Colliding Flows

This problem involves colliding two ellipsoidal cylinders essentially head on where the separation between the left and right flows is an arbitrary interface defined by a normal and a series of sine waves.

Run Parameters

These runs were all performed on a cube of length 44 pc

Run	Density	Velocity	Temperature	Resolution	Angle	Run Time	Sink Particles
A	1	21	5014	64	10	10.7 Myr	0
B	1	2.1	5014	64	10	100.7 Myr	0
C	10	21	160.6	64	10	10.7 Myr	1
D	10	2.1	160.6	64	10	100.7 Myr	1
E	20	21	100	64	10	10.7 Myr	1
F	4	21	471.9	64	10	10.7 Myr	0

Computational scales follow from a length scale of 1 pc, a Temperature scale of 1 K, and a Density scale of 1 part/cc

TIMESCALE	339631473950335
LSCALE	3.085680300000000E+018
RSCALE	1.672621580000000E-024
VELSCALE	9085.37793659026
PSCALE	1.380650300000000E-016
NSCALE	1.00000000000000
BSCALE	3.314644173409178E-009
TEMPSCALE	1.00000000000000
SCALEGRAV	1.287482066849589E-002

Results

A	B	C	D
movie​	movie​	movie​	movie​
movie​	movie​	movie​	movie​

A	F	C	E
movie​	movie​	movie​	movie​
movie​	movie​	movie​	movie​

Discussion

All of these runs had poorly resolved cooling lengths (fractions of a cell). The fastest growing modes were therefore at the nyquist frequency. This is however much larger than the cooling length of the shocked layer. I suspect that at higher resolutions the size of the condensations from the TI will much smaller and more prone to evaporation (or clump destruction) in the turbulent background flow…

Given the density of the flow and its velocity, we can calculate the shocked materials temperature, cooling length, jeans length, etc…

The cooling time is approximated by the shocked temperature as well as the instantaneous cooling rate at the shocked temperature and density. T_{cool}=\frac{nK_BT_{shock}}{(n\Gamma - n^2 \Lambda(T))(\gamma-1)}

Here we've plotted the cooling time as a function of density and ram pressure (in units of Kelvin/cc)

We can then calculate the cooling length of the shock L_{cool}=T_{cool} v_{shock} or the cooling length of the thermal instability \lambda_{TI}=T_{cool} c_s \approx L_{cool} since c_s \approx v_{shock} Here are plots of the cooling length as well as the thermal instability length scale.

We can also calculate the free fall time for the condensations t_{ff}=\sqrt{\frac{3 \pi}{32 G \rho}} as well as the Jeans length \lambda_J= c_s\sqrt{\frac{\pi}{G\rho}} plotted below

Finally given the density and temperature of the shocked material we can estimate the density contrasts of the thermally unstable clumps \chi=T_{shock}/T_{eq} and then calculate the clump destruction time assuming it is of size \lambda_{TI} embedded in a background flow of velocity v_{shock}. t_{cc}=\frac{\sqrt{\chi} \lambda_{TI}}{v_{shock}}

Combining these two time scales gives a clump survivability \xi=\frac{t_{cc}}{t_{ff}}

Image(nPTICollapsibility.png), width=400 which peaks at about .1

Plotting the same quantity in n vs V space we have

Image(nVTICollapsibility.png), width=400

we can see that optimal parameters are somewhere around a density of 20 and a velocity of 16 km/s although we still need clumps to survive for ~ 10 cloud crushing times before collapsing… Of course if the wind turns off then clumps will be able to survive longer and collapse. It might be better therefore to use finite wind durations…