Effect of Richardson number on Kelvin-Helmholtz Instability

A series of runs were performed with the SAM code to investigate the effect
of Richardson number on the morphology of the KH instability. Both 2D and
3D runs have been performed using both uniform stratification and a thermal
duct.

Series 1: 2D, sin*tanh velocity profile, uniform stratification, Re=2500

The runs in this series employ uniform stratification and use the
sine-modulated tanh velocity profile. The Reynolds number is 2500 and the
mesh contains 567 x 1152 points on a 4*pi*h x 8*pi*h box. Each case is seeded
with white noise having an amplitude 1% of the maximum mean velocity (except
where noted). Multiple runs were performed using unique noise fields for
each Richardson number. Although quantitative differences were observed
among the various realizations the basic morphology was consistent from case
to case.

All of the animations for this series use the same color scale for the
vorticity magnitude and only the middle half of the domain extent is shown
(since the bottom and top portions are uninteresting).

Perhaps non-intuitively, the Ri=0.00 case does not produce very high
vorticity values and does not undergo a turbulent-like instability during
the time simulated. The reason for this is that the secondary instability
mechanism for this case is 3D in nature and is thus precluded in the 2D
simulation. The 3D companion case (shown below) leads to small-scale
turbulent breakdown during the same time period simulated here.

The Ri=0.15 case with 1% initial npise is even more pathalogical with the
shear layer not even rolling up during the simulated time. A second case
with 10% initial noise does lead to roll-up however.

Series 2: 2D, sin*tanh velocity profile, uniform stratification, Re=6000

The runs in this series are similar to those in Series 1, but the Reynolds
number has been increased to 6000. The box size and initial conditions
are unchanged, but the number of mesh points was doubled in the Ri=0.00
case, yielding 1152 x 2304 points. It is clear that the increase in Reynolds
number for this case does not lead to the formation of secondary KHI
structures.
The number of mesh points for the Ri=0.15 case remained at 576 x 1152, but
several different values of the initial noise amplitude were considered.
Noise levels of 1 and 3% appear insufficient to induce secondary instability
whereas 4% is sufficient.

Series 3: 3D, sin*tanh velocity profile, uniform stratification, Re=2500

The runs in this series are 3D, employ uniform stratification, and use the
sine-modulated tanh velocity profile. The Reynolds number is 2500 and the
mesh contains 567 x 144 x 1152 points on a 4*pi*h x pi x 8*pi*h box.
Each case is seeded with white noise having an amplitude 1% of the maximum
mean velocity and is run as 2D from time 0 to about time 60 h/U. At this time
the run is restarted as 3D and 1% random noise is again added. The 2D initial
run saves significant computer time during the period where the primary (2D)
instability develops and leads to billow formation.

3D, uniform stratification, Ri=0.00, Re=2500

Series 4: 2D, sin*tanh velocity profile, stability duct

The runs in this series are 2D, use the sine-modulated tanh velocity profile
and contain a stability duct with half-width h. The duct parameters are
the minimum Richardson number (which occurs at the duct center) and the
ratio of N^2 outside the duct to the value at its center. Four different
Reynolds numbers were considered; Re=2500, 4000, 6000, and 10000.
The cases at Re=2500 use 576 x 1152 mesh points, the Re=4000 and 6000
cases use 1152 x 2304 points, and the Re=10000 cases use 1440 x 2880 points.
1% white noise is added to the intial conditions unless otherwise noted.

2D, duct with Ri=0.05, N^2_max/N^2_min=8, Re=2500

2D, duct with Ri=0.05, N^2_max/N^2_min=8, Re=4000

2D, duct with Ri=0.05, N^2_max/N^2_min=8, Re=6000

2D, duct with Ri=0.05, N^2_max/N^2_min=8, Re=10000

2D, duct with Ri=0.10, N^2_max/N^2_min=8, Re=2500

2D, duct with Ri=0.10, N^2_max/N^2_min=8, Re=6000

2D, duct with Ri=0.10, N^2_max/N^2_min=8, Re=10000

Series 5: 3D, sin*tanh velocity profile, stability duct

The runs in this series are 3D, use the sine-modulated tanh velocity profile
and contain a stability duct with half-width h. The duct parameters are
the minimum Richardson number (which occurs at the duct center) and the
ratio of N^2 outside the duct to the value at its center. Simulations
were perfomed for Reynolds numbers 2500 and 4000.
The Re=2500 cases use 576 x 144 x 1152 mesh points, whereas the Re=4000
cases use 1152 x 288 x 2304 points.
1% white noise is added to the intial conditions unless otherwise noted.