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.

Mixing Problem

To understand the mixing properties of colliding flows we focused on non-shearing interactions. The gas density is fixed at 3 particles/cc (free fall time of 20.5 Myr) and the temperature at the equilibrium value of (730 K) giving a sound speed of 3.17 km/s and a Jeans length of 116 pc.

Fixed Parameters
rho	3 particles/cc
T	730 K
cs	3.17 km/s
tff	20.5 Myr
LJ	116 pc

2D Studies

There are 6 runs total, in which we vary the diameter of the flows between 20, 40, and 60 pc as well as the velocity from Mach 1.5 to Mach 3.

Left panel is log rho, right panel is mixing fraction \eta=\frac{2\min{(T_L,T_R)}}{\rho} (The density in the colliding region should equal T_L+T_R). 1 is evenly mixed.

2D Runs
	20 pc	40 pc	60 pc
	movie​	movie​	movie​
1.5
	movie​	movie​	movie​
3
	Reran 20_3 but with a 75 pc box: movie​

Mixing Histograms

	20 pc	40 pc	60 pc
	movie​	movie​	movie​
1.5
	movie​	movie​	movie​
3

Fixed a few bugs and added multipole expansion for phi

Previously only the first particle did any accreting. Successive particles remained massless

Resolution study for 40 pc mach 1.5

Particles are ejected at high velocity. Need to determine why this is happening. #157

movie​

Here's the mixing ratio histogram both volume weighted (red) and mass weighted (blue) for the various resolutions at a constant time (before the first particle forms). Note that the particles form at frame 13 for the lowest resolution run, and then at frames 14, 15, & 16 as the resolution increases. The brighter color the higher the resolution. Notice that the lower resolution has larger bins because there are few data points to sample. In general the number of bins is proportional to the resolution (and not the number of cells as might be expected). This is because the structure that develops and the region containing non-zero mixing ratios is essentially 1D.

movie​

Also looked at 2D PDFs of Density and Pressure as well as Density vs Mixing (Upper left and lower right are mass weighted, while lower left and upper right are volume weighted)

Log Density vs. Log Pressure	Log Density vs. Mixing
movie​	movie​

Embedding clumps

Also looked at embedding clumps in flows that are in pressure equilibrium. Here the flow density sets the clump density and the clump density contrast. For a flow density of 1 part/cc the clumps have to be at 132 part/cc. For a flow density of 3 part/cc the contrast is around 10 so the clumps will be at 30 part/cc.

Given the clump density we can calculate the jeans length for the clumps which effectively sets their maximum gravitational stable size. For a density of 1 part/cc this gives a jeans length of about 11 pc. Setting the clump radius to be < .1 Jeans length ensures that the clumps won't collapse.

And finally the desired mean density of the flow sets the filling fraction of clumps. For example for a flow density of 1 part/cc, a filling fraction of 2% will yield a mean flow density of 3.6.

First run was an flow density of 1 which gives a clump density of 132 and a clump jeans length of about 11 pc. The radius was set to .1 jeans length so the clumps have a radius of 1 pc. And the mean density was set to 3 part/cc so the filling fraction is ~ 2%

IX

movie​

Next we increased the mean density to 3 part/cc so that the density contrast would be 10 instead of ~100. However the mean density had to be increased to 5 to get a reasonable filling fraction of 5%

X

movie​

Here is the same run as above but with the bug fixed and with a clump size of .05 Jeans length

XI

movie​

The little blips of low density material are actually the result of the clump objects exiting the grid. The clumps themselves actually get disrupted before crossing the grid - but currently the clumps step on cells whenever they overlap with the physical boundary… This should be fairly straightforward to fix…

Extending to 3D

First we ran two 40 pc diameter streams at mach 1.5.

Here are the column densities along the axis and normal to the axis

movie​

And here are contour plots of density

movie​

And here is the 2D pdf of density vs. pressure

movie​

And the same for density vs Mixing Ratio

movie​

The lack of dynamics may be due to the initial interface perturbation being two gradual…

Rescaling the wavelengths may allow for more interesting dynamics and less pancaking at the interface.

Live Updates

(These images are the column density projected along the x y and z planes.)

3D Runs
	20 pc	40 pc	60 pc

Column density along flow direction (x)

	​movie	​movie	​movie
1.5
	​movie	​movie	​movie
3

Column density projected along y axis

	20 pc	40 pc	60 pc
	​movie	​movie	​movie
1.5
	​movie	​movie	​movie
3

Column density projected along z axis

	20 pc	40 pc	60 pc
	​movie	​movie	​movie
1.5
	​movie	​movie	​movie
3

Previous Sets of Runs

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

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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​

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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}}

which peaks at about .1

Plotting the same quantity in n vs V space we have

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…

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Miscellaneous

Run Times

Here is an image showing the frame production rate for a 2D 1024² fixed grid vs a 64²+4 colliding flow run.

PhiDot bug fixed

Here are screenshots from before and after the PhiDot fix:

Note the density artifacts at the top of the interface are still present (probably due to the pressure gradients caused by the lack of resolution of gradients in vx).