velocity $U_z$')
plt.xlim([0,75])
plt.legend()
plt.tight_layout()
plt.savefig('plot_Uz_vs_t.svg')

Evolution of the axial coordinate (script)

import numpy as np
import matplotlib.pyplot as plt
	
n = 64
ax = plt.figure()
data1 = np.loadtxt('curve1.txt', delimiter=' ', usecols=[0,4])
data2 = np.loadtxt('curve2.txt', delimiter=' ', usecols=[0,4])
plt.plot(data1[:,0], data1[:,1], label='Vortex 1')
plt.plot(data2[:,0], data2[:,1], label='Vortex 2')

plt.xlabel(r'Time $t$')
plt.ylabel(r'Axial coordinate $z$')
plt.xlim([0,75])
plt.legend()
plt.tight_layout()
plt.savefig('plot_z_vs_t.svg')

Leapfrogging is also reflected on the radial coordinate

import numpy as np
import matplotlib.pyplot as plt
	
n = 64
ax = plt.figure()
data1 = np.loadtxt('curve1.txt', delimiter=' ', usecols=[0,2,3])
data2 = np.loadtxt('curve2.txt', delimiter=' ', usecols=[0,2,3])
plt.plot(data1[:,0], np.sqrt(data1[:,1]**2+data1[:,2]**2), label='Vortex 1')
plt.plot(data2[:,0], np.sqrt(data2[:,1]**2+data2[:,2]**2), label='Vortex 2')

plt.xlabel(r'Time $t$')
plt.ylabel(r'Radial coordinate $r$')
plt.xlim([0,75])
plt.legend()
plt.tight_layout()
plt.savefig('plot_r_vs_t.svg')

Evolution of the radial coordinate (script)

Strecthing of the cores is also visible,

import numpy as np
import matplotlib.pyplot as plt
	
n = 64
ax = plt.figure()
data1 = np.loadtxt('curve1.txt', delimiter=' ', usecols=[0,17])
data2 = np.loadtxt('curve2.txt', delimiter=' ', usecols=[0,17])
plt.plot(data1[:,0], data1[:,1], label='Vortex 1')
plt.plot(data2[:,0], data2[:,1], label='Vortex 2')

plt.xlabel(r'Time $t$')
plt.ylabel(r'Core size $a$')
plt.xlim([0,75])
plt.legend()
plt.tight_layout()
plt.savefig('plot_a_vs_t.svg')

References

[castillo2022]	A. Castillo-Castellanos and S. Le Dizès. Closely spaced co-rotating helical vortices: long-wave instability. Journal of Fluid Mechanics, 946:A10, 2022. [ DOI ]
[castillo2021]	A. Castillo-Castellanos, S. Le Dizès, and E. Durán Venegas. Closely spaced corotating helical vortices: General solutions. Phys. Rev. Fluids, 6:114701, Nov 2021. [ DOI \| http ]
[duran2019]	E. Durán Venegas and S. Le Dizès. Generalized helical vortex pairs. Journal of Fluid Mechanics, 865:523–545, 2019. [ DOI ]