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DSMC simulations of Rb gas flow A. Petrenko, G. Plyushchev AWAKE PEB 12.12.2014 CERN Direct Simulation Monte Carlo method (developed by G. A. Bird in the 60’s): • First divide the gas sample into cells smaller than the mean free path between collisions. • Select time step smaller than typical time between collisions. Repeat this cycle over and over again: 1) Move particles as in vacuum. 2) Apply boundary conditions. 3) Calculate collisions in each cell depending only on local gas density and particle relative velocities. Slightly modified Matlab scripts from A. L. Garcia’s book “Numerical Methods for Physics” were used. More details are given in this article: F. J. Alexander and A. L. Garcia. “The Direct Simulation Monte Carlo Method”. Computers in Physics, Vol. 11. Num. 6, 1997. 1D DSMC simulation of Rb density profile after 10 ms for the baseline density and for the free molecular flow: n0 = 7·1014 1/cm3: n0 = 7·1010 1/cm3: 3 m = 300 m/s * 10 ms t=0 t=0 t = 10 ms (m) Basic parameters: T = 200 °C, Rb atom diameter = 0.6 nm, 300 m/s, m/s, The most probable thermal velocity = 300 Mean free path between collisions = 0.9 mm (for n0 = 7·1014 1/cm3). t = 10 ms (m) The upper limit on the gas outflow, i. e. the worst possible case. 1D DSMC simulation of Rb density profile after 10 ms for the free molecular flow: n0 = 7·1010 1/cm3: Red line is the analytical calculation for the free molecular flow regime: Using the Maxwell distribution of velocity vector: 𝑚 𝑚𝑣𝑥 2 𝑓𝑣 𝑣𝑥 = exp − 2𝜋𝑘𝑇 2𝑘𝑇 t=0 𝑥+𝛿 Where 𝑣𝑥 = 𝑡 . Integrating over 𝛿 (or over 𝑣𝑥 ), one can obtain the final distribution: 𝑛0 𝑥 𝑚 𝑓𝑛 𝑥, 𝑡 = 1 + erf − 2 𝑡 2𝑘𝑇 t = 10 ms (m) The upper limit on the gas outflow, i. e. the worst possible case. 1D DSMC simulation of Rb density profile after 10 ms for the baseline density: n0 = 7·1014 1/cm3: Red line is the analytical calculation: 1D hydrodynamic expansion speed (from Euler equations, Landau & Lifshitz): 2 𝑥 𝑣𝑥 = 𝑐0 + 𝛾+1 𝑡 𝛾−1 Local speed of sound: 𝑐 = 𝑐0 − 𝑣𝑥 t=0 2 Where speed of sound is equal: = t = 10 ms 𝛾 2 from temperature one can get density: 2𝑘𝑇 𝑚 𝑛 𝑛0 , and = 𝑇 𝛾−1 𝑇0 Thus: (m) 𝑛 2 = 𝑛0 𝛾 𝛾−1 2 𝛾 𝛾−1 𝑚 𝑥 − 𝛾 + 1 2 𝛾 + 1 2𝑘𝑇 𝑡 2 𝛾−1 2D DSMC simulation up to 0.6 ms (Rb flow in a 4 cm wide slit) 2D DSMC simulation up to 3 ms: Pretty close to the simple estimate: 300 m/s * 3 ms = 90 cm On-axis injection requirements From our recent article: K.V. Lotov, A.P. Sosedkin, A.V. Petrenko, L.D. Amorim, J. Vieira, R.A. Fonseca, L.O. Silva, E. Gschwendtner, P. Muggli. Electron trapping and acceleration by the plasma wakefield of a self-modulating proton beam. Plasma density 0 L0 z There is simply no way to make plasma density ramp length L0 to be less than 10 cm with a valve opening time of several ms. (3 ms * 300 m/s = 90 cm!) Conclusions • DSMC method is easy to apply in our case (from free molecular flow to density of a few times 1015 1/cm3) using simple Matlab or C++ programs. • 1D model confirmed by analytical calculations shows that density drop travels with the speed of sound (i.e. 3 m in 10 ms @ 200 °C). • 2D model (Rb flow in a 4 cm wide slit) shows no significant difference from the 1D model so far (this is preliminary). • Next steps to implement in simulations: – – – – Moving valve (easy) Irises (easy) 3D model (easy but computationally expensive) Equilibrium density profile with hot/cold spots in the plasma pipe. • The only straightforward solution to the density ramp problem seems to be an additional lightweight and super-fast shutter (as already suggested by Mikhail Martyanov). On-axis injection needs <10 cm long ramp => 1 cm wide shutter must open in less than 0.1 m / 300 m/s = 0.3 ms => the speed of shutter plate will be 30 m/s => maybe something feasible with lightweight conducting/magnetic materials in a pulsed magnetic field? Spinning wheel with a hole? There is no need for this additional shutter to be leak tight! 8