The generation of Tollmien-Schlichting waves by free-stream turbulence

This paper analyses the response and receptivity of the hypersonic boundary layer over a wedge to free-stream disturbances including acoustic, vortical and entropy fluctuations. Due to the presence of an attached oblique shock, the boundary layer is known to support viscous instability modes whose eigenfunctions are oscillatory in the far field. These modes acquire a triple-deck structure. Any of three elementary types of disturbances with frequency and wavelength on the triple-deck scales interacts with the shock to generate a slow acoustic perturbation, which is reflected between the shock and the wall. Through this induced acoustic perturbation, vortical and entropy free-stream disturbances drive significant velocity and temperature fluctuations within the boundary layer, which is impossible when the shock is absent. A quasi-resonance was identified, due to which the boundary layer exhibits a strong response to a continuum of high-frequency disturbances within a narrow band of streamwise wavenumbers. Most importantly, in the vicinity of the lower-branch neutral curve the slow acoustic perturbation induced by a disturbance of suitable frequency and wavenumbers is in exact resonance with a neutral eigen mode. As a result, the latter can be generated directly by each of three types of free-stream disturbances without involving any surface roughness element. The amplitude of the instability mode is determined by analysing the disturbance evolution through the resonant region. The fluctuation associated with the eigen mode turns out to be much stronger than free-stream disturbances due to the resonant nature of excitation and in the case of acoustic disturbances, to the well-known amplification effect of a strong shock. Moreover, excitation at the neutral position means that the instability mode grows immediately without undergoing any decay, or missing any portion of the unstable region. All these indicate that this new mechanism is particularly efficient. The boundary-layer response and coupling coefficients are calculated for typical values of parameters.

“…The corresponding velocity field is denoted by (u, v, w). The total flow field can be written as 10) where…”

Section: Unsteady Disturbancesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Response and receptivity of the hypersonic boundary layer past a wedge to free-stream acoustic, vortical and entropy disturbances

Qin

2016

“…The local mean-flow distortion interacts with free-stream acoustic or vortical disturbances within a suitable frequency band to generate instability modes. In the case of an isolated roughness, the asymptotic theories, which reveal the essential mechanisms, were formulated first by Ruban (1984) and Goldstein (1985) for acoustic disturbances, and by Duck, Ruban & Zhikharev (1996) for vortical disturbances; see also Wu (2001a), who gave a second-order theory. The corresponding theory for distributed roughness was presented by Wu (2001b).…”

Section: Introductionmentioning

confidence: 99%

A local scattering theory for the effects of isolated roughness on boundary-layer instability and transition: transmission coefficient as an eigenvalue

Dong

2016

This paper is concerned with the rather broad issue of the impact of abrupt changes (such as isolated roughness, gaps and local suctions) on boundary-layer transition. To fix the idea, we consider the influence of a two-dimensional localized hump (or indentation) on an oncoming Tollmien-Schlichting (T-S) wave. We show that when the length scale of the former is comparable with the characteristic wavelength of the latter, the key physical mechanism to affect transition is through scattering of T-S waves by the roughness-induced mean-flow distortion. An appropriate mathematical theory, consisting of the boundary-value problem governing the local scattering, is formulated based on triple deck formalism. The transmission coefficient, defined as the ratio of the amplitude of the T-S wave downstream the roughness to that upstream, serves to characterize the impact on transition. The transmission coefficient appears as the eigenvalue of the discretized boundary-value problem. The latter is solved numerically, and the dependence of the eigenvalue on the height and width of the roughness and the frequency of the T-S wave is investigated. For a roughness element without causing separation, the transmission coefficient is found to be about 1.5 for typical frequencies, indicating a moderate but appreciable destabilizing effect. For a roughness causing incipient separation, the transmission coefficient can be as large as O(10), suggesting that immediate transition may take place at the roughness site. A roughness element with a fixed height produces the strongest impact when its width is comparable with the T-S wavelength, in which case the traditional linear stability theory is invalid. The latter however holds approximately when the roughness width is sufficiently large. By studying the two-hump case, a criterion when two roughness elements can be regarded as being isolated is suggested. The transmission coefficient can be converted to an equivalent N-factor increment, by making use of which the e N -method can be extended to predict transition in the presence of multiple roughness elements.

“…In the presence of low-level FST, transition is caused by amplification of Tollmien-Schlichting waves (Kachanov 1994), which correspond to discrete modes of the Orr-Sommerfeld (O-S) equation. FST influences what is referred to as natural transition through receptivity (Saric et al 2002;Duck et al 1996;Wu 2001). If the FST level exceeds a critical value about 1%, low-frequency streaks appear in the boundary layer (Klebanoff 1971;Kendall 1990;Westin et al 1994), and they amplify and break down via a secondary instability (Matsubara & Alfredsson 2001).…”

Section: Introductionmentioning

confidence: 99%

Entrainment of short-wavelength free-stream vortical disturbances in compressible and incompressible boundary layers

Dong

2016

The fundamental difference between continuous modes of the Orr-Sommerfeld/Squire equations and the entrainment of free-stream vortical disturbances (FSVD) into the boundary layer has been investigated in a recent paper (Dong & Wu 2013, J. Fluid Mech.). It was shown there that the non-parallel-flow effect plays a leading-order role in the entrainment, and neglecting it at outset, as is done in the continuous-mode formulation, leads to non-physical features of 'Fourier entanglement' and abnormal anisotropy. The analysis, which was for incompressible boundary layers and for FSVD with a characteristic wavelength of the order of the local boundary-layer thickness, is extended in this paper to compressible boundary layers and FSVD with even shorter wavelengths, which are comparable with the width of the so-called edge layer. Non-parallelism remains a leading-order effect in the present scaling, which turns out to be more general in that the equations and solutions in the previous paper are recovered in the appropriate limit. Appropriate asymptotic solutions in the main and edge layers are obtained to characterize the entrainment. It is found that when the Prandtl number Pr < 1, free-stream vortical disturbances of relatively low frequency generate very strong temperature fluctuations within the edge layer, leading to formation of thermal streaks. A composite solution, uniformly valid across the entire boundary layer, is constructed, and it can be used in receptivity studies and as inlet conditions for direct numerical simulations of bypass transition. For compressible boundary layers, continuous spectra of the disturbance equations linearised about a parallel base flow exhibit entanglement between vortical and entropy modes, namely, a vortical mode necessarily induces an entropy disturbance in the free stream and vice versa, and this amounts to a further nonphysical behaviour. High-Reynolds-number asymptotic analysis yields the relations between the amplitudes of entangled modes.