Abstract
Transition prediction is of great importance for the design of long distance flying vehicles. It starts from the problem of receptivity, i.e., how external disturbances trigger instability waves in the boundary layer. For super/hypersonic boundary layers, the external disturbances first interact with the shock ahead of the flying vehicles before entering the boundary layer. Since direct numerical simulation (DNS) is the only available tool for its comprehensive and detailed investigation, an important problem arises whether the numerical scheme, especially the shock-capturing method, can faithfully reproduce the interaction of the external disturbances with the shock, which is so far unknown. This paper is aimed to provide the answer. The interaction of weak disturbances with an oblique shock is investigated, which has a known theoretical solution. Numerical simulation using the shock-capturing method is conducted, and results are compared with those given by theoretical analysis, which shows that the adopted numerical method can faithfully reproduce the interaction of weak external disturbances with the shock.
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Abbreviations
- Ma :
-
Mach number
- u :
-
velocity normal to the shock
- v :
-
velocity tangential to the shock
- ρ :
-
density
- p :
-
pressure
- ω :
-
frequency
- k :
-
wave vector with components (k x , k y )
- λ x , λ y :
-
wave lengths in x- and y-axes
- θ 1 :
-
angle of incidence of wave relative to x-axis
- θ 2 :
-
angle of divergence of acoustic wave relative to x-axis
- θ 3 :
-
angle of divergence of entropy/vortical wave relative to x-axis
- ß :
-
angle between flow velocity and shock
- c :
-
sound speed
- v g :
-
group velocity
- γ :
-
ratio of the specific heat capacities
- \(\overline {\left( \bullet \right)} \) :
-
unperturbed flow quantities
- δ(·):
-
flow fluctuations
- (·)1,2 :
-
quantities at upstream and downstream of the shock
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Acknowledgements
The first author would like to thank Professor Heng ZHOU in Tianjin University for his continuous encouragement and Professor Jisheng LUO in Tianjin University for his inspiring discussion and helpful suggestions.
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Project supported by the National Natural Science Foundation of China (Nos. 11472188 and 11332007) and the National Key Research and Development Program of China (No. 2016YFA0401200)
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Su, C., Geng, J. Interaction of weak free-stream disturbance with an oblique shock: validation of the shock-capturing method. Appl. Math. Mech.-Engl. Ed. 38, 1601–1612 (2017). https://doi.org/10.1007/s10483-017-2279-9
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DOI: https://doi.org/10.1007/s10483-017-2279-9