Ray theory solution for finite amplitude effects in a two‐dimensional hard‐walled waveguide

1989 ◽  
Vol 86 (2) ◽  
pp. 771-776 ◽  
Author(s):  
Kun‐Tien Shu ◽  
Jerry H. Ginsberg
Keyword(s):  
Finite Amplitude ◽  
Two Dimensional ◽  
Ray Theory ◽  
Theory Solution

The anatomy of the mixing transition in homogeneous and stratified free shear layers

2000 ◽  
Vol 413 ◽  
pp. 1-47 ◽  
Author(s):  
C. P. CAULFIELD ◽  
W. R. PELTIER
Keyword(s):  
Shear Layer ◽  
Three Dimensional ◽  
Streamwise Vortex ◽  
Shear Layers ◽  
Finite Amplitude ◽  
Small Scale ◽  
Two Dimensional ◽  
Streamwise Vortices ◽  
Free Shear Layers ◽  
Nonlinear Amplification

We investigate the detailed nature of the ‘mixing transition’ through which turbulence may develop in both homogeneous and stratified free shear layers. Our focus is upon the fundamental role in transition, and in particular the associated ‘mixing’ (i.e. small-scale motions which lead to an irreversible increase in the total potential energy of the flow) that is played by streamwise vortex streaks, which develop once the primary and typically two-dimensional Kelvin–Helmholtz (KH) billow saturates at finite amplitude.Saturated KH billows are susceptible to a family of three-dimensional secondary instabilities. In homogeneous fluid, secondary stability analyses predict that the stream-wise vortex streaks originate through a ‘hyperbolic’ instability that is localized in the vorticity braids that develop between billow cores. In sufficiently strongly stratified fluid, the secondary instability mechanism is fundamentally different, and is associated with convective destabilization of the statically unstable sublayers that are created as the KH billows roll up.We test the validity of these theoretical predictions by performing a sequence of three-dimensional direct numerical simulations of shear layer evolution, with the flow Reynolds number (defined on the basis of shear layer half-depth and half the velocity difference) Re = 750, the Prandtl number of the fluid Pr = 1, and the minimum gradient Richardson number Ri(0) varying between 0 and 0.1. These simulations quantitatively verify the predictions of our stability analysis, both as to the spanwise wavelength and the spatial localization of the streamwise vortex streaks. We track the nonlinear amplification of these secondary coherent structures, and investigate the nature of the process which actually triggers mixing. Both in stratified and unstratified shear layers, the subsequent nonlinear amplification of the initially localized streamwise vortex streaks is driven by the vertical shear in the evolving mean flow. The two-dimensional flow associated with the primary KH billow plays an essentially catalytic role. Vortex stretching causes the streamwise vortices to extend beyond their initially localized regions, and leads eventually to a streamwise-aligned collision between the streamwise vortices that are initially associated with adjacent cores.It is through this collision of neighbouring streamwise vortex streaks that a final and violent finite-amplitude subcritical transition occurs in both stratified and unstratified shear layers, which drives the mixing process. In a stratified flow with appropriate initial characteristics, the irreversible small-scale mixing of the density which is triggered by this transition leads to the development of a third layer within the flow of relatively well-mixed fluid that is of an intermediate density, bounded by narrow regions of strong density gradient.

Instability modelling of drumlin formation incorporating lee-side cavity growth

2009 ◽  
Vol 465 (2109) ◽  
pp. 2681-2702 ◽  
Author(s):  
A. C. Fowler
Keyword(s):  
Mathematical Model ◽  
Free Boundary ◽  
Numerical Method ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Cavity Growth ◽  
Finite Amplitude ◽  
Two Dimensional

It is proposed that the formation of the subglacial bedforms known as drumlins occurs through an instability associated with the flow of ice over a wet deformable till. We pose a mathematical model that describes this instability, and we solve a simplified version of the model numerically in order to establish the form of finite-amplitude two-dimensional waveforms. A feature of the solutions is that cavities frequently form downstream of the bedforms; we allow the model to cater for this possibility and we provide an efficient numerical method to solve the resulting free boundary problem.

The wall jet in a rotating fluid

1997 ◽  
Vol 335 ◽  
pp. 1-28 ◽  
Author(s):  
MELVIN E. STERN ◽  
ERIC P. CHASSIGNET ◽  
J. A. WHITEHEAD
Keyword(s):  
Vertical Wall ◽  
Three Dimensional ◽  
Point Vortex ◽  
Separation Distance ◽  
Finite Amplitude ◽  
Large Reynolds Number ◽  
Spatial Evolution ◽  
Two Dimensional ◽  
Rotating Fluid ◽  
Coriolis Parameter

The previously observed spatial evolution of the two-dimensional turbulent flow from a source on the vertical wall of a shallow layer of rapidly rotating fluid is strikingly different from the non-rotating three-dimensional counterpart, insofar as the instability eddies generated in the former case cause the flow to separate completely from the wall at a finite downstream distance. In seeking an explanation of this, we first compute the temporal evolution of two-dimensional finite-amplitude waves on an unstable laminar jet using a finite difference calculation at large Reynolds number. This yields a dipolar vorticity pattern which propagates normal to the wall, while leaving some of the near-wall vorticity (negative) of the basic flow behind. The residual far-field eddy therefore contains a net positive circulation and this property is incorporated in a heuristic point-vortex model of the spatial evolution of the instability eddies observed in a laboratory experiment of a flow emerging from a source on a vertical wall in a rotating tank. The model parameterizes the effect of Ekman bottom friction in decreasing the circulation of eddies which are periodically emitted from the source flow on the wall. Further downstream, the point vortices of the model merge and separate abruptly from the wall; the statistics suggest that the downstream separation distance scales with the Ekman spin-up time (inversely proportional to the square root of the Coriolis parameter f) and with the mean source velocity. When the latter is small and f is large, qualitative support is obtained from laboratory experiments.

The theory of symmetrical gravity waves of finite amplitude. I

1951 ◽  
Vol 208 (1095) ◽  
pp. 475-486 ◽  
Keyword(s):  
Boundary Condition ◽  
Gravity Waves ◽  
Finite Amplitude ◽  
Breaking Wave ◽  
Two Dimensional ◽  
Linear Boundary ◽  
Dimensional Problem ◽  
Infinite Depth ◽  
Linear Boundary Condition ◽  
Small Parameters

The two-dimensional problem of symmetric finite amplitude gravity waves in an incompressible fluid of infinite depth is treated by a method which first involves satisfying a non-linear boundary condition exactly. The higher approximations are obtained by the method of small parameters. The breaking-wave conditions are discussed and expressions are given for the free-surface equation, the kinetic and the potential energies of the fluid.

Stability of two-dimensional finite-amplitude envelope solitons

1987 ◽  
Vol 35 (10) ◽  
pp. 4273-4279 ◽  
Author(s):  
R. Blaha ◽  
E. W. Laedke ◽  
K. H. Spatschek
Keyword(s):  
Finite Amplitude ◽  
Two Dimensional ◽  
Amplitude Envelope ◽  
Envelope Solitons

The effect of distant side-walls on the evolution and stability of finite-amplitude Rayleigh-Bénard convection

1981 ◽  
Vol 378 (1775) ◽  
pp. 539-566 ◽  
Keyword(s):  
Rayleigh Number ◽  
Prandtl Number ◽  
Finite Amplitude ◽  
Two Dimensional ◽  
Bénard Convection ◽  
Benard Convection ◽  
Side Walls ◽  
Rayleigh Benard Convection ◽  
Rayleigh Numbers ◽  
Rayleigh Benard

A recent study by Cross et al . (1980) has described a class of finite-amplitude phase-winding solutions of the problem of two-dimensional Rayleigh-Bénard convection in a shallow fluid layer of aspect ratio 2 L (≫ 1) confined laterally by rigid side-walls. These solutions arise at Rayleigh numbers R = R 0 + O ( L -1 ) where R 0 is the critical Rayleigh number for the corresponding infinite layer. Nonlinear solutions of constant phase exist for Rayleigh numbers R = R 0 + O ( L -2 ) but of these only the two that bifurcate at the lowest value of R are stable to two-dimensional linearized disturbances in this range (Daniels 1978). In the present paper one set of the class of phase-winding solutions is found to be stable to two-dimensional disturbances. For certain values of the Prandtl number of the fluid and for stress-free horizontal boundaries the results predict that to preserve stability there must be a continual readjustment of the roll pattern as the Rayleigh number is raised, with a corresponding increase in wavelength proportional to R - R 0 . These solutions also exhibit hysteresis as the Rayleigh number is raised and lowered. For other values of the Prandtl number the number of rolls remains unchanged as the Rayleigh number is raised, and the wavelength remains close to its critical value. It is proposed that the complete evolution of the flow pattern from a static state must take place on a number of different time scales of which t = O(( R - R 0 ) -1 ) and t = O(( R - R 0 ) -2 ) are the most significant. When t = O(( R - R 0 ) -1 ) the amplitude of convection rises from zero to its steady-state value, but the final lateral positioning of the rolls is only completed on the much longer time scale t = O(( R - R 0 ) -2 ).

Development of finite-amplitude two-dimensional and three-dimensional disturbances in jet flows

1986 ◽  
Vol 20 (5) ◽  
pp. 668-679
Author(s):  
S. Ya. Gertsenshtein ◽  
I. I. Olaru ◽  
A. Ya. Hudnitokii ◽  
A. N. Sukhorukov
Keyword(s):  
Three Dimensional ◽  
Finite Amplitude ◽  
Jet Flows ◽  
Two Dimensional

An experimental investigation of breaking waves produced by a towed hydrofoil

1981 ◽  
Vol 377 (1770) ◽  
pp. 331-348 ◽  
Keyword(s):  
Surface Profile ◽  
Phase Speed ◽  
Breaking Waves ◽  
Finite Amplitude ◽  
Stokes Wave ◽  
Two Dimensional ◽  
Vertical Thickness ◽  
Turbulent Momentum ◽  
The Vertical Distribution ◽  
Turbulent Water

This paper presents the results of experiments on breaking waves produced by towing a submerged, two-dimensional hydrofoil at constant depth and speed. The wave field consists of a breaker followed by a train of lower, non-breaking waves. The breaker has a small zone of turbulent water riding its forward slope; this zone is called the breaking region. Measurements were made of surface height profiles, the vertical distribution of mean horizontal velocity in the wake of the wave, and the vertical thickness of the wake. The results support the hypothesis that the breaking region imparts a shearing force along the forward slope equal to the component of its weight in that direction. The force produces a turbulent, momentum-deficient wake similar to the wake of a towed, two-dimensional body in an infinite fluid. The vertical thickness of the wake grows in proportion to the square root of distance behind the breaker. The momentum deficit is approximately equal to the maximum momentum flux of a Stokes wave with the same phase speed as the breaker. The surface profile measurements yield several results: the proper independent variables describing the wave are its speed and the slope of its forward face. The relation between breaking wavelength and speed follows the finite-amplitude Stokes wave equation. The amplitude and the vertical extent of the breaking region are both proportional to the phase speed squared; however, they are not functions of the slope of the forward face of the wave. The breaking region has a small oscillation in its length with a regular period of 4.4 the period of a wave with phase speed equal to the hydrofoil speed. The amplitude of the oscillation diminishes with time. It is believed that this oscillation is due to wave components produced when the foil is started from rest.

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