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Interaction of weak free-stream disturbance with an oblique shock: validation of the shock-capturing method

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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

References

  1. Reshotko, E. Transition issues for atmospheric entry. Journal of Spacecraft and Rockets, 45(2), 161–164 (2008)

    Article  Google Scholar 

  2. Fedorov, A. V. Transition and stability of high-speed boundary layers. Annual Review of Fluid Mechanics, 43, 79–95 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  3. Zhong, X. and Wang, X. Direct numerical simulation on the receptivity, instability and transition of hypersonic boundary layers. Annual Review of Fluid Mechanics, 44, 527–561 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  4. Qin, F. and Wu, X. Response and receptivity of the hypersonic boundary layer past a wedge to free-stream acoustic, vortical and entropy disturbances. Journal of Fluid Mechanics, 797, 874–915 (2016)

    Article  MATH  Google Scholar 

  5. Zhang, H. X. The similarity law for real gas flow (in Chinese). Acta Aerodynamica Sinica, 8(1), 1–8 (1990)

    Google Scholar 

  6. Zhong, X. Leading-edge receptivity to free-stream disturbance waves for hypersonic flow over a parabola. Journal of Fluid Mechanics, 441, 315–367 (2001)

    Article  MATH  Google Scholar 

  7. Ma, Y. and Zhong, X. Receptivity of a supersonic boundary layer over a flat plate, part 1, wave structures and interactions. Journal of Fluid Mechanics, 488, 31–78 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  8. Ma, Y. and Zhong, X. Receptivity of a supersonic boundary layer over a flat plate, part 2, receptivity to freestream sound. Journal of Fluid Mechanics, 488, 79–121 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  9. Ma, Y. and Zhong, X. Receptivity of a supersonic boundary layer over a flat plate, part 3, effects of different types of free-stream disturbances. Journal of Fluid Mechanics, 532, 63–109 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  10. Ma, Y. and Zhong, X. Boundary-layer receptivity of Mach 7.99 flow over a blunt cone to free-stream acoustic waves. Journal of Fluid Mechanics, 556(1), 55–103 (2006)

    MATH  Google Scholar 

  11. Bonfiglioli, A., Grottadaurea, M., Paciorri, R., and Sabetta, F. An unstructured, three-dimensional, shock-fitting solver for hypersonic flows. Computers and Fluids, 73, 162–174 (2013)

    Article  MATH  Google Scholar 

  12. Balakumar, P. Receptivity of a supersonic boundary layer to acoustic disturbances. AIAA Journal, 47(5), 1069–1078 (2009)

    Article  Google Scholar 

  13. Balakumar, P. and Kegerise, M. A. Receptivity of hypersonic boundary layers over straight and flared cones. AIAA Journal, 53 (2010)

    Google Scholar 

  14. Soudakov, V. G. Numerical simulation of hypersonic boundary layer receptivity to entropy and vorticity waves. TsAGI Science Journal, 44(2), 205–217 (2013)

    Article  Google Scholar 

  15. Zhou, H. and Zhang, H. X. Two problems in the transition and turbulence for near space hypersonic flying vehicles (in Chinese). Acta Aerodynamica Sinica, 35(2), 151–155 (2017)

    Google Scholar 

  16. Kovasznay, L. S. G. Turbulence in supersonic flow. Thermal Engineering, 20(10), 657–674 (1953)

    MATH  Google Scholar 

  17. McKenzie, J. F. and Westphal, K. O. Interaction of linear waves with oblique shock waves. Physics of Fluids, 11(11), 2350–2362 (1968)

    Article  MATH  Google Scholar 

  18. Anyiwo, J. C. and Bushnell, D. M. Turbulence amplification in shock-wave boundary-layer interaction. AIAA Journal, 20(7), 893–899 (1982)

    Article  Google Scholar 

Download references

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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Correspondence to Caihong Su.

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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

Key words

Chinese Library Classification

2010 Mathematics Subject Classification

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