G. Abbate, B.J. Thijsse, and C.R. Kleijn
is in different conditions: in the left tank it is at a pressure P
1
= 30 Pa and at a
temperature T
1
= 12000 K, while in the right tank and in the tube it is at a pres-
sure P
2
= 3 Pa and at a temperature T
2
= 2000 K. At t = 0 the membrane breaks
and the fluid can flow from one region to the other. Two different waves will start
travelling from the left to the right with two different velocities: a shock wave and a
contact discontinuity. The shock wave produces a rapid increase of the temperature
and pressure of the gas passing through it, while through the contact discontinuity,
the flow undergoes only a temperature, and not a pressure, variation [25, 26].
The thermodynamic conditions inside the infinitely large tanks remain constant. For
this reason the two tanks can be modeled with an inlet and an outlet boundary con-
dition.
Inside the tube, we suppose that the flow is one-dimensional. Upstream (left) from
the shock, the gas has a high temperature and relatively high pressure, and gradient
length scales are small. Downstream (right) from the shock, both temperature and
pressure are much lower, and gradient length scales are large. As a result, the contin-
uum breakdown parameter Kn
max
(using local values of Q
ref
) is high upstream from
the shock, and low downstream of it. In the hybrid DSMC-CFD approach, DSMC
is therefore applied upstream, and CFD is applied downstream. The continuum grid
is composed of 100 cells in the x-direction and 1 cell in the y-direction, while the
code automatically refines the mesh in the DSMC region to fulfil its requirements.
The coupling time step is
∆
t
coupling
= 4.0×10
−6
sec. and ensemble averages of the
DSMC solution were made on 30 repeated runs.
In figs. 5 and 6 the pressure (a), temperature (b), and velocity (c) inside the tube
after 1.5 ×10
−5
sec. and 3.0 ×10
−5
sec. respectively, evaluated with the coupled
DSMC-CFD method, are compared to the results of a full DSMC simulation. The
latter was feasible because of the 1-D nature of the problem. Results obtained with a
full CFD simulation are shown as well. The full DSMC solution is considered to be
the most accurate of the three. In figs. 5(d) and 6(d) also the continuum breakdown
parameter, computed using the coupled method, is compared to that same parameter
computed with the full CFD simulation.
From the results shown in figs. 5 and 6, it is clear that the full CFD approach fails
due to the high values of the local Kn number caused by the presence of the shock.
It predicts a shock thickness of ≈2 cm, which is unrealistic since even in continuum
conditions the shock thickness is one order of magnitude greater than the mean free
path (
λ
≈ 1 cm) [37]. In the full DSMC approach, therefore, the shock is smeared
over almost 10 cm. The results obtained with the hybrid approach are virtually iden-
tical to those obtained with the full DSMC solver (maximum difference < 1.5%)...,
but were obtained in less than one fifth of the CPU time.
Comparing figs. 5 and 6 it is also possible to see how the DSMC and CFD regions
adapt in time to the flow field evolution.
422