naca2414-starting3D.c

Wingtip vortex of a 3D NACA2414 wing at Re=1000

Wingtip vortex of a 3D NACA2414 wing at Re=1000

This test case is inspired from the Gerris test case wingtip. Wingtip vortices are tubes of circulating air which are left behind a wing as it generates lift. One wingtip vortex trails from the tip of each wing. The cores of vortices spin at very high speed and are regions of very low pressure. Vortex-lines cannot end in fluid, therefore in three dimensions, the starting vortex shed by the trailing edge on take-off remains joined to the wing by counterrotating wingtip vortices, which are associated with the escape of air from the underside to above via the wingtips, following the negative pressure difference from below to above which must be present if the wing is generating lift. This vortex-line consisting of the starting vortex and pair of wingtip vortices is notionally continued and closed by the ‘bound vortex’ inside the wing, associated with the circulation around the wing, and proportional to its lift in accordance with the Kutta–Joukowsky theorem. As the flow over the wing evolves from start-up to steady-state, the starting vortex recedes and the wing-tip vortices extend back to infinity.

We solve here the Navier-Stokes equations and add the NACA2414 using an embedded boundary.

#include "grid/octree.h"
#include "../myembed.h"
#include "../mycentered.h"
#include "../myembed-moving.h"
#include "../myperfs.h"
#include "view.h"
#include "lambda2.h"

Reference solution

#define chord   (1.) // NACA2414 chord length
#define wing    (2.) // Length of the wing
#define p_theta (6.*M_PI/180.) // Incidence of the NACA2414 airfoil
#define Re      (1000.) // Reynolds number based on the cord length
#define uref    (1.) // Reference velocity, uref
#define tref    ((chord)/(uref)) // Reference time, tref

We also define the shape of the domain.

#define naca00xx(x,y,a) (sq (y) - sq (5.*(a)*(0.2969*sqrt   ((x))	\
					      - 0.1260*((x))		\
					      - 0.3516*sq   ((x))	\
					      + 0.2843*cube ((x))	\
					      - 0.1036*pow  ((x), 4.)))) // -0.1015 or -0.1036

void p_shape (scalar c, face vector f, coord p)
{
  // NACA2414 parameters
  double mm = 0.02;
  double pp = 0.4;
  double tt = 0.14;

  // Rotation parameters around the position p,
  // located at the position cc in the airfoil referential
  double theta = (p_theta);
  coord cc = {0.25*(chord), 0., 0.};
  
  vertex scalar phi[];
  foreach_vertex() {

    // Coordinates with respect to the center of rotation of the airfoil p
    // where the head of the airfoil is identified as xrot = 0, yrot = 0
    double xrot = cc.x + (x - p.x)*cos (theta) - (y - p.y)*sin (theta);
    double yrot = cc.y + (x - p.x)*sin (theta) + (y - p.y)*cos (theta);
    double zrot =        (z - p.z);

    if (xrot >= 0. && xrot <= (chord) && fabs (zrot) <= (wing)) {
      
      // Camber line coordinates, adimensional
      double xc = xrot/(chord), yc = yrot/(chord), thetac = 0.;
      if (xc < pp) {
	yc     -= mm/sq (pp)*(2.*pp*xc - sq (xc));
	thetac = atan (2.*mm/sq (pp)*(pp - xc));
      }
      else {
	yc     -= mm/sq (1. - pp)*(1. - 2.*pp + 2.*pp*xc - sq (xc));
	thetac = atan (2.*mm/sq (1. - pp)*(pp - xc));
      }
      
      // Thickness
      phi[] = (naca00xx (xc, yc, tt*cos (thetac)));
    }
    else
      phi[] = 1.;
  }
  boundary ({phi});
  fractions (phi, c, f);
  fractions_cleanup (c, f,
		     smin = 1.e-14, cmin = 1.e-14);
}

Setup

We need a field for viscosity so that the embedded boundary metric can be taken into account.

face vector muv[];

We define the mesh adaptation parameters.

#define lmin (5) // Min mesh refinement level (l=5 is 4pt/c)
#define lmax (9) // Max mesh refinement level (l=9 is 64pt/c)
#define cmax (1.e-2*(uref)) // Absolute refinement criteria for the velocity field

int main ()
{

The domain is 8\times 8\times 8.

  L0 = 8.;
  size (L0);
  origin (-L0/2., -L0/2., -L0/2.);

We set the maximum timestep.

  DT = 1.e-2*(tref);

We set the tolerance of the Poisson solver.

  TOLERANCE    = 1.e-4;
  TOLERANCE_MU = 1.e-4*(uref);

We initialize the grid.

  N = 1 << (lmin);
  init_grid (N);

  run ();
}

Boundary conditions

We give boundary conditions for the face velocity to “potentially” improve the convergence of the multigrid Poisson solver.

uf.n[left]   = 0;
uf.n[right]  = 0;
uf.n[bottom] = 0;
uf.n[top]    = 0;
uf.n[back]   = 0;
uf.n[front]  = 0;

Properties

event properties (i++)
{
  foreach_face()
    muv.x[] = (uref)*(chord)/(Re)*fm.x[];
  boundary ((scalar *) {muv});
}

Initial conditions

event init (i = 0)
{

We set the viscosity field in the event properties.

  mu = muv;

We use “third-order” face flux interpolation.

#if ORDER2
  for (scalar s in {u, p, pf})
    s.third = false;
#else
  for (scalar s in {u, p, pf})
    s.third = true;
#endif // ORDER2

We use a slope-limiter to reduce the errors made in small-cells.

#if SLOPELIMITER
  for (scalar s in {u, p, pf}) {
    s.gradient = minmod2;
  }
#endif // SLOPELIMITER
  
#if TREE

When using TREE and in the presence of embedded boundaries, we should also define the gradient of u at the cell center of cut-cells.

#endif // TREE

We initialize the embedded boundary.

#if TREE

When using TREE, we refine the mesh around the embedded boundary.

  astats ss;
  int ic = 0;
  do {
    ic++;
    p_shape (cs, fs, p_p);
    ss = adapt_wavelet ({cs}, (double[]) {1.e-30},
			maxlevel = (lmax), minlevel = (1));
  } while ((ss.nf || ss.nc) && ic < 100);
#endif // TREE
  
  p_shape (cs, fs, p_p);

We initialize the particle’s speed and accelerating.

  p_u.x  = -(uref);
}

Embedded boundaries

The particle’s position is advanced to time t + \Delta t.

event advection_term (i++)
{
  p_p.x += (p_u.x)*(dt);
}

#if TREE
event adapt (i++)
{
  adapt_wavelet ({cs,u}, (double[]) {1.e-2,(cmax),(cmax),(cmax)},
  		 maxlevel = (lmax), minlevel = (1));

We do not need here to reset the embedded fractions to avoid interpolation errors on the geometry as the is already done when moving the embedded boundaries. It might be necessary to do this however if surface forces are computed around the embedded boundaries.

}
#endif // TREE

Outputs

event logfile (i++; t <= (0.5)*(tref))
{
  coord Fp, Fmu;
  embed_force (p, u, mu, &Fp, &Fmu);

  double CD =  (Fp.x + Fmu.x)/(0.5*sq ((uref))*pi*sq ((chord)/2.));
  double CLy = (Fp.y + Fmu.y)/(0.5*sq ((uref))*pi*sq ((chord)/2.));
  double CLz = (Fp.z + Fmu.z)/(0.5*sq ((uref))*pi*sq ((chord)/2.));

  fprintf (stderr, "%d %g %g %d %d %d %d %d %d %g %g %g %g %g %g %g %g\n",
	   i, (t)/(tref), dt/(tref),
	   mgp.i, mgp.nrelax, mgp.minlevel,
	   mgu.i, mgu.nrelax, mgu.minlevel,
	   mgp.resb, mgp.resa,
	   mgu.resb, mgu.resa,
	   p_p.x/(chord),
	   CD, CLy, CLz);
  fflush (stderr);
}

Animation

event movie (i += 25)
{
  scalar l2[];
  lambda2 (u, l2);

  view (fov = 13.8568,
	quat = {-0.204453,0.521493,0.0481286,0.826999},
	tx = 0.0823552, ty = -0.0249205, bg = {0.3,0.4,0.6},
	width = 600, height = 600, samples = 1);
  
  draw_vof ("cs", "fs");
  isosurface ("l2", -0.4);
  cells (n = {0,0,1}, alpha = 1.e-12);
  save ("l2.mp4", opt = "-r 10");
}

Results

Isosurface \lambda_2 = -0.4