On the generation of waves by wind

1980 ◽  
Vol 298 (1441) ◽  
pp. 451-494 ◽  
Keyword(s):  
Surface Waves ◽  
Linear Stability ◽  
Small Amplitude ◽  
Nonlinear Effects ◽  
Amplitude Equation ◽  
Basic Flow ◽  
Parallel Plates ◽  
Cubic Nonlinearity ◽  
Linear Effects ◽  
Layer Flow

The fully developed laminar flow of air over water confined between two infinite parallel plates was used to study nonlinear effects in the generation of surface waves. A linear stability analysis of the basic flow was made and the conditions at which small amplitude surface waves first begin to grow were determined. Then, following Stewartson & Stuart (1971), the nonlinear stability of the flow was examined and the usual parabolic equation with cubic nonlinearity obtained for the amplitude of the disturbances. The calculation of the linear stability characteristics and the coefficients appearing in the amplitude equation was a lengthy computational task, with most interest centred on the coefficient of the nonlinear terms in the amplitude equation. In two profiles, used as crude models of a boundary layer flow of air over water, the calculations indicated that, over a range of parameters, the non-linear effects would reduce the growth rate of the surface waves and hence lead to equilibrium amplitude waves.

The Onset of Instabilities in Mixed Convection Boundary Layer Flow Over a Heated Horizontal Circular Cylinder

2019 ◽  
Vol 11 (5) ◽  
Author(s):  
Smail Mouloud ◽  
Faïçal Nait-Bouda ◽  
Djamel Sadaoui ◽  
Fatsah Mendil
Keyword(s):  
Boundary Layer ◽  
Mixed Convection ◽  
Linear Stability ◽  
Boundary Layer Flow ◽  
Richardson Number ◽  
Basic Flow ◽  
Convection Boundary Layer ◽  
Wide Range ◽  
Convection Boundary ◽  
Layer Flow

The purpose of this study is to examine the instabilities of a two-dimensional mixed convection boundary layer flow induced by an impinging ascending flow on a heated horizontal cylinder. A significant amount of works is done in recent years on this problem because of its wide range of applications. However, they did not check the stability of the flow in the face of small disturbances that occur in reality. For this, we adopt the linear stability theory by first solving the steady basic flow and then solving the linear perturbed problem. Thus, the governing equations of the basic flow are reduced to two coupled partial differential equations and solved numerically with the Keller-Box method. The corresponding steady solution is obtained, by varying the position along the cylinder’s surface, for different values of Richardson number (λ) and Prandtl number (Pr), up to, respectively, 3000 and 20. To examine the onset of thermal instabilities, the linear stability analysis is done using the normal mode decomposition with small harmonic disturbances. The Richardson number λ is chosen as the control parameter of these instabilities. The resulting eigenvalue problem is solved numerically by the use of the pseudospectral method based on the Laguerre polynomials. The computed results for neutral and temporal growth curves are depicted and discussed in detail through graphs for various parametric conditions. The critical conditions are illustrated graphically to show from which thermodynamic state, the flow begins to become unstable. As a main result, from ξ = 0 to ξ ≈ π/3, we found that forced and mixed convection flow cases are linearly stable in this region. However, in free convection case (λ > 100), it appears that the stagnation zone is the most unstable one and then the instability decreases along the cylinder’s surface up to the limit of its first third, thus giving the most stable zone of the cylinder. Beyond ξ ≈ 1.2, strong instabilities are noted also for low values of Richardson number, and the flow tends to an unstable state even in the absence of thermal effect, i.e., hydrodynamically unstable Ri = 0, probably due to the occurring of the boundary layer separation.

Transcritical two-layer flow over topography

1987 ◽  
Vol 178 ◽  
pp. 31-52 ◽  
Author(s):  
W. K. Melville ◽  
Karl R. Helfrich
Keyword(s):  
Steady Flow ◽  
Solitary Waves ◽  
Numerical Solutions ◽  
Kdv Equation ◽  
Single Layer ◽  
Cubic Nonlinearity ◽  
Arbitrary Length ◽  
Weakly Nonlinear ◽  
Flow Over Topography ◽  
Layer Flow

The evolution of weakly-nonlinear two-layer flow over topography is considered. The governing equations are formulated to consider the effects of quadratic and cubic nonlinearity in the transcritical regime of the internal mode. In the absence of cubic nonlinearity an inhomogeneous Korteweg-de Vries equation describes the interfacial displacement. Numerical solutions of this equation exhibit undular bores or sequences of Boussinesq solitary waves upstream in a transcritical regime. For sufficiently large supercritical Froude numbers, a locally steady flow is attained over the topography. In that regime in which both quadratic and cubic nonlinearity are comparable, the evolution of the interface is described by an inhomogeneous extended Kortewegde Vries (EKdV) equation. This equation displays undular bores upstream in a subcritical regime, but monotonic bores in a transcritical regime. The monotonic bores are solitary wave solutions of the corresponding homogeneous EKdV equation. Again, locally steady flow is attained for sufficiently large supercritical Froude numbers. The predictions of the numerical solutions are compared with laboratory experiments which show good agreement with the solutions of the forced EKdV equation for some range of parameters. It is shown that a recent result of Miles (1986), which predicts an unsteady transcritical regime for single-layer flows, may readily be extended to two-layer flows (described by the forced KdV equation) and is in agreement with the results presented here.Numerical experiments exploiting the symmetry of the homogeneous EKdV equation show that solitary waves of fixed amplitude but arbitrary length may be generated in systems described by the inhomogeneous EKdV equation.

scholarly journals On the linear stability of compressible plane Couette flow

1994 ◽  
Vol 258 ◽  
pp. 131-165 ◽  
Author(s):  
Peter W. Duck ◽  
Gordon Erlebacher ◽  
M. Yousuff Hussaini
Keyword(s):  
Linear Stability ◽  
Couette Flow ◽  
Small Amplitude ◽  
Reynolds Numbers ◽  
Inviscid Limit ◽  
Plane Couette Flow ◽  
Moving Wall ◽  
Type Boundary ◽  
The Stability ◽  
Radiation Type

The linear stability of compressible plane Couette flow is investigated. The appropriate basic velocity and temperature distributions are perturbed by a small-amplitude normal-mode disturbance. The full small-amplitude disturbance equations are solved numerically at finite Reynolds numbers, and the inviscid limit of these equations is then investigated in some detail. It is found that instabilities can occur, although the corresponding growth rates are often quite small; the stability characteristics of the flow are quite different from unbounded flows. The effects of viscosity are also calculated, asymptotically, and shown to have a stabilizing role in all the cases investigated. Exceptional regimes to the problem occur when the wave speed of the disturbances approaches the velocity of either of the walls, and these regimes are also analysed in some detail. Finally, the effect of imposing radiation-type boundary conditions on the upper (moving) wall (in place of impermeability) is investigated, and shown to yield results common to both bounded and unbounded flows.

scholarly journals “Extraordinary” modulation instability in optics and hydrodynamics

2021 ◽  
Vol 118 (14) ◽  
pp. e2019348118
Author(s):  
Guillaume Vanderhaegen ◽  
Corentin Naveau ◽  
Pascal Szriftgiser ◽  
Alexandre Kudlinski ◽  
Matteo Conforti ◽  
...  
Keyword(s):  
Wave Propagation ◽  
Stability Analysis ◽  
Linear Stability ◽  
Nonlinear Theory ◽  
Linear Stability Analysis ◽  
Water Waves ◽  
Modulation Instability ◽  
Nonlinear Effects ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory

The classical theory of modulation instability (MI) attributed to Bespalov–Talanov in optics and Benjamin–Feir for water waves is just a linear approximation of nonlinear effects and has limitations that have been corrected using the exact weakly nonlinear theory of wave propagation. We report results of experiments in both optics and hydrodynamics, which are in excellent agreement with nonlinear theory. These observations clearly demonstrate that MI has a wider band of unstable frequencies than predicted by the linear stability analysis. The range of areas where the nonlinear theory of MI can be applied is actually much larger than considered here.

scholarly journals The transformation of an interfacial solitary wave of elevation at a bottom step

2009 ◽  
Vol 16 (1) ◽  
pp. 33-42 ◽  
Author(s):  
V. Maderich ◽  
T. Talipova ◽  
R. Grimshaw ◽  
E. Pelinovsky ◽  
B. H. Choi ◽  
...  
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Numerical Study ◽  
Nonlinear Effects ◽  
Ocean Model ◽  
Gardner Equation ◽  
Fully Nonlinear ◽  
Layer Flow ◽  
The One ◽  
Bottom Step

Abstract. In this paper we study the transformation of an internal solitary wave at a bottom step in the framework of two-layer flow, for the case when the interface lies close to the bottom, and so the solitary waves are elevation waves. The outcome is the formation of solitary waves and dispersive wave trains in both the reflected and transmitted fields. We use a two-pronged approach, based on numerical simulations of the fully nonlinear equations using a version of the Princeton Ocean Model on the one hand, and a theoretical and numerical study of the Gardner equation on the other hand. In the numerical experiments, the ratio of the initial wave amplitude to the layer thickness is varied up one-half, and nonlinear effects are then essential. In general, the characteristics of the generated solitary waves obtained in the fully nonlinear simulations are in reasonable agreement with the predictions of our theoretical model, which is based on matching linear shallow-water theory in the vicinity of a step with solutions of the Gardner equation for waves far from the step.

A porous-wavemaker theory

1983 ◽  
Vol 132 ◽  
pp. 395-406 ◽  
Author(s):  
Allen T. Chwang
Keyword(s):  
Surface Wave ◽  
Surface Waves ◽  
Small Amplitude ◽  
Vertical Plate ◽  
Hydrodynamic Pressure ◽  
Finite Depth ◽  
Wave Profile ◽  
Total Force ◽  
Wave Effect ◽  
Effect Parameter

A porous-wavemaker theory is developed to analyse small-amplitude surface waves on water of finite depth, produced by horizontal oscillations of a porous vertical plate. Analytical solutions in closed forms are obtained for the surface-wave profile, the hydrodynamic-pressure distribution and the total force on the wavemaker. The influence of the wave-effect parameter C and the porous-effect parameter G, both being dimensionless, on the surface waves and on the hydrodynamic pressures is discussed in detail.

The Linear Stability of Finite Amplitude Surface Waves on a Linear Shearing Flow

1989 ◽  
Vol 58 (7) ◽  
pp. 2386-2396 ◽  
Author(s):  
Makoto Okamura ◽  
Masayuki Oikawa
Keyword(s):  
Surface Waves ◽  
Linear Stability ◽  
Finite Amplitude ◽  
Shearing Flow

Linear Stability of Momentum Boundary Layer Flow and Heat Transfer Over a Moving Wedge

2020 ◽  
Vol 142 (6) ◽  
Author(s):  
Ramesh B. Kudenatti ◽  
Noor E. Misbah ◽  
M. C. Bharathi
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Linear Stability ◽  
Boundary Layer Flow ◽  
Shooting Methods ◽  
Flow And Heat Transfer ◽  
Moving Wedge ◽  
Layer Flow ◽  
Steady Solutions ◽  
Similar Solutions

Abstract This paper studies the linear stability of the unsteady boundary-layer flow and heat transfer over a moving wedge. Both mainstream flow outside the boundary layer and the wedge velocities are approximated by the power of the distance along the wedge wall. In a similar manner, the temperature of the wedge is approximated by the power of the distance that leads to a wall exponent temperature parameter. The governing boundary layer equations admit a class of self-similar solutions under these approximations. The Chebyshev collocation and shooting methods are utilized to predict the upper and lower branch solutions for various parameters. For these two solutions, the velocity, temperature profiles, wall shear-stress, and temperature gradient are entirely different and need to be assessed for their stability as to which of these solutions is practically realizable. It is shown that algebraically growing steady solutions do exist and their effects are significant in the unsteady context. The resulting eigenvalue problem determines whether or not the steady solutions are stable. There are interesting results that are linked to bypass an important class of boundary layer flow and heat transfer. The hydrodynamics behind these results are discussed in some detail.

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