11-Control of second mode





E-mail: celepm@coria.fr ,*E-mail: ahadjadj@coria.fr ,**E-mail: msshadloo@coria.fr

CORIA-UMR 6614, CNRS-University, INSA of Rouen and Normandy University, 76800, Saint-Étienne-du-Rouvray, France


***E-mail: markus.kloker@iag.uni-stuttgart.de

Institute of Aerodynamics and Gas Dynamics, University of Stuttgart, 70550 Stuttgart, Germany



Key Words: Transition control, second-mode instability, velocity streaks.



In high-speed flows, aerodynamic bodies are naturally subject to high viscous drag and thermal loads. These loads become more severe when the flow transits from the laminar to the turbulent regime. The transition leads to a drastic increase in skin friction and aerodynamic heating, which can be up to an order of magnitude higher depending on the location of the transition onset.

The resulting thermal load may cause significant mechanical damage and possible fatigue-related failures. To prevent such impacts, thermal protection systems must be integrated into the vehicle’s design, which can pose technical and economic complications.

To mitigate these issues, a comprehensive understanding of transition mechanisms and the development of control systems to maintain laminar flow as long as possible are crucial.


Various active and passive control techniques can be used to control transition, among which the streak employment method shows great promise. This method involves generating narrowly-spaced, streamwise-decaying streaks that represent alternating high/low-speed regions in the flow field.

While the method has been successfully tested numerically and experimentally for incompressible flows, its application in super/hypersonic flow regimes has been limited to recent numerical investigations.

Direct-numerical-simulation (DNS) studies at Mach 2.0 found that streaks with four-to-five times the spanwise wavenumber of the most amplified oblique first-mode disturbance are the most effective in suppressing the instability, both for a single mode pair as well as broad-band disturbances [1, 2].

Furthermore, a numerical study utilizing nonlinear Parabolic Stability Equations (PSE) extended the application of streaks to high-supersonic and hypersonic regimes. The study showed that transiently growing sub-optimal streaks can effectively stabilize the two-dimensional second mode instability both in supersonic/hypersonic boundary layers [3].

In a concurrent study of hypersonic flow over a circular cone, weakly non-linear optimally growing streaks were found to be stabilizing the planar 2nd mode, while the stability of the oblique 2nd modes was highly dependent on the streak amplitude [4].

In both of these studies the mean-flow-deformation (MFD) was found to be responsible for stabilizing the planar mode.

In a recent DNS study, the stability of the flow distorted by optimally growing streaks was investigated under the effect of white-noise forcing [5]. Confirming the findings of Paredes et al. [4], the stabilization of the oblique 2nd mode was achieved below a certain streak amplitude, except for the perturbations carrying half the spanwise wavenumber of the streaks. This wavenumber family of disturbances amplifies irrespective of the streak amplitude while exhibiting a secondary peak frequency in the energy spectrum.


In high-speed flows, the assumption of adiabatic flow can be maintained in experimental facilities when the stagnation temperature is low. However, at flight conditions an aerodynamic body experiences cooling due to surface radiation increasing with Mach number. Considering the stabilizing effect of cooling on the 1st mode and the destabilizing effect on the 2nd mode [6], the latter is expected to play a major role in hypersonic regime.

In a low-disturbance, flight-like environment, numerical [7] and experimental investigations [8] have shown that the 2nd mode (Mack mode) has been found to be the predominant instability for Mach numbers higher than about 4. As the two-dimensional 2nd mode amplifies the most, attention has been mostly given to the fundamental and subharmonic resonance mechanisms. However, Linear Stability Theory (LST) also indicates that oblique waves with low oblique wave angles can amplify almost as strong as the two-dimensional 2nd mode, leading to flow transition [9].


This study is aimed at investigating the use of the streak employment technique in controlling the 2nd-mode induced full laminar breakdown using DNS.

The transition is intentionally initiated by a single pair of selected perturbations, and a blowing/suction strip at the wall is applied to trigger the control streaks. The initial investigations focus on the interaction between the velocity streaks with a pair of 2nd-mode oblique waves with a low obliqueness angle in the linear and fully non-linear regimes.

Figure 1 compares the perturbed flow in the absence and presence of the control streaks, having four times the (low) spanwise wavenumber of the oblique wave pair, demonstrating successful suppression of the unsteady disturbance and preservation of the laminar boundary layer in the domain.

Further analysis is currently being performed to examine the evolution of the streaks since their amplification/attenuation depends on their initial amplitude. Once the basic physical mechanisms of the control-streak behaviour are understood, a similar study will be conducted in a flow where the breakdown is initiated by fundamental resonance involving two- and three-dimensional 2nd-mode disturbances, and subsequently broad-band disturbances.



The authors acknowledge the access to French HPC resources provided by the French regional computing center for Normandy (CRIANN) (Grant No. 2017002). The first author would like to express his gratitude for the funding provided by the Region of Normandy, France, for his doctoral studies.



[1] S. Sharma, M. S. Shadloo, A. Hadjadj and M. J. Kloker, “Control of oblique-type breakdown in a supersonic boundary layer employing streaks,” Journal of Fluid Mechanics, vol. 873, pp. 1072-1089, 2019.
[2] S. Kneer, Z. Guo and M. J. Kloker, “Control of laminar breakdown in a supersonic boundary layer employing streaks,” Journal of Fluid Mechanics, vol. 932, p. A53, 2022.
[3] J. Ren, S. Fu and A. Hanifi, “Stabilization of the hypersonic boundary layer by finite-amplitude streaks,” Physics of Fluids, vol. 28, no. 2, p. 024110, 2016.
[4] P. Paredes, M. M. Choudhari and F. Li, “Transition due to streamwise streaks in a supersonic flat plate boundary layer,” Physical Review Fluids, vol. 1, no. 8, p. 083601, 2016.
[5] C. Caillaud, G. Lehnasch, P. Jordan, E. Martini, E. Goncalves and L. Hallo, “Linear and non-linear dynamics of streak breakdown in a hypersonic boundary layer,” in 12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12), Osaka, Japan, 2022.
[6] L. M. Mack, “Boundar-layer linear stability theory,” California, 1984.
[7] C. Hader and H. F. Fasel, “Direct numerical simulations of hypersonic boundary-layer transition for a flared cone: fundamental breakdown,” Journal of Fluid Mechanics, vol. 869, pp. 341-384, 2019.
[8] K. Stetson, E. Thompson, J. Donaldson and L. Siler, “Laminar boundary layer stability experiments on a cone at Mach 8. V-Tests with a cooled model,” in 20th Fluid Dynamics, Plasma Dynamics and Lasers Conference, 1989.
[9] T. Zhou, Z. Liu, Y. Lu, Y. Wang and C. Yan, “Direct numerical simulation of complete transition to turbulence via first- and second-mode oblique breakdown at a high-speed boundary layer,” Physics of Fluids, vol. 34, p. 074101, 2022.