Instability of a stratified boundary layer and its coupling with internal gravity waves. Part 2. Coupling with internal gravity waves via topography

Self Cite

formal asymptotic procedure was developed to calculate the sound radiated by unsteady boundary-layer flows that are described by the triple-deck theory. That approach requires lengthy calculations, and so is now improved to construct a simpler composite theory, which retains the capacity of systematically identifying and approximating the relevant sources, but also naturally includes the effect of mean-flow refraction and more importantly the back action of the emitted sound on the source itself. The combined effect of refraction and back action is represented by an 'impedance coefficient', and the present analysis yields an analytical expression for this parameter, which was usually introduced on a semi-empirical basis. The expression indicates that for Mach number M = O(1), the mean-flow refraction and back action of the sound have a leading-order effect on the acoustic field within the shallow angles to the streamwise directions. A parametric study suggests that the back effect of sound is actually appreciable in a sizeable portion of the acoustic field for M > 0.5, becomes more pronounced, and eventually influences the entire acoustic field in the transonic limit. In the supersonic regime, the acoustic field is characterized by distinctive Machwave beams, which exert a leading-order influence on the source. The analysis also indicates that acoustic radiation in the subsonic and supersonic regimes is fundamentally different. In the subsonic regime, the sound is produced by smallwavenumber components of the hydrodynamic motion, and can be characterized by acoustic multipoles, whereas in the supersonic regime, broadband finite-wavenumber components of the hydrodynamic motion contribute and the concept of a multipolar source becomes untenable. The global acoustic feedback loop is investigated using a model consisting of two well-separated roughness elements, in which the sound wave emitted due to the scattering of a Tollmien-Schlichting (T-S) wave by the downstream roughness propagates upstream and impinges on the upstream roughness to regenerate the T-S wave. Numerical calculations suggest that at high Reynolds numbers and for moderate roughness heights, the long-range acoustic coupling may lead to global instability, which is characterized by self-sustained oscillations at discrete frequencies. † Email address for correspondence: x.wu@ic.ac.uk 280 X. Wu The dominant peak frequency may jump from one value to another as the Reynolds number or the distance between the roughness elements is varied gradually.

Section: Transonic Limitmentioning

confidence: 71%

On generation of sound in wall-bounded shear flows: back action of sound and global acoustic coupling

2011

Self Cite

“…This justifies a posteriori the local parallel-flow assumption that we have made in neglecting the spatial development of the flow. Nevertheless, it would be interesting to perform a spatial stability analysis (in which the frequency is fixed, and one of the wavenumbers unknown) to gain information on the spatial development of the instability from a localized excitation, as done in Wu & Zhang (2008b).…”

Section: Discussionmentioning

confidence: 99%

Instability of a boundary layer flow on a vertical wall in a stably stratified fluid

Bai

Dizès

2016

The stability of a horizontal boundary layer flow on a vertical wall in a viscous stably stratified fluid is considered in this work. A temporal stability analysis is performed for a tanh velocity profile as a function of the Reynolds number Re = UL/ν and the Froude number F = U/(LN) where U is the main stream velocity, L the boundary layer thickness, N the buoyancy frequency and ν the kinematic viscosity. The diffusion of density is neglected. The boundary layer flow is found to be unstable with respect to two instabilities. The first one is the classical viscous instability which gives rise to Tollmien-Schlichting (TS) waves. We demonstrate that, even in the presence of stratification, the most unstable TS wave remains two-dimensional and therefore independent of the Froude number. The other instability is three-dimensional, inviscid in nature and associated with the stratification. It corresponds to the so-called radiative instability. We show that this instability appears first for Re Re (r) c ≈ 1995 for a Froude number close to 1.5 whereas the viscous instability develops for Re Re (v) c ≈ 3980. For large Reynolds numbers, the radiative instability is also shown to exhibit a much larger growth rate than the viscous instability in a large Froude number interval. We argue that this instability could develop in experimental facilities as well as in geophysical situations encountered in ocean and atmosphere.

“…This makes sound waves on the triple-deck scale a more eective receptivity agent than usual acoustic waves. A similar feature arises for gravity waves on the triple-deck scale (Wu & Zhang 2008). The velocitiesÛ 1 ,V 1 andŴ 1 would be governed by the fully nonlinear triple-deck equations if ε s = O(R −2/8 ).…”

Section: Signature Of Sound Wavesmentioning

confidence: 69%

“…depending on the branch considered. The solution for the velocities can be obtained from (A 1) as (Smith et al 1977, Sykes 1978, Wu & Zhang 2008)…”

Section: Appendix a Mean-ow Distortion Equations In The Lower Deckmentioning

confidence: 99%

Receptivity of supersonic boundary layers over smooth and wavy surfaces to impinging slow acoustic waves

Hernández

2019

Self Cite

In this paper, we investigate the receptivity of a supersonic boundary layer to impinging acoustic waves. Unlike previous studies of acoustic receptivity, where the sound waves have phase speeds comparable with or larger than the free-stream velocity $U_{\infty }$, the acoustic waves here have much slower ($O(R^{-1/8}U_{\infty })$) phase velocity, and their characteristic wavelength and frequency are of $O(R^{-3/8}L)$ and $O(R^{1/4}U_{\infty }/L)$ respectively, compatible with the triple-deck structure, where $L$ is the distance to the leading edge and $R$ the Reynolds number based on $L$ and $U_{\infty }$. A significant feature of a sound wave on the triple-deck scale is that an $O(\unicode[STIX]{x1D700}_{s})$ perturbation in the free stream generates much stronger ($O(\unicode[STIX]{x1D700}_{s}R^{1/8})$) velocity fluctuations in the boundary layer. Two receptivity mechanisms are considered. The first is new, involving the interaction of two such acoustic waves and operating in a boundary layer over a smooth wall. The second involves the interaction between an acoustic wave and the steady perturbation induced by a wavy wall. The sound–sound, or sound–roughness, interactions generate a forcing in resonance with a neutral Tollmien–Schlichting (T–S) wave. The latter is thus excited near the lower branch of the neutral curve, and subsequently undergoes exponential amplification. The excitation through sound–sound interaction may offer a possible explanation for the appearance of instability modes downstream of their neutral locations as was observed in a supersonic boundary layer over a smooth wall. The triple-deck formalism is adopted to describe impingement and reflection of the acoustic waves, and ensuing receptivity, allowing the coupling coefficient to be calculated. The two receptivity processes with the acoustic waves on the triple-deck scale are much more effective compared with those involving usual sound waves, with the coupling coefficient being greater by a factor of $O(R^{1/4})$ and $O(R^{1/8})$ in the sound–sound and sound–roughness interactions, respectively. A parametric study for both the reflection and coupling coefficients is conducted for representative T–S waves, to assess the influence of the streamwise and spanwise wavenumbers, and the phase speed (or frequency) of the acoustic wave.