Receptivity of a high-speed boundary layer on a flat plate to acoustic disturbances is investigated using a combined numerical and asymptotic approach. The leading-edge receptivity problem is discussed with emphasis on physical mechanisms associated with scattering and diffraction of acoustic waves. Analytical solutions provide insight into the interplay of these mechanisms as a function of the angle of incidence of external acoustic waves. The theoretical predictions are in good agreement with the wind-tunnel experimental data of Maslov et al. obtained at free-stream Mach number 6. The leading-edge receptivity model is incorporated into the multiple-modes method to account for the inter-modal exchange downstream from the leading edge. This combined modelling resembles basic features of the direct numerical simulation of Ma & Zhong. A comparative analysis of the leading-edge receptivity and the inter-modal exchange associated with non-parallel effects is presented. The theory allows fast evaluation of the receptivity coefficients and clarifies the physics of the receptivity process. The theoretical results may guide further direct numerical simulations and experimental studies of boundary layer receptivity at supersonic and hypersonic speeds.

Receptivity of a high-speed boundary layer to acoustic disturbances

A pair of oblique waves at low amplitudes is introduced in a supersonic flat-plate boundary layer at Mach 3. Its downstream development and the concomitant process of laminar to turbulent transition is then investigated numerically using linear-stability theory, parabolized stability equations and direct numerical simulations (DNS). In the present paper, the linear regime is studied first in great detail. The focus of the second part is the early and late nonlinear regimes. It is shown how the disturbance wave spectrum is filled up by nonlinear interactions and which flow structures arise and how these structures locally break down to small scales. Finally, the study answers the question whether a fully developed turbulent boundary layer can be reached by oblique breakdown. It is shown that the skin friction develops such as is typical of transitional and turbulent boundary layers. Initially, the skin friction coefficient increases in the streamwise direction in the transitional region and finally decays when the early turbulent state is reached. Downstream of the maximum in the skin friction, the flow loses its periodicity in time and possesses characteristic mean-flow and spectral properties of a turbulent boundary layer. The DNS data clearly demonstrate that oblique breakdown can lead to a fully developed turbulent boundary layer and therefore it is a relevant mechanism for transition in two-dimensional supersonic boundary layers.

Direct numerical simulation of complete transition to turbulence via oblique breakdown at Mach 3

As a first step towards understanding the role of free-stream turbulence in laminar–turbulent transition, we calculate the fluctuations induced by free-stream vortical disturbances in a compressible laminar boundary layer. As with the incompressible case investigated by Leib et al. (J. Fluid Mech. vol. 380, 1999, p. 169), attention is focused on components with long streamwise wavelength. The boundary-layer response is governed by the linearized unsteady boundary-region equations in the typical streamwise region where the local boundary-layer thickness δ* iscomparable with the spanwise length scale Λ of the disturbances. The compressible boundary-region equations are solved numerically for a single Fourier component to obtain the boundary-layer signature. The root-mean-square of the velocity and mass-flux fluctuations induced by a continuous spectrum of free-stream disturbances are computed by an appropriate superposition of the individual Fourier components.Low-frequency vortical disturbances penetrate into the boundary layer to form slowly modulating streamwise-elongated velocity streaks. In the compressible regime, vortical disturbances are found to induce substantial temperature fluctuations so that ‘thermalstreaks’ also form. They may have a significant effect on the secondary instability. The calculations indicate that for a vortical disturbance with a relatively large Λ, the induced boundary-layer fluctuation ultimately evolves into an amplifying wave. This is due to a receptivity mechanism, in which a vortical disturbance first excites a decaying quasi-three-dimensional Lam–Rott eigensolution. The latter then undergoes wavelength shortening to generate a spanwise pressure gradient, which eventually converts the Lam–Rott mode into an exponentially growing mode. The latter is recognized to bea highly oblique Tollmien–Schlichting wave. A parametric study suggests that this receptivity mechanism could be significant when the free-stream Mach number is larger than 0.8.

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Response of a compressible laminar boundary layer to free-stream vortical disturbances

Transition to turbulence of a Mach 6 flat plate boundary layer over a porous surface is investigated by direct numerical simulation considering two Reynolds numbers based on the laminar boundary layer displacement thickness, namely, Reδ∗=6000 and Reδ∗=20 000. The transition was initiated by perturbing the laminar boundary layer with small random disturbances and was followed all the way to the turbulent state. The porous geometry was modeled by directly resolving the flow within the pores and the damping of the primary Mack mode of instability was verified. The presence of a porous surface was found to reduce the secondary instability growth rate by reducing the amplitude of the second mode saturation. In particular, the pores suppress the growth of the secondary wave in the near wall region, so that the secondary instability mainly happens near the critical layer. Besides the secondary instabilities Fourier analysis shows additional modes growing at the same rate as the primary instability, consistent wi...

Direct numerical simulation of breakdown to turbulence in a Mach 6 boundary layer over a porous surface

The nonlinear response of a compressible boundary layer to unsteady free-stream vortical fluctuations of the convected-gust type is investigated theoretically and numerically. The free-stream Mach number is assumed to be of O(1) and the effects of compressibility, including aerodynamic heating and heat transfer at the wall, are taken into account. Attention is focused on low-frequency perturbations, which induce strong streamwise-elongated components of the boundary-layer disturbances, known as streaks or Klebanoff modes. The amplitude of the disturbances is intense enough for nonlinear interactions to occur within the boundary layer. The generation and nonlinear evolution of the streaks, which acquire an O(1) magnitude, are described on a self-consistent and first-principle basis using the mathematical framework of the nonlinear unsteady compressible boundary-region equations, which are derived herein for the first time. The free-stream flow is studied by including the boundary-layer displacement effect and the solution is matched asymptotically with the boundary-layer flow. The nonlinear interactions inside the boundary layer drive an unsteady two-dimensional flow of acoustic nature in the outer inviscid region through the displacement effect. A close analogy with the flow over a thin oscillating airfoil is exploited to find analytical solutions. This analogy has been widely employed to investigate steady flows over boundary layers, but is considered herein for the first time for unsteady boundary layers. In the subsonic regime the perturbation is felt from the plate in all directions, while at supersonic speeds the disturbance only propagates within the dihedron defined by the Mach line. Numerical computations are performed for carefully chosen parameters that characterize three practical applications: turbomachinery systems, supersonic flight conditions and wind tunnel experiments. The results show that nonlinearity plays a marked stabilizing role on the velocity and temperature streaks, and this is found to be the case for low-disturbance environments such as flight conditions. Increasing the free-stream Mach number inhibits the kinematic fluctuations but enhances the thermal streaks, relative to the free-stream velocity and temperature respectively, and the overall effect of nonlinearity becomes weaker. An abrupt deviation of the nonlinear solution from the linear one is observed in the case pertaining to a supersonic wind tunnel. Large-amplitude thermal streaks and the strong abrupt stabilizing effect of nonlinearity are two new features of supersonic flows. The present study provides an accurate signature of nonlinear streaks in compressible boundary layers, which is indispensable for the secondary instability analysis of unsteady streaky boundary-layer flows.

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Nonlinear unsteady streaks engendered by the interaction of free-stream vorticity with a compressible boundary layer

The results of an experimental study of stability, receptivity and transition of the flat-plate laminar boundary layer at Mach 3 are discussed. With a relatively low free-stream disturbance level (∼0.1%), spectra, growth rates and amplitude distributions of naturally occurring boundary layer waves were measured using hot wires. Physical (mass-flux) amplitudes in the boundary layer and free stream are reported and provide stability and receptivity results against which predictions can be directly compared. Comparisons are made between measurements of growth rates of unstable high-frequency waves and theoretical predictions based on a non-parallel, mode-averaging stability theory and receptivity assumptions; good agreement is found. In contrast, it was found that linear stability theory does not account for the measured growth of low-frequency disturbances. A detailed investigation of the disturbance fields in the free stream and on the nozzle walls provides the basis for a discussion of the source and the development of the measured boundary layer waves. Attention is drawn to the close matching in streamwise wavelengths for instability waves and the free-stream acoustic disturbances. It was also found that a calibration of the hot wire in the free stream yields a double-peak boundary layer disturbance amplitude distribution, as has been found by previous investigators, which is not consistent with the predictions of linear stability theory. This double peak was found to be an experimental anomaly which resulted from assumptions that are frequently made in the free-stream calibration procedure. A single-peak amplitude distribution across the boundary layer was established only when the hot-wire voltage was calibrated against the mean boundary layer profile. Finally, the late stages of transition, at a higher Reynolds number with a higher free-stream disturbance level, were explored. Calibrated amplitude levels are provided at locations where nonlinearities are first detected and where the mean boundary layer profile is first observed to depart from the laminar similarity solution. A qualitative discussion of the character of ensuing nonlinearities is also included.

Experiments on stability and transition at Mach 3

Experiments on Stability and Transition at Mach 3

: A high quality flow has been established in a low turbulence variable geometry (LTVG) 8' x 8' supersonic wind-tunnel at Princeton. In particular, the facility is being operated at very low stagnation pressures (- = 4 psia) giving a large region of laminar flow on a flat plate with laminar boundary layers up to approximately 3mm thick. Mean profiles, amplitude distributions, two wire correlations and spectra have been measured in detail through a Mach 3 boundary layer along the centerline of the flat plate. The initial results are for "naturally" occurring instability and transition and they are being followed by measurements of the forced response from a point source (spark) .1.

Paolo Graziosi

Papers

Experiments on stability and transition at Mach 3

Experiments on Stability and Transition at Mach 3

Compressible Boundary Layer Receptivity and Instability