Continuously forced transient growth in oblique breakdown for supersonic boundary layers

Abstract

The early nonlinear transition process initiated by a small-amplitude pair of oblique waves is studied using both temporal numerical simulation and theoretical considerations. This investigation is performed under the flow conditions of the experiments by Corke et al. (AIAA J., vol. 40, 2002, pp. 1015–1018) who investigated a sharp $7^{\circ }$ cone in the NASA Mach 3.5 Quiet Tunnel. In particular, both the linear and the nonlinear mechanisms prior to transition onset are investigated in great detail as the physics of this regime predetermine the flow topology of the nonlinear transition zone. The objective of this study is (i) to advance the understanding of the underlying physical mechanisms relevant for the early nonlinear transition regime of oblique breakdown and (ii) to make the connection to oblique transition, the incompressible scenario for bypass transition investigated by Schmid & Henningson (Phys. Fluids A, vol. 4, 1992, pp. 1986–1989). The dominance of the longitudinal vortex mode in oblique breakdown is shown to be a consequence of a constantly forced transient growth instability. In particular, the primary pair of oblique waves serves as an ‘actuator’ that is permanently introducing disturbances into the longitudinal mode where these disturbances exhibit transient growth. The effect of the transient growth instability on the longitudinal mode is to raise its amplitude rather than change the growth rate of this mode.

Transition scenario in hypersonic axisymmetrical compression ramp flow

“…A linear streak growth triggered by the nonlinear interaction of oblique modes was already observed for a supersonic boundary layer by Laible & Fasel (2016), who concluded that the nonlinear interaction of oblique modes acted as an 'actuator' that forces component-type non-normal growth of the streaks, in the same way as it was described by Schmid & Henningson (1992) for an incompressible channel flow. This mechanism is known as one of the fastest ways to transition in attached boundary layers according to the studies of Franko & Lele (2013).…”

“…Even with these discrepancies, it is still interesting to notice that the linear amplification of streaks is increasingly stronger in the mixing layer and at the beginning of the reattachment region than in the boundary layer, due to an increasingly stronger linear mechanism. As previously discussed, oblique breakdown is already known to be one of the fastest ways to create turbulence in attached boundary layers (Franko & Lele 2013, 2014; Laible & Fasel 2016), the fact that linear mechanisms associated with streaks become stronger after the separation point makes it even more relevant for SBLI flow.

Figure 25.Three-dimensional reconstruction (isosurface of equal positive and negative density forcing) of the optimal forcing linked to streak amplification for the reattachment at and .

Figure 26.Three-dimensional reconstruction (isosurface of equal positive and negative density fluctuations) of ( a ) the leading SPOD mode (), ( b ) the optimal response of the reattachment region at and , showing elongated streaks.

Figure 27.Percentage of energy contained in the first four SPOD modes for the reattachment at depending on azimuthal wavenumber.

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“…Even with these discrepancies, it is still interesting to notice that the linear amplification of streaks is increasingly stronger in the mixing layer and at the beginning of the reattachment region than in the boundary layer, due to an increasingly stronger linear mechanism. As previously discussed, oblique breakdown is already known to be one of the fastest ways to create turbulence in attached boundary layers (Franko & Lele 2013Laible & Fasel 2016), the fact that linear mechanisms associated with streaks become stronger after the separation point makes it even more relevant for SBLI flow. To summarise these findings, in the reattachment region, the streaks caused by a nonlinear interaction in the boundary which are linearly amplified in the mixing layer are further amplified by a linear mechanism.…”

Mechanism for frustum transition over blunt cones at hypersonic speeds

“…For the nonlinear computations, the five components (ρ,ũ,ṽ,w,T) of the linear non-modal disturbance field from figure 4 are extracted at the inflow location of ξ = 0.03 m. Specifically, an inflow disturbance composed of a pair of oblique travelling waves with f = 250 kHz and m = ±50 is used to investigate the possibility of oblique-mode breakdown. The salient features of oblique-mode breakdown involving linearly unstable modes have been reported both for incompressible boundary layers (where the nonlinear interaction between a pair of oblique Tollmien-Schlichting waves (Berlin, Lundbladh & Henningson 1994) drives the breakdown process) and for supersonic boundary layers (that involve the interaction between a pair of oblique first-mode waves (Thumm 1991;Joslin, Streett & Chang 1992;Adams & Kleiser 1993;Fasel, Thumm & Bestek 1993;Chang & Malik 1994;Laible, Mayer & Fasel 2009;Mayer, Von Terzi & Fasel 2011;Laible & Fasel 2016;Panina et al 2017) and between a pair of oblique Mack-mode waves (Sivasubramanian & Fasel 2013)). The present oblique-mode breakdown differs from the aforementioned cases because it is initiated by the interaction between a pair of oblique travelling waves that peak within the entropy layer beyond the boundary-layer edge.…”

“…The stationary streaks, also referred to as the longitudinal or streamwise vortex mode in oblique-mode transition studies (e.g. Schmid & Henningson 1992;Berlin et al 1994;Chang & Malik 1994), increase in amplitude along the downstream direction, as the result of a continuously forced transient growth mechanism (Laible & Fasel 2016). The evolution of boundary-layer streaks in the present computation shares some similarities with the studies of Siconolfi, Camarri & Fransson (2015) and Martin & Paredes (2016), where a pair of stationary vortices are introduced in an incompressible flat-plate boundary-layer flow at selected distances from the boundary-layer edge to study the generation of stationary streaks and their attenuation effect on Tollmien-Schlichting waves.…”