A Family of Wave‐Mean Shear Interactions and Their Instability to Longitudinal Vortex Form

Phillips, W. A.; Shen, Qing

doi:10.1002/sapm1996962143

Cited by 14 publications

(32 citation statements)

References 25 publications

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“…Comparisons with experiments by Gong, Taylor & Dörnbrack (1996) further indicate that CLg is physically realizable (Phillips, Wu & Lumley 1996). But unlike CL2, where instability is assured in a neutral wavy disturbance only when differential drift and shear are of the same sense (for temporal wavy disturbances see Phillips 2002Phillips , 2003, instability to CLg must satisfy the necessary but not sufficient Craik-Phillips-Shen criterion (Craik 1982;Phillips & Shen 1996). This criterion states that an O(1) shear flow bounded by rigid wavy walls is unstable to CLg if, from the reference frame of the waves and in the direction of increasing mean shear, the relative increase in mean velocity exceeds the relative increase in wave amplitude; specifically (in terms of later defined variables), ϑdU/dz > Udϑ/dz where ϑ = α|Φ|.…”

Section: Introductionmentioning

confidence: 93%

On the spacing of Langmuir circulation in strong shear

Phillips¹

2005

J. Fluid Mech.

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The inviscid instability of O(ε) two-dimensional free-surface gravity waves propagating along an O (1) parallel shear flow is considered. The modes of instability involve spanwise-periodic longitudinal vortices resembling oceanic Langmuir circulation. Here, not only are wave-induced mean effects important but also wave modulation, caused by velocity anomalies which develop in the streamwise direction. The former are described by a generalized Lagrangian-mean formulation and the latter by a modified Rayleigh equation. Since both effects are essential, the instability may be called 'generalized' Craik-Leibovich (CLg). Of specific interest is whether spanwise distortion of the wave field, both at the free surface and in the interior, acts to enhance or inhibit instability to longitudinal vortices. Also of interest is whether the instability gives rise to a preferred spacing for the vortices and whether that spacing concurs well or poorly with experiment. The layer depth is varied from much less than the e-folding depth of the O(ε) wave motion to infinity. Relative to an identical shear flow with rigid though wavy top boundary, it is found, inter alia, that wave modulation acts in concert with the free surface, at some wavenumbers, to increase the maximum growth rate of the instability. Indeed, two preferred spanwise spacings occur, one which gives rise to longitudinal vortices through a convective oscillatory bifurcation and a second, at higher wavenumber and growth rate, through a stationary bifurcation. The preferred spacings set by the stationary bifurcation concur well with those observed in laboratory experiments, with the implication that the instability acting in the experiments is very likely to be CLg.

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Section: Introductionmentioning

confidence: 93%

On the spacing of Langmuir circulation in strong shear

Phillips¹

2005

J. Fluid Mech.

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“…Finally, note that while d i and p i are quadratic averages of the interacting fluctuating velocityȗ j and displacement fields ξ j , and are specified by GLM, the fieldsȗ j and ξ j themselves are necessarily solutions to NS given u i and appropriate boundary conditions. Examples in whichȗ j , ξ j and d i , p i are determined for given u i can be found in [11,16,17].…”

Section: Formulationmentioning

confidence: 99%

“…The drift and pseudomomentum are defined in terms of correlations that involve velocity fluctuations of a fluid particle and its displacement from a defined mean position [2] and can, for single wave trains, be evaluated in closed form. That said, the complexity of the calculation increases when the waves interact with an aligned shear flow, from zero or weak shear [4,9], to moderate shear [6], to strong shear [10,11]. Evaluation of the correlations poses even further difficulty for a spectrum of waves, even a discrete symmetric spectrum of irrotational waves in weak shear [12] and is formidable for a similar spectrum of rotational waves in strong shear [3].…”

Section: Introductionmentioning

confidence: 99%

“…In strong shear, the velocity scale of the streaks far exceeds that of the rolls [10,17], which means the streaks act to modulate the fluctuating field. Thus, while retaining the seminal idea of the Craik-Leibovich [12] work, the CLg instability is markedly different to CL2 [10,11]. Accordingly the underlying theory is more complicated [17], a feature which, in addition to the aforementioned secularity, confronted those [14,15,44] who first sought to take such notions further.…”

Section: Introductionmentioning

confidence: 99%

“…Accordingly the underlying theory is more complicated [17], a feature which, in addition to the aforementioned secularity, confronted those [14,15,44] who first sought to take such notions further. Craik [10] and Phillips & Shen [11] were the first to solve this theory analytically, albeit in an asymptotic limit in the simpler context of irrotational wave fields in uniform shear [10] and rotational wave fields in nonuniform shear [11]; others [5,45,46] have considered more complicated cases numerically. As mentioned above, CLg theory requires the drift and pseudomomentum as input.…”

Section: Introductionmentioning

confidence: 99%

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Drift and pseudomomentum in bounded turbulent shear flows

Phillips

2015

Phys. Rev. E

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This paper is concerned with the evaluation of two Lagrangian measures which arise in oscillatory or fluctuating shear flows when the fluctuating field is rotational and the spectrum of wave numbers which comprise it is continuous. The measures are the drift and pseudomomentum. Phillips [J. Fluid Mech. 430, 209 (2001)] has shown that the measures are, in such instances, succinctly expressed in terms of Lagrangian integrals of Eulerian space-time correlations. But they are difficult to interpret, and the present work begins by expressing them in a more insightful form. This is achieved by assuming the space-time correlations are separable as magnitude, determined by one-point velocity correlations, and spatial diminution. The measures then parse into terms comprised of the mean Eulerian velocity, one-point velocity correlations, and a family of integrals of spatial diminution, which in turn define a series of Lagrangian time and velocity scales. The pseudomomentum is seen to be strictly negative and related to the turbulence kinetic energy, while the drift is mixed and strongly influenced by the Reynolds stress. Both are calculated for turbulent channel flow for a range of Reynolds numbers and appear, as the Reynolds number increases, to approach a terminal form. At all Reynolds numbers studied, the pseudomomentum has a sole peak located in wall units in the low teens, while at the highest Reynolds number studied, Re(τ)=5200, the drift is negative in the vicinity of that peak, positive elsewhere, and largest near the rigid boundary. In contrast, the time and velocity scales grow almost logarithmically over much of the layer. Finally, the drift and pseudomomentum are discussed in the context of coherent wall layer structures with which they are intricately linked.

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The multi-flow framework and the generalized Lagrangian mean theory

Viúdez¹

2003

Fluid Dyn. Res.

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A mathematical framework for describing di erent ows of the same uid and their relationships is introduced. This theory, named the Multi-ow Framework (MfF), considers the referential, spatial, and relative descriptions of ÿelds of both pure and mixed distributions. A mixed distribution, as opposed to a pure distribution, results when the transformation between descriptions of some quantity in one ow is carried out using the particle or position distributions from another ow. The basic kinematics of the mixed distributions is developed with emphasis on those ÿelds giving the circulation and ux in one ow per unit of length and area of another ow. These ÿelds are called here the relative circulation and the relative ux ÿelds, respectively.The MfF is applied to both the material and the spatial ensemble average theories. The results in the material average framework are then applied to obtain a new formulation and interpretation of the conservation of mass, in terms of the mean-ow density, and balance of linear momentum given in the Generalized Lagrangian Mean theory. The MfF puts the GLM mean-ow density into an exact correspondence with the reference mass density, of every ow at the average position, per unit of spatial volume of the average ow relative to the reference (initial) volume. The MfF formulates the GLM momentum equation as the spatial ensemble average of the relative circulation balance of momentum relative to the actual ow in the material average ow, expressed in terms of the rate of change of the relative circulation velocity.

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A Family of Wave‐Mean Shear Interactions and Their Instability to Longitudinal Vortex Form

Cited by 14 publications

References 25 publications

On the spacing of Langmuir circulation in strong shear

On the spacing of Langmuir circulation in strong shear

Drift and pseudomomentum in bounded turbulent shear flows

The multi-flow framework and the generalized Lagrangian mean theory

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