Changes between Version 17 and Version 18 of ShockedClumps
- Timestamp:
- 05/20/11 12:50:23 (14 years ago)
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ShockedClumps
v17 v18 1 1 = Shocked Clumps = 2 Common params 2 3 == Common params == 3 4 || n,,amb,, || 250 cc^-1^ || 4 5 || T,,amb,, || 100 K || … … 6 7 7 8 8 Run Params 9 == Run Params == 9 10 10 11 || Run || v,,wind,, (km/s) || [[latex($\chi$)]] || final time (yr) || T,,bow,, (K)|| T,,trans,, (K)|| L,,bow,, (R,,c,,) || L,,trans,, (R,,c,,) || … … 16 17 || F || 200 || 100 || 142.2 yr || 9.02e+05 || 9.02e+04 || 3.93e+01 || 1.41e+00 || 17 18 18 [[latex($T_{bow}=\frac{3 m_H v_{wind}^2}{16 k_b}$)]] 19 == Shock Structure == 20 21 === Bow Shock === 22 For the bow shock temperature, we assume that this shock is stationary with respect to the flow in which case the material flowing into the shock will see a shock travelling at [[latex($v_{wind}$)]] so [[latex($T_{bow}=\frac{3 \mu v_{wind}^2}{16 k_b}$)]] 23 24 === Wind Shock === 25 26 However, before the wind reaches the clump it first shocks against the stationary ambient and produces a wind shock. To solve for the speed and temperature of this wind shock we first switch to a reference frame in which both the wind and the ambient are colliding at [[latex($v_{wind}/2$)]] 27 28 If we shift to this reference frame, we then know from symmetry that the postshock velocity must be 0. Using this along with the pre-shock conditions we can solve for the post-shock density, pressure, and the shock speed by Applying the Rankine-Hugoniot jump conditions at the interface 29 {{{ 30 #!latex 31 \begin{eqnarray} 32 s(\rho_2-\rho_1)= \rho_2 u_2 - \rho_1 u_1 \\ 33 s(\rho_2u_2-\rho_1u_1)=(\rho_2u_2^2+p_2) - (\rho_1u_1^2+p_1) \\ 34 e_1+\frac{1}{2}u_1^2+\frac{p_1}{\rho_1}=e_2+\frac{1}{2}u_2^2+\frac{p_2}{\rho_2} 35 \end{eqnarray} 36 }}} 37 38 Or we can assume that the shock is adiabatic with a high mach number which then gives us that [[latex($\rho_2 = 4\rho_1$)]] in which case [[latex($s=-1/3u_1$)]] or in this case [[latex($s=+1/6v_{wind}$)]] or in the ambient frame of reference [[latex($s=2/3v_{wind}$)]]. The backward facing shock would then have velocity [[latex($s=1/3v_{wind}$)]] so the forward shock would be travelling at twice the speed as the backward shock and the shocked region would be as large as the unshocked inflow region - which is visible in the non-cooling bow runs. In any event the velocity of the wind shock is then [[latex($T_{windshock}=4/9T_{bow}$)]]... On a side note, if you run material into a wall at a high supersonic velocity [[latex($v$)]], the shocked temperature ends up being [[latex($\frac{16}{9} \times \frac{3\mu v^2}{16k_b}$)]]. 39 40 41 Eventually this wind shock slams into the shock and should produce a reflected shock. Using the same analysis we get that the reflected shock temperature within the windshock should be [[latex($\frac{16}{9} \frac{T_bow}{4}=\frac{4}{9}T_{bow}$)]] however we have ignored the fact that this windshock material is already shocked, so this reflected shock would be fairly weak and some of the assumptions used above break down. Eventually this reflected shock breaks out of the windshock and into the ambient where it now sees cold material flowing in at [[latex($v_wind$)]] and forms the bow shock. 42 43 === Transmitted Shock === 44 45 And finally we have 19 46 20 47 [[latex($T_{trans}=T_{bow}/\chi$)]] 21 48 49 50 == Temperature lineout for non-cooling case == 51 Below is a snapshot from [attachment:Temp_LineOut.gif this movie] that shows the evolution of the shock structure along the axis. Also plotted are the temperatures for the bow, wind, and transmitted shocks calculated above. 52 [[Image(Temp_LineOut0009.png, width=800)]] 53 54 == Cooling Lengths == 22 55 23 56 A naive calculation of L,,shock,, using the instantaneous cooling rate at T,,shock,, would implies that run A is bow only, run D is trans only, run E is neither, and that run F is marginally trans only. However it does appear that run F has both a cooling bow shock and transmitted shock. To understand the discrepancy we should consider the following figure where we have plotted the cooling lengths in units of clump radius as a function of wind velocity for the bow shock as well as for the transmitted shocks with [[latex($\chi$)]] = 10 & 100 … … 32 65 Run E which is also very curious because while neither the bow shock nor the transmitted shock initially is cooling, the intersection of the transmitted shock with the wrap around shock triggers dramatic cooling and fragmentation. From the intersection of the red line with 800 km/s we see that the transmitted shock is on the same region of the cooling curve as the bow shock was for run F and once it begins to cool the cooling length drops from 10 R,,c,, to .01 R,,c,,. 33 66 67 == Images == 68 34 69 || || A 50 10 (bow only) || D 400 10 (clump only) || E 800 10 (neither **) || F 200 100 (both) || 35 70 || || [attachment:movie_50_10.gif movie] || [attachment:movie_400_10.gif movie] [attachment:zoom_400_10.gif zoom] || [attachment:movie_800_10.gif movie] || [attachment:movie.gif movie] [attachment:zoom.gif zoom]||
