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