scholarly journals Interaction between mountain waves and shear flow in an inertial layer

2017 ◽  
Vol 816 ◽  
pp. 352-380 ◽  
Author(s):  
Jin-Han Xie ◽  
Jacques Vanneste
Keyword(s):  
Shear Flow ◽  
Inviscid Limit ◽  
Mean Flow ◽  
Mountain Waves ◽  
Planetary Rotation ◽  
Thin Region ◽  
Linear Shear ◽  
Propagation Regime ◽  
Finite Distance ◽  
The Mean

Mountain-generated inertia–gravity waves (IGWs) affect the dynamics of both the atmosphere and the ocean through the mean force they exert as they interact with the flow. A key to this interaction is the presence of critical-level singularities or, when planetary rotation is taken into account, inertial-level singularities, where the Doppler-shifted wave frequency matches the local Coriolis frequency. We examine the role of the latter singularities by studying the steady wavepacket generated by a multiscale mountain in a rotating linear shear flow at low Rossby number. Using a combination of Wentzel–Kramers–Brillouin (WKB) and saddle-point approximations, we provide an explicit description of the form of the wavepacket, of the mean forcing it induces and of the mean-flow response. We identify two distinguished regimes of wave propagation: Regime I applies far enough from a dominant inertial level for the standard ray-tracing approximation to be valid; Regime II applies to a thin region where the wavepacket structure is controlled by the inertial-level singularities. The wave–mean-flow interaction is governed by the change in Eliassen–Palm (or pseudomomentum) flux. This change is localised in a thin inertial layer where the wavepacket takes a limiting form of that found in Regime II. We solve a quasi-geostrophic potential-vorticity equation forced by the divergence of the Eliassen–Palm flux to compute the wave-induced mean flow. Our results, obtained in an inviscid limit, show that the wavepacket reaches a large-but-finite distance downstream of the mountain (specifically, a distance of order$(k_{\ast }\unicode[STIX]{x1D6E5})^{1/2}\unicode[STIX]{x1D6E5}$, where$k_{\ast }^{-1}$and$\unicode[STIX]{x1D6E5}$measure the wave and envelope scales of the mountain) and extends horizontally over a similar scale.

scholarly journals Divergent versus Nondivergent Instabilities of Piecewise Uniform Shear Flows on the f Plane

2009 ◽  
Vol 39 (7) ◽  
pp. 1685-1699
Author(s):  
Nathan Paldor ◽  
Yona Dvorkin ◽  
Eyal Heifetz
Keyword(s):  
Numerical Solutions ◽  
Delta Function ◽  
Relative Vorticity ◽  
Linear Instability ◽  
Large Radius ◽  
Mean Flow ◽  
Uniform Shear ◽  
Linear Shear ◽  
The Mean ◽  
Inner Region

Abstract The linear instability of a piecewise uniform shear flow is classically formulated for nondivergent perturbations on a 2D barotropic mean flow with linear shear, bounded on both sides by semi-infinite half-planes where the mean flows are uniform. The problem remains unchanged on the f plane because for nondivergent perturbations the instability is driven by vorticity gradient at the edges of the inner, linear shear region, whereas the vorticity itself does not affect it. The instability of the unbounded case is recovered when the outer regions of uniform velocity are bounded, provided that these regions are at least twice as wide as the inner region of nonzero shear. The numerical calculations demonstrate that this simple scenario is greatly modified when the perturbations’ divergence and the variation of the mean height (which geostrophically balances the mean flow) are retained in the governing equations. Although a finite deformation radius exists on the shallow water f plane, the mean vorticity gradient that governs the instability in the nondivergent case remains unchanged, so it is not obvious how the instability is modified by the inclusion of divergence in the numerical solutions of the equations. The results here show that the longwave instability of nondivergent flows is recovered by the numerical solution for divergent flows only when the radius of deformation is at least one order of magnitude larger than the width of the inner uniform shear region. Nevertheless, even at this large radius of deformation both the amplitude of the velocity eigenfunction and the distribution of vorticity and divergence differ significantly from those of nondivergent perturbations and vary strongly in the cross-stream direction. Whereas for nondivergent flows the vorticity and divergence both have a delta-function structure located at the boundaries of the inner region, in divergent flows they are spread out and attain their maximum away from the boundaries (either in the inner region or in the outer regions) in some range of the mean shear. In contrast to nondivergent flows for which the mean shear is merely a multiplicative factor of the growth rates, in divergent flows new unstable modes exist for sufficiently large mean shear with no shortwave cutoff. This unstable mode is strongly affected by the sign of the mean shear (i.e., the sign of the mean relative vorticity).

The Reflection of a Stationary Gravity Wave by a Viscous Boundary Layer

2007 ◽  
Vol 64 (9) ◽  
pp. 3363-3371 ◽  
Author(s):  
François Lott
Keyword(s):  
Boundary Layer ◽  
Gravity Wave ◽  
Richardson Number ◽  
Critical Level ◽  
Inviscid Limit ◽  
Mean Flow ◽  
Viscous Boundary Layer ◽  
Lee Waves ◽  
The Mean ◽  
Viscous Boundary

Abstract The backward reflection of a stationary gravity wave (GW) propagating toward the ground is examined in the linear viscous case and for large Reynolds numbers (Re). In this case, the stationary GW presents a critical level at the ground because the mean wind is null there. When the mean flow Richardson number at the surface (J) is below 0.25, the GW reflection by the viscous boundary layer is total in the inviscid limit Re → ∞. The GW is a little absorbed when Re is finite, and the reflection decreases when both the dissipation and J increase. When J > 0.25, the GW is absorbed for all values of the Reynolds number, with a general tendency for the GW reflection to decrease when J increases. As a large ground reflection favors the downstream development of a trapped lee wave, the fact that it decreases when J increases explains why the more unstable boundary layers favor the onset of mountain lee waves. It is also shown that the GW reflection when J > 0.25 is substantially larger than that predicted by the conventional inviscid critical level theory and larger than that predicted when the dissipations are represented by Rayleigh friction and Newtonian cooling. The fact that the GW reflection depends strongly on the Richardson number indicates that there is some correspondence between the dynamics of trapped lee waves and the dynamics of Kelvin–Helmholtz instabilities. Accordingly, and in one classical example, it is shown that some among the neutral modes for Kelvin–Helmholtz instabilities that exist in an unbounded flow when J < 0.25 can also be stationary trapped-wave solutions when there is a ground and in the inviscid limit Re → ∞. When Re is finite, these solutions are affected by the dissipation in the boundary layer and decay in the downstream direction. Interestingly, their decay rate increases when both the dissipation and J increase, as does the GW absorption by the viscous boundary layer.

Stabilization and destabilization of turbulent shear flow in a rotating fluid

1992 ◽  
Vol 241 ◽  
pp. 503-523 ◽  
Author(s):  
D. J. Tritton
Keyword(s):  
Experimental Data ◽  
Shear Flow ◽  
Reynolds Stress ◽  
Rotation Rate ◽  
Turbulent Shear ◽  
Rotating Fluid ◽  
Mean Flow ◽  
Rotation Rates ◽  
Principal Axes ◽  
The Mean

We consider turbulent shear flows in a rotating fluid, with the rotation axis parallel or antiparallel to the mean flow vorticity. It is already known that rotation such that the shear becomes cyclonic is stabilizing (with reference to the non-rotating case), whereas the opposite rotation is destabilizing for low rotation rates and restabilizing for higher. The arguments leading to and quantifying these statement are heuristic. Their status and limitations require clarification. Also, it is useful to formulate them in ways that permit direct comparison of the underlying concepts with experimental data.An extension of a displaced particle analysis, given by Tritton & Davies (1981) indicates changes with the rotation rate of the orientation of the motion directly generated by the shear/Coriolis instability occurring in the destabilized range.The ‘simplified Reynolds stress equations scheme’, proposed by Johnston, Halleen & Lezius (1972), has been reformulated in terms of two angles, representing the orientation of the principal axes of the Reynolds stress tensor (αa) and the orientation of the Reynolds stress generating processes (αb), that are approximately equal according to the scheme. The scheme necessarily fails at large rotation rates because of internal inconsistency, additional to the fact that it is inapplicable to two-dimensional turbulence. However, it has a wide range of potential applicability, which may be tested with experimental data. αa and αb have been evaluated from numerical data for homogeneous shear flow (Bertoglio 1982) and laboratory data for a wake (Witt & Joubert 1985) and a free shear layer (Bidokhti & Tritton 1992). The trends with varying rotation rate are notably similar for the three cases. There is a significant range of near equality of αa and αb. An extension of the scheme, allowing for evolution of the flow, relates to the observation of energy transfer from the turbulence to the mean flow.

Hydraulic control of homogeneous shear flows

2003 ◽  
Vol 475 ◽  
pp. 163-172 ◽  
Author(s):  
CHRIS GARRETT ◽  
FRANK GERDES
Keyword(s):  
Shear Flow ◽  
Velocity Profile ◽  
Water Waves ◽  
Flow Speed ◽  
Hydraulic Control ◽  
Mean Flow ◽  
Shallow Water Waves ◽  
Homogeneous Fluid ◽  
Apparent Paradox ◽  
The Mean

If a shear flow of a homogeneous fluid preserves the shape of its velocity profile, a standard formula for the condition for hydraulic control suggests that this is achieved when the depth-averaged flow speed is less than (gh)1/2. On the other hand, shallow-water waves have a speed relative to the mean flow of more than (gh)1/2, suggesting that information could propagate upstream. This apparent paradox is resolved by showing that the internal stress required to maintain a constant velocity profile depends on flow derivatives along the channel, thus altering the wave speed without introducing damping. By contrast, an inviscid shear flow does not maintain the same profile shape, but it can be shown that long waves are stationary at a position of hydraulic control.

The trailing vorticity field behind a line source in two-dimensional incompressible linear shear flow

2013 ◽  
Vol 720 ◽  
pp. 618-636 ◽  
Author(s):  
Sjoerd W. Rienstra ◽  
Mirela Darau ◽  
Edward J. Brambley
Keyword(s):  
Shear Flow ◽  
Free Field ◽  
Two Dimensional ◽  
Mean Flow ◽  
Mass Source ◽  
Transform Method ◽  
Impedance Boundary ◽  
Duct Acoustics ◽  
Dominant Feature ◽  
Linear Shear

AbstractThe explicit exact analytic solution for harmonic perturbations from a line mass source in an incompressible inviscid two-dimensional linear shear is derived using a Fourier transform method. The two cases of an infinite shear flow and a semi-infinite shear flow over an impedance boundary are considered. For the free-field and hard-wall configurations, the pressure field is (in general) logarithmically diverging and its Fourier representation involves a diverging integral that is interpreted as an integral of generalized functions; this divergent behaviour is not present for a finite impedance boundary or if the frequency equals the mean flow shear rate. The dominant feature of the solution is the hydrodynamic wake caused by the shed vorticity of the source. For linear shear over an impedance boundary, in addition to the wake, (at most) two surface modes along the wall are excited. The implications for duct acoustics with flow over an impedance wall are discussed.

Outline of a theory of turbulent shear flow

1956 ◽  
Vol 1 (5) ◽  
pp. 521-539 ◽  
Author(s):  
W. V. R. Malkus
Keyword(s):  
Shear Flow ◽  
Velocity Profile ◽  
Boundary Conditions ◽  
Momentum Transport ◽  
Turbulent Shear ◽  
Spectral Structure ◽  
Turbulent Shear Flow ◽  
Mean Flow ◽  
Mean Velocity ◽  
The Mean

In this paper the spatial variations and spectral structure of steady-state turbulent shear flow in channels are investigated without the introduction of empirical parameters. This is made possible by the assumption that the non-linear momentum transport has only stabilizing effects on the mean field of flow. Two constraints on the possible momentum transport are drawn from this assumption: first, that the mean flow will be statistically stable if an Orr-Sommerfeld type equation is satisfied by fluctuations of the mean; second, that the smallest scale of motion that can be present in the spectrum of the momentum transport is the scale of the marginally stable fluctuations of the mean. Within these two constraints, and for a given mass transport, an upper limit is sought for the rate of dissipation of potential energy into heat. Solutions of the stability equation depend upon the shape of the mean velocity profile. In turn, the mean velocity profile depends upon the spatial spectrum of the momentum transport. A variational technique is used to determine that momentum transport spectrum which is both marginally stable and produces a maximum dissipation rate. The resulting spectrum determines the velocity profile and its dependence on the boundary conditions. Past experimental work has disclosed laminar, ‘transitional’, logarithmic and parabolic regions of the velocity profile. Several experimental laws and their accompanying constants relate the extent of these regions to the boundary conditions. The theoretical profile contains each feature and law that is observed. First approximations to the constants are found, and give, in particular, a value for the logarithmic slope (von Kármán's constant) which is within the experimental error. However, the theoretical boundary constant is smaller than the observed value. Turbulent channel flow seems to achieve the extreme state found here, but a more decisive quantitative comparison of theory and experiment requires improvement in the solutions of the classical laminar stability problem.

Multiple linear instability of layered stratified shear flow

1994 ◽  
Vol 258 ◽  
pp. 255-285 ◽  
Author(s):  
Colm-Cille P. Caulfield
Keyword(s):  
Shear Flow ◽  
Intermediate Layer ◽  
Linear Instability ◽  
Mean Flow ◽  
Relative Significance ◽  
Wide Range ◽  
Different Types ◽  
Stratified Shear Flow ◽  
New Type ◽  
The Mean

We develop a simple model for the behaviour of an inviscid stratified shear flow with a thin mixed layer of intermediate fluid. We find that the flow is simultaneously unstable to oscillatory disturbances that are a generalization of those discussed by Holmboe (1962), purely unstable modes analogous to those considered by Taylor (1931), and a new type of oscillatory disturbance at large wavelength. The relative significance of these different types of instability depends on the ratio R of the depth of the intermediate layer to the depth of the shear layer. For small values of R, the new type of oscillatory wave has both the largest growthrate for given bulk Richardson number Ri0, and is also primarily unstable to disturbances propagating at an angle to the mean flow, i.e. such modes violate the conditions of Squire's theorem (1933), and thus the assumption of initial two dimensionality of such flows is invalid. For intermediate values of R, the Holmboe-type modes and the Taylor-types modes may have wavelengths and phase speeds conducive to the formation of a resonant triad over a wide range of Ri0. Thus the presence of an intermediate layer in a stratified shear flow markedly changes its stability properties.

scholarly journals On the nonlinear destabilization of stably stratified shear flow

2013 ◽  
Vol 731 ◽  
pp. 443-460 ◽  
Author(s):  
Nadia Mkhinini ◽  
Thomas Dubos ◽  
Philippe Drobinski
Keyword(s):  
Shear Flow ◽  
Nonlinear Analysis ◽  
Base Flow ◽  
Mean Flow ◽  
Fully Nonlinear ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Analysis ◽  
Ekman Flow ◽  
The Mean ◽  
Parallel Shear Flow

AbstractA weakly nonlinear analysis of the bifurcation of the stratified Ekman boundary-layer flow near a critical bulk Richardson number is conducted and compared to a similar analysis of a continuously stratified parallel shear flow subject to Kelvin–Helmholtz instability. Previous work based on asymptotic expansions and predicting supercritical bifurcation at Prandtl number $Pr\lt 1$ and subcritical bifurcation at $Pr\gt 1$ for the parallel base flow is confirmed numerically and through fully nonlinear temporal simulations. When applied to the non-parallel Ekman flow, weakly nonlinear analysis and fully nonlinear calculations confirm that the nature of the bifurcation is dominantly controlled by $Pr$, although a sharp threshold at $Pr= 1$ is not found. In both flows the underlying physical mechanism is that the mean flow adjusts so as to induce a viscous (respectively diffusive) flux of momentum (respectively buoyancy) that balances the vertical flux induced by the developing instability, leading to a weakening of the mean shear and mean stratification. The competition between the former nonlinear feedback, which tends to be stabilizing, and the latter, which is destabilizing and strongly amplified as $Pr$ increases, determines the supercritical or subcritical character of the bifurcation. That essentially the same competition is at play in both the parallel shear flow and the Ekman flow suggests that the underlying mechanism is valid for complex, non-parallel stratified shear flows.

scholarly journals Modulation of near-wall turbulence in the transitionally rough regime

2019 ◽  
Vol 865 ◽  
pp. 1042-1071 ◽  
Author(s):  
Nabil Abderrahaman-Elena ◽  
Chris T. Fairhall ◽  
Ricardo García-Mayoral
Keyword(s):  
Shear Flow ◽  
Flow Instability ◽  
Direct Numerical Simulations ◽  
Wall Turbulence ◽  
Mean Flow ◽  
Displacement Height ◽  
Background Turbulence ◽  
The Mean ◽  
Shear Flow Instability ◽  
Near Wall Turbulence

Direct numerical simulations of turbulent channels with rough walls are conducted in the transitionally rough regime. The effect that roughness produces on the overlying turbulence is studied using a modified triple decomposition of the flow. This decomposition separates the roughness-induced contribution from the background turbulence, with the latter essentially free of any texture footprint. For small roughness, the background turbulence is not significantly altered, but merely displaced closer to the roughness crests, with the change in drag being proportional to this displacement. As the roughness size increases, the background turbulence begins to be modified, notably by the increase of energy for short, wide wavelengths, which is consistent with the appearance of a shear-flow instability of the mean flow. A laminar model is presented to estimate the roughness-coherent contribution, as well as the displacement height and the velocity at the roughness crests. Based on the effects observed in the background turbulence, the roughness function is decomposed into different terms to analyse different contributions to the change in drag, laying the foundations for a predictive model.

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