Wave breakdown in stratified shear flows

1977 ◽  
Vol 79 (3) ◽  
pp. 481-497 ◽  
Author(s):  
M. T. Landahl ◽  
W. O. Criminale
Keyword(s):  
Shear Flows ◽  
Flow Instability ◽  
Stability Boundary ◽  
Small Scale ◽  
Mean Flow ◽  
Order Unity ◽  
Stable Regime ◽  
Density Interface ◽  
Density Interfaces ◽  
The Stability

The wave-mechanical condition (Landahl 1972) for breakdown of an unsteady laminar flow into strong small-scale secondary instabilities is applied to some simple stratified inviscid shear flows. The cases considered have one or two discrete density interfaces and simple discontinuous or continuous velocity profiles. A primary wavelike disturbance to such a flow produces a perturbation velocity that is discontinuous at a density interface. The resulting instantaneous system, defined as the sum of the mean flow and the primary oscillation, develops a local secondary shear-flow instability that has a group velocity equal to the arithmetic mean of the instantaneous velocities on the two sides of the interface. According to the breakdown criterion, the disturbed flow will become critical whenever this velocity reaches a value equal to the phase velocity of the primary wave. The calculations show that for a single density interface breakdown may occur for low to moderate wave amplitudes in a fairly narrow range of Richardson numbers on the stable side of the stability boundary. On the other hand, in the unstable regime maximum wave slopes of order unity may be reached before breakdown occurs; this conclusion is in qualitative agreement with experiments. When the system includes two density interfaces, it is found that there exists a range of high Richardson numbers far into the stable regime for which breakdown may take place even for very small and zero wave interface deflexions.

Frontal instabilities and waves in a differentially rotating fluid

2011 ◽  
Vol 685 ◽  
pp. 532-542 ◽  
Author(s):  
J.-B. Flór ◽  
H. Scolan ◽  
J. Gula
Keyword(s):  
Baroclinic Instability ◽  
Phase Speed ◽  
Small Scale ◽  
Immiscible Fluid ◽  
Mean Flow ◽  
Disk Rotation ◽  
Fluid Layers ◽  
The Stability ◽  
Almost All ◽  
Instability Regime

AbstractWe present an experimental investigation of the stability of a baroclinic front in a rotating two-layer salt-stratified fluid. A front is generated by the spin-up of a differentially rotating lid at the fluid surface. In the parameter space set by rotational Froude number, $F$, dissipation number, $d$ (i.e. the ratio between disk rotation time and Ekman spin-down time) and flow Rossby number, a new instability is observed that occurs for Burger numbers larger than the critical Burger number for baroclinic instability. This instability has a much smaller wavelength than the baroclinic instability, and saturates at a relatively small amplitude. The experimental results for the instability regime and the phase speed show overall a reasonable agreement with the numerical results of Gula, Zeitlin & Plougonven (J. Fluid Mech., vol. 638, 2009, pp. 27–47), suggesting that this instability is the Rossby–Kelvin instability that is due to the resonance between Rossby and Kelvin waves. Comparison with the results of Williams, Haines & Read (J. Fluid Mech., vol. 528, 2005, pp. 1–22) and Hart (Geophys. Fluid Dyn., vol. 3, 1972, pp. 181–209) for immiscible fluid layers in a small experimental configuration shows continuity in stability regimes in $(F, d)$ space, but the baroclinic instability occurs at a higher Burger number than predicted according to linear theory. Small-scale perturbations are observed in almost all regimes, either locally or globally. Their non-zero phase speed with respect to the mean flow, cusped-shaped appearance in the density field and the high values of the Richardson number for the observed wavelengths suggest that these perturbations are in many cases due to Hölmböe instability.

Instability in Stratified Shear Flow: Review of a Physical Interpretation Based on Interacting Waves

2011 ◽  
Vol 64 (6) ◽  
Author(s):  
Jeffrey R. Carpenter ◽  
Edmund W. Tedford ◽  
Eyal Heifetz ◽  
Gregory A. Lawrence
Keyword(s):  
Shear Flow ◽  
Physical Interpretation ◽  
Shear Flows ◽  
Flow Instability ◽  
Phase Locking ◽  
River Estuary ◽  
Wave Interaction ◽  
The Stability ◽  
Interaction Approach ◽  
The Waves

Instability in homogeneous and density stratified shear flows may be interpreted in terms of the interaction of two (or more) otherwise free waves in the velocity and density profiles. These waves exist on gradients of vorticity and density, and instability results when two fundamental conditions are satisfied: (I) the phase speeds of the waves are stationary with respect to each other (“phase-locking“), and (II) the relative phase of the waves is such that a mutual growth occurs. The advantage of the wave interaction approach is that it provides a physical interpretation to shear flow instability. This paper is largely intended to purvey the basics of this physical interpretation to the reader, while both reviewing and consolidating previous work on the topic. The interpretation is shown to provide a framework for understanding many classical and nonintuitive results from the stability of stratified shear flows, such as the Rayleigh and Fjørtoft theorems, and the destabilizing effect of an otherwise stable density stratification. Finally, we describe an application of the theory to a geophysical-scale flow in the Fraser River estuary.

The onset of meandering in a barotropic jet

2001 ◽  
Vol 449 ◽  
pp. 85-114 ◽  
Author(s):  
N. J. BALMFORTH ◽  
C. PICCOLO
Keyword(s):  
Nonlinear Theory ◽  
Stability Boundary ◽  
Wave Model ◽  
Inviscid Limit ◽  
Reduced Model ◽  
Mean Flow ◽  
Single Wave ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory ◽  
The Stability

This study explores the dynamics of an unstable jet of two-dimensional, incompressible fluid on the beta-plane. In the inviscid limit, standard weakly nonlinear theory fails to give a low-order description of this problem, partly because the simple shape of the unstable normal mode is insufficient to capture the structure of the forming pattern. That pattern takes the form of ‘cat's eyes’ in the vorticity distribution which develop inside the modal critical layers (slender regions to either side of the jet's axis surrounding the levels where the modal wave speed matches the mean flow). Asymptotic expansions furnish a reduced model which is a version of what is known as the single-wave model in plasma physics. The reduced model predicts that the amplitude of the unstable mode saturates at a relatively low level and is not steady. Rather, the amplitude evolves aperiodically about the saturation level, a result with implications for Lagrangian transport theories. The aperiodic amplitude ‘bounces’ are intimately connected with sporadic deformations of the vortices within the cat's eyes. The theory is compared with numerical simulations of the original governing equations. Slightly asymmetrical jets are also studied. In this case the neutral modes along the stability boundary become singular; an extension of the weakly nonlinear theory is presented for these modes.

Centre modes in inviscid swirling flows and their application to the stability of the Batchelor vortex

2007 ◽  
Vol 576 ◽  
pp. 325-348 ◽  
Author(s):  
C. J. HEATON
Keyword(s):  
Continuous Spectrum ◽  
Single Point ◽  
Stability Boundary ◽  
Infinite Family ◽  
Neutral Curve ◽  
Swirling Flows ◽  
Mean Flow ◽  
Linearized Euler Equations ◽  
Numerical Studies ◽  
The Stability

We identify a family of centre-mode disturbances to inviscid swirling flows such as jets, wakes and other vortices. The centre modes form an infinite family of modes, increasingly concentrated near to the symmetry axis of the mean flow, and whose frequencies accumulate to a single point in the complex plane. This asymptotic accumulation allows analytical progress to be made, including a theoretical stability boundary, inO(1) parameter regimes. The modes are located close to the continuous spectrum of the linearized Euler equations, and the theory is closely related to that of the continuous spectrum. We illustrate our analysis with the inviscid Batchelor vortex, defined by swirl parameterq. We show that the inviscid instabilities found in previous numerical studies are in fact the first members of an infinite set of centre modes of the type we describe. We investigate the inviscid neutral curve, and find good agreement of the neutral curve predicted by the analysis with the results of numerical computations. We find that the unstable region is larger than previously reported. In particular, the value ofqabove which the inviscid vortex stabilizes is significantly larger than previously reported and in agreement with a long-standing theoretical prediction.

Convection in a rotating annulus uniformly heated from below

1971 ◽  
Vol 46 (1) ◽  
pp. 65-81 ◽  
Author(s):  
Robert P. Davies-Jones ◽  
Peter A. Gilman
Keyword(s):  
Thermal Stresses ◽  
Stability Boundary ◽  
Second Order ◽  
Infinite Plane ◽  
Reynolds Stresses ◽  
Order Unity ◽  
Prandtl Numbers ◽  
The Stability ◽  
Boussinesq Fluid ◽  
Outer Half

We present a linear stability analysis, to second order in initial amplitude, of Bénard convection of a Boussinesq fluid in a thin rotating annulus for modest Taylor numbers T ([les ] 104). The work is motivated in part by the desire to study further a mechanism for maintaining, through horizontal Reynolds stresses induced in the convection, the sun's ‘equatorial acceleration’, which has been demonstrated for a rotating convecting spherical shell by Busse & Durney. The annulus is assumed to have stress free, perfectly conducting top and bottom (which allows separation of the equations) and non-conducting non-slip sides. A laboratory experiment which fulfills these conditions (except perhaps the free bottom) is being developed with H. Snyder.We study primarily annuli with gap-width to depth ratios a of order unity. The close, non-slip side-walls produce a number of effects not present in the infinite plane case, including overstability at high Prandtl numbers P, and multiple minima in Rayleigh number R on the stability boundary. The latter may give rise to vacillation. The eigenfunctions for stationary convection for a = 2, T [lsim ] 2000 clearly show momentum of the same sense as the rotation is transported from the inner to the outer half of the annulus, corresponding to transport toward equatorial latitudes on the sphere. The complete second-order solutions for the induced circulations indeed give faster rotation in the outer half, except for large P (> 102), in which case thermal stresses dominate. At all P, this differential rotation is qualitatively a thermal wind. Overstable convective cells, and stationary cells at higher T, induce more complicated differential rotations.

Numerical studies of the stability of inviscid stratified shear flows

1972 ◽  
Vol 51 (1) ◽  
pp. 39-61 ◽  
Author(s):  
Philip Hazel
Keyword(s):  
Richardson Number ◽  
Shear Flows ◽  
Stability Boundary ◽  
Phase Speed ◽  
Neutral Curve ◽  
Free Shear Layer ◽  
Density Profiles ◽  
Complex Phase ◽  
Numerical Studies ◽  
The Stability

The infinitesimal stability of inviscid, parallel, stratified shear flows to two-dimensional disturbances is described by the Taylor-Goldstein equation. Instability can only occur when the Richardson number is less than 1/4 somewhere in the flow. We consider cases where the Richardson number is everywhere non- negative. The eigenvalue problem is expressed in terms of four parameters,Ja ‘typical’ Richardson number, α the (real) wavenumber andcthe complex phase speed of the disturbance. Two computer programs are developed to integrate the stability equation and to solve for eigenvalues: the first findscgiven α andJ, the second finds α andJwhenc≡ 0 (i.e. it computes the stationary neutral curve for the flow). This is sometimes,but not always, the stability boundary in the α,Jplane. The second program works only for cases where the velocity and density profiles are antisymmetric about the velocity inflexion point. By means of these two programs, several configurations of velocity and density have been investigated, both of the free-shear-layer type and the jet type. Calculations of temporal growth rates for particular profiles have been made.

Stability of spatially developing boundary layers in pressure gradients

1995 ◽  
Vol 300 ◽  
pp. 117-147 ◽  
Author(s):  
Rama Govindarajan ◽  
R. Narasimha
Keyword(s):  
Boundary Layer ◽  
Stability Boundary ◽  
Streamwise Velocity ◽  
Present Approach ◽  
Pressure Gradients ◽  
Mean Flow ◽  
Rational Analysis ◽  
Displacement Effect ◽  
The Stability ◽  
New Formulation

A new formulation of the stability of boundary-layer flows in pressure gradients is presented, taking into account the spatial development of the flow and utilizing a special coordinate transformation. The formulation assumes that disturbance wavelength and eigenfunction vary downstream no more rapidly than the boundary-layer thickness, and includes all terms nominally of orderR−1in the boundary-layer Reynolds numberR. In Blasius flow, the present approach is consistent with that of Bertolottiet al.(1992) toO(R−1) but simpler (i.e. has fewer terms), and may best be seen as providing a parametric differential equation which can be solved without having to march in space. The computed neutral boundaries depend strongly on distance from the surface, but the one corresponding to the inner maximum of the streamwise velocity perturbation happens to be close to the parallel flow (Orr-Sommerfeld) boundary. For this quantity, solutions for the Falkner-Skan flows show the effects of spatial growth to be striking only in the presence of strong adverse pressure gradients. As a rational analysis toO(R−1) demands inclusion of higher-order corrections on the mean flow, an illustrative calculation of one such correction, due to the displacement effect of the boundary layer, is made, and shown to have a significant destabilizing influence on the stability boundary in strong adverse pressure gradients. The effect of non-parallelism on the growth of relatively high frequencies can be significant at low Reynolds numbers, but is marginal in other cases. As an extension of the present approach, a method of dealing with non-similar flows is also presented and illustrated.However, inherent in the transformation underlying the present approach is a lower-order non-parallel theory, which is obtained by dropping all terms of nominal orderR−1except those required for obtaining the lowest-order solution in the critical and wall layers. It is shown that a reduced Orr-Sommerfeld equation (in transformed coordinates) already contains the major effects of non-parallelism.

scholarly journals Mean flow evolution of saturated forced shear flows in polytropic atmospheres

2019 ◽  
Vol 82 ◽  
pp. 259-269
Author(s):  
V. Witzke ◽  
L.J. Silvers
Keyword(s):  
Large Scale ◽  
Shear Flows ◽  
Three Dimensional ◽  
Relaxation Method ◽  
Marginal Stability ◽  
Small Scale ◽  
Mean Flow ◽  
Physical Processes ◽  
Flow Evolution ◽  
Stellar Interiors

In stellar interiors shear flows play an important role in many physical processes. So far helioseismology provides only large-scale measurements, and so the small-scale dynamics remains insufficiently understood. To draw a connection between observations and three-dimensional DNS of shear driven turbulence, we investigate horizontally averaged profiles of the numerically obtained mean state. We focus here on just one of the possible methods that can maintain a shear flow, namely the average relaxation method. We show that although some systems saturate by restoring linear marginal stability this is not a general trend. Finally, we discuss the reason that the results are more complex than expected.

Theoretical Study on the Flow Instability of Supercritical Water in the Parallel Channels

2012 ◽  
Author(s):  
Yali Su ◽  
Jian Feng ◽  
Wenxi Tian ◽  
Suizheng Qiu ◽  
Guanghui Su
Keyword(s):  
Supercritical Water ◽  
Flow Instability ◽  
Stability Boundary ◽  
Physical Models ◽  
Marginal Stability ◽  
Inlet Temperature ◽  
Flow And Heat Transfer ◽  
System Pressure ◽  
Parallel Channels ◽  
The Stability

For the flow of the supercritical water (SCW), the fierce variation of density and specific volume possibly cause flow instability. Based on the structure of parallel channels, mathematical and physical models were established to simulate the flow and heat transfer characteristics of the supercritical water in the parallel channels with semi-implicit scheme and staggered mesh scheme. Flow instability of super-critical water was obtained by using the little perturbation method. Pseudo-subcooling number (NSUB) and pseudo-phase change number (NPCH) are defined based on the property of SCW. The marginal stability boundary (MSB) is obtained with using the NSUB and NPCH. The effects of mass flow rate, inlet temperature and system pressure on the flow instability boundary were also investigated. When increasing the mass flows and system pressure, decreasing the heat flux, the stability in the parallel channels increases. The effect of inlet temperature in the low pseudo-subcooling number region is different from that in high pseudo-subcooling number region.

scholarly journals A Geometric Interpretation of Zonostrophic Instability

2020 ◽  
Vol 50 (9) ◽  
pp. 2759-2779
Author(s):  
Georgios Kontogiannis ◽  
Nikolaos A. Bakas
Keyword(s):  
Stress Tensor ◽  
Stability Boundary ◽  
Geometric Interpretation ◽  
Mean Flow ◽  
Ginzburg Landau ◽  
Zonal Jets ◽  
Beta Plane ◽  
Geometric Decomposition ◽  
Eddy Dynamics ◽  
The Stability

AbstractThe zonostrophic instability that leads to the emergence of zonal jets in barotropic beta-plane turbulence is analyzed through a geometric decomposition of the eddy stress tensor. The stress tensor is visualized by an eddy variance ellipse whose characteristics are related to eddy properties. The tilt of the ellipse principal axis is the tilt of the eddies with respect to the shear, and the eccentricity of the ellipse is related to the eddy anisotropy, and its size is related to the eddy kinetic energy. Changes of these characteristics are directly related to the vorticity fluxes forcing the mean flow. The statistical state dynamics of the turbulent flow closed at second order is employed as it provides an analytic expression for both the zonostrophic instability and the stress tensor. For the linear phase of the instability, the stress tensor is analytically calculated at the stability boundary. For the nonlinear equilibration of the instability the tensor is calculated in the limit of small supercriticality in which the amplitude of the jet velocity follows Ginzburg–Landau dynamics. It is found that, dependent on the characteristics of the forcing, the jet is accelerated either because the jet primarily anisotropizes the eddies so as to produce upgradient fluxes, or because the jet changes the eddy tilt. The instability equilibrates as these changes are partially reversed by the nonlinear jet–eddy dynamics.

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