Nonlinear Interactions of Waves on Shallow Water

1972 ◽  
Vol 50 (21) ◽  
pp. 2698-2711
Author(s):  
G. N. Ionides ◽  
F. L. Curzon
Keyword(s):  
Shallow Water ◽  
Second Order ◽  
Navier Stokes ◽  
Experimental Investigations ◽  
Free Fluid ◽  
Nonlinear Interactions ◽  
Order Mode ◽  
Navier Stokes Equation ◽  
Time Periodic ◽  
Free Fluid Surface

This paper presents the results of theoretical and experimental investigations of second-order nonlinear interactions between standing surface waves on shallow water. The nonlinearity is introduced by the nonlinear terms in the Navier–Stokes equation and in the boundary conditions. The wave amplitudes are kept small enough for amplitude dispersion to be negligible, and the surface is treated as a resonantly driven, damped, harmonic oscillator. A pure first-order mode is excited by applying time periodic electrical stresses onto the free fluid surface, and this mode self-interacts to drive resonantly a second-order mode. The amplitudes are ultimately limited by dissipative processes in the fluid. These processes are taken into account in the theory, and the experimental results justify the assumptions made. Wave amplitudes could be monitored to a spatial resolution of 5 × 10−4 cm by an optical technique.

scholarly journals CAUCHY PROBLEM FOR VISCOUS SHALLOW WATER EQUATIONS WITH A TERM OF CAPILLARITY

2010 ◽  
Vol 20 (07) ◽  
pp. 1049-1087 ◽  
Author(s):  
BORIS HASPOT
Keyword(s):  
Shallow Water ◽  
Existence Of Solutions ◽  
Stokes Equations ◽  
Variable Viscosity ◽  
Water System ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Viscosity Coefficients ◽  
Global Existence Of Solutions ◽  
Dependent Viscosity

In this paper, we consider the compressible Navier–Stokes equation with density-dependent viscosity coefficients and a term of capillarity introduced formally by van der Waals in Ref. 51. This model includes at the same time the barotropic Navier–Stokes equations with variable viscosity coefficients, shallow-water system and the model introduced by Rohde in Ref. 46. We first study the well-posedness of the model in critical regularity spaces with respect to the scaling of the associated equations. In a functional setting as close as possible to the physical energy spaces, we prove global existence of solutions close to a stable equilibrium, and local in time existence of solutions with general initial data. Uniqueness is also obtained.

Numerical and Experimental Investigations of Ship Maneuvers in Waves

2016 ◽  
Author(s):  
Ould el Moctar ◽  
Florian Sprenger ◽  
Thomas E. Schellin ◽  
Apostolos Papanikolaou
Keyword(s):  
Second Order ◽  
Navier Stokes ◽  
Experimental Investigations ◽  
Added Resistance ◽  
Numerical Investigations ◽  
Ship Maneuvers ◽  
Funded Project ◽  
Strong Winds ◽  
Energy Efficiency Design Index ◽  
Fundamental Requirement

Assuring a ship’s maneuverability under diverse conditions is a fundamental requirement for safe and economic ship operations. Considering the introduction of the Energy Efficiency Design Index (EEDI) for new ships and the related decreasing installed power on ships, the necessity arose to more accurately predict the maneuverability of ships in severe seas, strong winds, and confined waters. To address these issues, extensive experimental and numerical investigations were performed within the European funded Project SHOPERA. Here, second order forces and moments for a containership and a tanker were measured in model tests and computed by solving the Reynolds-Averaged Navier-Stokes (RANS) equations. Generally, these measured and computed second order loads (drift forces and yaw moments, added resistance) compared favorably. Furthermore, the effects of waves on zig-zag and turning circle maneuvers were investigated.

scholarly journals DECAY PROPERTIES OF SOLUTIONS TO THE LINEARIZED COMPRESSIBLE NAVIER–STOKES EQUATION AROUND TIME-PERIODIC PARALLEL FLOW

2012 ◽  
Vol 22 (07) ◽  
pp. 1250007 ◽  
Author(s):  
JAN BŘEZINA ◽  
YOSHIYUKI KAGEI
Keyword(s):  
Periodic Function ◽  
Stokes Equation ◽  
Parallel Flow ◽  
Navier Stokes ◽  
Decay Estimates ◽  
Convective Term ◽  
Navier Stokes Equation ◽  
Decay Properties ◽  
Linearized Problem ◽  
Time Periodic

Decay estimates on solutions to the linearized compressible Navier–Stokes equation around time-periodic parallel flow are established. It is shown that if the Reynolds and Mach numbers are sufficiently small, solutions of the linearized problem decay in L2 norm as an (n - 1)-dimensional heat kernel. Furthermore, it is proved that the asymptotic leading part of solutions is given by solutions of an (n - 1)-dimensional linear heat equation with a convective term multiplied by time-periodic function.

scholarly journals On the Spectral Properties for the Linearized Problem around Space-Time-Periodic States of the Compressible Navier–Stokes Equations

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 696
Author(s):  
Mohamad Nor Azlan ◽  
Shota Enomoto ◽  
Yoshiyuki Kagei
Keyword(s):  
Spectral Properties ◽  
Stokes Equations ◽  
Space Time ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Navier Stokes Equations ◽  
Infinite Layer ◽  
Periodic State ◽  
Linearized Problem ◽  
Time Periodic

This paper studies the linearized problem for the compressible Navier-Stokes equation around space-time periodic state in an infinite layer of Rn (n=2,3), and the spectral properties of the linearized evolution operator is investigated. It is shown that if the Reynolds and Mach numbers are sufficiently small, then the asymptotic expansions of the Floquet exponents near the imaginary axis for the Bloch transformed linearized problem are obtained for small Bloch parameters, which would give the asymptotic leading part of the linearized solution operator as t→∞.

Formulation of Boundary Conditions at the Surface of a Porous Medium

1974 ◽  
Vol 14 (05) ◽  
pp. 434-436 ◽  
Author(s):  
Graham Neale ◽  
Walter Nader
Keyword(s):  
Porous Medium ◽  
Boundary Conditions ◽  
Open Space ◽  
Tangential Velocity ◽  
Petroleum Engineering ◽  
Navier Stokes ◽  
Free Fluid ◽  
Engineering Practice ◽  
Navier Stokes Equation ◽  
Flow Equations

In petroleum engineering practice it is sometimes necessary to predict fluid flow occurring within adjacent regions of porous medium and open space (for example, fractured porous media, 1–3 vuggy media, 4 porous granules5). In such situations the governing flow equations are well known: Darcy's law and the Navier-Stokes equation. However, the set of boundary conditions at the permeable interfaces is seldom obvious. The usual conditions assumed are that the mass flux normal to the surface IS continuous, that the pressure is continuous across the surface, and that the tangential velocity in the free fluid tends to zero at the surface. The first two conditions are completely satisfactory; however, the third is clearly only an approximation. 6

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