Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-21T04:56:45.985Z Has data issue: false hasContentIssue false

Interactions between vortical, acoustic and thermal components during hypersonic transition

Published online by Cambridge University Press:  16 April 2019

S. Unnikrishnan*
Affiliation:
Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH 43210, USA
Datta V. Gaitonde
Affiliation:
Mechanical and Aerospace Engineering, The Ohio State University, Columbus, OH 43210, USA
*
Email address for correspondence: sasidharannair.1@osu.edu

Abstract

Discrete unstable modes of hypersonic laminar boundary layers, obtained from an eigenvalue analysis, provide insight into key transition scenarios. The character of such modes near the leading edge is often identified with the corresponding asymptotic free-stream behaviour of acoustic, vortical or entropic (thermal) content, which we designate fluid-thermodynamic (FT) components. In downstream regions, however, this direct one-to-one correspondence between discrete modes and FT components does not hold, since FT components interact in well-defined ways with the basic state and with each other (even under linear scenarios). In the present work, we perform an FT decomposition of discrete modes using momentum potential theory, to yield a physics-based analysis that complements linear stability theory in the linear regime, and seamlessly extends to the nonlinear domain where direct numerical simulations are appropriate. Linear and nonlinear saturated disturbance effects, different forcing types and wall thermal conditions are considered, with emphasis on phenomena occurring near stability-mode synchronization locations. The results show that, in the linear regime, each discrete mode contains all FT components, whose relative amplitudes vary with streamwise distance. Vortical components are always the largest, followed by thermal and acoustic components. These latter two show distinct fore and aft signatures near mode synchronization. The vortical component displays a series of rope-shaped recirculation-cell patterns across the generalized inflection point. However, both acoustic and thermal components display ‘trapped’ structures. The former contains an alternating monopole array between the wall and the critical layer, while the latter is confined to an undulating region between the wall and a wavy locus straddling the generalized inflection point. Nonlinear saturation in the region of Mack-mode growth further strengthens the rope-shaped structures in the vortical component and higher harmonics appear, whose form and location depend on the specific component. Wall cooling modifies the eigenfunctions such that the acoustic component accounts for more of its composition, consistent with its destabilization. Analysis of energy interactions among the FT components indicates that, even though the vorticity component is the largest, the thermal component induces the most significant source term for the growth of acoustic perturbations, possibly due to the trapped nature of both.

Type
JFM Papers
Copyright
© 2019 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anderson, J. D. 2000 Hypersonic and High Temperature Gas Dynamics. AIAA.Google Scholar
Bitter, N. P. & Shepherd, J. E. 2015 Stability of highly cooled hypervelocity boundary layers. J. Fluid Mech. 778, 586620.Google Scholar
Chu, B. T. & Kovásznay, L. S. G. 1958 Non-linear interactions in a viscous heat-conducting compressible gas. J. Fluid Mech. 3 (5), 494514.Google Scholar
Daviller, G., Jordan, P. & Comte, P. 2009 Flow decompositions for the study of source mechanisms. In 15th AIAA/CEAS Aeroacoustique Conference and Exhibit (30th AIAA Aeroacoustique Conference), Miami, Florida. AIAA Paper 2009–3305. AIAA.Google Scholar
Doak, P. E. 1989 Momentum potential theory of energy flux carried by momentum fluctuations. J. Sound Vib. 131 (1), 6790.Google Scholar
Doak, P. E. 1998 Fluctuating total enthalpy as the basic generalized acoustic field. Theor. Comput. Fluid Dyn. 10 (1–4), 115133.Google Scholar
Duck, P. W., Lasseigne, D. G. & Hussaini, M. Y. 1995 On the interaction between the shock wave attached to a wedge and freestream disturbances. Theor. Comput. Fluid Dyn. 7 (2), 119139.Google Scholar
Egorov, I. V., Fedorov, A. V. & Soudakov, V. G. 2006 Direct numerical simulation of disturbances generated by periodic suction–blowing in a hypersonic boundary layer. Theor. Comput. Fluid Dyn. 20 (1), 4154.Google Scholar
Egorov, I. V., Fedorov, A. V. & Soudakov, V. G. 2008 Receptivity of a hypersonic boundary layer over a flat plate with a porous coating. J. Fluid Mech. 601, 165187.Google Scholar
Erlebacher, G. & Hussaini, M. Y. 1990 Numerical experiments in supersonic boundary-layer stability. Phys. Fluids A 2 (1), 94104.Google Scholar
Fedorov, A. & Tumin, A. 2011 High-speed boundary-layer instability: old terminology and a new framework. AIAA J. 49 (8), 16471657.Google Scholar
Fedorov, A. V. 2003 Receptivity of a high-speed boundary layer to acoustic disturbances. J. Fluid Mech. 491, 101129.Google Scholar
Fedorov, A. V. 2011 Transition and stability of high-speed boundary layers. Annu. Rev. Fluid Mech. 43, 7995.Google Scholar
Fedorov, A. V. & Khokhlov, A. P. 1991 Excitation of unstable modes in a supersonic boundary layer by acoustic waves. Fluid Dyn. 26 (4), 531537.Google Scholar
Fedorov, A. V. & Khokhlov, A. P. 1992 Excitation and evolution of unstable disturbances in supersonic boundary layer. In Proceedings of ASME Fluids Engineering Conference, vol. 151, pp. 113. ASME.Google Scholar
Fedorov, A. V. & Tumin, A. 2003 Initial-value problem for hypersonic boundary-layer flows. AIAA J. 41 (3), 379389.Google Scholar
Kennedy, R. E., Laurence, S. J., Smith, M. S. & Marineau, E. C. 2018 Visualization of the second-mode instability on a sharp cone at Mach 14. In AIAA Aerospace Sciences Meeting. AIAA Paper 2018-2083. AIAA.Google Scholar
Kosinov, A. D., Maslov, A. A. & Shevelkov, S. G. 1990 Experiments on the stability of supersonic laminar boundary layers. J. Fluid Mech. 219, 621633.Google Scholar
Kovásznay, L. S. G. 1953 Turbulence in supersonic flow. J. Aeronaut. Sci. 20 (10), 657674.Google Scholar
Laurence, S. J., Wagner, A. & Hannemann, K. 2016 Experimental study of second-mode instability growth and breakdown in a hypersonic boundary layer using high-speed Schlieren visualization. J. Fluid Mech. 797, 471503.Google Scholar
van Leer, B. 1979 Towards the ultimate conservation difference scheme V. A second-order sequel to Godunov’s method. J. Comput. Phys. 32, 101136.Google Scholar
Lees, L. & Lin, C. C.1946 Investigation of the stability of the laminar boundary layer in a compressible fluid. NACA Tech. Note No. 1115. NACA.Google Scholar
Liang, X., Li, X., Fu, D. & Ma, Y. 2010 Effects of wall temperature on boundary layer stability over a blunt cone at Mach 7.99. Comput. Fluids 39 (2), 359371.Google Scholar
Lysenko, V. I. & Maslov, A. A. 1984 The effect of cooling on supersonic boundary-layer stability. J. Fluid Mech. 147, 3952.Google Scholar
Ma, Y. & Zhong, X. 2003a Receptivity of a supersonic boundary layer over a flat plate. Part 1. Wave structures and interactions. J. Fluid Mech. 488, 3178.Google Scholar
Ma, Y. & Zhong, X. 2003b Receptivity of a supersonic boundary layer over a flat plate. Part 2. Receptivity to free-stream sound. J. Fluid Mech. 488, 79121.Google Scholar
Ma, Y. & Zhong, X. 2005 Receptivity of a supersonic boundary layer over a flat plate. Part 3. Effects of different types of free-stream disturbances. J. Fluid Mech. 532, 63109.Google Scholar
Mack, L. M. 1975 Linear stability theory and the problem of supersonic boundary-layer transition. AIAA J. 13 (3), 278289.Google Scholar
Mack, L. M.1984 Boundary-layer linear stability theory. AGARD Rep. 709. AGARD.Google Scholar
Malik, M. R. 1990 Numerical methods for hypersonic boundary layer stability. J. Comput. Phys. 86 (2), 376413.Google Scholar
Masad, J. A., Nayfeh, A. H. & Al-Maaitah, A. A. 1992 Effect of heat transfer on the stability of compressible boundary layers. Comput. Fluids 21 (1), 4361.Google Scholar
Özgen, S. & Kırcalı, S. A. 2008 Linear stability analysis in compressible, flat-plate boundary-layers. Theor. Comput. Fluid Dyn. 22 (1), 120.Google Scholar
Pruett, C. D. & Chang, C.-L. 1995 Spatial direct numerical simulation of high-speed boundary-layer flows. Part II: Transition on a cone in Mach 8 flow. Theor. Comput. Fluid Dyn. 7 (5), 397424.Google Scholar
Reshotko, E. 1976 Boundary-layer stability and transition. Annu. Rev. Fluid Mech. 8 (1), 311349.Google Scholar
Reshotko, E. 2008 Transition issues for atmospheric entry. J. Spacecr. Rockets 45 (2), 161164.Google Scholar
Roe, P. L. 1981 Approximate Riemann solvers, parameter vectors and difference schemes. J. Comput. Phys. 43, 357372.Google Scholar
Schmid, P. J. 2010 Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 528.Google Scholar
Schneider, S. P. 2001 Effects of high-speed tunnel noise on laminar–turbulent transition. J. Spacecr. Rockets 38 (3), 323333.Google Scholar
Shu, C. W. & Osher, S. 1988 Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77 (2), 439471.Google Scholar
Sivasubramanian, J. & Fasel, H. F. 2014 Numerical investigation of the development of three-dimensional wavepackets in a sharp cone boundary layer at Mach 6. J. Fluid Mech. 756, 600649.Google Scholar
Sivasubramanian, J. & Fasel, H. F. 2015 Direct numerical simulation of transition in a sharp cone boundary layer at Mach 6: fundamental breakdown. J. Fluid Mech. 768, 175218.Google Scholar
Soudakov, V. G., Egorov, I. V., Fedorov, A. V. & Novikov, A. V. 2016 Numerical simulation of receptivity and stability of a supersonic boundary layer. In 27th International Congress of the Aeronautical Sciences. ICAS.Google Scholar
Stetson, K. & Kimmel, R. 1993 On the breakdown of a hypersonic laminar boundary layer. In 31st Aerospace Sciences Meeting. AIAA Paper 1993-0896. AIAA.Google Scholar
Stetson, K. F. & Kimmel, R. 1992 On hypersonic boundary-layer stability. In 30th Aerospace Sciences Meeting and Exhibit, Reno, NV. AIAA Paper 1992-737. AIAA.Google Scholar
Stetson, K. F., Thompson, E. R., Donaldson, J. C. & Siler, L. G. 1989 Laminar boundary layer stability experiments on a cone at Mach 8, Part 5: Tests with a cooled model. In 20th Fluid Dynamics, Plasma Dynamics and Lasers Conference, Buffalo, New York. AIAA Paper 1989-1895. AIAA.Google Scholar
Taira, K., Brunton, S. L., Dawson, S. T. M., Rowley, C. W., Colonius, T., McKeon, B. J., Schmidt, O. T., Gordeyev, S., Theofilis, V. & Ukeiley, L. S. 2017 Modal analysis of fluid flows: an overview. AIAA J. 55 (12), 40134041.Google Scholar
Towne, A., Schmidt, O. T. & Colonius, T. 2018 Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis. J. Fluid Mech. 847, 821867.Google Scholar
Unnikrishnan, S. & Gaitonde, D. V. 2016 Acoustic, hydrodynamic and thermal modes in a supersonic cold jet. J. Fluid Mech. 800, 387432.Google Scholar
Wang, X. & Zhong, X. 2009 Effect of wall perturbations on the receptivity of a hypersonic boundary layer. Phys. Fluids 21 (4), 044101.Google Scholar
Wang, X., Zhong, X. & Ma, Y. 2011 Response of a hypersonic boundary layer to wall blowing–suction. AIAA J. 49 (7), 13361353.Google Scholar
Yao, Y., Krishnan, L., Sandham, N. D. & Roberts, G. T. 2007 The effect of Mach number on unstable disturbances in shock/boundary-layer interactions. Phys. Fluids 19 (5), 054104.Google Scholar
Zhang, S., Liu, J. & Luo, J. 2016 Effect of wall-cooling on Mack-mode instability in high speed flat-plate boundary layers. Z. Angew. Math. Mech. Appl. Math. Mech. 37 (9), 12191230.Google Scholar
Zhong, X. 2001 Leading-edge receptivity to free-stream disturbance waves for hypersonic flow over a parabola. J. Fluid Mech. 441, 315367.Google Scholar
Zhong, X. & Wang, X. 2012 Direct numerical simulation on the receptivity, instability, and transition of hypersonic boundary layers. Annu. Rev. Fluid Mech. 44, 527561.Google Scholar
Zhou, X. C. & Gore, J. P. 1998 Experimental estimation of thermal expansion and vorticity distribution in a buoyant diffusion flame. Symposium (International) Combustion 27 (2), 27672773.Google Scholar