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Free-stream receptivity of a hypersonic blunt cone using input–output analysis and a shock-kinematic boundary condition

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Abstract

Traditional stability tools have done much in the last few decades to demonstrate the significance of modal instabilities as a pathway for laminar to turbulent transition in hypersonic flows, but are less effective at predicting transition in flows with significant streamwise variation and strong shock waves. Because of this, most stability analyses over blunt cones tend to focus on the growth of instabilities in regions of the flow away from the blunt tip and downstream of any strong shock waves. We develop a new shock-kinematic boundary condition which is compatible with both the finite-volume method and input–output analysis. This boundary condition enables analysis of the receptivity of blunt cones to disturbances in the free stream by careful treatment of linear interactions of small disturbances with the shock. In particular, a Mach 5.8 flow over a 7\(^{\circ }\) half-angle cone with a 0.15" nose radius is analyzed, showing significant amplification of disturbances along the cone frustum in a 5–15 kHz bandwidth due to the destabilization of a slow acoustic boundary layer mode, and significant amplification of entropy layer instabilities between 100 and 180 kHz due to rotation/deceleration of entropy/vorticity waves. These mechanisms are receptive to free-stream disturbances in very localized positions upstream of the bow shock.

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References

  1. Boeing. Boeing debuts first passenger-carrying hypersonic vehicle concept. https://www.boeing.com/features/2018/06/hypersonic-concept-vehicle.page. Accessed 26 June 2018

  2. Morkovin, M.V.: Viscous Drag Reduction. Springer, Berlin (1968)

    Google Scholar 

  3. Lin, T.C.: Influence of laminar boundary-layer transition on entry vehicle design. J. Spacecr. Rockets 45(2), 165 (2008)

    Article  Google Scholar 

  4. Mack, L.M.: The inviscid stability of the compressible laminar boundary layer. Technical Report, pp. 37–23. Space Programs Summary (1963)

  5. Mack, L.M.: Boundary-layer linear stability theory. Technical Report, p. 709, Part 3, AGARD Report (1984)

  6. Malik, M.R.: Numerical methods for hypersonic boundary layer stability. J. Comput. Phys. 86(2), 376 (1990)

    Article  Google Scholar 

  7. Malik, M.R., Spall, R.E.: On the stability of compressible flow past axisymmetric bodies. J. Fluid Mech. 228, 443 (1991)

    MATH  Google Scholar 

  8. Herbert, T.: Parabolized stability equations. Annu. Rev. Fluid Mech. 29(1), 245 (1997)

    Article  MathSciNet  Google Scholar 

  9. Stetson, K.F.: Hypersonic Boundary Layer Transition Experiments. AF Wright Aeronautical Laboratories (1980)

  10. Rufer, S.J.: Hot-wire measurements of instability waves on sharp and blunt cones at mach 6. Ph.D. thesis, Ann Arbor, MI (2005)

  11. Robarge, T.W.: Laminar boundary-layer instabilities on hypersonic cones: computations for benchmark experiments. Master’s thesis, Ann Arbor, MI (2006)

  12. Jewell, J.S., Kimmel, R.L.: In: AIAA 54th Aerospace Sciences Meeting. San Diego, CA (2016)

  13. Lui, J., Zhang, S., Fu, S.: Linear spatial instability analysis in 3D boundary layers using plane-marching 3D-LPSE. Appl. Math. Mech. 37(8), 1013 (2016)

    Article  MathSciNet  Google Scholar 

  14. Paredes, P., Choudhari, M.M., Li, F., Chang, C.: Optimal growth in hypersonic boundary layers. AIAA J. 54(10), 3050 (2016)

    Article  Google Scholar 

  15. Paredes, P., Choudhari, M.M., Li, F.: Blunt-body paradox and transient growth on a hypersonic spherical forebody. Phys. Rev. Fluids 2(5), 053903 (2017)

    Article  Google Scholar 

  16. Paredes, P., Choudhari, M.M., Li, F.: In: AIAA SciTech Forum. Kissimmee, FL (2018)

  17. Paredes, P., Jewell, J.S., Kimmel, R.L.: Non-modal growth of traveling waves on blunt cones at hypersonic speeds. AIAA J. 57(11), 4738 (2019)

    Article  Google Scholar 

  18. Federov, A., Khokhlov, A.P.: Prehistory of instability in a hypersonic boundary layer. Theor. Comput. Fluid Mech. 14(6), 359 (2001)

    Article  Google Scholar 

  19. Federov, A., Khokhlov, A.P.: Receptivity of hypersonic boundary layer to wall disturbances. Theor. Comput. Fluid Mech. 15(4), 231 (2002)

    Article  Google Scholar 

  20. Federov, A.: Receptivity of a high-speed boundary layer to acoustic disturbances. J. Fluid Mech. 491, 101 (2003)

    Article  Google Scholar 

  21. Federov, A.: Transition and stability of high-speed boundary layers. Annu. Rev. Fluid Mech. 43(1), 79 (2011)

    Article  MathSciNet  Google Scholar 

  22. Cook, D.A., et al.: In: AIAA Aviation Forum. Atlanta, GA (2018)

  23. Barrett, A.N., et al.: In: AIAA SciTech Forum. San Diego, CA (2019)

  24. McKenzie, J.F., Westphal, K.O.: Interaction of linear waves with oblique shock waves. Phys. Fluids 11(11), 2350 (1968)

    Article  Google Scholar 

  25. Huang, Z., Wang, H.: Linear interaction of two-dimensional free-stream disturbances with an oblique shock wave. J. Fluid Mech. 837, 1179 (2019)

    Article  MathSciNet  Google Scholar 

  26. Lubchich, A.A., Pudovkin, M.I.: Interaction of small perturbations with shock waves. Phys. Fluids 16(12), 4489 (2004)

    Article  MathSciNet  Google Scholar 

  27. Robinet, J.C., Casalis, G.: Critical interaction of a shock wave with an acoustic wave. Phys. Fluids 13(4), 1047 (2001)

    Article  MathSciNet  Google Scholar 

  28. Andreopoulos, Y., Agui, J.H., Briassulis, G.: Shock wave-turbulence interactions. Annu. Rev. Fluids 32(1), 309 (2000)

    Article  MathSciNet  Google Scholar 

  29. Grube, N.E.: Shock wave-turbulence interactions. Ph.D. thesis, Princeton University (2020)

  30. Mahesh, K., Lee, S., Lele, S.K., Moin, P.: The interaction of an isotropic field of acoustic waves with a shock wave. J. Fluid Mech. 300, 383 (1995)

    Article  MathSciNet  Google Scholar 

  31. MacCormack, R.W.: Numerical computation of compressible and viscous flow. AIAA (2014)

  32. McKeon, B.J., Sharma, A.S., Jacobi, I.: Experimental manipulation of wall turbulence: a systems approach. Phys. Fluids 25, 3 (2013)

    Article  Google Scholar 

  33. Schmid, P.J., Henningson, D.S.: Stability and Transition in Shear Flows. Springer, Berlin (2001)

    Book  Google Scholar 

  34. Hanifi, A., Schmid, P.J., Henningson, D.S.: Transient growth in compressible boundary layer flow. Phys. Fluids 8(3), 826 (1996)

    Article  MathSciNet  Google Scholar 

  35. Nompelis, I.: Computational study of hypersonic double-cone experiments for code validation. Ph.D. thesis, University of Minnesota, Minneapolis MN (2004)

  36. Steger, J.L., Warming, R.F.: Flux vector splitting of the inviscid gasdynamic equations with application to finite-difference methods. J. Comput. Phys. 40(2), 263 (1981)

    Article  MathSciNet  Google Scholar 

  37. MacCormack, R.W., Candler, G.V.: The solution of the Navier–Stokes equations using Gauss–Seidel line relaxation. Comput. Fluids 17, 135 (1989)

    Article  Google Scholar 

  38. Roe, P.L.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 135(2), 250 (1997)

    Article  MathSciNet  Google Scholar 

  39. van Leer, B.: Toward the ultimate conservative difference scheme. J. Comput. Phys. 135(2), 229 (1997)

    Article  Google Scholar 

  40. van Albada, G.D., van Leer, B., Roberts, W.W.: A comparative study of computational methods in cosmic gas dynamics. Astron. Astrophys. 108(1), 76 (1982)

    MATH  Google Scholar 

  41. Wright, M.J., Candler, G.V., Prampolini, M.: Data-parallel lower-upper relaxation method for the Navier–Stokes equations. AIAA J. 34(7), 1371 (1996)

    Article  Google Scholar 

  42. Johnson, E., Larsson, J., Bhangatwala, A.V., Cabot, W.H., Moin, P., Olson, B.J., Rawar, P.S., Shankar, S.K., Sjogreen, B., Yee, H.C., Zhong, X., Lele, S.K.: Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves. J. Comput. Phys. 229(4), 1213 (2010)

    Article  MathSciNet  Google Scholar 

  43. Rawat, P.S., Zhong, X.: On high-order shock-fitting and front-tracking schemes for numerical simulation of shock-disturbance interactions. J. Comput. Phys. 229(19), 6744 (2010)

    Article  MathSciNet  Google Scholar 

  44. MATLAB, version 9.7.0.1319299 (R2019b) The MathWorks Inc., Natick, Massachusetts (2019)

  45. Juniper, M.P.: The effect of confinement on the stability of two-dimensional shear flows. J. Fluid Mech. 565, 171 (2006)

    Article  Google Scholar 

  46. Rees, S.J., Juniper, M.P.: The effect of confinement on the stability of viscous planar jets and wakes. J. Fluid Mech. 656, 309 (2010)

    Article  MathSciNet  Google Scholar 

  47. Healey, J.J.: Inviscid axisymmetric absolute instability of swirling jets. J. Fluid Mech. 613, 1 (2008)

    Article  MathSciNet  Google Scholar 

  48. Healey, J.J.: Enhancing the absolute instability of a boundary layer by adding a far-away plate. J. Fluid Mech. 579, 29 (2007)

    Article  MathSciNet  Google Scholar 

  49. Paredes, P., Choudhari, M.M., Li, F.: Mechanism for frustum transition over blunt cones at hypersonic speeds. J. Fluid Mech. 894, A22 (2020)

    Article  MathSciNet  Google Scholar 

  50. Dwivedi, A., Hildebrand, N., Nichols, J.W., Candler, G.V., Jovanovic, M.R.: Transient growth analysis of oblique shock-wave/boundary-layer interactions at Mach 5.92. Phys. Rev. Fluids 5(6), 063904 (2020)

  51. Sidharth, G.S., Dwivedi, A., Candler, G.V., Nichols, J.W.: Onset of three-dimensionality in supersonic flow over a slender double wedge. Phys. Rev. Fluids 3(9), 093901 (2018)

    Article  Google Scholar 

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Acknowledgements

Support from ONR grant number N00014-19-1-2037 is gratefully acknowledged. The authors thank Prof. Graham Candler for helpful comments on an early version of this manuscript. We are also grateful to an anonymous reviewer for extensive, thoughtful comments that helped significantly improve this manuscript.

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  1. University of Minnesota, Minneapolis, USA

    David A. Cook & Joseph W. Nichols

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  1. David A. Cook
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  2. Joseph W. Nichols
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Correspondence to David A. Cook.

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The datasets generated during and/or analysed during the current study are available from the corresponding author on reasonable request.

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Communicated by Pino Martin.

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The authors gratefully acknowledge support for this research by the Office of Naval Research, Grant No. N00014-19-1-2037.

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Cook, D.A., Nichols, J.W. Free-stream receptivity of a hypersonic blunt cone using input–output analysis and a shock-kinematic boundary condition. Theor. Comput. Fluid Dyn. 36, 155–180 (2022). https://doi.org/10.1007/s00162-021-00597-5

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