4.6 Simulation Results
4.6.7 Shock Perturbations
Upon interaction with the interface perturbation, the reflected and transmitted shocks de- velop perturbations of their own. These perturbations are stable (Landau and Lifshitz, 1959; Erpenbeck, 1962), unlike the interface perturbation, and are expected to decay with time. This is confirmed by the simulation results, which show that the perturbation am- plitude oscillates within a decaying envelope at both the reflected and transmitted shocks.
Also notable is that the amplitude perturbation passes through zero multiple times as it decays, and that the reflected shock amplitude is initially of opposite sign to the transmitted shock amplitude.
The collapse of the interface perturbation amplitude curves when plotted against time scaled by the perturbation wave number (kt) suggests that a similar scaling may apply to the shock perturbations. For the transmitted shock perturbation, this is generally true, illustrated by the examples in Figure 4.25, where collapse is very good except for some of the lowest Reynolds number cases. On the other hand, the reflected shock perturbation does not appear to collapse at all when plotted against kt (Figure 4.26). The spatial frequency of the reflected shock perturbation is necessarilyk by the form of the linearization, so it is speculated that collapse may occur for a different scaling of the formkαtβ. The reason for a higher-order dependence on k and/or t for the reflected shock and not the transmitted shock is unclear.
0 1 2 3 4 5 x 104 1.0
1.1 1.2 1.3
t amplitude h(t)/h(0+ )
Re = 296, ! = 0.063 Re = 1185, ! = 0.25 Re = 2370, ! = 0.51 Re = 4740, ! = 1.0
(a) A = 0.2
0 1 2 3 4 5
x 104 1.0
1.2 1.4 1.6 1.8
t amplitude h(t)/h(0+ )
Re = 445, ! = 0.063 Re = 1781, ! = 0.25 Re = 3563, ! = 0.51 Re = 7125, ! = 1.0
(b) A = 0.6
Figure 4.10: Plots of perturbation amplitudeh(t)/h(0+) for MI= 1.05 with varying wave number.
0 0.05 0.10 0.15 0.20 0
0.05 0.10 0.15 0.20
A+!U k t
normalized amplitude h(t)/h(0+ ) − 1
Re = 296, " = 0.063 Re = 1185, " = 0.25 Re = 2370, " = 0.51 Re = 4740, " = 1.0 Impulsive model
(a) A = 0.2
0 0.1 0.2 0.3 0.4
0 0.1 0.2 0.3 0.4
A+!U k t
normalized amplitude h(t)/h(0+ ) − 1
Re = 445, " = 0.063 Re = 1781, " = 0.25 Re = 3563, " = 0.51 Re = 7125, " = 1.0 Impulsive model
(b) A = 0.6
Figure 4.11: Plots of normalized perturbation amplitudeh(t)/h(0+)−1 against scaled time A+∆U ktforMI= 1.05, with the dashed line showing asymptotic impulsive model growth.
The time axis scaling is chosen such that the impulsive model has unity slope.
2 4 6 8 10 12 0
0.005 0.010 0.015 0.020
k t scaled growth rate h’(t)/k h(0+ )
Re = 1185, ! = 0.25 Re = 2370, ! = 0.51 Re = 4740, ! = 1.0 Re = 9481, ! = 2.0 Impulsive model
(a) A = 0.2
2 4 6 8 10 12
0 0.01 0.02 0.03 0.04
k t scaled growth rate h’(t)/k h(0+ )
Re = 1781, ! = 0.25 Re = 3563, ! = 0.51 Re = 7125, ! = 1.0 Re = 14250, ! = 2.0 Impulsive model
(b) A = 0.6
Figure 4.12: Plots of perturbation amplitude growth rate ˙h(t)/kh(0+) against scaled time ktforMI= 1.05, with the dashed line showing asymptotic impulsive model growth in each case. Very early time values (kt < 1) are erratic due to the initial ampltiude ambiguity discussed in Section 4.5.2.
0 1 2 3 4 5 6 0
0.02 0.04 0.06 0.08
t/!
normalized amplitude h(t)/h(0+ ) − 1
Re = 1185, " = 0.25 Re = 2370, " = 0.51 Re = 4740, " = 1.0 Re = 9481, " = 2.0 Impulsive model
(a) A = 0.2
0 1 2 3 4 5 6
0 0.1 0.2 0.3
t/!
normalized amplitude h(t)/h(0+ ) − 1
Re = 1781, " = 0.25 Re = 3563, " = 0.51 Re = 7125, " = 1.0 Re = 14250, " = 2.0 Impulsive model
(b) A = 0.6
Figure 4.13: Plots of normalized perturbation amplitudeh(t)/h(0+)−1 against time scaled by the start-up time τ from Lombardini (2008), for MI = 1.05. The dashed line shows asymptotic impulsive model growth shifted to begin att=τ.
0 2000 4000 6000 8000 10000 12000 1.0
1.5 2.0 2.5 3.0
t amplitude h(t)/h(0+ )
Re = 746, ! = 0.13 Re = 1492, ! = 0.27 Re = 2984, ! = 0.53 Re = 5967, ! = 1.1
(a) A = 0.2
0 2000 4000 6000 8000 10000 12000
1 2 3 4 5 6
t amplitude h(t)/h(0+ )
Re = 1184, ! = 0.13 Re = 2368, ! = 0.27 Re = 4736, ! = 0.53 Re = 9472, ! = 1.1
(b) A = 0.6
Figure 4.14: Plots of perturbation amplitudeh(t)/h(0+) for MI= 1.21 with varying wave number.
0 0.5 1.0 1.5 0
0.5 1.0 1.5
A+!U k t
normalized amplitude h(t)/h(0+ ) − 1
Re = 746, " = 0.13 Re = 1492, " = 0.27 Re = 2984, " = 0.53 Re = 5967, " = 1.1 Impulsive model
(a) A = 0.2
0 1 2 3
0 1 2 3
A+!U k t
normalized amplitude h(t)/h(0+ ) − 1
Re = 1184, " = 0.13 Re = 2368, " = 0.27 Re = 4736, " = 0.53 Re = 9472, " = 1.1 Impulsive model
(b) A = 0.6
Figure 4.15: Plots of normalized perturbation amplitudeh(t)/h(0+)−1 against scaled time A+∆U ktforMI= 1.21, with the dashed line showing asymptotic impulsive model growth.
0 1 2 3 4 5 6 7 0
0.1 0.2 0.3 0.4
t/!
normalized amplitude h(t)/h(0+ ) − 1
Re = 1492, " = 0.27 Re = 2984, " = 0.53 Re = 5967, " = 1.1 Re = 11935, " = 2.1 Impulsive model
(a) A = 0.2
0 1 2 3 4 5 6 7
0 0.5 1.0 1.5
t/!
normalized amplitude h(t)/h(0+ ) − 1
Re = 2368, " = 0.27 Re = 4736, " = 0.53 Re = 9472, " = 1.1 Re = 18943, " = 2.1 Impulsive model
(b) A = 0.6
Figure 4.16: Plots of normalized perturbation amplitudeh(t)/h(0+)−1 against time scaled by the start-up time τ from Lombardini (2008), for MI = 1.21. The dashed line shows asymptotic impulsive model growth shifted to begin att=τ.
5 10 15 20 25 30 0
0.01 0.02 0.03 0.04 0.05 0.06
k t scaled growth rate h’(t)/k h(0+ )
Re = 1492, ! = 0.27 Re = 2984, ! = 0.53 Re = 5967, ! = 1.1 Impulsive model Wouchuk model
(a) A = 0.2
0 5 10 15 20 25
0 0.02 0.04 0.06 0.08 0.10 0.12 0.14
k t scaled growth rate h’(t)/k h(0+ )
Re = 2368, ! = 0.27 Re = 4736, ! = 0.53 Re = 9472, ! = 1.1 Impulsive model Wouchuk model
(b) A = 0.6
Figure 4.17: Plots of perturbation amplitude growth rate ˙h(t)/kh(0+) against scaled time ktforMI= 1.21. The dashed and dot-dashed lines show asymptotic growth rates predicted by the impulsive and Wouchuk (2001a) models, respectively.
0 2000 4000 6000 8000 10000 12000 0.02
0.04 0.06 0.08
t scaled growth rate h’(t)/k h(0+ )
Re = 746, ! = 0.13 Carles model, Re = 746 Re = 1492, ! = 0.27 Carles model, Re = 1492 Impulsive model
Figure 4.18: Plots of perturbation amplitude growth rate ˙h(t)/kh(0+) against scaled time kt for MI = 1.21 and A = 0.2, for low-ReI cases affected by viscosity. Dashed lines show the growth rate predictions of the Carl`es and Popinet (2001) model, and the dot-dashed line shows the impulsive model prediction for reference.
0 2000 4000 6000 8000 10000 12000
1 2 3 4
t amplitude h(t)/h(0+ )
Re = 239, ! = 0.085 Re = 477, ! = 0.17 Re = 955, ! = 0.34 Re = 7638, ! = 2.7
Figure 4.19: Plots of perturbation amplitude forMI= 1.21 andη= 0.02, showing Reynolds number effects. Dashed lines show the predictions of the Carl`es and Popinet (2001) model for each case. At low ReI, viscosity attenuates growth significantly compared to the impulsive model.
0 2000 4000 6000 8000 10000 0
0.5 1.0 1.5 2.0 2.5 3.0
t amplitude h(t)/h(0− )
Re = 4994, ! = 0.49 Re = 9988, ! = 0.97 Re = 19976, ! = 1.9 Re = 79904, ! = 7.8
(a) A = 0.2
0 2000 4000 6000 8000 10000
0 1 2 3 4 5 6
t amplitude h(t)/h(0− )
Re = 8737, ! = 0.49 Re = 17474, ! = 0.97 Re = 34948, ! = 1.9 Re = 69896, ! = 3.9
(b) A = 0.6
Figure 4.20: Plots of perturbation amplitudeh(t)/h(0+) for MI= 2.20 with varying wave number. Dashed lines show the asymptotic growth predicted by the asymptotic model of Wouchuk (2001a) for each case.
5 10 15 20 25 30
−0.05 0 0.05 0.10 0.15 0.20
k t scaled growth rate h’(t)/k h(0+ )
Re = 4994, ! = 0.49 Re = 9988, ! = 0.97 Re = 19976, ! = 1.9 Impulsive model Wouchuk model
(a) A = 0.2
10 20 30 40
0 0.1 0.2 0.3 0.4 0.5
k t scaled growth rate h’(t)/k h(0+ )
Re = 8737, ! = 0.49 Re = 17474, ! = 0.97 Re = 34948, ! = 1.9 Impulsive model Wouchuk model
(b) A = 0.6
Figure 4.21: Plots of perturbation amplitude growth rate ˙h(t)/kh(0+) against scaled time kt for MI = 2.20, with the dashed and dot-dashed lines showing asymptotic growth rates for the impulsive and Wouchuk (2001a) models in each case, respectively.
0 2 4 6 8 10 0
1 2 3
t/!
normalized amplitude h(t)/h(0+ ) − 1
Re = 4994, " = 0.49 Re = 9988, " = 0.97 Re = 19976, " = 1.9 Re = 39952, " = 3.9 Impulsive model Wouchuk model
(a) A = 0.2
0 1 2 3 4 5
0 1 2 3 4 5
t/!
normalized amplitude h(t)/h(0+ ) − 1
Re = 8737, " = 0.49 Re = 17474, " = 0.97 Re = 34948, " = 1.9 Re = 69896, " = 3.9 Impulsive model Wouchuk model
(b) A = 0.6
Figure 4.22: Plots of normalized perturbation amplitudeh(t)/h(0+)−1 against time scaled by the start-up time τ from Lombardini (2008), for MI = 2.20. The dashed line shows asymptotic impulsive model growth shifted to begin at t=τ, and the dot-dashed line the asymptotic prediction of Wouchuk (2001a).
0 2000 4000 6000 8000 10000 12000 0
50 100 150 200 250
t interface thickness ! C(t)
Re = 746, " = 0.13 Re = 2984, " = 0.53 Re = 5967, " = 1.1 Re = 11935, " = 2.1
(a)MI= 1.21, A = 0.2
0 1 2 3 4
x 104 0
100 200 300 400 500 600
t interface thickness ! C(t)
Re = 296, " = 0.063 Re = 2370, " = 0.51 Re = 9481, " = 2.0 Re = 18962, " = 4.0
(b)MI= 1.05, A = 0.2
0 5000 10000 15000
0 50 100 150 200
t interface thickness ! C(t)
Re = 1184, " = 0.13 Re = 4736, " = 0.53 Re = 9472, " = 1.1 Re = 18943, " = 2.1
(c) MI= 1.21, A = 0.6
0 2000 4000 6000 8000 10000
0 10 20 30 40 50 60 70
t interface thickness ! C(t)
Re = 8737, " = 0.49 Re = 17474, " = 0.97 Re = 34948, " = 1.9 Re = 69896, " = 3.9
(d)MI= 2.20, A = 0.6
Figure 4.23: Plots of interface perturbation thickness for a range of initial problem param- eters. The weak influence of pre-shock thickness (indicated byζ) is clear for all cases; note for plot (b) that the ratio of thickest to thinnest initial thickness is 64.
0 1 2 3 4 x 104 0
100 200 300 400
t interface thickness ! C(t)
Re = 445, " = 0.063 Re = 1781, " = 0.25 Re = 7125, " = 1.0 Re = 14250, " = 2.0 Diffusion model
(a) ∆C(t)
0 1 2 3 4
x 104 0
5 10 15
x 104
t interface thickness squared ! C2 (t)
Re = 445, " = 0.063 Re = 1781, " = 0.25 Re = 7125, " = 1.0 Re = 14250, " = 2.0 Diffusion model
(b) ∆2C(t)
Figure 4.24: Plots of interface thickness for the case MI= 1.05, A = 0.6 with the diffusion model for comparison. A linear fit for ∆2C plotted against time indicates that the diffusive model is appropriate.
0 10 20 30 40
−0.15
−0.10
−0.05 0
k t
T−shock perturbation amplitude Re = 746
Re = 1492 Re = 2984 Re = 5967
(a) MI= 1.21, A = 0.2
0 10 20 30
−0.4
−0.3
−0.2
−0.1 0 0.1 0.2
k t
T−shock perturbation amplitude Re = 8737
Re = 17474 Re = 34948 Re = 69896
(b) MI= 2.20, A = 0.6
Figure 4.25: Transmitted shock perturbation amplitude plotted against scaled time kt, showing collapse of the curves across Reynolds number.
0 10 20 30 40
−0.5 0 0.5 1.0 1.5 2.0
k t
R−shock perturbation amplitude
Re = 746 Re = 1492 Re = 2984 Re = 5967
(a) MI= 1.21, A = 0.2
0 10 20 30
−0.2 0 0.2 0.4 0.6 0.8 1
k t
R−shock perturbation amplitude
Re = 8737 Re = 17474 Re = 34948 Re = 69896
(b) MI= 2.20, A = 0.6
Figure 4.26: Reflected shock perturbation amplitude plotted against scaled timekt, showing that collapse does not occur with this scaling.