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Nonlinearly most dangerous disturbance for high-speed boundary-layer transition

  • Reza Jahanbakhshi (a1) and Tamer A. Zaki (a1)


Laminar-to-turbulent transition in a zero-pressure-gradient boundary layer at Mach 4.5 is studied using direct numerical simulations. For a given level of total disturbance energy, the inflow spectrum was designed to correspond to the nonlinearly most dangerous condition that leads to the earliest possible transition Reynolds number. The synthesis of the inlet disturbance is formulated as a constrained optimization, where the control vector is comprised of the amplitudes and relative phases of the inlet modes; the constraints are the prescribed total energy and that the flow evolution satisfies the full nonlinear compressible Navier–Stokes equations; the cost function is defined in terms of the mean skin-friction coefficient and, once maximized, ensures the earliest possible transition location. An ensemble-variational (EnVar) technique is developed to solve the optimization problem. Starting from an initial guess, here a broadband disturbance, EnVar updates the estimate of the control vector at the end of each iteration using the gradient of the cost function, which is evaluated from the outcomes of an ensemble of possible solutions. Two inflow conditions are computed, each corresponding to a different level of energy, and their spectra are contrasted: the lower-energy case includes two normal acoustic waves and one oblique vorticity perturbation, whereas the higher-energy condition consists of oblique acoustic and vorticity waves. The focus is placed on the former case because it cannot be categorized as any of the classical breakdown scenarios (fundamental, subharmonic or oblique), while the higher-energy condition undergoes a second-mode oblique transition. At the lower energy level, the instability wave that initiates the rapid breakdown to turbulence is not present at the inlet plane. Instead, it appears at a downstream location after a series of nonlinear interactions that spur the fastest onset of turbulence. The results from the nonlinearly most potent inflow condition are also compared to other inlet disturbances that are selected solely based on linear theory, and which all yield relatively delayed transition onset.


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Adrian, R. J. 2007 Hairpin vortex organization in wall turbulence. Phys. Fluids 19 (4), 041301.
Bertolotti, F. P., Herbert, T. & Spalart, P. R. 1992 Linear and nonlinear stability of the Blasius boundary layer. J. Fluid Mech. 242, 441474.
Bewley, T. R., Moin, P. & Temam, R. 2001 DNS-based predictive control of turbulence: an optimal benchmark for feedback algorithms. J. Fluid Mech. 447, 179225.
Casper, K. M., Beresh, S. J., Henfling, J. F., Spillers, R. W., Pruett, B. O. M. & Schneider, S. P. 2016 Hypersonic wind-tunnel measurements of boundary-layer transition on a slender cone. AIAA J. 54 (1), 12501263.
Chang, C.-L. & Malik, M. R. 1994 Oblique-mode breakdown and secondary instability in supersonic boundary layers. J. Fluid Mech. 273, 323360.
Cherubini, S., De Palma, P., Robinet, J.-C. & Bottaro, A. 2011 The minimal seed of turbulent transition in the boundary layer. J. Fluid Mech. 689, 221253.
Cherubini, S., Robinet, J.-C. & De Palma, P. 2013 Nonlinear control of unsteady finite-amplitude perturbations in the Blasius boundary-layer flow. J. Fluid Mech. 737, 440465.
Cheung, L. C. & Zaki, T. A. 2010 Linear and nonlinear instability waves in spatially developing two-phase mixing layers. Phys. Fluids 22 (5), 052103.
Colburn, C. H., Cessna, J. B. & Bewley, T. R. 2011 State estimation in wall-bounded flow systems. Part 3. The ensemble Kalman filter. J. Fluid Mech. 682, 289303.
Evensen, G. 2009 Data Assimilation: The Ensemble Kalman Filter. Springer.
Farano, M., Cherubini, S., Robinet, J.-C. & De Palma, P. 2015 Hairpin-like optimal perturbations in plane poiseuille flow. J. Fluid Mech. 775, R2.
Fedorov, A. 2011 Transition and stability of high-speed boundary layers. Annu. Rev. Fluid Mech. 43, 7995.
Franko, K. J. & Lele, S. 2014 Effect of adverse pressure gradient on high speed boundary layer transition. Phys. Fluids 26 (2), 024106.
Franko, K. J. & Lele, S. K. 2013 Breakdown mechanisms and heat transfer overshoot in hypersonic zero pressure gradient boundary layers. J. Fluid Mech. 730, 491532.
Gao, X., Wang, Y., Overton, N., Zupanski, M. & Tu, X. 2017 Data-assimilated computational fluid dynamics modeling of convection-diffusion-reaction problems. J. Comput. Sci. 21, 3859.
Haack, A., Gerding, M. & Lübken, F.-J. 2014 Characteristics of stratospheric turbulent layers measured by LITOS and their relation to the Richardson number. J. Geophys. Res. 119 (18).
Harvey, W. D.1978 Influence of free-stream disturbances on boundary-layer transition, NASA Tech. Memorandum 78635.
Herbert, T. 1988 Secondary instability of boundary layers. Annu. Rev. Fluid Mech. 20 (1), 487526.
Herbert, T. 1997 Parabolized stability equations. Annu. Rev. Fluid Mech. 29 (1), 245283.
Jiang, L., Choudhari, M., Chang, C.-L. & Liu, C. 2006 Numerical simulations of laminar-turbulent transition in supersonic boundary layer. In 36th AIAA Fluid Dynamics Conference and Exhibit held on 05–08 June 2006 in San Francisco, California, p. 3224. AIAA Aerospace Research Central.
Johnsen, E., Larsson, J., Bhagatwala, A. V., Cabot, W. H., Moin, P., Olson, B. J., Rawat, P. S., Shankar, S. K., Sjögreen, B., Yee, H. C. et al. 2010 Assessment of high-resolution methods for numerical simulations of compressible turbulence with shock waves. J. Comput. Phys. 229 (4), 12131237.
Juliano, T. J., Adamczak, D. & Kimmel, R. L. 2015 HIFiRE-5 flight test results. J. Spacecr. Rockets 52 (3), 650663.
Kato, H., Yoshizawa, A., Ueno, G. & Obayashi, S. 2015 A data assimilation methodology for reconstructing turbulent flows around aircraft. J. Comput. Phys. 283, 559581.
Kawai, S. & Larsson, J. 2012 Wall-modeling in large eddy simulation: length scales, grid resolution, and accuracy. Phys. Fluids 24 (1), 015105.
Kennedy, R. E., Laurence, S. J., Smith, M. S. & Marineau, E. C. 2018 Investigation of the second-mode instability at Mach 14 using calibrated Schlieren. J. Fluid Mech. 845.
Kimmel, R. L., Adamczak, D. W., Hartley, D., Alesi, H., Frost, M. A., Pietsch, R., Shannon, J. & Silvester, T. 2018 Hypersonic international flight research experimentation-5b flight overview. J. Spacecr. Rockets 112.
Larsson, J. & Lele, S. K. 2009 Direct numerical simulation of canonical shock/turbulence interaction. Phys. Fluids 21 (12), 126101.
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.
Leyva, I. A. 2017 The relentless pursuit of hypersonic flight. Phys. Today 70 (11), 3036.
Luchini, P. & Bottaro, A. 2014 Adjoint equations in stability analysis. Annu. Rev. Fluid Mech. 46, 493517.
Mack, L. M.1984 Boundary-layer linear stability theory. Tech. Rep., California Institute of Technology, Jet Propulsion Laboratory.
Marxen, O. & Zaki, T. A. 2019 Turbulence in intermittent transitional boundary layers and in turbulence spots. J. Fluid Mech. 860, 350383.
Mayer, C. S. J., Von Terzi, D. A. & Fasel, H. F. 2011 Direct numerical simulation of complete transition to turbulence via oblique breakdown at Mach 3. J. Fluid Mech. 674, 542.
Mons, V., Chassaing, J.-C., Gomez, T. & Sagaut, P. 2016 Reconstruction of unsteady viscous flows using data assimilation schemes. J. Comput. Phys. 316, 255280.
Narasimha, R. 1985 The laminar-turbulent transition zone in the boundary layer. Prog. Aerosp. Sci. 22 (1), 2980.
Nocedal, J. & Wright, S. 2006 Numerical Optimization. Springer.
Novikov, A. V. & Egorov, I. 2016 Direct numerical simulations of transitional boundary layer over a flat plate in hypersonic free-stream. In 46th AIAA Fluid Dynamics Conference, p. 3952.
Park, J. & Zaki, T. A. 2019 Sensitivity of high-speed boundary-layer stability to base-flow distortion. J. Fluid Mech. 859, 476515.
Pringle, C. C. T. & Kerswell, R. R. 2010 Using nonlinear transient growth to construct the minimal seed for shear flow turbulence. Phys. Rev. Lett. 105 (15), 154502.
Rabin, S. M. E., Caulfield, C. P. & Kerswell, R. R. 2012 Triggering turbulence efficiently in plane Couette flow. J. Fluid Mech. 712, 244272.
Schneider, S. P. 1999 Flight data for boundary-layer transition at hypersonic and supersonic speeds. J. Spacecr. Rockets 36 (1), 820.
Schneider, S. P. 2015 Developing mechanism-based methods for estimating hypersonic boundary-layer transition in flight: the role of quiet tunnels. Prog. Aerosp. Sci. 72, 1729.
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.
Sivasubramanian, J. & Fasel, H. F. 2016 Direct numerical simulation of laminar-turbulent transition in a flared cone boundary layer at Mach 6. In 54th AIAA Aerospace Sciences Meeting, p. 0846.
Stanfield, S. A., Kimmel, R. L., Adamczak, D. & Juliano, T. J. 2015 Boundary-layer transition experiment during reentry of HIFiRE-1. J. Spacecr. Rockets 52 (3), 637649.
Sutherland, W. 1893 LII. The viscosity of gases and molecular force. The London, Edinburgh, Dublin Philos. Mag. J. Sci. 36 (223), 507531.
Suzuki, T. 2012 Reduced-order Kalman-filtered hybrid simulation combining particle tracking velocimetry and direct numerical simulation. J. Fluid Mech. 709, 249288.
Thumm, A., Wolz, W. & Fasel, H. 1990 Numerical simulation of spatially growing three-dimensional disturbance waves in compressible boundary layers. In IUTAM Symposium on Laminar-Turbulent Transition, pp. 303308. Springer.
Wächter, A. & Biegler, L. T. 2006 On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. Math. Program. 106 (1), 2557.
Xiao, D. & Papadakis, G. 2017 Nonlinear optimal control of bypass transition in a boundary layer flow. Phys. Fluids 29 (5), 054103.
Yang, Y., Robinson, C., Heitz, D. & Mémin, E. 2015 Enhanced ensemble-based 4DVar scheme for data assimilation. Comput. Fluids 115, 201210.
Zaki, T. A. & Durbin, P. A. 2005 Mode interaction and the bypass route to transition. J. Fluid Mech. 531, 85111.
Zaki, T. A. & Durbin, P. A. 2006 Continuous mode transition and the effects of pressure gradient. J. Fluid Mech. 563, 357388.
Zhang, C., Tang, Q. & Lee, C. 2013 Hypersonic boundary-layer transition on a flared cone. Acta Mechanica Sin. 29 (1), 4854.
Zhang, C., Zhu, Y., Chen, X., Yuan, H., Wu, J., Chen, S., Lee, C. & Gad-el Hak, M. 2015 Transition in hypersonic boundary layers. AIP Advances 5 (10), 107137.
Zhao, R., Wen, C. Y., Tian, X. D., Long, T. H. & Yuan, W. 2018 Numerical simulation of local wall heating and cooling effect on the stability of a hypersonic boundary layer. Intl J. Heat Mass Transfer 121, 986998.
Zhong, X. & Wang, X. 2012 Direct numerical simulation on the receptivity, instability, and transition of hypersonic boundary layers. Annu. Rev. Fluid Mech. 44, 527561.
Zhu, Y., Chen, X., Wu, J., Chen, S., Lee, C. & Gad-el Hak, M. 2018 Aerodynamic heating in transitional hypersonic boundary layers: role of second-mode instability. Phys. Fluids 30 (1), 011701.
Zhu, Y., Zhang, C., Chen, X., Yuan, H., Wu, J., Chen, S., Lee, C. & Gad-el Hak, M. 2016 Transition in hypersonic boundary layers: Role of dilatational waves. AIAA J. 30393049.
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Jahanbakhshi and Zaki supplementary movie 1
Side view of numerical Schlieren contours from case E1N at $z = Lz/2$.

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Jahanbakhshi and Zaki supplementary movie 2
Side view of numerical Schlieren contours from case E2N at $z = Lz/4$.

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Nonlinearly most dangerous disturbance for high-speed boundary-layer transition

  • Reza Jahanbakhshi (a1) and Tamer A. Zaki (a1)


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