The critical layer in linear-shear boundary layers over acoustic linings

2012 ◽  
Vol 710 ◽  
pp. 545-568 ◽  
Author(s):  
E. J. Brambley ◽  
M. Darau ◽  
S. W. Rienstra
Keyword(s):  
Boundary Layers ◽  
Phase Speed ◽  
Sound Field ◽  
Critical Layer ◽  
Uniform Density ◽  
Mean Flow ◽  
Linearized Euler Equations ◽  
Sheared Flow ◽  
Linear Shear ◽  
Critical Layers

AbstractAcoustics within mean flows are governed by the linearized Euler equations, which possess a singularity wherever the local mean flow velocity is equal to the phase speed of the disturbance. Such locations are termed critical layers, and are usually ignored when estimating the sound field, with their contribution assumed to be negligible. This paper studies fully both numerically and analytically a simple yet typical sheared ducted flow in order to investigate the influence of the critical layer, and shows that the neglect of critical layers is sometimes, but certainly not always, justified. The model is that of a linear-then-constant velocity profile with uniform density in a cylindrical duct, allowing exact Green’s function solutions in terms of Bessel functions and Frobenius expansions. For sources outside the sheared flow, the contribution of the critical layer is found to decay algebraically along the duct as $O(1/ {x}^{4} )$, where $x$ is the distance downstream of the source. For sources within the sheared flow, the contribution from the critical layer is found to consist of a non-modal disturbance of constant amplitude and a disturbance decaying algebraically as $O(1/ {x}^{3} )$. For thin boundary layers, these disturbances trigger the inherent convective instability of the flow. Extra care is required for high frequencies as the critical layer can be neglected only in combination with a particular downstream pole. The advantages of Frobenius expansions over direct numerical calculation are also demonstrated, especially with regard to spurious modes around the critical layer.

The role of nonlinear critical layers in boundary layer transition

1995 ◽  
Vol 352 (1700) ◽  
pp. 425-442 ◽  
Keyword(s):  
Asymptotic Methods ◽  
Nonlinear Effects ◽  
Boundary Layer Transition ◽  
Critical Layer ◽  
Mean Flow ◽  
Layer Transition ◽  
Instability Waves ◽  
Laminar Boundary Layers ◽  
Instability Modes ◽  
Critical Layers

Asymptotic methods are used to describe the nonlinear self-interaction between pairs of oblique instability modes that eventually develops when initially linear spatially growing instability waves evolve downstream in nominally two-dimensional laminar boundary layers. The first nonlinear reaction takes place locally within a so-called ‘critical layer’, with the flow outside this layer consisting of a locally parallel mean flow plus a pair of oblique instability waves - which may or may not be accompanied by an associated plane wave. The amplitudes of these waves, which are completely determined by nonlinear effects within the critical layer, satisfy either a single integro-differential equation or a pair of integro-differential equations with quadratic to quartic-type nonlinearities. The physical implications of these equations are discussed.

scholarly journals Physics of tidal dissipation in early-type stars and white dwarfs: hydrodynamical simulations of internal gravity wave breaking in stellar envelopes

2020 ◽  
Vol 495 (1) ◽  
pp. 1239-1251 ◽  
Author(s):  
Yubo Su ◽  
Daniel Lecoanet ◽  
Dong Lai
Keyword(s):  
Wave Breaking ◽  
White Dwarfs ◽  
Phase Speed ◽  
Critical Layer ◽  
Linear Wave ◽  
Early Type ◽  
Mean Flow ◽  
Layer Width ◽  
Early Type Stars ◽  
Hydrodynamical Simulations

ABSTRACT In binaries composed of either early-type stars or white dwarfs, the dominant tidal process involves the excitation of internal gravity waves (IGWs), which propagate towards the stellar surface, and their dissipation via non-linear wave breaking. We perform 2D hydrodynamical simulations of this wave breaking process in a stratified, isothermal atmosphere. We find that, after an initial transient phase, the dissipation of the IGWs naturally generates a sharp critical layer, separating the lower stationary region (with no mean flow) and the upper ‘synchronized’ region (with the mean flow velocity equal to the horizontal wave phase speed). While the critical layer is steepened by absorption of these waves, it is simultaneously broadened by Kelvin–Helmholtz instabilities such that, in steady state, the critical layer width is determined by the Richardson criterion. We study the absorption and reflection of incident waves off the critical layer and provide analytical formulae describing its long-term evolution. The result of this study is important for characterizing the evolution of tidally heated white dwarfs and other binary stars.

A theoretical model analysis of the sudden narrow temperature-layer formation observed in the ALOHA-93 Campaign

2002 ◽  
Vol 80 (12) ◽  
pp. 1543-1558 ◽  
Author(s):  
H Hur ◽  
T Y Huang ◽  
Z Zhao ◽  
P Karunanayaka ◽  
T F Tuan
Keyword(s):  
Energy Dissipation ◽  
Gravity Wave ◽  
Temperature Rise ◽  
Critical Level ◽  
Inversion Layer ◽  
Phase Speed ◽  
Temperature Inversion ◽  
Critical Layer ◽  
Mean Flow ◽  
Wind Profiles

The behavior of temperature and wind profiles observed on 21 October 1993 in the ALOHA-93 Campaign is theoretically and numerically analyzed. A sudden temperature rise took place in a very narrow vertical region (3–4 km) at about 87 km. Simultaneously observed radar wind profiles and mesospheric airglow wave structures that show a horizontal phase speed of 35 m/s and a period of about half an hour strongly suggest that a critical level may occur in the proximity of that altitude and that the energy dissipation due to the interaction of the gravity wave with the critical level causes the temperature rise. The numerical model used is a solution to the gravity wave – mean-flow interaction in the critical layer, including a simple cooling mechanism and a wave-energy dissipation simulated by the "optical model" technique. The solutions for the temperature variations so obtained show good agreement with the observed temperature profiles at different times, providing a quantitative explanation for the temperature inversion layer as a phenomenon of gravity wave – critical layer interaction. PACS Nos.: 91.10V, 94.10D

Laboratory Observations of Gravity Wave, Critical Layer Flows Using Single and Double Wave Forcing

1994 ◽  
Vol 47 (6S) ◽  
pp. S113-S117
Author(s):  
Donald P. Delisi ◽  
Timothy J. Dunkerton
Keyword(s):  
Gravity Wave ◽  
Wave Breaking ◽  
Critical Level ◽  
Phase Speed ◽  
Vertical Shear ◽  
Critical Layer ◽  
Small Scale ◽  
Mean Flow ◽  
Wave Forcing ◽  
Internal Mixing

Laboratory measurements of gravity wave, critical layer flows are presented. The measurements are obtained in a salt-stratified annular tank, with a vertical shear profile. Internal gravity waves are generated at the floor of the tank and propagate vertically upward into the fluid. At a depth where the phase speed of the wave equals the mean flow speed, defined as a critical level, the waves break down, under the right forcing conditions, generating small scale turbulence. Two cases are presented. In the first case, the wave forcing is a single, monochromatic wave. In this case, the early wave breaking is characterized as Kelvin-Helmholtz breaking at depths below the critical level. Later wave breaking is characterized by weak overturning in the upper part of the tank and regular, internal mixing regions in the lower part of the tank. In the second case, the wave forcing is two monochromatic waves, each propagating with a different phase speed. In this case, the early wave breaking is again Kelvin-Helmholtz in nature, but later wave breaking is characterized by sustained overturning in the upper part of the tank with internal mixing regions in the lower part of the tank. Mean velocity profiles are obtained both before and during the experiments.

Quantifying Residual, Eddy, and Mean Flow Effects on Mixing in an Idealized Circumpolar Current

2017 ◽  
Vol 47 (8) ◽  
pp. 1897-1920 ◽  
Author(s):  
Phillip J. Wolfram ◽  
Todd D. Ringler
Keyword(s):  
High Performance ◽  
Phase Speed ◽  
Critical Layer ◽  
Linear Wave ◽  
Mean Flow ◽  
Wave Regime ◽  
Linear Waves ◽  
Parameter Ratio ◽  
Low Pass ◽  
Circumpolar Current

AbstractMeridional diffusivity is assessed for a baroclinically unstable jet in a high-latitude idealized circumpolar current (ICC) using the Model for Prediction across Scales Ocean (MPAS-O) and the online Lagrangian in Situ Global High-Performance Particle Tracking (LIGHT) diagnostic via space–time dispersion of particle clusters over 120 monthly realizations of O(106) particles on 11 potential density surfaces. Diffusivity in the jet reaches values of O(6000) m2 s−1 and is largest near the critical layer supporting mixing suppression and critical layer theory. Values in the vicinity of the shelf break are suppressed to O(100) m2 s−1 because of the presence of westward slope front currents. Diffusivity attenuates less rapidly with depth in the jet than both eddy velocity and kinetic energy scalings would suggest. Removal of the mean flow via high-pass filtering shifts the nonlinear parameter (ratio of the eddy velocity to eddy phase speed) into the linear wave regime by increasing the eddy phase speed via the depth-mean flow. Low-pass filtering, in contrast, quantifies the effect of mean shear. Diffusivity is decomposed into mean flow shear, linear waves, and the residual nonhomogeneous turbulence components, where turbulence dominates and eddy-produced filamentation strained by background mean shear enhances mixing, accounting for ≥80% of the total diffusivity relative to mean shear [O(100) m2 s−1], linear waves [O(1000) m2 s−1], and undecomposed full diffusivity [O(6000) m2 s−1]. Diffusivity parameterizations accounting for both the nonhomogeneous turbulence residual and depth variability are needed.

scholarly journals The aeroacoustics of slowly diverging supersonic jets

2008 ◽  
Vol 600 ◽  
pp. 291-337 ◽  
Author(s):  
M. E. GOLDSTEIN ◽  
S. J. LEIB
Keyword(s):  
Flow Analysis ◽  
Practical Interest ◽  
Sound Field ◽  
Supersonic Jets ◽  
Parallel Flow ◽  
Critical Layer ◽  
Acoustic Analogy ◽  
Mean Flow ◽  
Retarded Time ◽  
Number Range

This paper is concerned with utilizing the acoustic analogy approach to predict the sound from unheated supersonic jets. Previous attempts have been unsuccessful at making such predictions over the Mach number range of practical interest. The present paper, therefore, focuses on implementing the refinements needed to accomplish this objective. The important effects influencing peak supersonic noise are found to be source convection, mean flow refraction, mean flow amplification, and source non-compactness. It appears that the last two effects have not been adequately dealt with in the literature. For the first of these this is because the usual parallel flow models produce most of the amplification in the so-called critical layer where the solution becomes singular and, therefore, causes the predicted sound field to become infinite. We deal with this by introducing a new weakly non-parallel flow analysis that eliminates the critical layer singularity. This has a strong effect on the shape of the peak noise spectrum. The last effect places severe demands on the source models at the higher Mach numbers because the retarded-time variations significantly increase the sensitivity of the radiated sound to the source structure in this case. A highly refined (non-separable) source model is, therefore, introduced in this paper.

Troposphere-Stratosphere Coupling and the Role of Critical Layer Nonlinearity

2020 ◽  
Author(s):  
Imogen Dell
Keyword(s):  
Rossby Waves ◽  
Phase Speed ◽  
Nonlinear Effects ◽  
Critical Layer ◽  
Coupling Mechanism ◽  
Mathematical Framework ◽  
Mean Flow ◽  
Weather And Climate ◽  
Dynamical Coupling

<p>There exists a coupling mechanism between the troposphere and the stratosphere, which plays a fundamental role in weather and climate. The coupling is highly complex and rests upon radiative and chemical feedbacks, as well as dynamical coupling by Rossby waves. The troposphere acts as a source of Rossby waves which propagate upwards in to the stratosphere, affecting the zonal mean flow. Rossby waves are also likely to play a significant role in downward communication of information via reflection from the stratosphere in to the troposphere. A mechanism for this reflection could be from a so-called critical layer. A shear flow exhibits a critical layer where the phase speed equals the flow velocity, where viscous and nonlinear effects become important. A wave incident upon a critical layer may be absorbed, reflected or overreflected, whereby the amplitude of the reflected wave is larger than that of the incident wave. In the case of troposphere-stratosphere coupling, the concept of critical layer overreflection is key to understanding atmospheric instability.</p><p>Motivated by this, a mathematical framework for understanding the coupling will be presented together with an investigation in to the role of nonlinearity versus viscosity inside the critical layer.</p>

Sensitivity of high-speed boundary-layer stability to base-flow distortion

2018 ◽  
Vol 859 ◽  
pp. 476-515 ◽  
Author(s):  
J. Park ◽  
T. A. Zaki
Keyword(s):  
Boundary Layer ◽  
Boundary Layers ◽  
High Speed ◽  
Phase Speed ◽  
Streamwise Velocity ◽  
Base Flow ◽  
Critical Layer ◽  
Base Temperature ◽  
Flow Distortion ◽  
Lagrangian Optimization

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 largest within the boundary layer away from the wall. Near the critical layer, where the phase speed of the mode equals the base streamwise velocity, the sensitivity to the base streamwise velocity is negative. For the mode F, there is a discontinuous jump in the sensitivity when the phase speed is below unity, and a critical layer is established. The sensitivity of the two modes increases with the Reynolds number, but there is a sudden drop and a jump in the sensitivities of the modes S and F, respectively, near the synchronization point where the phase speeds of the two modes are equal. Furthermore, the maximum uncertainty bounds are obtained for the distorted base state that maximizes the destabilization or stabilization of the modes by solving the Lagrangian optimization problem for the sensitivity. The sensitivity of the flow stability to surface heating is then studied, and changes in growth rate and the$N$-factor are evaluated. The formulation provides a clear physical interpretation of these changes, and establishes uncertainty bounds for stability predictions for a given level of uncertainty in wall temperature.

Broadband liner impedance eduction for multimodal acoustic propagation in the presence of a mean flow

2016 ◽  
Vol 11 (2) ◽  
pp. 150-155
Author(s):  
R. Troian ◽  
D. Dragna ◽  
C. Bailly ◽  
M.-A. Galland
Keyword(s):  
Numerical Model ◽  
Flow Characteristics ◽  
Acoustic Propagation ◽  
Rational Fraction ◽  
Physical Parameters ◽  
Mean Flow ◽  
Linearized Euler Equations ◽  
Grazing Flow ◽  
Propagation Medium ◽  
Difference Time

Modeling of acoustic propagation in a duct with absorbing treatment is considered. The surface impedance of the treatment is sought in the form of a rational fraction. The numerical model is based on a resolution of the linearized Euler equations by finite difference time domain for the calculation of the acoustic propagation under a grazing flow. Sensitivity analysis of the considered numerical model is performed. The uncertainty of the physical parameters is taken into account to determine the most influential input parameters. The robustness of the solution vis-a-vis changes of the flow characteristics and the propagation medium is studied.

Low-Reynolds-Number Turbulent Boundary Layers in Zero and Favorable Pressure Gradients

1983 ◽  
Vol 27 (03) ◽  
pp. 147-157 ◽  
Author(s):  
A. J. Smits ◽  
N. Matheson ◽  
P. N. Joubert
Keyword(s):  
Friction Coefficient ◽  
Skin Friction ◽  
Boundary Layers ◽  
Leading Edge ◽  
Reynolds Numbers ◽  
Skin Friction Coefficient ◽  
Turbulent Boundary Layers ◽  
Pressure Gradients ◽  
Mean Flow ◽  
Low Reynolds

This paper reports the results of an extensive experimental investigation into the mean flow properties of turbulent boundary layers with momentum-thickness Reynolds numbers less than 3000. Zero pressure gradient and favorable pressure gradients were studied. The velocity profiles displayed a logarithmic region even at very low Reynolds numbers (as low as Rθ = 261). The results were independent of the leading-edge shape, and the pin-type turbulent stimulators performed well. It was found that the shape and Clauser parameters were a little higher than the correlation proposed by Coles [10], and the skin friction coefficient was a little lower. The skin friction coefficient behavior could be fitted well by a simple power-law relationship in both zero and favorable pressure gradients.

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