Nonlinear Transient Growth and Boundary Layer Transition

Abstract

Parabolized stability equations (PSE) are used in a variational approach to study the optimal, non-modal disturbance growth in a Mach 3 flat plate boundary layer and a Mach 6 circular cone boundary layer. As noted in previous works, the optimal initial disturbances correspond to steady counter-rotating streamwise vortices, which subsequently lead to the formation of streamwise-elongated structures, i.e., streaks, via a lift-up effect. The nonlinear evolution of the linearly optimal stationary perturbations is computed using the nonlinear plane-marching PSE for stationary perturbations. A fully implicit marching technique is used to facilitate the computation of nonlinear streaks with large amplitudes. To assess the effect of the finite-amplitude streaks on transition, the linear form of planemarching PSE is used to investigate the instability of the boundary layer flow modified by spanwise periodic streaks. The onset of bypass transition is estimated by using an Nfactor criterion based on the amplification of the streak instabilities. Results show that, for both flow configurations of interest, streaks of sufficiently large amplitude can lead to significantly earlier onset of transition than that in an unperturbed boundary layer without any streaks.

Transition mechanisms in cross-flow-dominated hypersonic flows with free-stream acoustic noise

“…In the case of free-stream high-amplitude disturbances, nonlinearities can lead to a rapid transition to turbulence. In some conditions, transition occurs as a result of transient growth, whose related mechanism in hypersonic flows, similarly to what was found in low-speed flows (Andersson et al 2001;Brandt & Henningson 2002), is associated with the conversion of streamwise vorticity into streamwise streaks via a lift-up effect, as was shown by Paredes, Choudhari & Li (2016a), Paredes et al (2016b,c) for flat-plate and circular-cone flows at hypersonic Mach numbers. Another possibility is based on oblique transition (Berlin & Henningson 1999), in which the response to oblique free-stream waves in combination with a nonlinear mechanism was found to be a powerful process, characterised by non-modal growth of the disturbances and requiring lower initial amplitudes compared to the case of transition caused by two-dimensional Tollmien-Schlichting (TS) and second mode instabilities.…”

An alternative method to calculate cross-flow instabilities

“…An alternative is to use more sophisticated techniques, such as 2D eigenfunctions [8], Floquet analysis or linear and nonlinear Parabolized Stability Equations (PSE), see e.g. [9]. However, these techniques may find difficulties in predicting mode interactions.…”

Optimal Growth in Hypersonic Boundary Layers

“…The ratio of kinetic energy to total energy of the disturbance, K∕E, is studied for the optimal spanwise wave numbers for a variety of freestream conditions. Because the secondary instability of transient growth disturbances is mostly driven by streamwise velocity shear [52,53], a greater value of this ratio is likely to enhance the growth of shear layer instabilities and possibly result in an earlier onset of bypass transition associated with a nonlinear disturbance. As the Mach number increases, K∕E is found to decrease with Mach number and increase with T w ∕T ad .…”