Mathematical Derivation of Viscous Shallow-Water Equations with Zero Surface Tension Didier Bresch, Pascal Noble
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Well-Balanced Schemes for the Shallow Water Equations with Coriolis Forces
Well-balanced schemes for the shallow water equations with Coriolis forces Alina Chertock, Michael Dudzinski, Alexander Kurganov & Mária Lukáčová- Medvid’ová Numerische Mathematik ISSN 0029-599X Volume 138 Number 4 Numer. Math. (2018) 138:939-973 DOI 10.1007/s00211-017-0928-0 1 23 Your article is protected by copyright and all rights are held exclusively by Springer- Verlag GmbH Germany, part of Springer Nature. This e-offprint is for personal use only and shall not be self-archived in electronic repositories. If you wish to self-archive your article, please use the accepted manuscript version for posting on your own website. You may further deposit the accepted manuscript version in any repository, provided it is only made publicly available 12 months after official publication or later and provided acknowledgement is given to the original source of publication and a link is inserted to the published article on Springer's website. The link must be accompanied by the following text: "The final publication is available at link.springer.com”. 1 23 Author's personal copy Numer. Math. (2018) 138:939–973 Numerische https://doi.org/10.1007/s00211-017-0928-0 Mathematik Well-balanced schemes for the shallow water equations with Coriolis forces Alina Chertock1 · Michael Dudzinski2 · Alexander Kurganov3,4 · Mária Lukáˇcová-Medvid’ová5 Received: 28 April 2014 / Revised: 19 September 2017 / Published online: 2 December 2017 © Springer-Verlag GmbH Germany, part of Springer Nature 2017 Abstract In the present paper we study shallow water equations with bottom topog- raphy and Coriolis forces. The latter yield non-local potential operators that need to be taken into account in order to derive a well-balanced numerical scheme. -
Fluid Mechanics
FLUID MECHANICS PROF. DR. METİN GÜNER COMPILER ANKARA UNIVERSITY FACULTY OF AGRICULTURE DEPARTMENT OF AGRICULTURAL MACHINERY AND TECHNOLOGIES ENGINEERING 1 1. INTRODUCTION Mechanics is the oldest physical science that deals with both stationary and moving bodies under the influence of forces. Mechanics is divided into three groups: a) Mechanics of rigid bodies, b) Mechanics of deformable bodies, c) Fluid mechanics Fluid mechanics deals with the behavior of fluids at rest (fluid statics) or in motion (fluid dynamics), and the interaction of fluids with solids or other fluids at the boundaries (Fig.1.1.). Fluid mechanics is the branch of physics which involves the study of fluids (liquids, gases, and plasmas) and the forces on them. Fluid mechanics can be divided into two. a)Fluid Statics b)Fluid Dynamics Fluid statics or hydrostatics is the branch of fluid mechanics that studies fluids at rest. It embraces the study of the conditions under which fluids are at rest in stable equilibrium Hydrostatics is fundamental to hydraulics, the engineering of equipment for storing, transporting and using fluids. Hydrostatics offers physical explanations for many phenomena of everyday life, such as why atmospheric pressure changes with altitude, why wood and oil float on water, and why the surface of water is always flat and horizontal whatever the shape of its container. Fluid dynamics is a subdiscipline of fluid mechanics that deals with fluid flow— the natural science of fluids (liquids and gases) in motion. It has several subdisciplines itself, including aerodynamics (the study of air and other gases in motion) and hydrodynamics (the study of liquids in motion). -
Fluid Viscosity and the Attenuation of Surface Waves: a Derivation Based on Conservation of Energy
INSTITUTE OF PHYSICS PUBLISHING EUROPEAN JOURNAL OF PHYSICS Eur. J. Phys. 25 (2004) 115–122 PII: S0143-0807(04)64480-1 Fluid viscosity and the attenuation of surface waves: a derivation based on conservation of energy FBehroozi Department of Physics, University of Northern Iowa, Cedar Falls, IA 50614, USA Received 9 June 2003 Published 25 November 2003 Online at stacks.iop.org/EJP/25/115 (DOI: 10.1088/0143-0807/25/1/014) Abstract More than a century ago, Stokes (1819–1903)pointed out that the attenuation of surface waves could be exploited to measure viscosity. This paper provides the link between fluid viscosity and the attenuation of surface waves by invoking the conservation of energy. First we calculate the power loss per unit area due to viscous dissipation. Next we calculate the power loss per unit area as manifested in the decay of the wave amplitude. By equating these two quantities, we derive the relationship between the fluid viscosity and the decay coefficient of the surface waves in a transparent way. 1. Introduction Surface tension and gravity govern the propagation of surface waves on fluids while viscosity determines the wave attenuation. In this paper we focus on the relation between viscosity and attenuation of surface waves. More than a century ago, Stokes (1819–1903) pointed out that the attenuation of surface waves could be exploited to measure viscosity [1]. Since then, the determination of viscosity from the damping of surface waves has receivedmuch attention [2–10], particularly because the method presents the possibility of measuring viscosity noninvasively. In his attempt to obtain the functionalrelationship between viscosity and wave attenuation, Stokes observed that the harmonic solutions obtained by solving the Laplace equation for the velocity potential in the absence of viscosity also satisfy the linearized Navier–Stokes equation [11]. -
Part II-1 Water Wave Mechanics
Chapter 1 EM 1110-2-1100 WATER WAVE MECHANICS (Part II) 1 August 2008 (Change 2) Table of Contents Page II-1-1. Introduction ............................................................II-1-1 II-1-2. Regular Waves .........................................................II-1-3 a. Introduction ...........................................................II-1-3 b. Definition of wave parameters .............................................II-1-4 c. Linear wave theory ......................................................II-1-5 (1) Introduction .......................................................II-1-5 (2) Wave celerity, length, and period.......................................II-1-6 (3) The sinusoidal wave profile...........................................II-1-9 (4) Some useful functions ...............................................II-1-9 (5) Local fluid velocities and accelerations .................................II-1-12 (6) Water particle displacements .........................................II-1-13 (7) Subsurface pressure ................................................II-1-21 (8) Group velocity ....................................................II-1-22 (9) Wave energy and power.............................................II-1-26 (10)Summary of linear wave theory.......................................II-1-29 d. Nonlinear wave theories .................................................II-1-30 (1) Introduction ......................................................II-1-30 (2) Stokes finite-amplitude wave theory ...................................II-1-32 -
Lecture 18 Ocean General Circulation Modeling
Lecture 18 Ocean General Circulation Modeling 9.1 The equations of motion: Navier-Stokes The governing equations for a real fluid are the Navier-Stokes equations (con servation of linear momentum and mass mass) along with conservation of salt, conservation of heat (the first law of thermodynamics) and an equation of state. However, these equations support fast acoustic modes and involve nonlinearities in many terms that makes solving them both difficult and ex pensive and particularly ill suited for long time scale calculations. Instead we make a series of approximations to simplify the Navier-Stokes equations to yield the “primitive equations” which are the basis of most general circu lations models. In a rotating frame of reference and in the absence of sources and sinks of mass or salt the Navier-Stokes equations are @ �~v + �~v~v + 2�~ �~v + g�kˆ + p = ~ρ (9.1) t r · ^ r r · @ � + �~v = 0 (9.2) t r · @ �S + �S~v = 0 (9.3) t r · 1 @t �ζ + �ζ~v = ω (9.4) r · cpS r · F � = �(ζ; S; p) (9.5) Where � is the fluid density, ~v is the velocity, p is the pressure, S is the salinity and ζ is the potential temperature which add up to seven dependent variables. 115 12.950 Atmospheric and Oceanic Modeling, Spring '04 116 The constants are �~ the rotation vector of the sphere, g the gravitational acceleration and cp the specific heat capacity at constant pressure. ~ρ is the stress tensor and ω are non-advective heat fluxes (such as heat exchange across the sea-surface).F 9.2 Acoustic modes Notice that there is no prognostic equation for pressure, p, but there are two equations for density, �; one prognostic and one diagnostic. -
Modelling Viscous Free Surface Flow
Chapter 8 Modelling Viscous Free Surface Flow Free surface flows, where a boundary of a fluid body is free to move constrained only by forces across the surface, are possibly the most commonly observed flow phenomenon, with the motion of the free surface readily allowing the observation of flow of the fluid. Flows that are commonly encountered by the layperson include the motion of the surface of a river, the waves on the surface of the ocean, and the more personal flow of that in a cup of tea. From an engineering perspective, flows of interest include the open channel flow of rivers and canals, the erosive forces of waves on the shoreline, and the wakes generated by ships when under way. When modelling these flows, the problems associated with the solution of the Navier–Stokes equations are compounded by the free motion of the surface of the fluid, with the boundaries of the flow domain being a function of the flow structure, and thus an unknown which must be calculated along with the flow field. The movement of the free surface therefore both aids the observation of the flow, and hinders it’s modelling. In this chapter an attempt is made to model the steady flow around a ships hull. The modelling of such a flow is of great interest to Naval Architects, with the ability to predict the resistance of ships allowing the design of more efficient hull forms, whilst the accurate modelling of the ships wake allows the design of hulls that create a smaller disturbance in confined waterways. -
Fluid Simulation Using Shallow Water Equation
Fluid Simulation using Shallow Water Equation Dhruv Kore Itika Gupta Undergraduate Student Graduate Student University of Illinois at Chicago University of Illinois at Chicago [email protected] [email protected] 847-345-9745 312-478-0764 ABSTRACT rivers and channel. As the name suggest, the main In this paper, we present a technique, which shows how characteristic of shallow Water flows is that the vertical waves once generated from a small drop continue to ripple. dimension is much smaller as compared to the horizontal Waves interact with each other and on collision change the dimension. Naïve Stroke Equation defines the motion of form and direction. Once the waves strike the boundary, fluids and from these equations we derive SWE. they return with the same speed and in sometime, depending on the delay, you can see continuous ripples in In Next section we talk about the related works under the the surface. We use shallow water equation to achieve the heading Literature Review. Then we will explain the desired output. framework and other concepts necessary to understand the SWE and wave equation. In the section followed by it, we Author Keywords show the results achieved using our implementation. Then Naïve Stroke Equation; Shallow Water Equation; Fluid finally we talk about conclusion and future work. Simulation; WebGL; Quadratic function. LITERATURE REVIEW INTRODUCTION In [2] author in detail explains the Shallow water equation In early years of fluid simulation, procedural surface with its derivation. Since then a lot of authors has used generation was used to represent waves as presented by SWE to present the formation of waves in water. -
A Simple and Unified Linear Solver for Free-Surface and Pressurized
water Technical Note A Simple and Unified Linear Solver for Free-Surface and Pressurized Mixed Flows in Hydraulic Systems Dechao Hu 1 , Songping Li 1, Shiming Yao 2 and Zhongwu Jin 2,* 1 School of Hydropower and Information Engineering, Huazhong University of Science and Technology, Wuhan 430074, China; [email protected] (D.H.); [email protected] (S.L.) 2 Yangtze River Scientific Research Institute, Wuhan 430010, China; [email protected] * Correspondence: [email protected]; Tel.: +86-027-8282-9873 Received: 19 August 2019; Accepted: 19 September 2019; Published: 23 September 2019 Abstract: A semi–implicit numerical model with a linear solver is proposed for the free-surface and pressurized mixed flows in hydraulic systems. It solves the two flow regimes within a unified formulation, and is much simpler than existing similar models for mixed flows. Using a local linearization and an Eulerian–Lagrangian method, the new model only needs to solve a tridiagonal linear system (arising from velocity-pressure coupling) and is free of iterations. The model is tested using various types of mixed flows, where the simulation results agree with analytical solutions, experiment data and the results reported by former researchers. Sensitivity studies of grid scales and time steps are both performed, where a common grid scale provides grid-independent results and a common time step provides time-step-independent results. Moreover, the model is revealed to achieve stable and accurate simulations at large time steps for which the CFL is greater than 1. In simulations of a challenging case (mixed flows characterized by frequent flow-regime conversions and a closed pipe with wide-top cross-sections), an artificial slot (A-slot) technique is proposed to cope with possible instabilities related to the discontinuous main-diagonal coefficients of the linear system. -
Shallow Water Waves and Solitary Waves Article Outline Glossary
Shallow Water Waves and Solitary Waves Willy Hereman Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden, Colorado, USA Article Outline Glossary I. Definition of the Subject II. Introduction{Historical Perspective III. Completely Integrable Shallow Water Wave Equations IV. Shallow Water Wave Equations of Geophysical Fluid Dynamics V. Computation of Solitary Wave Solutions VI. Water Wave Experiments and Observations VII. Future Directions VIII. Bibliography Glossary Deep water A surface wave is said to be in deep water if its wavelength is much shorter than the local water depth. Internal wave A internal wave travels within the interior of a fluid. The maximum velocity and maximum amplitude occur within the fluid or at an internal boundary (interface). Internal waves depend on the density-stratification of the fluid. Shallow water A surface wave is said to be in shallow water if its wavelength is much larger than the local water depth. Shallow water waves Shallow water waves correspond to the flow at the free surface of a body of shallow water under the force of gravity, or to the flow below a horizontal pressure surface in a fluid. Shallow water wave equations Shallow water wave equations are a set of partial differential equations that describe shallow water waves. 1 Solitary wave A solitary wave is a localized gravity wave that maintains its coherence and, hence, its visi- bility through properties of nonlinear hydrodynamics. Solitary waves have finite amplitude and propagate with constant speed and constant shape. Soliton Solitons are solitary waves that have an elastic scattering property: they retain their shape and speed after colliding with each other. -
Shallow-Water Equations and Related Topics
CHAPTER 1 Shallow-Water Equations and Related Topics Didier Bresch UMR 5127 CNRS, LAMA, Universite´ de Savoie, 73376 Le Bourget-du-Lac, France Contents 1. Preface .................................................... 3 2. Introduction ................................................. 4 3. A friction shallow-water system ...................................... 5 3.1. Conservation of potential vorticity .................................. 5 3.2. The inviscid shallow-water equations ................................. 7 3.3. LERAY solutions ........................................... 11 3.3.1. A new mathematical entropy: The BD entropy ........................ 12 3.3.2. Weak solutions with drag terms ................................ 16 3.3.3. Forgetting drag terms – Stability ............................... 19 3.3.4. Bounded domains ....................................... 21 3.4. Strong solutions ............................................ 23 3.5. Other viscous terms in the literature ................................. 23 3.6. Low Froude number limits ...................................... 25 3.6.1. The quasi-geostrophic model ................................. 25 3.6.2. The lake equations ...................................... 34 3.7. An interesting open problem: Open sea boundary conditions .................... 41 3.8. Multi-level and multi-layers models ................................. 43 3.9. Friction shallow-water equations derivation ............................. 44 3.9.1. Formal derivation ....................................... 44 3.10. -
Waves in Water from Ripples to Tsunamis and to Rogue Waves
Waves in Water from ripples to tsunamis and to rogue waves Vera Mikyoung Hur (Mathematics) (Image from Flickr) The motion of a fluid can be very complicated as we know whenever we see waves break on a beach, (Image from the Internet) The motion of a fluid can be very complicated as we know whenever we fly in an airplane, (Image from the Internet) The motion of a fluid can be very complicated as we know whenever we look at a lake on a windy day. (Image from the Internet) Euler in the 1750s proposed a mathematical model of an incompressible fluid. @u + (u · r)u + rP = F, @t r · u = 0. Here, u(x; t) is the velocity of the fluid at the point x and time t, P(x; t) is the pressure, and F(x; t) is an outside force. The Navier-Stokes equations (adding ν∆u) allow the fluid to be viscous. It is concise and captures the essence of fluid behavior. The theory of fluids has provided source and inspiration to many branches of mathematics, e.g. Cauchy's complex function theory. Difficulties of understanding fluids are profound. e.g. the global-in-time solution of the Navier-Stokes equations in 3 dimensions is a Clay Millennium Problem! Waves, jets, drops come to mind when thinking of fluids. They involve one or more fluids separated by an unknown surface. In the mathematical community, they go by free boundary problems. (Image from the Internet) Free boundaries are mathematically challenging in their own right. They occur in many other situations, such as melting of ice stretching a membrane over an obstacle. -
Modified Shallow Water Equations for Significantly Varying Seabeds Denys Dutykh, Didier Clamond
Modified Shallow Water Equations for significantly varying seabeds Denys Dutykh, Didier Clamond To cite this version: Denys Dutykh, Didier Clamond. Modified Shallow Water Equations for significantly varying seabeds: Modified Shallow Water Equations. Applied Mathematical Modelling, Elsevier, 2016, 40 (23-24), pp.9767-9787. 10.1016/j.apm.2016.06.033. hal-00675209v6 HAL Id: hal-00675209 https://hal.archives-ouvertes.fr/hal-00675209v6 Submitted on 26 Apr 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Distributed under a Creative Commons Attribution - NonCommercial| 4.0 International License Denys Dutykh CNRS, Université Savoie Mont Blanc, France Didier Clamond Université de Nice – Sophia Antipolis, France Modified Shallow Water Equations for significantly varying seabeds arXiv.org / hal Last modified: April 26, 2016 Modified Shallow Water Equations for significantly varying seabeds Denys Dutykh∗ and Didier Clamond Abstract. In the present study, we propose a modified version of the Nonlinear Shal- low Water Equations (Saint-Venant or NSWE) for irrotational surface waves in the case when the bottom undergoes some significant variations in space and time. The model is derived from a variational principle by choosing an appropriate shallow water ansatz and imposing some constraints.