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 with optionally embedded clumps

Fixed Parameters
average density of flows	1.0 particles/cc
vflow	8.25 km/s
Mass flux	523 Msun/Myr
Ram pressure	8245 particles K/cc
Ram density	271.5 particles/cc
Ram temp	30.4 K
Ram Jeans Length	6.7 pc
Flow diameter	40 pc
Xmu	1.0
gamma	5/3

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Uniform density flow
rho	1.0 particles/cc
T	5013 K
cs	6.5 km/s
tff	51 Myr
LJ	1427 pc

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Clumpy flow
	Ambient	Clumps
rho	0.25 particles/cc	15.1825 particles/cc
T	6857 K	112 K
cs	7.5 km/s	.964 km/s
tff	102 Myr	13.2 Myr
LJ	3339 pc	55 pc
radius	N/A	.55 pc

Here are the results from the first round of runs using the old cooling curve. Most statistics are from hockey puck region centered over the flow with a 40 pc diameter, and a 20 pc width. Also the column density looking along the flow direction is showing only contributions from matter inside of this hockey puck. This avoids confusion in the clumpy runs due to the large number of clumps between the edge of the domain and the interaction region…

Dashed lines are for clumpy flow, smooth lines for uniform flow

The clumpy flow initially has no clumps within a few pc of the interface - so the mass is only about 1 quarter of the the smooth run initially. Overall materially is getting splashed vertically out of the region - but in the clumpy flow I think some additional material escapes out the other side of the hockey puck region at about 2 Myr, but then eventually gets brought back in…

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The Mach numbers for the flows are approximately sonic - but the thermal energy ends up being 3x the kinetic for a M=1 flow. The initial thermal energy density is higher for the smooth flow because the material is at 1 part/cc at the peak of the equilibrium curve - while for the clumpy flow, the material (both the ambient and the clumps) are at a lower pressure. Also plotted are the theoretical growth curves for the thermal energies and the kinetic energy (just looking at the fluxes into the region - ie assuming no conversion of energies). Overall the thermal energies behave similarly - though the unperturbed material flowing in has a thermal energy density that is 2.92 higher. The Kinetic energy of the smooth run also grows for a very short time before quickly beginning to dissipate in what I imagine is a reflected shock working it's way outward before the material has a chance to cool and this shock collapses. In the movies of the smooth flows - you can faintly see this reverse shock until about 2 Myr when the interior has cooled into dense structures and the support falls out… This doesn't happen in the clumpy flow run since most of the kinetic energy is in the dense clumps which plow through the opposing flow…

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Here is also another curve showing the longer time development when the flows are shut-off. The clumpy flow in this plot was generated using a wider grid - and the resolution was slightly different… and the largest effect would be on the gravitational energy since the gravitational well is deeper because of the additional material surrounding the hockey puck… But you do see the rapid dissipation of kinetic energy and thermal energy and the stalling of the mass influx (since the gravitational energy flattens) but then you get very rapid collapse. Then the gravitational energy of the gas appears to lessen but that is because material is getting accreted into sink particles…

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Movie from side ColumnDensity.gif​ ColumnDensity.AVI​

Movie looking down barrel of hockey-puck region ColumnDensityAlong1.gif​ ColumnDensityAlong1.AVI​

Movie of density-pressure distribution pdfs.gif​ pdfs.AVI​

Movie of MixingRatioHistogram MixingRatio.gif​ MixingRatio.AVI​

Move of density histogram DensityHistogram.gif​ DensityHistogram.AVI​

The smooth Run is completed out to 30 Myr and the clumpy run should be finished soon. The plan is then to restart these runs at 10 or 20 Myr but with the flow shut off to see if the lack of kinetic energy injection leads to collapse and star formation…

Modified Cooling Params

We also decided to modify the cooling curve to favor core formation (see johannjc04082012)

Also here is a plot showing the equilibrium cooling curve along with the isotherms at 10 and 10,000 K. Also are shown are lines of constant jeans length. The thermally unstable gas needs to be collected in structures that are a few pc in size… If you have ram pressure

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Addtionally we raised the mean particle mass to 1.27 to be more consistent with the cooling curve

Fixed Parameters
average density of flows	1.0 particles/cc
vflow	8.25 km/s
Mass flux	665 Msun/Myr
Ram pressure	10471 particles K/cc
Ram density	1055 particles/cc
Ram temp	9.924 K
Ram Jeans Length	1.5 pc
Flow diameter	40 pc
Xmu	1.27
gamma	5/3

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Spectra

First looked at 1D models of colliding flows. The initial velocity field is essentially a heaviside function - and the window used is a cosine window. While these results are for 1D models - the initial flow is 3D although there could be subtleties associated with the windowing in the y and z direction - though probably not at small wave numbers.

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Before the spectra are produced - the data is mapped onto a fixed grid with a resolution equal to the highest level of data. So I looked at the initial velocity field by processing the first frame of the smooth case - but I modified the highest level of data to go from the base grid (level 0) to level 4 - and I used a cube window that was 40x40x40.

The next figure shows the resulting spectra when only considering the level 0 data, then the level 1 data, and so on out to level 4. (There are 5 levels of data but I need to transfer the data files to a larger cluster to have enough memory to handle the FFT's). Also shown are the Nyquist frequency cutoffs which correspond to the highest wavenumber mode that can be resolved by level's 0, 1, 2, 3, & 4. This wavelength will actually be 2*the cell size (which will correspond with the next coarser level resolution). The spectra continue passed this point but quickly drop off because of incomplete coverage of those wave numbers. Imagine taking a cube and sampling spherical shells… Once the diameter of the shell is larger than the cube width, you will get contributions from only the corners of the cube and not uniformly throughout. So everything on the red line to the right of the first vertical line should be ignored. And so on…

It is good to see that the spectra agree quite well at resolved wave lengths and that with each additional level of data, you extend the spectra to higher wave numbers. Also the rise at the end of the spectra (as well as the smaller one to the left) coincide with the maximum resolvable wavelength which coincides with the cell size of the next coarser grid. Regions that have gradients but are not resolved to the finest level will have a stair-step signal, where each step has 2 points (or 4, or 8 points). This will lead to power with wavelengths corresponding to the nyquist frequency, or twice, four times, etc… On the first frame - the only place where there are gradients that are not completely refined - are at the edges of the colliding flow (circled below) - where there is a good amount of shear. This is verified below where the initial spike at the nyquist frequency is dominated by solenoidal terms and not compressive terms.

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The following were taken from a region 80x80x20 with a radial cosine window (ellipsoid) so it includes shear with ambient… Also the fact that the box is not cubic - leads to different wavenumbers (normalized to the longest mode 80 pc). While the ky and kz wavenumbers go like 1, 2, 3, 4… the kx wave numbers go like 4, 8, 12, 16, and so on… This means that if we sample radial bins with a width of 1, we will have over sampling the kx modes - which leads to the sawtooth pattern that was not seen above. This can be corrected if we switch to a 40x40x40 window.

Also for reference here are all the pertinent scales…

k=1 corresponds to 80 pc wave

kmin	1	80 pc
clump radius	145.45	.55 pc
clump spacing	33.33	2.4 pc
smooth_distance for colliding flow	20	4 pc
interface distance between flows	40	2 pc
interface fluctuations	1.414	56.56 pc	with a spectra of -2
grid resolution	51.2/102/205/410/819/1638	.05 pc

Kinetic Energy Spectra gif​ and avi​ (Left is smooth and Right is clumpy. Lines are solenoidal kinetic energy, compressive kinetic energy, the x-contribution to the kinetic energy, and the total kinetic energy spectra.)

0 Myr
10 Myr
20 Myr
27 Myr

Velocity Spectra gif​ and avi​ (Same as for Kinetic Energy Spectra)

0 Myr
10 Myr
20 Myr
27 Myr

Mass and Grav Energy Spectra gif​ and avi​

(Left is Mass and Right is GravEnergy. Blue line is clumpy and red line is smooth.)

0 Myr
10 Myr
20 Myr
27 Myr

Previous runs

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…

Here is a live update of the latest 3D version

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
Column density along flow direction (x)
	20 pc	40 pc	60 pc
	​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).