Boundary layer on a rotating disk

Von Kármán, 1921 showed that the steady flow of an incompressible liquid of kinematic viscosity ν induced by an infinite plane disk rotating at angular velocity Ω can be described by a similarity solution. In effect, using ζ=zΩ/ν and setting the axial velocity U, radial velocity V and azimuthal velocity W as U=νΩF(ζ)V=ΩrH(ζ)W=ΩrG(ζ) the Navier-Stokes equations reduce to a couple of ODEs: FFFʺ+Fʹ2/2+2G2=0andGʺFGʹ+GFʹ=0 with boundary conditions F(0)=Fʹ(0)=0G(0)=1.andFʹ()=G()=0. where the prime denotes differentiation with respect to ζ.

To reproduce this solution numerically, we use the axisymmetric Navier–Stokes solver with azimuthal velocity (swirl).

#include "grid/multigrid.h"
#include "axi.h"
#include "navier-stokes/centered.h"
#include "navier-stokes/swirl.h"

The left boundary is the rotating disk with Ω=1 and a no-slip condition for the tangential velocity i.e.

u.t[left] = dirichlet(0);
w[left]   = dirichlet(y);

We use an open (outflow) boundary condition for the right boundary.

u.n[right] = neumann(0);
p[right] = dirichlet(0);
pf[right] = dirichlet(0);

The top boundary condition is more tricky but the following seems to work.

u.n[top] = neumann(0);
p[top] = neumann(0);

We use a constant viscosity but it needs to be weighted by the (axisymmetric) metric.

face vector muv[];

event properties (i++) {
    muv.x[] = 0.2*fm.x[];

The computational domain is 12×12 and we limit the timestep.

int main()
  size (12);
  N = 128;
  μ = muv;
  DT = 2e-2;

We wait until the boundary layer is fully developed and quasi-stationary. We only consider values close to the origin to minimize the influence of boundaries (von Kármán’s solution is valid in an infinite domain).

event end (t = 20)
    if (x*x + y*y < 8)
      fprintf (stderr, "%g %g %.4g %.4g\n", x, y, u.x[], w[]);
Axial F(\zeta) and azimuthal G(\zeta) dimensionless velocity components

Axial F(ζ) and azimuthal G(ζ) dimensionless velocity components



Th V Karman. Über laminare und turbulente reibung. ZAMM-Journal of Applied Mathematics and Mechanics/Zeitschrift für Angewandte Mathematik und Mechanik, 1(4):233-252, 1921.

See also