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| #include "filaments.h"
double test_vorticity_filament0(coord pcar, int n_seg, double a, double* t0, coord* c, coord* tvec, coord* nvec, coord* bvec, int period){
/* Compute the local coordinates required for the vorticity field.
Each position P(x,y,z) is projected into the local Frenet-Serret frame to
obtain a set of local coordinates, such that:
i) (P - X . T = 0
ii) (P - X) . N = x_n
ii) (P - X) . B = x_b
This requires finding the value of X(t0) along each space curve that verifies
i) through a minization process.
Then, we use the local coordinates (x_n, x_b) to define a radial coordinate
rho required to compute the vorticity of a Lamb-Oseen vortex as
omega = Gamma/(pi a²) exp(-rho²/a²) . T
where Gamma is the circulation and a the core size.
*/
/* First, we approximate the minimal distance between the point P and each
segment of the curve. */
double dmin = 1e30, tmin=0;
double rho_loc;
for (int i = 0; i < n_seg; i++){
if (vecdist2(pcar, c[i]) < dmin){
dmin = vecdist2(pcar, c[i]);
tmin = t0[i];
}
}
if (period != 0)
tmin = fmod(tmin + period*2*pi, period*2*pi);
// If P is close to the vortex, we refine the initial guess
coord ploc, ccar, frenet[3];
double tq = frenet_projection_min(n_seg, a, t0, c, tvec, nvec, bvec, pcar, tmin);
ccar = gsl_interp1d( n_seg, t0, c, tq);
// Then, compute the local coordinates for the vortex
frenet[0] = gsl_interp1d( n_seg, t0, tvec, tq);
frenet[1] = gsl_interp1d( n_seg, t0, nvec, tq);
frenet[2] = gsl_interp1d( n_seg, t0, bvec, tq);
ploc.x = vecdot(vecdiff(pcar, ccar), frenet[0]);
ploc.y = vecdot(vecdiff(pcar, ccar), frenet[1]);
ploc.z = vecdot(vecdiff(pcar, ccar), frenet[2]);
rho_loc = sqrt(vecdot(ploc, ploc));
return rho_loc;
}
coord test_vorticity_filament1(coord pcar, int n_seg, double a, double* t0, coord* c, coord* tvec, coord* nvec, coord* bvec, int period){
double dmin = 1e30, tmin=0;
for (int i = 0; i < n_seg; i++){
if (vecdist2(pcar, c[i]) < dmin){
dmin = vecdist2(pcar, c[i]);
tmin = t0[i];
}
}
if (period != 0)
tmin = fmod(tmin + period*2*pi, period*2*pi);
// If P is close to the vortex, we refine the initial guess
coord ccar, frenet[1];
double tq = frenet_projection_min(n_seg, a, t0, c, tvec, nvec, bvec, pcar, tmin);
ccar = gsl_interp1d( n_seg, t0, c, tq);
// Then, compute the local coordinates for the vortex
frenet[0] = gsl_interp1d( n_seg, t0, tvec, tq);
return (coord) {frenet[0].x, frenet[0].y, frenet[0].z};
}
coord get_vorticity_filament(coord pcar, int n_seg, double a, double* t0, coord* c, coord* tvec, coord* nvec, coord* bvec, int period){
/* Compute the local coordinates required for the vorticity field.
Each position P(x,y,z) is projected into the local Frenet-Serret frame to
obtain a set of local coordinates, such that:
i) (P - X . T = 0
ii) (P - X) . N = x_n
ii) (P - X) . B = x_b
This requires finding the value of X(t0) along each space curve that verifies
i) through a minization process.
Then, we use the local coordinates (x_n, x_b) to define a radial coordinate
rho required to compute the vorticity of a Lamb-Oseen vortex as
omega = Gamma/(pi a²) exp(-rho²/a²) . T
where Gamma is the circulation and a the core size.
*/
/* First, we approximate the minimal distance between the point P and each
segment of the curve. */
coord omega;
double dmin = 1e30, tmin=0;
double rho_loc, omega_mag;
for (int i = 0; i < n_seg; i++){
if (vecdist2(pcar, c[i]) < dmin){
dmin = vecdist2(pcar, c[i]);
tmin = t0[i];
}
}
if (period != 0)
tmin = fmod(tmin + period*2*pi, period*2*pi);
if (dmin < L0){
// If P is close to the vortex, we refine the initial guess
coord ploc, ccar, frenet[3];
double tq = frenet_projection_min(n_seg, a, t0, c, tvec, nvec, bvec, pcar, tmin);
ccar = gsl_interp1d( n_seg, t0, c, tq);
// Then, compute the local coordinates for the vortex
frenet[0] = gsl_interp1d( n_seg, t0, tvec, tq);
frenet[1] = gsl_interp1d( n_seg, t0, nvec, tq);
frenet[2] = gsl_interp1d( n_seg, t0, bvec, tq);
ploc.x = vecdot(vecdiff(pcar, ccar), frenet[0]);
ploc.y = vecdot(vecdiff(pcar, ccar), frenet[1]);
ploc.z = vecdot(vecdiff(pcar, ccar), frenet[2]);
rho_loc = sqrt(vecdot(ploc, ploc));
// Last, we compute the vorticity for the vortex 1
omega_mag = exp(-sq(rho_loc)/sq(a))/(pi*sq(a));
omega = (coord) {omega_mag * frenet[0].x, omega_mag * frenet[0].y, omega_mag * frenet[0].z};
}
else {
// Otherwise, if the point is too far, we set the vorticity to zero.
foreach_dimension()
omega.x = 0;
}
return omega;
}
coord get_vorticity_filament2(double Gamma, double Uc, coord pcar, int n_seg, double a, double* t0, coord* c, coord* tvec, coord* nvec, coord* bvec, int period){
/* Compute the local coordinates required for the vorticity field.
Each position P(x,y,z) is projected into the local Frenet-Serret frame to
obtain a set of local coordinates, such that:
i) (P - X . T = 0
ii) (P - X) . N = x_n
ii) (P - X) . B = x_b
This requires finding the value of X(t0) along each space curve that verifies
i) through a minization process.
Then, we use the local coordinates (x_n, x_b) to define a radial coordinate
rho required to compute the vorticity of a Lamb-Oseen vortex as
omega = Gamma/(pi a²) exp(-rho²/a²) . T + 2 Uc/a (rho/a) exp(-rho²/a²)
where Gamma is the circulation and a the core size and Uc is the
axial centerline velocity.
*/
/* First, we approximate the minimal distance between the point P and each
segment of the curve. */
coord omega;
double dmin = 1e30, tmin=0;
double rho_loc, omega_mag;
for (int i = 0; i < n_seg; i++){
if (vecdist2(pcar, c[i]) < dmin){
dmin = vecdist2(pcar, c[i]);
tmin = t0[i];
}
}
if (period != 0)
tmin = fmod(tmin + period*2*pi, period*2*pi);
if (dmin < L0){
// If P is close to the vortex, we refine the initial guess
coord ploc, ccar, frenet[3];
double tq = frenet_projection_min(n_seg, a, t0, c, tvec, nvec, bvec, pcar, tmin);
ccar = gsl_interp1d( n_seg, t0, c, tq);
// Then, compute the local coordinates for the vortex
frenet[0] = gsl_interp1d( n_seg, t0, tvec, tq);
frenet[1] = gsl_interp1d( n_seg, t0, nvec, tq);
frenet[2] = gsl_interp1d( n_seg, t0, bvec, tq);
ploc.x = vecdot(vecdiff(pcar, ccar), frenet[0]);
ploc.y = vecdot(vecdiff(pcar, ccar), frenet[1]);
ploc.z = vecdot(vecdiff(pcar, ccar), frenet[2]);
rho_loc = sqrt(vecdot(ploc, ploc));
// Last, we compute the vorticity for the vortex 1
omega_mag = exp(-sq(rho_loc)/sq(a))/(sq(a));
omega.x = omega_mag * (Gamma/pi * frenet[0].x + (2.0 * Uc) * (-ploc.z * frenet[1].x + ploc.y * frenet[2].x));
omega.y = omega_mag * (Gamma/pi * frenet[0].y + (2.0 * Uc) * (-ploc.z * frenet[1].y + ploc.y * frenet[2].y));
omega.z = omega_mag * (Gamma/pi * frenet[0].z + (2.0 * Uc) * (-ploc.z * frenet[1].z + ploc.y * frenet[2].z));
}
else {
// Otherwise, if the point is too far, we set the vorticity to zero.
foreach_dimension()
omega.x = 0;
}
return omega;
}
|
