The runs are performed by solving the Navier-Stokes equations
without pressure and no energy equation but with all viscous
terms in the radial and azimuthal equation included.
We start the evolution imposing the analytic solution
at a viscous time of t=0.016.
The ring is located initially at r=1 and
the inner and outer radii are located at r=0.2 and r=1.8.
The interval is typically covered by 128 radial gridcells.
One dimensional runs
Before we go to the two-dimensional runs we present cases
where only the radial dependence has been taken into account
(only one gridcell in the azimuthal direction).
For a low (dimensionless) viscosity the results for different
viscous times are shown in Fig. 1.
The results agree well with the analytic solution.
For larger viscosity however, one finds instabilities in the flow.
Three different test cases solving the following dimensionless system
of equations
were performed. Note that we use two different viscosities in the radial
and angular momentum equation. The initial conditions are the same as above
and the dimensionless viscosity is 1.0e-5, i.e. ten times larger as
above. The only parameter varied in the three test cases is the magnitude
of the radial viscosity. In the first (model 1) it is switched off, in the
second (Model 2) it is equal to the angular viscosity, and in the third
(model 3) it is 1.5 times larger.
The solid lines represent the analytic solution (at about 47 orbital periods)
which refers to a viscous time of about 0.036, and crosses/dashed-lines
to the numerical solution. Note, that the initial time
of 0.016 viscous timescales refers to about 21 orbital periods for this
viscosity. For each of the runs about 4000 timesteps were used.
It can be seen that without radial viscosity an instability exists which
can be damped by increasing the radial viscosity coefficient. This is
exactly the Viscous Overstability which was described
first by Kato (1978, MN, 185, 629), for a one-dimensional radial disk by
Papaloizou & Stanley (1986, MN, 220, 593), and for a two-dimensional
(r-z) disk by Kley, Papaloizou & Lin (1993, ApJ, 409, 739).
We remark that decreasing the resolution also damps the instability
because the smallest wavelengths show the largest growth rates.