scholarly journals An exact theory of nonlinear waves on a Lagrangian-mean flow

1978 ◽  
Vol 89 (4) ◽  
pp. 609-646 ◽  
Author(s):  
D. G. Andrews ◽  
M. E. Mcintyre
Keyword(s):  
Water Waves ◽  
Finite Amplitude ◽  
Mean Flow ◽  
Radiation Stress ◽  
Initial State ◽  
Mean Velocity ◽  
Generalized Lagrangian ◽  
Flow Evolution ◽  
The Mean ◽  
The Waves

An exact and very general Lagrangian-mean description of the back effect of oscillatory disturbances upon the mean state is given. The basic formalism applies to any problem whose governing equations are given in the usual Eulerian form, and irrespective of whether spatial, temporal, ensemble, or ‘two-timing’ averages are appropriate. The generalized Lagrangian-mean velocity cannot be defined exactly as the ‘mean following a single fluid particle’, but in cases where spatial averages are taken can easily be visualized, for instance, as the motion of the centre of mass of a tube of fluid particles which lay along the direction of averaging in a hypothetical initial state of no disturbance.The equations for the Lagrangian-mean flow are more useful than their Eulerian-mean counterparts in significant respects, for instance in explicitly representing the effect upon mean-flow evolution of wave dissipation or forcing. Applications to irrotational acoustic or water waves, and to astrogeophysical problems of waves on axisymmetric mean flows are discussed. In the latter context the equations embody generalizations of the Eliassen-Palm and Charney-Drazin theorems showing the effects on the mean flow of departures from steady, conservative waves, for arbitrary, finite-amplitude disturbances to a stratified, rotating fluid, with allowance for self-gravitation as well as for an external gravitational field.The equations show generally how the pseudomomentum (or wave ‘momentum’) enters problems of mean-flow evolution. They also indicate the extent to which the net effect of the waves on the mean flow can be described by a ‘radiation stress’, and provide a general framework for explaining the asymmetry of radiation-stress tensors along the lines proposed by Jones (1973).

On the stability of an inviscid shear layer which is periodic in space and time

1967 ◽  
Vol 27 (4) ◽  
pp. 657-689 ◽  
Author(s):  
R. E. Kelly
Keyword(s):  
Growth Rate ◽  
Shear Layer ◽  
Finite Amplitude ◽  
Periodic Component ◽  
Periodic Flow ◽  
Flow Profile ◽  
Mean Flow ◽  
Mean Velocity ◽  
The Mean ◽  
The Stability

In experiments concerning the instability of free shear layers, oscillations have been observed in the downstream flow which have a frequency exactly half that of the dominant oscillation closer to the origin of the layer. The present analysis indicates that the phenomenon is due to a secondary instability associated with the nearly periodic flow which arises from the finite-amplitude growth of the fundamental disturbance.At first, however, the stability of inviscid shear flows, consisting of a non-zero mean component, together with a component periodic in the direction of flow and with time, is investigated fairly generally. It is found that the periodic component can serve as a means by which waves with twice the wavelength of the periodic component can be reinforced. The dependence of the growth rate of the subharmonic wave upon the amplitude of the periodic component is found for the case when the mean flow profile is of the hyperbolic-tangent type. In order that the subharmonic growth rate may exceed that of the most unstable disturbance associated with the mean flow, the amplitude of the streamwise component of the periodic flow is required to be about 12 % of the mean velocity difference across the shear layer. This represents order-of-magnitude agreement with experiment.Other possibilities of interaction between disturbances and the periodic flow are discussed, and the concluding section contains a discussion of the interactions on the basis of the energy equation.

The evolution of strained turbulent plane wakes

2002 ◽  
Vol 463 ◽  
pp. 53-120 ◽  
Author(s):  
MICHAEL M. ROGERS
Keyword(s):  
Wake Flow ◽  
Strain Rates ◽  
Mean Flow ◽  
Mean Velocity ◽  
Velocity Deficit ◽  
Self Similar Solution ◽  
Flow Evolution ◽  
Self Similar ◽  
The Mean ◽  
Pseudo Spectral

Direct numerical simulations of ten turbulent time-evolving strained wakes have been generated using a pseudo-spectral numerical method. In all the simulations, the strain was applied to the same (previously generated) initial developed self-similar wake flow field. The cases include flows in which the wake is subjected to various orientations of the applied mean strain, including both plane and axisymmetric strain configurations. In addition, for one particular strain geometry, cases with differing strain rates were considered. Although classical self-similar analysis does yield a self-similar solution for strained wakes, this solution does not describe the observed flow evolution. Instead, the wake mean velocity profiles evolve according to a different ‘equilibrium similarity solution’, with the strained wake width being determined by the straining in the inhomogeneous cross-stream direction. Wakes that are compressed in this direction eventually exhibit constant widths, whereas wakes in cases with expansive cross-stream strain ultimately spread at the same rate as the distortion caused by the applied strain. The shape of the wake mean velocity deficit profile is nearly universal. Although the effect of the strain on the mean flow is pronounced and rapid, the response of the turbulence to the strain occurs more slowly. Changes in the turbulence intensity cannot keep pace with changes in the mean wake velocity deficit, even for relatively low strain rates.

Nonlinear internal gravity waves in a rotating fluid

1975 ◽  
Vol 71 (3) ◽  
pp. 497-512 ◽  
Author(s):  
R. Grimshaw
Keyword(s):  
Gravity Waves ◽  
Wave Structure ◽  
Internal Gravity Waves ◽  
Mean Flow ◽  
Mean Velocity ◽  
Main Effect ◽  
Local Wave ◽  
The Mean ◽  
The Waves ◽  
Circulation Theorem

The interaction between internal gravity waves in a rotating frame and the mean flow is discussed for the case when the properties of the mean flow vary slowly on a scale determined by the local wave structure. The principle of conservation of wave action is established. It is shown that the main effect of the waves on the Lagrangian mean velocity is due to an appropriate ‘radiation stress’ tensor. A circulation theorem and a potential-vorticity equation are derived for the mean velocity.

On internal gravity waves in an accelerating shear flow

1978 ◽  
Vol 88 (4) ◽  
pp. 623-639 ◽  
Author(s):  
S. A. Thorpe
Keyword(s):  
Gravity Waves ◽  
Phase Distribution ◽  
Phase Speed ◽  
Internal Gravity Waves ◽  
Initial Energy ◽  
Constant Density ◽  
Mean Flow ◽  
Mean Velocity ◽  
The Mean ◽  
The Waves

The investigation of the effects which a changing mean flow has on a uniform train of internal gravity waves (Thorpe 1978a) is continued by considering waves in a uniformly accelerating stratified plane Couette flow with constant density gradient. Experiments reveal a change in the mode structure and phase distribution of the waves, and their eventual breaking near the boundary where the mean flow is greatest, the phase speed of the waves being positive. A linear numerical model is devised which accurately describes the waves up to the onset of their breaking, and this is used to investigate their energetics. The working of the Reynolds stress against the mean velocity gradient results in a very rapid transfer of energy from the waves to the mean flow, so that by the time breaking occurs only a small fraction of their initial energy remains for possible transfer into potential energy of the fluid.The consequences have important applications in oceanography and meteorology, to flow stability and flow generation, and explain some earlier laboratory observations.

A Wave Amplitude Transition in a Quasi-Linear Model with Radiative Forcing and Surface Drag

2009 ◽  
Vol 66 (11) ◽  
pp. 3479-3490 ◽  
Author(s):  
Orli Lachmy ◽  
Nili Harnik
Keyword(s):  
Potential Vorticity ◽  
Radiative Forcing ◽  
Wave Amplitude ◽  
Finite Amplitude ◽  
Vertical Shear ◽  
Mean Flow ◽  
Horizontal Scale ◽  
Zonal Wavenumber ◽  
The Mean ◽  
The Waves

Abstract A quasi-linear two-layer quasigeostrophic β-plane model of the interaction between a baroclinic jet and a single zonal wavenumber perturbation is used to study the mechanics leading to a wave amplitude bifurcation—in particular, the role of the critical surfaces in the upper-tropospheric jet flanks. The jet is forced by Newtonian heating toward a radiative equilibrium state, and Ekman damping is applied at the surface. When the typical horizontal scale is approximately the Rossby radius of deformation, the waves equilibrate at a finite amplitude that is comparable to the mean flow. This state is obtained as a result of a wave-induced temporary destabilization of the mean flow, during which the waves grow to their finite-equilibrium amplitude. When the typical horizontal scale is wider, the model also supports a state in which the waves equilibrate at negligible amplitudes. The transition from small to finite-amplitude waves, which occurs at weak instabilities, is abrupt as the parameters of the system are gradually varied, and in a certain range of parameter values both equilibrated states are supported. The simple two-layer quasi-linear setting of the model allows a detailed examination of the temporary destabilization process inherent in the large-amplitude equilibration. As the waves grow they reduce the baroclinic growth by reducing the vertical shear of the mean flow, and reduce the barotropic decay by reducing the mean potential vorticity gradient at the inner sides of the upper-layer critical levels. Temporary destabilization occurs when the reduction in barotropic decay is larger than the reduction in baroclinic growth, leading to a larger total growth rate. Ekman friction and radiative damping are found to play a major role in sustaining the vertical shear of the mean flow and enabling the baroclinic growth to continue. By controlling the mean flow potential vorticity gradient near the critical level, the model evolution can be changed from one type of equilibration to the other.

scholarly journals Quantifying the Effects of Wave—Current Interactions on Tidal Energy Resource at Sites in the English Channel Using Coupled Numerical Simulations

Energies ◽  
2021 ◽  
Vol 14 (12) ◽  
pp. 3625
Author(s):  
Jon Hardwick ◽  
Ed B. L. Mackay ◽  
Ian G. C. Ashton ◽  
Helen C. M. Smith ◽  
Philipp R. Thies
Keyword(s):  
Wave Model ◽  
Tidal Energy ◽  
English Channel ◽  
Flow Conditions ◽  
Mean Flow ◽  
Radiation Stress ◽  
Large Wave ◽  
Stress Gradients ◽  
The Mean ◽  
Flow Power

Numerical modeling of currents and waves is used throughout the marine energy industry for resource assessment. This study compared the output of numerical flow simulations run both as a standalone model and as a two-way coupled wave–current simulation. A regional coupled flow-wave model was established covering the English Channel using the Delft D-Flow 2D model coupled with a SWAN spectral wave model. Outputs were analyzed at three tidal energy sites: Alderney Race, Big Roussel (Guernsey), and PTEC (Isle of Wight). The difference in the power in the tidal flow between coupled and standalone model runs was strongly correlated to the relative direction of the waves and currents. The net difference between the coupled and standalone runs was less than 2.5%. However, when wave and current directions were aligned, the mean flow power was increased by up to 7%, whereas, when the directions were opposed, the mean flow power was reduced by as much as 9.6%. The D-Flow Flexible Mesh model incorporates the effects of waves into the flow calculations in three areas: Stokes drift, forcing by radiation stress gradients, and enhancement of the bed shear stress. Each of these mechanisms is discussed. Forcing from radiation stress gradients is shown to be the dominant mechanism affecting the flow conditions at the sites considered, primarily caused by dissipation of wave energy due to white-capping. Wave action is an important consideration at tidal energy sites. Although the net impact on the flow power was found to be small for the present sites, the effect is site specific and may be significant at sites with large wave exposure or strong asymmetry in the flow conditions and should thus be considered for detailed resource and engineering assessments.

scholarly journals Influence of the Monsoon Trough on Westward-Propagating Tropical Waves over the Western North Pacific. Part II: Energetics and Numerical Experiments

2015 ◽  
Vol 28 (23) ◽  
pp. 9332-9349 ◽  
Author(s):  
Liang Wu ◽  
Zhiping Wen ◽  
Renguang Wu
Keyword(s):  
North Pacific ◽  
Numerical Experiments ◽  
Western North Pacific ◽  
Monsoon Trough ◽  
Water Model ◽  
Mean Flow ◽  
Horizontal Structure ◽  
Rotational Flows ◽  
The Mean ◽  
The Waves

Abstract Part I of this study examined the modulation of the monsoon trough (MT) on tropical depression (TD)-type–mixed Rossby–gravity (MRG) and equatorial Rossby (ER) waves over the western North Pacific based on observations. This part investigates the interaction of these waves with the MT through a diagnostics of energy conversion that separates the effect of the MT on TD–MRG and ER waves. It is found that the barotropic conversion associated with the MT is the most important mechanism for the growth of eddy energy in both TD–MRG and ER waves. The large rotational flows help to maintain the rapid growth and tilted horizontal structure of the lower-tropospheric waves through a positive feedback between the wave growth and horizontal structure. The baroclinic conversion process associated with the MT contributes a smaller part for TD–MRG waves, but is of importance comparable to barotropic conversion for ER waves as it can produce the tilted vertical structure. The growth rates of the waves are much larger during strong MT years than during weak MT years. Numerical experiments are conducted for an idealized MRG or ER wave using a linear shallow-water model. The results confirm that the monsoon background flow can lead to an MRG-to-TD transition and the ER wave amplifies along the axis of the MT and is more active in the strong MT state. Those results are consistent with the findings in Part I. This indicates that the mean flow of the MT provides a favorable background condition for the development of the waves and acts as a key energy source.

The structure of a highly decelerated axisymmetric turbulent boundary layer

2021 ◽  
Vol 929 ◽  
Author(s):  
N. Agastya Balantrapu ◽  
Christopher Hickling ◽  
W. Nathan Alexander ◽  
William Devenport
Keyword(s):  
Boundary Layer ◽  
Pressure Gradient ◽  
Shear Layer ◽  
Layer Thickness ◽  
Boundary Layer Thickness ◽  
Large Scale ◽  
Convection Velocity ◽  
Mean Flow ◽  
Mean Velocity ◽  
The Mean

Experiments were performed over a body of revolution at a length-based Reynolds number of 1.9 million. While the lateral curvature parameters are moderate ( $\delta /r_s < 2, r_s^+>500$ , where $\delta$ is the boundary layer thickness and r s is the radius of curvature), the pressure gradient is increasingly adverse ( $\beta _{C} \in [5 \text {--} 18]$ where $\beta_{C}$ is Clauser’s pressure gradient parameter), representative of vehicle-relevant conditions. The mean flow in the outer regions of this fully attached boundary layer displays some properties of a free-shear layer, with the mean-velocity and turbulence intensity profiles attaining self-similarity with the ‘embedded shear layer’ scaling (Schatzman & Thomas, J. Fluid Mech., vol. 815, 2017, pp. 592–642). Spectral analysis of the streamwise turbulence revealed that, as the mean flow decelerates, the large-scale motions energize across the boundary layer, growing proportionally with the boundary layer thickness. When scaled with the shear layer parameters, the distribution of the energy in the low-frequency region is approximately self-similar, emphasizing the role of the embedded shear layer in the large-scale motions. The correlation structure of the boundary layer is discussed at length to supply information towards the development of turbulence and aeroacoustic models. One major finding is that the estimation of integral turbulence length scales from single-point measurements, via Taylor's hypothesis, requires significant corrections to the convection velocity in the inner 50 % of the boundary layer. The apparent convection velocity (estimated from the ratio of integral length scale to the time scale), is approximately 40 % greater than the local mean velocity, suggesting the turbulence is convected much faster than previously thought. Closer to the wall even higher corrections are required.

scholarly journals Self-consistent triple decomposition of the turbulent flow over a backward-facing step under finite amplitude harmonic forcing

2019 ◽  
Vol 475 (2225) ◽  
pp. 20190018
Author(s):  
E. Yim ◽  
P. Meliga ◽  
F. Gallaire
Keyword(s):  
Turbulent Flow ◽  
Finite Amplitude ◽  
Navier Stokes ◽  
Reynolds Stresses ◽  
Mean Flow ◽  
Backward Facing Step ◽  
Eddy Viscosity Model ◽  
Coherent Interaction ◽  
Self Consistent ◽  
The Mean

We investigate the saturation of harmonically forced disturbances in the turbulent flow over a backward-facing step subjected to a finite amplitude forcing. The analysis relies on a triple decomposition of the unsteady flow into mean, coherent and incoherent components. The coherent–incoherent interaction is lumped into a Reynolds averaged Navier–Stokes (RANS) eddy viscosity model, and the mean–coherent interaction is analysed via a semi-linear resolvent analysis building on the laminar approach by Mantič-Lugo & Gallaire (2016 J. Fluid Mech. 793 , 777–797. ( doi:10.1017/jfm.2016.109 )). This provides a self-consistent modelling of the interaction between all three components, in the sense that the coherent perturbation structures selected by the resolvent analysis are those whose Reynolds stresses force the mean flow in such a way that the mean flow generates exactly the aforementioned perturbations, while also accounting for the effect of the incoherent scale. The model does not require any input from numerical or experimental data, and accurately predicts the saturation of the forced coherent disturbances, as established from comparison to time-averages of unsteady RANS simulation data.

A Mean Flow Model for Polymer and Fiber Turbulent Drag Reduction

2005 ◽  
Vol 15 (6) ◽  
pp. 370-389 ◽  
Author(s):  
Anshuman Roy ◽  
Ronald G. Larson
Keyword(s):  
Drag Reduction ◽  
Weissenberg Number ◽  
Extensional Viscosity ◽  
Mean Flow ◽  
Dimensionless Numbers ◽  
Zero Shear Viscosity ◽  
Mean Velocity ◽  
Fiber Concentration ◽  
Turbulent Drag ◽  
The Mean

Abstract We present a one-parameter model that fits quantitatively the mean velocity profiles from experiments and numerical simulations of drag-reduced wall-bounded flows of dilute solutions of polymers and non-Brownian fibers in the low and modest drag reduction regime. The model is based on a viscous mechanism of drag reduction, in which either extended polymers or non-Brownian fibers increase the extensional viscosity of the fluid and thereby suppress both small and large turbulent eddies and reduce momentum transfer to the wall, resulting in drag reduction. Our model provides a rheological interpretation of the upward parallel shift S+ in the mean velocity profile upon addition of polymer, observed by Virk. We show that Virk’s correlations for the dependence on polymer molecular weight and concentration of the onset wall shear stress and slope increment on the Prandtl-Karman plot can be translated to two dimensionless numbers, namely an onset Weissenberg number and an asymptotic Trouton ratio of maximum extensional viscosity to zero-shear viscosity. We believe that our model, while simple, captures the essential features of drag reduction that are universal to flexible polymers and fibers, and, unlike the Virk phenomenology, can easily be extended to flows with inhomogeneous polymer or fiber concentration fields.

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