Rayleigh–Taylor and Kelvin–Helmholtz instability studied in the frame of a dimension-reduced model

2020 ◽  
Vol 378 (2174) ◽  
pp. 20190508
Author(s):  
Michael Bestehorn
Keyword(s):  
Numerical Solutions ◽  
Lower Layer ◽  
Reduced Model ◽  
Viscous Layer ◽  
Theme Issue ◽  
Layer System ◽  
Weakly Nonlinear ◽  
Long Wave ◽  
Set Up ◽  
Long Wave Approximation

Introducing an extension of a recently derived dimension-reduced model for an infinitely deep inviscid and irrotational layer, a two-layer system is examined in the present paper. A second thin viscous layer is added on top of the original one-layer system. The set-up is a combination of a long-wave approximation (upper layer) and a deep-water approximation (lower layer). Linear stability analysis shows the emergency of Rayleigh–Taylor and Kelvin–Helmholtz instabilities. Finally, numerical solutions of the model reveal spatial and temporal pattern formation in the weakly nonlinear regime of both instabilities. This article is part of the theme issue ‘Stokes at 200 (Part 1)’.

Weakly nonlinear internal waves in a two-fluid system

1996 ◽  
Vol 313 ◽  
pp. 83-103 ◽  
Author(s):  
Wooyoung Choi ◽  
Roberto Camassa
Keyword(s):  
Internal Waves ◽  
Numerical Solutions ◽  
Lower Layer ◽  
Bottom Topography ◽  
Ocean Dynamics ◽  
Two Dimensional ◽  
Weakly Nonlinear ◽  
One Dimensional ◽  
Characteristic Wavelength ◽  
Upper Boundary

We derive general evolution equations for two-dimensional weakly nonlinear waves at the free surface in a system of two fluids of different densities. The thickness of the upper fluid layer is assumed to be small compared with the characteristic wavelength, but no restrictions are imposed on the thickness of the lower layer. We consider the case of a free upper boundary for its relevance in applications to ocean dynamics problems and the case of a non-uniform rigid upper boundary for applications to atmospheric problems. For the special case of shallow water, the new set of equations reduces to the Boussinesq equations for two-dimensional internal waves, whilst, for great and infinite lower-layer depth, we can recover the well-known Intermediate Long Wave and Benjamin–Ono models, respectively, for one-dimensional uni-directional wave propagation. Some numerical solutions of the model for one-dimensional waves in deep water are presented and compared with the known solutions of the uni-directional model. Finally, the effects of finite-amplitude slowly varying bottom topography are included in a model appropriate to the situation when the dependence on one of the horizontal coordinates is weak.

scholarly journals New approximations, and policy implications, from a delayed dynamic model of a fast pandemic

2020 ◽  
Author(s):  
C. P. Vyasarayani ◽  
Anindya Chatterjee
Keyword(s):  
Numerical Solutions ◽  
Initial Conditions ◽  
Reduced Order Model ◽  
Policy Implications ◽  
Galerkin Projection ◽  
Order Model ◽  
Reduced Order ◽  
Wave Approximation ◽  
Long Wave ◽  
Long Wave Approximation

AbstractWe study an SEIQR (Susceptible-Exposed-Infectious-Quarantined-Recovered) model for an infectious disease, with time delays for latency and an asymptomatic phase. For fast pandemics where nobody has prior immunity and everyone has immunity after recovery, the SEIQR model decouples into two nonlinear delay differential equations (DDEs) with five parameters. One parameter is set to unity by scaling time. The subcase of perfect quarantining and zero self-recovery before quarantine, with two free parameters, is examined first. The method of multiple scales yields a hyperbolic tangent solution; and a long-wave approximation yields a first order ordinary differential equation (ODE). With imperfect quarantining and nonzero self-recovery, the long-wave approximation is a second order ODE. These three approximations each capture the full outbreak, from infinitesimal initiation to final saturation. Low-dimensional dynamics in the DDEs is demonstrated using a six state non-delayed reduced order model obtained by Galerkin projection. Numerical solutions from the reduced order model match the DDE over a range of parameter choices and initial conditions. Finally, stability analysis and numerics show how correctly executed time-varying social distancing, within the present model, can cut the number of affected people by almost half. Alternatively, faster detection followed by near-certain quarantining can potentially be even more effective.

Geostrophic adjustment in a channel: nonlinear and dispersive effects

1992 ◽  
Vol 241 ◽  
pp. 23-57 ◽  
Author(s):  
G. G. Tomasson ◽  
W. K. Melville
Keyword(s):  
Internal Waves ◽  
Numerical Solutions ◽  
Boussinesq Equations ◽  
Kelvin Waves ◽  
Geostrophic Adjustment ◽  
Homogeneous Fluid ◽  
Layer System ◽  
Weakly Nonlinear ◽  
Resonant Interactions ◽  
Poincare Waves

We consider the general problem of geostrophic adjustment in a channel in the weakly nonlinear and dispersive (non-hydrostatic) limit. Governing equations of Boussinesq-type are derived, based on the assumption of weak nonlinear, dispersive and rotational effects, both for surface waves on a homogeneous fluid and internal waves in a two-layer system. Numerical solutions of the Boussinesq equations are presented, giving examples of the geostrophic adjustment in a channel for two different kinds of initial disturbances, both with non-zero perturbation potential vorticity. The timescales of rotational separation (that is, the separation of the Kelvin and Poincaré waves due to their dispersive properties) and that of nonlinear evolution are considered, with particular concern for the resonant interactions of nonlinear Kelvin waves and linear Poincaré waves described by Melville, Tomasson & Renouard (1989). A parameter measuring the ratio of the two timescales is used to predict when the free and forced Poincaré waves may be separated in the solution. It also distinguishes the cases in which the linear solutions are valid for the rotational separation from those requiring the full Boussinesq equations. Finally, solutions for the evolution of nonlinear internal waves in a sea strait are presented, and the effects of friction on the wavefront curvature of the nonlinear Kelvin waves are briefly considered.

Nonlinear Internal Waves in Stratified Fluid With Homogeneous Layer

2005 ◽  
Author(s):  
Nikolai I. Makarenko ◽  
Janna L. Maltseva
Keyword(s):  
Internal Waves ◽  
Lower Layer ◽  
Nonlinear Ordinary Differential Equation ◽  
Homogeneous Layer ◽  
Constant Density ◽  
Long Wave ◽  
Scaling Procedure ◽  
Internal Bores ◽  
Long Wave Approximation ◽  
Exponential Stratification

The problem of steady internal waves in a weakly stratified two-layered fluid is studied analytically. We discuss the model with a constant density in lower layer and exponential stratification in the other one. The long-wave approximation using a scaling procedure with small Boussinesq parameter is constructed. The nonlinear ordinary differential equation describing large amplitude solitary waves and internal bores is obtained.

Viscous film-flow coating the interior of a vertical tube. Part 2. Air-driven flow

2017 ◽  
Vol 825 ◽  
pp. 1056-1090 ◽  
Author(s):  
Roberto Camassa ◽  
H. Reed Ogrosky ◽  
Jeffrey Olander
Keyword(s):  
Free Surface ◽  
Numerical Solutions ◽  
Wave Model ◽  
Film Coating ◽  
Vertical Tube ◽  
Substantial Improvement ◽  
Surface Stresses ◽  
Long Wave ◽  
Set Up ◽  
Theoretical Results

The flow of a viscous liquid film coating the interior of a vertical tube is studied for the case when the film is driven upwards against gravity by a constant volume flux of air through the centre of the tube. A nonlinear model exploiting the slowly varying liquid–air interface is first developed to estimate the interfacial stresses created by the airflow. A comparison of the model with both experiments and previously developed theoretical results is conducted for two geometrical settings: channel and pipe flow. In both geometries, the model compares reasonably well with previous experiments. A long-wave asymptotic theory is then developed for the air–liquid interface taking into account the estimated free-surface stresses created by the airflow. The stability of small interfacial disturbances is studied analytically, and it is shown that the modelled free-surface stresses contribute to both an increased upwards disturbance velocity and a more rapid instability growth than those of a previously developed ‘locally Poiseuille’ model. Numerical solutions to the long-wave model exhibit saturated waves, whose profiles and velocities show substantial improvement with respect to the previous model predictions. The theoretical results are compared with new experiments for a modified version of the set-up described in Part 1.

Nonlinear dispersion hyperbolic models of continuous media

2007 ◽  
Vol 5 ◽  
pp. 273-278
Author(s):  
V.Yu Liapidevskii
Keyword(s):  
Boussinesq Equations ◽  
Nonlinear Dispersion ◽  
Order Of Accuracy ◽  
Flow Models ◽  
Wave Approximation ◽  
Long Wave ◽  
Primary Model ◽  
Nonequilibrium Flows ◽  
Internal Inertia ◽  
Long Wave Approximation

Nonequilibrium flows of an inhomogeneous liquid in channels and pipes are considered in the long-wave approximation. Nonlinear dispersion hyperbolic flow models are derived allowing taking into account the influence of internal inertia during the relative motion of phases upon the structure of nonlinear wave fronts. The asymptotic derivation of dispersion hyperbolic models is shown on the example of classical Boussinesq equations. It is shown that the hyperbolic approximation of the equations has the same order of accuracy as the primary model.

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.

Two-layer gap-leaping oceanic boundary currents: experimental investigation

2014 ◽  
Vol 740 ◽  
pp. 97-113 ◽  
Author(s):  
Joseph J. Kuehl ◽  
V. A. Sheremet
Keyword(s):  
Gulf Stream ◽  
Lower Layer ◽  
Kuroshio Current ◽  
Loop Current ◽  
Boundary Current ◽  
Layer System ◽  
Open Problems ◽  
Typical Structure ◽  
Boundary Currents ◽  
Rotating Table

AbstractThe problem of oceanic gap-traversing boundary currents, such as the Kuroshio current crossing the Luzon Strait or the Gulf Stream traversing the mouth of the Gulf of Mexico, is considered. Systems such as these are known to admit two dominant states: leaping across the gap or penetrating into the gap forming a loop current. Which state the system will assume and when transitions between states will occur are open problems. Sheremet (J. Phys. Oceanogr., vol. 31, 2001, pp. 1247–1259) proposed, based on idealized barotropic numerical results, that variation in the current’s inertia is responsible for these transitions and that the system admits multiple states. Generalized versions of these results have been confirmed by barotropic rotating-table experiments (Sheremet & Kuehl, J. Phys. Oceanogr., vol. 37, 2007, 1488–1495; Kuehl & Sheremet,J. Mar. Res., vol. 67, 2009, pp. 25–42). However, the typical structure of oceanic boundary currents, such as the Gulf Stream or Kuroshio, consists of an upper-layer intensified flow riding atop a weakly circulating lower layer. To more accurately address this oceanic situation, the present work extends the above findings by considering two-layer rotating table experiments. The flow is driven by pumping water through sponges and vertical seals, creating a Sverdrup interior circulation in the upper layer which impinges on a ridge where a boundary current is formed. The $\beta $ effect is incorporated in both layers by a sloping rigid lid as well as a sloping bottom and the flow is visualized with the particle image velocimetry method. The experimental set-up is found to produce boundary currents consistent with theory. The existence of multiple states and hysteresis, characterized by a cusp topology of solutions, is found to be robust to stratification and various properties of the two-layer system are explored.

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