One-dimensional validation and verification tests
5.2 Inviscid test cases
5.2.5 Shock-bubble interaction
Test case
This problem is a right-facing shock wave travelling in air and passing through an helium slab thus pro-ducing a complex series of transmitted and reflected shock waves inside the slab. This problem is more complex albeit more representative of numerous scenarios of practical interest such as shock-bubble inter-actions and Richtmyer-Meshkov instabilities. The problem is the one-dimensional version of the classical helium shock-bubble interaction problem presented experimentally by Haas and Sturtevant [122] and numerically by [13,143,198,250,259,270]. The two-dimensional version problem is discussed at length in
§.6.2.
The domain has a length of L = 0.325 [m] which corresponds to 6.5D, where D = 5 [cm] is the bubble diameter and spans −2.5 6 x/D 6 4. The upstream edge of the bubble is situated at x/D = 0, and the shock is initially positioned at x/D = −1 and is initialised as a perfect discontinuity over a single cell.
It travels for one diameter before impacting the bubble thus leaving some time for the numerical start-up errors (pair of low-frequency/low-amplitude waves moving start-upstream with the contact discontinuity speed) to disappear [129]. The shock is travelling at Ms= 1.22 in air. The helium slab and surrounding air are initially in thermal and mechanical equilibriums. According to the literature, the helium is not pure in the slab, but contaminated with 28 % of air by mass. The different states can be defined by their different vectors Q = [P/Pair, ρ/ρair, U/cair, γ/γair, Yair, YHe]T,
◦ Post-shock air (x/D ∈ [−2.5, −1.0]) and is characterised by,
Qshock= [1.5698, 1.3764, 0.3336, 1.0, 1.0, 0.0]T
◦ Bubble (x/D ∈ [0, 1]) defined as,
Qb= [1.0, 0.1829, 0.0, 1.18, 0.28, 0.72]T
◦ Pre-shock air (x/D ∈ [−1.0, 0]S[1, 4]) and is defined by,
Qair= [1.0, 1.0, 0.0, 1.0, 1.0, 0.0]T
where D = 50 [mm], ρair = 1.29 [kg/m3], Pair = 101325 [P a], γair = 1.4 and cair= 331.6 [m/s]. Details on the shock wave properties estimation for both calorically and thermally perfect EoS are given in Ap-pendixC.
0
Figure 5.15: Primitive variables for the O2/H2- PL/PR= 10 multi-species shock-tube computed with the thermally perfect EoS at t = 2 × 10−4[s] - Ref, QCVF, QCMF, FCMF
0
Figure 5.16: Primitive variables for the O2/H2- PL/PR= 50 multi-species shock-tube computed with the thermally perfect EoS at t = 2 × 10−4[s] - Ref, QCVF, QCMF, FCMF
1.26
Figure 5.17: γ variations for the O2/H2 - PL/PR= 50 multi-species shock-tube computed with the thermally perfect EoS at t = 2 × 10−4[s] - Ref, QCVF, QCMF, FCMF
0 5e-05 0.0001 0.00015 0.0002
EρE[−]
0 5e-05 0.0001 0.00015 0.0002
EρYk[−]
t [s]
(b) Total Mass
Figure 5.18: Conservation errors for the O2/H2multi-species shock-tube computed with the thermally perfect EoS - filled symbol for PL/PR= 50 and empty symbols for PL/PR= 10 - QCVF, QCMF, FCMF
Computations are run on a grid composed of N x = 1300 cells, with a cell size corresponding to
∆x/D = 0.005, or ∆x = 2.5 × 10−4[m]. Similarly to published results, the calorically perfect EoS is considered for the study, and CF L = 0.5 for the time marching. The final time has been chosen as tf = 5 × 10−4[s] to let some time to the different shocks to propagate.
Results
The results are presented in Fig. 5.19, where the complex system of shocks and expansion waves can be seen. The main shock is at the right of the domain after having been transmitted through the he-lium slab. A rarefaction wave at the left is propagating towards the left and results from the initial interaction between the shock and the slab. Numerous other shocks can be seen, resulting from several reflections/transmissions within the moving slab.
The three different methods employed here (QCMF, QCVF and FCMF) seem to converge towards a single solution, which is a good indication of the correctness of the different approaches considered.
The most noticeable feature indicating the failure of the FCMF in this type of problem are the two large spikes of temperature forming on both sides of the slab. Temperature is roughly bounded by 270 [K] 6 T 6 330 [K] without the spikes, and roughly between 250 [K] 6 T 6 500 [K] in the case of the
FCMF. This clearly cannot be tolerated if any temperature dependent processes have to be accounted for.
Fig.5.20depicts the pressure oscillations and temperature spikes obtained for different grid resolutions.
According to Karni [147], the rate of convergence of these pressure oscillations is extremely slow. This can be observed here, where a ratio of 16 exists between the coarsest and finest grids, and oscillations can still be observed at the finest level. A more worrisome observation is the increase of the temperature spikes magnitude as the mesh is being refined.
1
Figure 5.19: Primitive variables for the shock-bubble interaction computed with the calorically perfect EoS at t = 4 × 10−4[s] - Ref, QCVF, QCMF, FCMF
1.555 1.56 1.565 1.57 1.575
0 0.02 0.04 0.06 0.08 0.1
P/Pair[−]
x [m]
(a) Pressure (Zoom)
0.8 1 1.2 1.4 1.6 1.8 2 2.2
0 0.02 0.04 0.06 0.08 0.1
T/Tair[−]
x [m]
(b) Temperature (Zoom)
Figure 5.20: Grid convergence for the shock-bubble interaction computed with the calorically perfect EoS and FCMF model at t = 4 × 10−4[s] - N x = 325, N x = 650, N x = 1300, N x = 2600, N x = 5200
5.2.6 Discussion
Several observations can be made about the multi-component models tested. The FCMF generates sig-nificant oscillations of pressure and velocity at material interfaces as well as temperature spikes, due essentially to the numerical method employed. The FCVF is both unstable and unreliable in the simu-lation of multi-components flows. Furthermore, it does not capture shock-waves correctly. This model is thus discarded for the rest of the work. Both the QCMF and QCVF model eliminate pressure oscillations.
The QCMF is generally more dissipative than the QCVF, which can be potentially explained by the numerical methods used to solve both sets of equations. On the other hand, the QCVF does conserve mass, momentum and energy a lot better than the QCMF which features the largest energy errors. Based on the test cases of this section, both the QCMF and QCVF models will be used for the validation of the diffusive fluxes presented below.
Finally, a remark on the rather poor energy conservation featured by the QCMF model can be made.
It can be shown that most of the energy conservation errors occurring are due to the freezing of both γ and hm0 when the hyperbolic integration step artificially mixes species at sharp contact surfaces [6,133].
Abgrall and Karni [6] further estimated that energy conservation errors could be related to the maximum variation of γ between two consecutive cells. This implies that an initially smeared profile would feature smaller conservation errors. This is illustrated in Fig. 5.21, where the moving contact wave has been initialised with different smeared profiles of temperature and mass fractions following,
ϕi= ϕin+ ϕout
2 −ϕout− ϕin
2 tanh
Cs
D
2 − |xi− x0|
(5.2) where ϕinis the value inside the bubble, conversely ϕoutis the value outside, D is the bubble diameter and x0 the position of its centre. It can be seen that as Csdecreases and the smearing increases, conservation errors reduce up to EρE ≈ 10−5 for Cs = 50. A slight smearing will therefore be initially applied in the simulations presented in the next sections to reduce conservation errors.
10−8 10−7 10−6 10−5 10−4 10−3 10−2
0 0.002 0.004 0.006 0.008 0.01
EρE[−]
t [s]
Figure 5.21: QCMF energy conservation error as a function of the initial smearing of the interface for the moving contact wave problem computed with the thermally perfect EoS - Sharp profile, Cs= 50,
Cs= 500, Cs= 5000, Cs= 50000