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# Boussinesq approximation (water waves)

About: Boussinesq approximation (water waves) is a research topic. Over the lifetime, 3513 publications have been published within this topic receiving 90280 citations.

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01 Jun 1978TL;DR: In this paper, the authors evaluated the applicability of the standard κ-ϵ equations and other turbulence models with respect to their applicability in swirling, recirculating flows.

Abstract: The standard κ-ϵ equations and other turbulence models are evaluated with respect to their applicability in swirling, recirculating flows. The turbulence models are formulated on the basis of two separate viewpoints. The first perspective assumes that an isotropic eddy viscosity and the modified Boussinesq hypothesis adequately describe the stress distributions, and that the source of predictive error is a consequence of the modeled terms in the κ-ϵ equations. Both stabilizing and destabilizing Richardson number corrections are incorporated to investigate this line of reasoning. A second viewpoint proposes that the eddy viscosity approach is inherently inadequate and that a redistribution of the stress magnitudes is necessary. Investigation of higher-order closure is pursued on the level of an algebraic stress closure. Various turbulence model predictions are compared with experimental data from a variety of isothermal, confined studies. Supportive swirl comparisons are also performed for a laminar flow case, as well as reacting flow cases. Parallel predictions or contributions from other sources are also consulted where appropriate. Predictive accuracy was found to be a partial function of inlet boundary conditions and numerical diffusion. Despite prediction sensitivity to inlet conditions and numerics, the data comparisons delineate the relative advantages and disadvantages of the various modifications. Possible research avenues in the area of computational modeling of strongly swirling, recirculating flows are reviewed and discussed.

5,396 citations

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TL;DR: In this article, a number of ases in which these equations reduce to a one dimensional nonlinear Schrodinger (NLS) equation are enumerated, and several analytical solutions of NLS equations are presented, with discussion of their implications for describing the propagation of water waves.

Abstract: Equations governing modulations of weakly nonlinear water waves are described. The modulations are coupled with wave-induced mean flows except in the case of water deeper than the modulation length scale. Equations suitable for water depths of the order the modulation length scale are deduced from those derived by Davey and Stewartson [5] and Dysthe [6]. A number of ases in which these equations reduce to a one dimensional nonlinear Schrodinger (NLS) equation are enumerated.Several analytical solutions of NLS equations are presented, with discussion of some of their implications for describing the propagation of water waves. Some of the solutions have not been presented in detail, or in convenient form before. One is new, a “rational” solution describing an “amplitude peak” which is isolated in space-time. Ma's [13] soli ton is particularly relevant to the recurrence of uniform wave trains in the experiment of Lake et al.[10].In further discussion it is pointed out that although water waves are unstable to three-dimensional disturbances, an effective description of weakly nonlinear two-dimensional waves would be a useful step towards describing ocean wave propagation.

1,318 citations

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TL;DR: In this paper, a new form of the Boussinesq equations is derived using the velocity at an arbitrary distance from the still water level as the velocity variable instead of the commonly used depth-averaged velocity.

Abstract: Boussinesq‐type equations can be used to model the nonlinear transformation of surface waves in shallow water due to the effects of shoaling, refraction, diffraction, and reflection. Different linear dispersion relations can be obtained by expressing the equations in different velocity variables. In this paper, a new form of the Boussinesq equations is derived using the velocity at an arbitrary distance from the still water level as the velocity variable instead of the commonly used depth‐averaged velocity. This significantly improves the linear dispersion properties of the Boussinesq equations, making them applicable to a wider range of water depths. A finite difference method is used to solve the equations. Numerical and experimental results are compared for the propagation of regular and irregular waves on a constant slope beach. The results demonstrate that the new form of the equations can reasonably simulate several nonlinear effects that occur in the shoaling of surface waves from deep to shallow w...

1,112 citations

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01 Jan 1998

TL;DR: In this article, the authors present a survey of the main problems of self-adjoint EIGEN-value problems and propose a solution to solve them based on a simplified version of the standard EIGE algorithm.

Abstract: Preface List of Symbols CHAPTER 1. DIFFUSIVE FLUXES AND MATERIAL PROPERTIES 1.1 INTRODUCTION 1.2 BASIC CONSTITUTIVE EQUATIONS 1.3 DIFFUSIVITIES FOR ENERGY, SPECIES, AND MOMENTUM 1.4 MAGNITUDES OF TRANSPORT COEFFICIENTS 1.5 MOLECULAR INTERPRETATION OF TRANSPORT COEFFICIENTS 1.6 LIMITATIONS ON LENGTH AND TIME SCALES References Problems CHAPTER 2. FUNDAMENTALS OF HEAT AND MASS TRANSFER 2.1 INTRODUCTION 2.2 GENERAL FORMS OF CONSERVATION EQUATIONS 2.3 CONSERVATION OF MASS 2.4 CONSERVATION OF ENERGY: THERMAL EFFECTS 2.5 HEAT TRANSFER AT INTERFACES 2.6 CONSERVATION OF CHEMICAL SPECIES 2.7 MASS TRANSFER AT INTERFACES 2.8 MOLECULAR VIEW OF SPECIES CONSERVATION References Problems CHAPTER 3. FORMULATION AND APPROXIMATION 3.1 INTRODUCTION 3.2 ONE-DIMENSIONAL EXAMPLES 3.3 ORDER-OF-MAGNITUDE ESTIMATION AND SCALING 3.4 " IN MODELING 3.5 TIME SCALES IN MODELING References Problems CHAPTER 4. SOLUTION METHODS BASED ON SCALING CONCEPTS 4.1 INTRODUCTION 4.2 SIMILARITY METHOD 4.3 REGULAR PERTURBATION ANALYSIS 4.4 SINGULAR PERTURBATION ANALYSIS References Problems CHAPTER 5. SOLUTION METHODS FOR LINEAR PROBLEMS 5.1 INTRODUCTION 5.2 PROPERTIES OF LINEAR BOUNDARY-VALUE PROBLEMS 5.3 FINITE FOURIER TRANSFORM METHOD 5.4 BASIS FUNCTIONS 5.5 FOURIER SERIES 5.6 FFT SOLUTIONS FOR RECTANGULAR GEOMETRIES 5.7 FFT SOLUTIONS FOR CYLINDRICAL GEOMETRIES 5.8 FFT SOLUTIONS FOR SPHERICAL GEOMETRIES 5.9 POINT-SOURCE SOLUTIONS 5.10 MORE ON SELF-ADJOINT EIGENVALUE PROBLEMS AND FFT SOLUTIONS References Problems CHAPTER 6. FUNDAMENTALS OF FLUID MECHANICS 6.1 INTRODUCTION 6.2 CONSERVATION OF MOMENTUM 6.3 TOTAL STRESS, PRESSURE, AND VISCOUS STRESS 6.4 FLUID KINEMATICS 6.5 CONSTITUTIVE EQUATIONS FOR VISCOUS STRESS 6.6 FLUID MECHANICS AT INTERFACES 6.7 FORCE CALCULATIONS 6.8 STREAM FUNCTION 6.9 DIMENSIONLESS GROUPS AND FLOW REGIMES References Problems CHAPTER 7. UNIDIRECTIONAL AND NEARLY UNIDIRECTIONAL FLOW 7.1 INTRODUCTION 7.2 STEADY FLOW WITH A PRESSURE GRADIENT 7.3 STEADY FLOW WITH A MOVING SURFACE 7.4 TIME-DEPENDENT FLOW 7.5 LIMITATIONS OF EXACT SOLUTIONS 7.6 NEARLY UNIDIRECTIONAL FLOW References Problems CHAPTER 8. CREEPING FLOW 8.1 INTRODUCTION 8.2 GENERAL FEATURES OF LOW REYNOLDS NUMBER FLOW 8.3 UNIDIRECTIONAL AND NEARLY UNIDIRECTIONAL SOLUTIONS 8.4 STREAM-FUNCTION SOLUTIONS 8.5 POINT-FORCE SOLUTIONS 8.6 PARTICLES AND SUSPENSIONS 8.7 CORRECTIONS TO STOKES' LAW References Problems CHAPTER 9. LAMINAR FLOW AT HIGH REYNOLDS NUMBER 9.1 INTRODUCTION 9.2 GENERAL FEATURES OF HIGH REYNOLDS NUMBER FLOW 9.3 IRROTATIONAL FLOW 9.4 BOUNDARY LAYERS AT SOLID SURFACES 9.5 INTERNAL BOUNDARY LAYERS References Problems CHAPTER 10. FORCED-CONVECTION HEAT AND MASS TRANSFER IN CONFINED LAMINAR FLOWS 10.1 INTRODUCTION 10.2 PECLET NUMBER 10.3 NUSSELT AND SHERWOOD NUMBERS 10.4 ENTRANCE REGION 10.5 FULLY DEVELOPED REGION 10.6 CONSERVATION OF ENERGY: MECHANICAL EFFECTS 10.7 TAYLOR DISPERSION References Problems CHAPTER 11. FORCED-CONVECTION HEAT AND MASS TRANSFER IN UNCONFINED LAMINAR FLOWS 11.1 INTRODUCTION 11.2 HEAT AND MASS TRANSFER IN CREEPING FLOW 11.3 HEAT AND MASS TRANSFER IN LAMINAR BOUNDARY LAYERS 11.4 SCALING LAWS FOR NUSSELT AND SHERWOOD NUMBERS References Problems CHAPTER 12. TRANSPORT IN BUOYANCY-DRIVEN FLOW 12.1 INTRODUCTION 12.2 BUOYANCY AND THE BOUSSINESQ APPROXIMATION 12.3 CONFINED FLOWS 12.4 DIMENSIONAL ANALYSIS AND BOUNDARY-LAYER EQUATIONS 12.5 UNCONFINED FLOWS References Problems CHAPTER 13. TRANSPORT IN TURBULENT FLOW 13.1 INTRODUCTION 13.2 BASIC FEATURES OF TURBULENCE 13.3 TIME-SMOOTHED EQUATIONS 13.4 EDDY DIFFUSIVITY MODELS 13.5 OTHER APPROACHES FOR TURBULENT-FLOW CALCULATIONS References Problems CHAPTER 14. SIMULTANEOUS ENERGY AND MASS TRANSFER AND MULTICOMPONENT SYSTEMS 14.1 INTRODUCTION 14.2 CONSERVATION OF ENERGY: MULTICOMPONENT SYSTEMS 14.3 SIMULTANEOUS HEAT AND MASS TRANSFER 14.4 INTRODUCTION TO COUPLED FLUXES 14.5 STEFAN-MAXWELL EQUATIONS 14.6 GENERALIZED DIFFUSION IN DILUTE MIXTURES 14.7 GENERALIZED STEFAN-MAXWELL EQUATIONS References Problems CHAPTER 15. TRANSPORT IN ELECTROLYTE SOLUTIONS 15.1 INTRODUCTION 15.2 FORMULATION OF MACROSCOPIC PROBLEMS 15.3 MACROSCOPIC EXAMPLES 15.4 EQUILIBRIUM DOUBLE LAYERS 15.5 ELECTROKINETIC PHENOMENA References Problems APPENDIX A. VECTORS AND TENSORS A.1 INTRODUCTION A.2 REPRESENTATION OF VECTORS AND TENSORS A.3 VECTOR AND TENSOR PRODUCTS A.4 VECTOR-DIFFERENTIAL OPERATORS A.5 INTEGRAL TRANSFORMATIONS A.6 POSITION VECTORS A.7 ORTHOGONAL CURVILINEAR COORDINATES A.8 SURFACE GEOMETRY References APPENDIX B. ORDINARY DIFFERENTIAL EQUATIONS AND SPECIAL FUNCTIONS B.1 INTRODUCTION B.2 FIRST-ORDER EQUATIONS B.3 EQUATIONS WITH CONSTANT COEFFICIENTS B.4 BESSEL AND SPHERICAL BESSEL EQUATIONS B.5 OTHER EQUATIONS WITH VARIABLE COEFFICIENTS References Index

1,081 citations

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TL;DR: In this paper, a new method for obtaining approximate equations for natural convection flows is presented, which allows the specification of the conditions under which the traditional Boussinesq approximation applies to a given Newtonian liquid or gas.

Abstract: A new method for obtaining approximate equations for natural convection flows is presented. The systematic application of this method leads to explicit conditions for the neglect of various terms. It is shown that this method allows the specification of the conditions under which the traditional Boussinesq approximation applies to a given Newtonian liquid or gas. The method is applied to room temperature water and air.

940 citations