Abstract:The linear stability of high-speed boundary layers can be altered by distortions to the base velocity and temperature profiles. An analytic expression for the sensitivity is derived for parallel and spatially developing boundary layers, the latter using linear parabolized stability equations and their adjoint. Both the slow mode, S, and the fast mode, F, are investigated at Mach number 4.5. The mode S is more sensitive with respect to distortion in base velocity than in base temperature. The sensitivity is lar… Show more
“…These observations are consistent with the notion that the synchronization point of instability waves is a key modulator of stability in thermally treated boundary layers (Zhao et al. 2018; Park & Zaki 2019). Figure 4. Surface heat flux of the optimized solution after the final iteration in physical space, amplitudes of heat flux complex coefficients of the optimized solution in spectral space and surface heat flux of the reduced heat flux case in physical space.…”
Section: Resultssupporting
confidence: 91%
“…The spatial growth rates, streamwise wavenumbers and phase speeds of modes and at those frequencies are plotted as a function of Reynolds number in figure 5. The figure is evaluated using parallel linear theory, and shows the typical characteristics of slow and fast modes (Fedorov 2011; Fedorov & Tumin 2011; Park & Zaki 2019): (i) at low Reynolds number modes and appear, respectively, near the slow acoustic, , and fast acoustic, , branches of the spectrum; (ii) the former accelerates and the later decelerates with Reynolds number until they synchronize; and (iii) the profiles of mode show a discontinuity at the location where its phase speed approaches from either side which corresponds to crossing the continuous branch of entropy and vorticity modes. The synchronization Reynolds numbers for is at and for it is .…”
Section: Resultsmentioning
confidence: 99%
“…They included the thermal non-equilibrium effects in their analysis and showed that certain unstable modes can travel supersonically with respect to the free stream, which changes the nature of the dispersion curve and leads to instability over a much wider band of frequencies. Park & Zaki (2019) used adjoint techniques to analytically isolate the different contribution of wall-temperature changes onto stability, for example the direct effect of the temperature profile versus the indirect effect of the change in the mean-velocity profile that takes place when the temperature boundary condition is altered. Their sensitivity analysis at Mach 4.5 demonstrated that the change in temperature with wall heating is stabilizing although the change in the mean velocity is destabilizing.…”
“…These observations are consistent with the notion that the synchronization point of instability waves is a key modulator of stability in thermally treated boundary layers (Zhao et al. 2018; Park & Zaki 2019). Figure 4. Surface heat flux of the optimized solution after the final iteration in physical space, amplitudes of heat flux complex coefficients of the optimized solution in spectral space and surface heat flux of the reduced heat flux case in physical space.…”
Section: Resultssupporting
confidence: 91%
“…The spatial growth rates, streamwise wavenumbers and phase speeds of modes and at those frequencies are plotted as a function of Reynolds number in figure 5. The figure is evaluated using parallel linear theory, and shows the typical characteristics of slow and fast modes (Fedorov 2011; Fedorov & Tumin 2011; Park & Zaki 2019): (i) at low Reynolds number modes and appear, respectively, near the slow acoustic, , and fast acoustic, , branches of the spectrum; (ii) the former accelerates and the later decelerates with Reynolds number until they synchronize; and (iii) the profiles of mode show a discontinuity at the location where its phase speed approaches from either side which corresponds to crossing the continuous branch of entropy and vorticity modes. The synchronization Reynolds numbers for is at and for it is .…”
Section: Resultsmentioning
confidence: 99%
“…They included the thermal non-equilibrium effects in their analysis and showed that certain unstable modes can travel supersonically with respect to the free stream, which changes the nature of the dispersion curve and leads to instability over a much wider band of frequencies. Park & Zaki (2019) used adjoint techniques to analytically isolate the different contribution of wall-temperature changes onto stability, for example the direct effect of the temperature profile versus the indirect effect of the change in the mean-velocity profile that takes place when the temperature boundary condition is altered. Their sensitivity analysis at Mach 4.5 demonstrated that the change in temperature with wall heating is stabilizing although the change in the mean velocity is destabilizing.…”
“…In order to highlight the importance of the present nonlinear approach in determining the inflow disturbance spectrum, we examine other possible inflow conditions that are selected based on linear theory alone. The starting point is to evaluate the linear evolution of all the Orr–Sommerfeld and Squire instability waves that are part of the inlet condition, using the linear parabolized instability equations (Park & Zaki 2019). The -factor associated with every wave was obtained as follows: where is the energy of the mode at the inlet.…”
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. †
“…Lysenko & Maslov (1984) explained the deviations in measured and theoretical instability growth rates in cooled flows, in part, in terms of the angle of waves infiltrating the boundary layer. Park & Zaki (2019) performed adjoint sensitivity analyses that detail how uncertainty in mean-velocity and temperature profiles affect growth rates; they showed that the measured growth rates by Lysenko & Maslov (1984) can result from 1 to 2 % uncertainty in wall temperature.…”
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
