Two new techniques for the study of the linear and nonlinear instability in growing boundary layers are presented. The first technique employs partial differential equations of parabolic type exploiting the slow change of the mean flow, disturbance velocity profiles, wavelengths, and growth rates in the streamwise direction. The second technique solves the Navier–Stokes equation for spatially evolving disturbances using buffer zones adjacent to the inflow and outflow boundaries. Results of both techniques are in excellent agreement. The linear and nonlinear development of Tollmien–Schlichting (TS) waves in the Blasius boundary layer is investigated with both techniques and with a local procedure based on a system of ordinary differential equations. The results are compared with previous work and the effects of non-parallelism and nonlinearly are clarified. The effect of nonparallelism is confirmed to be weak and, consequently, not responsible for the discrepancies between measurements and theoretical results for parallel flow. Experimental uncertainties, the adopted definition of the growth rate, and the transient initial evolution of the TS wave in vibrating-ribbon experiments probably cause the discrepancies. The effect of nonlinearity is consistent with previous weakly nonlinear theories. White nonlinear effects are small near branch I of the neutral curve, they are significant near branch II and delay or event prevent the decay of the wave.

The observed nonlinear saturation of crossflow vortices in the DLR swept-plate transition experiment, followed by the onset of high-frequency signals, motivated us to compute nonlinear equilibrium solutions for this flow and investigate their instability to high-frequency disturbances. The equilibrium solutions are independent of receptivity, i.e. the way crossflow vortices are generated, and thus provide a unique characterization of the nonlinear flow prior to turbulence. Comparisons of these equilibrium solutions with experimental measurements exhibit strong similarities. Additional comparisons with results from the nonlinear parabolized stability equations (PSE) and spatial direct numerical simulations (DNS) reveal that the equilibrium solutions become unstable to steady, spatial oscillations with very long wavelengths following a bifurcation close to the leading edge. Such spatially oscillating solutions have been observed also in critical layer theory computations. The nature of the spatial behaviour is herein clarified and shown to be analogous to that encountered in temporal direct numerical simulations. We then employ Floquet theory to systematically study the dependence of the secondary, high-frequency instabilities on the saturation amplitude of the equilibrium solutions. With increasing amplitude, the most amplified instability mode can be clearly traced to spanwise inflectional shear layers that occur in the wake-like portions of the equilibrium solutions (Malik et al. 1994 call it ‘mode I’ instability). Both the frequency range and the eigenfunctions resemble recent experimental measurements of Kawakami et al. (1999). However, the lack of an explosive growth leads us to believe that additional self-sustaining processes are active at transition, including the possibility of an absolute instability of the high-frequency disturbances.

Two-and three-dimensional vortical modes that solve the linearized Navier-Stokes equations in the free stream are used in the present theory to represent some of the key features of low-level turbulence. Excluding the leading edge, the effect of these modes on the Blasius boundary layer is investigated using the parabolized stability equations ͑PSE͒. When the vortical modes are steady, or have low frequencies, the PSE analysis is started at a location x 0 from the solution to a new set of ordinary differential equations. This solution is able to satisfy the linearized Navier-Stokes equations in a rather large neighborhood of x 0 . When the vortical modes have frequencies equal to those of unstable Tollmien-Schlichting waves, the scattering of the vortical modes by surface undulation produces only a weak response in the boundary layer, in agreement with other investigations. However, when steady and low-frequency vortical modes are considered, the analysis yields results that successfully reproduce a number of the experimental measurements of Kendall ͓AIAA Paper 90-1504 ͑1990͔͒ on streaky structures, known as Klebanoff modes, that cause a periodic spanwise modulation of the streamwise velocity.

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We investigate the influence of rotational and vibrational energy relaxation on the stability of laminar boundary layers in supersonic flows by numerically solving the linearized equations of motion for a flow in thermal non-equilibrium. We model air as a mixture of nitrogen, oxygen and carbon dioxide, and derive accurate models for the relaxation rates from published experimental data in the field of physical chemistry. The influence of rotational relaxation is to dampen high-frequency instabilities, consistent with the well known damping effect of rotational relaxation on acoustical waves. The influence of rotational relaxation can be modelled with acceptable accuracy through the use of the bulk-viscosity approximation when the bulk viscosity is computed with a formula described herein. Vibrational relaxation affects the growth of disturbances by changing the characteristics of the laminar mean flow. The influence is strongest when the flow field contains a region at, or near, stagnation conditions, followed by a rapid expansion, such as inside wind tunnels and around bodies with a blunt leading edge, whereby the rapid expansion causes the internal energy to freeze in a distribution out of equilibrium. For flows at Mach 4.5 and stagnation temperature of 1000 K, the total amplification exhibited by boundary-layer disturbances over a sharp flat plate in wind-tunnel flows can reach a value that is fifty times as high as the value computed under the assumption of thermal equilibrium. The difference in amplification can be twice as high in the case of a blunt flat plate at atmospheric flight conditions.