zbMATH — the first resource for mathematics

3D global optimal forcing and response of the supersonic boundary layer. (English) Zbl 1453.76194
Summary: 3D optimal forcing and response of a 2D supersonic boundary layer are obtained by computing the largest singular value and the associated singular vectors of the global resolvent matrix. This approach allows to take into account both convective-type and component-type non-normalities responsible for the non-modal growth of perturbations in noise selective amplifier flows. It is moreover a fully non-parallel approach that does not require any particular assumptions on the baseflow. The numerical method is based on the explicit calculation of the Jacobian matrix proposed by [C. Mettot et al., J. Comput. Phys. 269, 234–258 (2014; Zbl 1349.76129)] for 2D perturbations. This strategy uses the numerical residual of the compressible Navier-Stokes equations imported from a finite-volume solver that is then linearised employing a finite difference method. Extension to 3D perturbations, which are expanded into modes of wave number, is here proposed by decomposing the Jacobian matrix according to the direction of the derivatives contained in its coefficients. Validation is performed on a Blasius boundary layer and a supersonic boundary layer, in comparison respectively to global and local results. Application of the method to a boundary layer at \(M = 4.5\) recovers three regions of receptivity in the frequency-transverse wave number space. Finally, the energy growth of each optimal response is studied and discussed.
Reviewer: Reviewer (Berlin)
76N20 Boundary-layer theory for compressible fluids and gas dynamics
76M12 Finite volume methods applied to problems in fluid mechanics
76J20 Supersonic flows
76N06 Compressible Navier-Stokes equations
Full Text: DOI
[1] Mettot, C.; Renac, F.; Sipp, D., Computation of eigenvalue sensitivity to base flow modifications in a discrete framework: application to open-loop control, J. Comput. Phys., 269, 234-258 (2014) · Zbl 1349.76129
[2] Huerre, P.; Monkewitz, P. A., Local and global instabilities in spatially developing flows, Annu. Rev. Fluid Mech., 22, 1, 473-537 (1990) · Zbl 0734.76021
[3] Schmid, P. J.; Henningson, D. S., Stability and Transition in Shear Flows, vol. 142 (2012), Springer Science & Business Media
[4] Mack, L. M., A numerical study of the temporal eigenvalue spectrum of the Blasius boundary layer, J. Fluid Mech., 73, 3, 497-520 (1976) · Zbl 0339.76030
[5] Monkewitz, P. A., The absolute and convective nature of instability in two-dimensional wakes at low Reynolds numbers, Phys. Fluids, 31, 5, 999-1006 (1988)
[6] Michalke, A., On spatially growing disturbances in an inviscid shear layer, J. Fluid Mech., 23, 3, 521-544 (1965)
[7] Huerre, P.; Monkewitz, P. A., Absolute and convective instabilities in free shear layers, J. Fluid Mech., 159, 151-168 (1985) · Zbl 0588.76067
[8] Lees, L.; Lin, C., Investigation of the Stability of the Laminar Boundary Layer in a Compressible Fluid (1946), National Advisory Committee for Aeronautics: National Advisory Committee for Aeronautics Washington, DC, Tech. Rep. · Zbl 0063.03471
[9] Dunn, D. W.; Lin, C. C., On the Stability of the Laminar Boundary Layer in a Compressible Fluid (1953), Massachusetts Institute of Technology, Ph.D. thesis
[10] Mack, L. M., Numerical Calculation of the Stability of the Compressible, Laminar Boundary Layer, Jet Propulsion Laboratory, California Institute of Technology (1960)
[11] Mack, L. M., Boundary-Layer Linear Stability Theory (1984), Tech. Rep., DTIC Document
[12] Squire, H. B., On the stability for three-dimensional disturbances of viscous fluid flow between parallel walls, Proc. R. Soc. Lond., a Contain. Pap. Math. Phys. Character, 142, 847, 621-628 (1933) · JFM 59.1458.02
[13] Mack, L. M., Linear stability theory and the problem of supersonic boundary-layer transition, AIAA J., 13, 3, 278-289 (1975)
[14] Kendall, J., Wind tunnel experiments relating to supersonic and hypersonic boundary-layer transition, AIAA J., 13, 3, 290-299 (1975)
[15] Stetson, K.; Kimmel, R., On hypersonic boundary-layer stability, (30th Aerospace Sciences Meeting and Exhibit (1992)), Article 737 pp.
[16] El-Hady, N. M., Nonparallel instability of supersonic and hypersonic boundary layers, Phys. Fluids A, Fluid Dyn., 3, 9, 2164-2178 (1991) · Zbl 0746.76058
[17] Chang, C. L.; Malik, M., Non-parallel stability of compressible boundary layers, (23rd Fluid Dynamics, Plasmadynamics, and Lasers Conference (1993)), 2912
[18] Farrell, B. F., Optimal excitation of perturbations in viscous shear flow, Phys. Fluids, 31, 8, 2093 (1988)
[19] Hanifi, A.; Schmid, P. J.; Henningson, D. S., Transient growth in compressible boundary layer flow, Phys. Fluids (1994 – present), 8, 3, 826-837 (1996) · Zbl 1025.76536
[20] Tumin, A.; Reshotko, E., Spatial theory of optimal disturbances in boundary layers, Phys. Fluids, 13, 7, 2097-2104 (2001) · Zbl 1184.76562
[21] Tumin, A.; Reshotko, E., Optimal disturbances in compressible boundary layers, AIAA J., 41, 12, 2357-2363 (2003)
[22] Zuccher, S.; Tumin, A.; Reshotko, E., Parabolic approach to optimal perturbations in compressible boundary layers, J. Fluid Mech., 556, 189-216 (2006) · Zbl 1147.76050
[23] Tempelmann, D.; Hanifi, A.; Henningson, D. S., Spatial optimal growth in three-dimensional compressible boundary layers, J. Fluid Mech., 704, 251-279 (2012) · Zbl 1246.76025
[24] Herbert, T., Parabolized stability equations, Annu. Rev. Fluid Mech., 29, 1, 245-283 (1997)
[25] Paredes, P.; Choudhari, M. M.; Li, F.; Chang, C. L., Transient growth analysis of compressible boundary layers with parabolized stability equations, AIAA Pap., 51, 2016 (2016)
[26] Paredes, P.; Choudhari, M. M.; Li, F.; Chang, C. L., Optimal growth in hypersonic boundary layers, AIAA J., 54, 10, 3050-3061 (2016)
[27] Theofilis, V., Advances in global linear instability analysis of nonparallel and three-dimensional flows, Prog. Aerosp. Sci., 39, 4, 249-315 (2003)
[28] Bres, G. A.; Colonius, T., Three-dimensional instabilities in compressible flow over open cavities, J. Fluid Mech., 599, 309-339 (2008) · Zbl 1151.76468
[29] Robinet, J. C., Bifurcations in shock-wave/laminar-boundary-layer interaction: global instability approach, J. Fluid Mech., 579, 85-112 (2007) · Zbl 1175.76065
[30] Crouch, J.; Garbaruk, A.; Magidov, D., Predicting the onset of flow unsteadiness based on global instability, J. Comput. Phys., 224, 2, 924-940 (2007) · Zbl 1123.76018
[31] Sipp, D.; Marquet, O.; Meliga, P.; Barbagallo, A., Dynamics and control of global instabilities in open-flows: a linearized approach, Appl. Mech. Rev., 63, 3, Article 030801 pp. (2010)
[32] Ehrenstein, U.; Gallaire, F., On two-dimensional temporal modes in spatially evolving open flows: the flat-plate boundary layer, J. Fluid Mech., 536, 209-218 (2005) · Zbl 1073.76027
[33] Alizard, F.; Robinet, J. C., Spatially convective global modes in a boundary layer, Phys. Fluids, 19, 11, Article 114105 pp. (2007) · Zbl 1182.76019
[34] Monokrousos, A.; Åkervik, E.; Brandt, L.; Henningson, D. S., Global three-dimensional optimal disturbances in the blasius boundary-layer flow using time-steppers, J. Fluid Mech., 650, 181-214 (2010) · Zbl 1189.76192
[35] Sipp, D.; Marquet, O., Characterization of noise amplifiers with global singular modes: the case of the leading-edge flat-plate boundary layer, Theor. Comput. Fluid Dyn., 27, 5, 617-635 (2013)
[36] Schmid, P. J., Nonmodal stability theory, Annu. Rev. Fluid Mech., 39, 129-162 (2007) · Zbl 1296.76055
[37] Chomaz, J. M., Global instabilities in spatially developing flows: non-normality and nonlinearity, Annu. Rev. Fluid Mech., 37, 357-392 (2005) · Zbl 1117.76027
[38] Marquet, O.; Lombardi, M.; Chomaz, J. M.; Sipp, D.; Jacquin, L., Direct and adjoint global modes of a recirculation bubble: lift-up and convective non-normalities, J. Fluid Mech., 622, 1-21 (2009) · Zbl 1165.76337
[39] Landahl, M., A note on an algebraic instability of inviscid parallel shear flows, J. Fluid Mech., 98, 2, 243-251 (1980) · Zbl 0428.76049
[40] Orr, W. M., The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Part I: a perfect liquid, (Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences. Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences, JSTOR, vol. 27 (1907)), 9-68
[41] Nichols, J. W.; Lele, S. K., Global modes and transient response of a cold supersonic jet, J. Fluid Mech., 669, 225-241 (2011) · Zbl 1225.76116
[42] Garnaud, X.; Lesshafft, L.; Schmid, P.; Huerre, P., Modal and transient dynamics of jet flows, Phys. Fluids, 25, 4, Article 044103 pp. (2013)
[43] Sartor, F.; Mettot, C.; Bur, R.; Sipp, D., Unsteadiness in transonic shock-wave/boundary-layer interactions: experimental investigation and global stability analysis, J. Fluid Mech., 781, 550-577 (2015)
[44] Sutherland, W., LII. The viscosity of gases and molecular force, Lond. Edinb. Dublin Philos. Mag., J. Sci., 36, 223, 507-531 (1893) · JFM 25.1544.01
[45] Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction (2013), Springer Science & Business Media
[46] Liou, M. S., A sequel to AUSM: AUSM+, J. Comput. Phys., 129, 2, 364-382 (1996) · Zbl 0870.76049
[47] Boin, J. P.; Robinet, J. C.; Corre, C.; Deniau, H., 3d steady and unsteady bifurcations in a shock-wave/laminar boundary layer interaction: a numerical study, Theor. Comput. Fluid Dyn., 20, 3, 163-180 (2006) · Zbl 1112.76350
[48] Jameson, A., Time dependent calculations using multigrid, with applications to unsteady flows past airfoils and wings, AIAA Pap., 1596, 1991 (1991)
[49] Mettot, C., Linear Stability, Sensitivity, and Passive Control of Turbulent Flows Using Finite Differences (2013), Palaiseau, Ecole polytechnique, Ph.D. thesis
[50] Knoll, D. A.; Keyes, D. E., Jacobian-free Newton-Krylov methods: a survey of approaches and applications, J. Comput. Phys., 193, 2, 357-397 (2004) · Zbl 1036.65045
[51] Chu, B. T., On the energy transfer to small disturbances in fluid flow (part i), Acta Mech., 1, 3, 215-234 (1965)
[52] Theofilis, V., Global linear instability, Annu. Rev. Fluid Mech., 43, 319-352 (2011) · Zbl 1299.76074
[53] Balay, S.; Abhyankar, S.; Adams, M. F.; Brown, J.; Brune, P.; Buschelman, K. (2016), PETSc Web page
[54] Amestoy, P. R.; Duff, I. S.; L’Excellent, J. Y.; Koster, J., A fully asynchronous multifrontal solver using distributed dynamic scheduling, SIAM J. Matrix Anal. Appl., 23, 1, 15-41 (2001) · Zbl 0992.65018
[55] Balay, S.; Abhyankar, S.; Adams, M. F.; Brown, J.; Brune, P.; Buschelman, K., PETSc users manual (2016), Argonne National Laboratory, Tech. Rep. ANL-95/11 - Revision 3.7
[57] Hildebrand, N.; Dwivedi, A.; Nichols, J. W.; Jovanović, M. R.; Candler, G. V., Simulation and stability analysis of oblique shock-wave/boundary-layer interactions at mach 5.92, Phys. Rev. Fluids, 3, 1, Article 013906 pp. (2018)
[58] Özgen, S.; Kırcalı, S. A., Linear stability analysis in compressible, flat-plate boundary-layers, Theor. Comput. Fluid Dyn., 22, 1, 1-20 (2008) · Zbl 1161.76471
[59] Mack, L. M., Review of linear compressible stability theory, (Stability of Time Dependent and Spatially Varying Flows (1987), Springer), 164-187
[60] Ellingsen, T.; Palm, E., Stability of linear flow, Phys. Fluids, 18, 4, 487-488 (1975) · Zbl 0308.76030
[61] Erlebacher, G.; Hussaini, M., Numerical experiments in supersonic boundary-layer stability, Phys. Fluids A, Fluid Dyn., 2, 1, 94-104 (1990)
[62] Malik, M. R.; Balakumar, P., Acoustic receptivity of mach 4.5 boundary layer with leading-edge bluntness, Theor. Comput. Fluid Dyn., 21, 5, 323-342 (2007) · Zbl 1161.76560
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.