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

Abstract

Laminar-to-turbulence transition in 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 spectra 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 the average wall friction which, once maximized, corresponds to the earliest possible transition location. An ensemble variational (EnVar) technique is developed to solve the optimization problem. EnVar updates the estimate of the control vector at the end of each iteration using the gradient of the cost function, which is computed from the outcome 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 can not be categorized as any of the classical breakdown scenarios, while the higher energy condition undergoes a typical 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. †

Observation-infused simulations of high-speed boundary-layer transition

“…In contrast, the EnVar technique provides the smallest wall-pressure errors, which concurs with the results in figure 4. In the region , nonlinear interactions lead to important energy exchanges among modes for this configuration (Jahanbakhshi & Zaki 2019). Within this region and also upstream, successive EnVar iterations progressively refine wall-pressure predictions and reduce the errors, until the point of transition to turbulence.…”

“…In order to generate the surrogate observations, the precursor simulation used the spectra of the nonlinearly most dangerous inflow disturbance that was reported by Jahanbakhshi & Zaki (2019) for a Mach 4.5 boundary layer, and scaled the total energy to half their value. This configuration should be contrasted to the classical scenarios where either subharmonic or fundamental oblique waves are introduced in order to initiate secondary instability of a primary two-dimensional second mode.…”

“…Simulation data provide the complete space–time solution of the Navier–Stokes equations; transition mechanisms can then be directly probed and clarified. Recently, for Mach 4.5 flow, Jahanbakhshi & Zaki (2019) computed the energy transfer among wavenumber triads, as done in Cheung & Zaki (2010), to detail the mode interactions that lead to the earliest possible transition on an adiabatic flat plate. Tightly controlled DNS also provides a framework for identifying plausible hypersonic transition mechanisms (Franko & Lele 2013, 2014; Hader & Fasel 2019) and discerning valid regimes of parallel linear stability theory, linear parabolized stability equations (LPSE) and their nonlinear extension (Chang et al.…”

Nonlinear input/output analysis: application to boundary layer transition

“…2011; Pringle et al. 2012), integral skin friction coefficient (Jahanbakhshi & Zaki 2019), dissipation (Monokrousos et al. 2011) and mean shear (Karp & Cohen 2017).…”

“…The optimal perturbation is calculated over a finite time interval and the one with the lowest energy is known as the minimal seed in the time domain. Similar methodology has been applied to study the transition mechanisms in thermoacoustic systems (Juniper 2011) and also recently has been extended to compressible flows (Jahanbakhshi & Zaki 2019;Huang & Hack 2020). The results still depend on the specific metric (cost function) used to measure the growth; common choices include perturbation kinetic energy (Cherubini et al 2011;Pringle et al 2012), integral skin friction coefficient (Jahanbakhshi & Zaki 2019), dissipation (Monokrousos et al 2011) and mean shear (Karp & Cohen 2017).…”