Propagation of finite-amplitude long-wave disturbances in aerosols

1981 ◽  
Vol 21 (5) ◽  
pp. 602-606 ◽  
Author(s):  
A. A. Borisov ◽  
A. F. Vakhgel't ◽  
V. E. Nakoryakov
Keyword(s):  
Finite Amplitude ◽  
Wave Disturbances ◽  
Long Wave

On stability of parallel flow of an incompressible fluid of variable density and viscosity

1962 ◽  
Vol 58 (4) ◽  
pp. 646-661 ◽  
Author(s):  
P. G. Drazin
Keyword(s):  
Incompressible Fluid ◽  
Water Waves ◽  
Salt Water ◽  
Reynolds Numbers ◽  
Variable Density ◽  
Stability Equation ◽  
Wave Disturbances ◽  
Long Wave ◽  
Vertical Variations ◽  
The Stability

ABSTRACTSome aspects of generation of water waves by wind and of turbulence in a heterogeneous fluid may be described by the theory of hydrodynamic stability. The technical difficulties of these problems of instability have led to obscurities in the literature, some of which are elucidated in this paper. The stability equation for a basic steady parallel horizontal flow under the influence of gravity is derived carefully, the undisturbed fluid having vertical variations of density and viscosity. Methods of solution of the equation for large Reynolds numbers and for long-wave disturbances are described. These methods are applied to simple models of wind blowing over water and of fresh water flowing over salt water.

On certain aspects of three-dimensional instability of parallel flows

1983 ◽  
Vol 385 (1788) ◽  
pp. 53-84 ◽  
Keyword(s):  
Three Dimensional ◽  
Periodic Variation ◽  
Finite Amplitude ◽  
Basic Flow ◽  
Spanwise Direction ◽  
Equilibrium Solutions ◽  
Wave Disturbances ◽  
Parallel Flows ◽  
Tollmien Schlichting ◽  
Dimensional Instability

The effect of small imperfections in shear flows on the development of finite-amplitude three-dimensional disturbances in the flow is examined. A model problem is studied, one in which the basic flow is plane Poiseuille flow in a channel and the small imperfection in the form of spanwise periodic variation of the basic flow is introduced from the channel boundaries. It is shown that this has a positive effect on the growth of larger Tollmien-Schlichting wave disturbances, which are in the form of standing waves in the spanwise direction. Equations for the amplitudes of these disturbances, based on Stuart-Watson-Eckhaus theory, are derived and over a range of Reynolds number, the regions in the wavenumber plane over which equilibrium solutions are possible are identified. The possibility of three-dimensional disturbances that are oscillatory in the streamwise direction but that may be growing exponentially in a spanwise direction is examined.

LARGE-SCALE DISTORTIONS OF SQUARE PATTERNS

1994 ◽  
Vol 04 (05) ◽  
pp. 1147-1154 ◽  
Author(s):  
ALEXANDER NEPOMNYASHCHY
Keyword(s):  
Stability Analysis ◽  
Linear Stability ◽  
Linear Stability Analysis ◽  
Large Scale ◽  
Instability Threshold ◽  
Amplitude Equations ◽  
Wave Disturbances ◽  
Long Wave

Stationary square patterns are typical in several instability problems. Near the instability threshold, the evolution of long-wave disturbances can be described by a system of amplitude equations resembling the Newell-Whitehead-Segel equations. These equations are used for the linear stability analysis and the investigation of the defects.

scholarly journals Topographically forced long waves on a sheared coastal current. Part 2. Finite amplitude waves

1997 ◽  
Vol 343 ◽  
pp. 153-168 ◽  
Author(s):  
S. R. CLARKE ◽  
E. R. JOHNSON
Keyword(s):  
Wave Equation ◽  
Initial Conditions ◽  
Finite Amplitude ◽  
Coastal Current ◽  
Constant Vorticity ◽  
Long Wave ◽  
Long Time ◽  
Wave Limit ◽  
Parameter Values ◽  
Upstream Flow

This paper analyses the finite-amplitude flow of a constant-vorticity current past coastal topography in the long-wave limit. A forced finite-amplitude long-wave equation is derived to describe the evolution of the vorticity interface. An analysis of this equation shows that three distinct near-critical regimes occur. In the first the upstream flow is restricted, with overturning of the vorticity interface for sufficiently large topography. In the second quasi-steady nonlinear waves form downstream of the topography with weak upstream influence. In the third regime the upstream rotational fluid is partially blocked. Blocking and overturning are enhanced at headlands with steep rear faces and decreased at headlands with steep forward faces. For certain parameter values both overturning and partially blocked solutions are possible and the long-time evolution is critically dependent on the initial conditions. The reduction of the problem to a one-dimensional nonlinear wave equation allows solutions to be followed to much longer times and parameter space to be explored more finely than in the related pioneering contour-dynamical integrations of Stern (1991).

The distortion of short internal waves produced by a long wave, with application to ocean boundary mixing

1989 ◽  
Vol 208 ◽  
pp. 395-415 ◽  
Author(s):  
S. A. Thorpe
Keyword(s):  
Density Gradient ◽  
Internal Waves ◽  
Numerical Solutions ◽  
Equations Of Motion ◽  
Finite Amplitude ◽  
Constant Density ◽  
Boundary Mixing ◽  
Short Waves ◽  
Long Wave ◽  
Boussinesq Fluid

The propagation of a train of short, small-amplitude, internal waves through a long, finite-amplitude, two-dimensional, internal wave is studied. An exact solution of the equations of motion for a Boussinesq fluid of constant density gradient is used to describe the long wave, and its distortion of the density gradient as well as its velocity field are accounted for in determining the propagation characteristics of the short waves. To illustrate the magnitude of the effects on the short waves, particular numerical solutions are found for short waves generated by an idealized flow induced by a long wave adjacent to sloping, sinusoidal topography in the ocean, and the results are compared with a laboratory experiment. The theory predicts that the long wave produces considerably distortion of the short waves, changing their amplitudes, wavenumbers and propagation directions by large factors, and in a way which is generally consistent with, but not fully tested by, the observations. It is suggested that short internal waves generated by the interaction of relatively long waves with a rough sloping topography may contribute to the mixing observed near continental slopes.

The initial-value problem for long waves of finite amplitude

1964 ◽  
Vol 20 (1) ◽  
pp. 161-170 ◽  
Author(s):  
Robert R. Long
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Integration ◽  
Initial Value Problem ◽  
Finite Amplitude ◽  
Long Waves ◽  
Initial Value ◽  
Partial Differential ◽  
Long Wave ◽  
Constant Slope

Derived herein is a set of partial differential equations governing the propagation of an arbitrary, long-wave disturbance of small, but finite amplitude. The equations reduce to that of Boussinesq (1872) when the assumption is made that the disturbance is propagating in one direction only. The equations are hyperbolic with characteristic curves of constant slope. The initial-value problem can be solved very readily by numerical integration along characteristics. A few examples are included.

Differential equations for long-period gravity waves on fluid of rapidly varying depth

1977 ◽  
Vol 83 (2) ◽  
pp. 289-310 ◽  
Author(s):  
James Hamilton
Keyword(s):  
Gravity Waves ◽  
Wave Equations ◽  
Velocity Potential ◽  
Taylor Series Expansion ◽  
Finite Amplitude ◽  
Necessary Condition ◽  
Exact Equation ◽  
Rapid Convergence ◽  
Long Period ◽  
Long Wave

The conventional long-wave equations for waves propagating over fluid of variable depth depend for their formal derivation on a Taylor series expansion of the velocity potential about the bottom. The expansion, however, is not possible if the depth is not an analytic function of the horizontal co-ordinates and it is a necessary condition for its rapid convergence that the depth is also slowly varying. We show that if in the case of two-dimensional motions the undisturbed fluid is first mapped conformally onto a uniform strip, before the Taylor expansion is made, the analytic condition is removed and the approximations implied in the lowest-order equations are much improved.In the limit of infinitesimal waves of very long period, consideration of the form of the error suggests that by modifying the coefficients of the reformulated equation we may find an equation exact for arbitrary depth profiles. We are thus able to calculate the reflexion coefficients for long-period waves incident on a step change in depth and a half-depth barrier. The forms of the coefficients of the exact equation are not simple; however, for these particular cases, comparison with the coefficients of the reformulated long-wave equation suggests that in most cases the latter may be adequate. This opens up the possibility of beginning to study finite amplitude and frequency effects on regions of rapidly varying depth.

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