Changes between Version 9 and Version 10 of CollidingFlows

Timestamp:

06/30/11 13:56:31 (14 years ago)

Author:

Jonathan

Comment:

—

CollidingFlows

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 || [[Image(233_1_Sinks.png, width=200)]] || [[Image(233_4_Sinks.png, width=200)]] || [[Image(233_10_Sinks.png, width=200)]] || [[Image(233_20_Sinks.png, width=200)]] || 
 
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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.  [[latex($T_{cool}=\frac{nK_BT_{shock}}{(n\Gamma - n^2 \Lambda(T))(\gamma-1)}$)]]
+
+Here we've plotted the cooling time as a function of density and ram pressure (in units of Kelvin/cc)
+[[Image(nPCoolingTime.png)]]
+
+We can then calculate the cooling length of the shock [[latex($L_{cool}=T_{cool} v_{shock}$)]] or the cooling length of the thermal instability [[latex($\lambda_{TI}=T_{cool} c_s \approx L_{cool}$)]] since [[latex($c_s \approx v_{shock}$)]]
+Here are plots of the cooling length as well as the thermal instability length scale.
+[[Image(nPCoolingLength.png, width=50%)]][[Image(nPTILength.png, width=50%)]]
+
+We can also calculate the free fall time for the condensations
+[[latex($t_{ff}=\sqrt{\frac{3 \pi}{32 G \rho}}$)]] as well as the Jeans length [[latex($\lambda_J= c_s\sqrt{\frac{\pi}{G\rho}}$)]] plotted below
+[[Image(nPFreeFallTime.png, width=50%)]][[Image(nPJeansLength.png, width=50%)]]
+
+
+Finally given the density and temperature of the shocked material we can estimate the density contrasts of the thermally unstable clumps [[latex($\chi=T_{shock}/T_{eq}$)]] and then calculate the clump destruction time assuming it is of size [[latex($\lambda_{TI}$)]] embedded in a background flow of velocity [[latex($v_{shock}$)]].  [[latex($t_{cc}=\frac{\sqrt{\chi} \lambda_{TI}}{v_{shock}}$)]]
+
+[[Image(nPTIDestructionTime.png)]]
+
+Combining these two time scales gives a clump survivability [[latex($\xi=\frac{t_{cc}}{t_{ff}}$)]]
+[[Image(nPTICollapsibility.png)]] which peaks at about .1
+
+Plotting the same quantity in n vs V space we have
+[[Image(nVTICollapsibility.png)]]
+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...