DOCSLIB.ORG

Chapter 3 FEM on Nonlinear Free-Surface Flow

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Chapter 3 FEM on Nonlinear Free-Surface Flow Numerical Models in Fluid-Structure Interaction Chapter 3 FEM on nonlinear free-surface flow K.J. Bai1 & J.W. Kim2 1Seoul National University, Korea 2American Bureau of Shipping, USA 3.1 Introduction Recently there has been a growing need for the treatment of the nonlinear water- wave problems. In treating the nonlinearity, there are basically two different approaches: One is based on the long-wave approximations or a shallow-water theory that gives an approximate formulation including the effect of the nonlinearity. The other is to treat the Laplace equation defined in the fluid domain with the exact nonlinear free-surface boundary conditions and the exact body-boundary condition if a body is present. The former is to treat a simplified nonlinear formulation defined only in a horizontal free-surface plane, which is one dimension less than the original fluid domain. The free-surface flow problems have been of interest to many naval architects and ocean and coastal engineers for a long time. The generation and evolution of the water waves and their interaction with the man-made structures are the main concerns in the problem. The theoretical investigations on the topics have been usually made in the scope of the potential theory by assuming the fluid is inviscid and incompressible. The most distinctive feature of this problem is the 83 WIT Transactions on State of the Art in Science and Engineering, Vol 18, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) doi:10.2495/978-1-85312-837-0/03 Numerical Models in Fluid-Structure Interaction presence of the free surface. This is one of the typical free-boundary problems where the part of the boundary should be obtained as a part of the solution. Due to this complexity the theoretical investigation on the problem has been restricted until recently. In the past, the linearized problem has been mainly treated. Sometimes, the nonlinear effect is considered by treating the second-order problem obtained in a systematic perturbation expansion. Another approach, where the leading nonlinearity is included, is to use approximate theoretical models such as the well known KdV and Boussinesq equations based on the long-wave approximation or shallow-water theory. Recently, the application of the Green–Naghdi equation is also proved to be an efficient method for treating nonlinear free-surface flow problems (Ertekin et al., 1986). In this chapter we will present the finite-element method applied to the nonlinear water-wave problems, i.e., in ship motions, wave resistance, lifting problems, and initial-value problems. The present numerical scheme can be used for the validation of the existing approximate theories, i.e., the KdV and the Boussinesq equations or Green–Naghdi equation as well as for a better prediction for more realistic physical models. In Section 3.2, we will describe the finite-element method applied to a two- dimensional hydrofoil problem. In this numerical scheme we introduced a buffer zone where the forced damping is introduced to reduce the wave elevation in matching the nonlinear numerical solution with the far-field linear solution. However, in three dimensions the nonlinear wave amplitude will reduce as it propagates to infinity. Thus the introduction of the fictitious damping term is not essential. In Section 3.3, we treat an initial-value problem for a physical model of an axi-symmetric container filled with water freely falling into a flat solid surface. In this problem we included the surface tension. In the numerical procedures, the finite-element subdivisions are made such that the free-surface elevation can be described as the multi-valued functions, since the surface tension as well as the gravity is important in the flow. In Section 3.4, the generation of solitons in a shallow-water towing-tank near the critical speed as well as a numerical towing- tank simulation for arbitrary tank conditions is discussed. In Section 3.5 a preliminary result of a sloshing problem by the same finite-element method used in Section 3.4 is discussed. In Section 3.6 a nonlinear diffraction problem is discussed. In Section 3.7, a brief concluding remark is given. 84 WIT Transactions on State of the Art in Science and Engineering, Vol 18, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Numerical Models in Fluid-Structure Interaction 3.2 Nonlinear steady waves due to a two-dimensional hydrofoil An application is described of the localized finite-element method to a steady nonlinear free-surface flow past a submerged two-dimensional hydrofoil at an arbitrary angle of attack. The earlier investigations with the linear free-surface boundary condition in Bai (1978) have shown some disagreement between the computed results and the experimental measurements for the cases of shallow submergence. The computational method of solution is the localized finite- element method based on the classical Hamilton’s principle. In the present study, a notable step is introduced in the matching procedure between the fully nonlinear and the linear sub-domains. Additional numerical results of wave resistance, lift force, and circulation strength are presented. Details can be found in Bai and Han (1994). It should be noted that in the earlier treatment of the linear hydrofoil problems, the velocity-potential formulation was used. However, in the following treatment of the nonlinear problem, a stream-function formulation is used since it has a distinct advantage in the numerical treatment. 3.2.1 Mathematical formulation We consider here a steady uniform flow past a fixed two-dimensional hydrofoil submerged in a fluid. The coordinate system is right-handed and rectangular as shown in Fig. 3.1. The y-axis is directed opposite to the force of gravity, and the x-axis coincides with the undisturbed free surface. The unit normal vector nr is always directed outward from the fluid domain. We neglect surface tension and assume that the fluid is inviscid, incompressible, and that the motions are irrotational. Here we assume the water depth, H, is constant. The steady two-dimensional flow is described by a total stream function Ψ(x, y) = Uy +ψ (x, y) , (3.1) where U is the incoming uniform flow velocity in the upstream and ψ (x, y) is the perturbation stream function. The total stream function Ψ of the incoming uniform flow is set to 0 on the undisturbed free surface and −UH on the bottom. The perturbation stream function ψ must satisfy 85 WIT Transactions on State of the Art in Science and Engineering, Vol 18, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Numerical Models in Fluid-Structure Interaction y S F 0 x S U 0 H c S TE S R1 R2 D S B Figure 3.1: Sketch of boundary configurations. ∇2ψ (x, y) = 0 (3.2) in the fluid domain D. The kinematic and dynamic boundary conditions on the free surface SF are as follows: ψ = −Uη or Uη +ψ = 0 x x 1 on S , y =η(x) , (3.3a,b) Uψ + gη + ()∇ψ ⋅∇ψ = 0 F y 2 where y =η(x) and g are the free-surface elevation and the gravitational acceleration, respectively. The boundary condition on the hydrofoil surface S0 can be Ψ(x, y) = C on S0 , (3.4) where the constant C will be determined as a part of the solution. To determine the constant C, we require at the trailing edge denoted by TE an additional condition, i.e., the Kutta condition, which states that the pressure on the upper and lower surfaces is continuous. Therefore, the tangential velocities on the upper and lower surfaces have the same magnitude at the trailing edge and bounded, that is, 86 WIT Transactions on State of the Art in Science and Engineering, Vol 18, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Numerical Models in Fluid-Structure Interaction ∂ ∂ ()Uy +ψ + = (Uy +ψ − ) at TE, (3.5) ∂n ∂n where ∂ / ∂n is the normal derivative and ψ + and ψ − are the perturbation stream function ψ on the upper and lower surfaces, respectively. The boundary condition on the bottom, SB , i.e., y = −H is ψ = 0 on SB . (3.6) y x x x 1 S 2 3 F 0 x S R1 S U 0 S H J J R2 c TE 2 3 J 1 D 1.1 D N DB D1.2 SB Figure 3.2: Subdomains in nonlinear calculation. As the radiation condition, we require that no disturbances exist far upstream, that is, ∇ψ → 0 as x → −∞ , (3.7) and that the perturbed flow velocity is bounded in the far downstream, i.e., ∇ψ is bounded as x → +∞ . (3.8) We hereby assume that the solution of eqns (3.2~3.8) can be determined uniquely. 3.2.2 Variational principle A mathematical formulation of the nonlinear free-surface flow phenomena of an incompressible ideal fluid can be described by variational principles based on the 87 WIT Transactions on State of the Art in Science and Engineering, Vol 18, © 2005 WIT Press www.witpress.com, ISSN 1755-8336 (on-line) Numerical Models in Fluid-Structure Interaction classical Hamilton’s principle as given in Kim and Bai (1990). Therefore, the formulation given in the previous section can be replaced by an equivalent variational functional formulation as follows. Now the nonlinear problem defined by eqns (3.2–3.8) is replaced by an equivalent Hamilton’s principle for the Lagrangian. The Lagrangian L is defined as L = T – V + W, (3.9) where T, V, and W are the kinetic energy, the potential energy, and the work done by external pressure, respectively.
Load more

Recommended publications
  • 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).
    [Show full text]
  • 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].
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • 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.
    [Show full text]
  • Sensitivity Analysis of Non-Linear Steep Waves Using VOF Method
    Tenth International Conference on ICCFD10-268 Computational Fluid Dynamics (ICCFD10), Barcelona, Spain, July 9-13, 2018 Sensitivity Analysis of Non-linear Steep Waves using VOF Method A. Khaware*, V. Gupta*, K. Srikanth *, and P. Sharkey ** Corresponding author: [email protected] * ANSYS Software Pvt Ltd, Pune, India. ** ANSYS UK Ltd, Milton Park, UK Abstract: The analysis and prediction of non-linear waves is a crucial part of ocean hydrodynamics. Sea waves are typically non-linear in nature, and whilst models exist to predict their behavior, limits exist in their applicability. In practice, as the waves become increasingly steeper, they approach a point beyond which the wave integrity cannot be maintained, and they 'break'. Understanding the limits of available models as waves approach these break conditions can significantly help to improve the accuracy of their potential impact in the field. Moreover, inaccurate modeling of wave kinematics can result in erroneous hydrodynamic forces being predicted. This paper investigates the sensitivity of non-linear wave modeling from both an analytical and a numerical perspective. Using a Volume of Fluid (VOF) method, coupled with the Open Channel Flow module in ANSYS Fluent, sensitivity studies are performed for a variety of non-linear wave scenarios with high steepness and high relative height. These scenarios are intended to mimic the near-break conditions of the wave. 5th order solitary wave models are applied to shallow wave scenarios with high relative heights, and 5th order Stokes wave models are applied to short gravity waves with high wave steepness. Stokes waves are further applied in the shallow regime at high wave steepness to examine the wave sensitivity under extreme conditions.
    [Show full text]
  • Waves and Structures
    WAVES AND STRUCTURES By Dr M C Deo Professor of Civil Engineering Indian Institute of Technology Bombay Powai, Mumbai 400 076 Contact: [email protected]; (+91) 22 2572 2377 (Please refer as follows, if you use any part of this book: Deo M C (2013): Waves and Structures, http://www.civil.iitb.ac.in/~mcdeo/waves.html) (Suggestions to improve/modify contents are welcome) 1 Content Chapter 1: Introduction 4 Chapter 2: Wave Theories 18 Chapter 3: Random Waves 47 Chapter 4: Wave Propagation 80 Chapter 5: Numerical Modeling of Waves 110 Chapter 6: Design Water Depth 115 Chapter 7: Wave Forces on Shore-Based Structures 132 Chapter 8: Wave Force On Small Diameter Members 150 Chapter 9: Maximum Wave Force on the Entire Structure 173 Chapter 10: Wave Forces on Large Diameter Members 187 Chapter 11: Spectral and Statistical Analysis of Wave Forces 209 Chapter 12: Wave Run Up 221 Chapter 13: Pipeline Hydrodynamics 234 Chapter 14: Statics of Floating Bodies 241 Chapter 15: Vibrations 268 Chapter 16: Motions of Freely Floating Bodies 283 Chapter 17: Motion Response of Compliant Structures 315 2 Notations 338 References 342 3 CHAPTER 1 INTRODUCTION 1.1 Introduction The knowledge of magnitude and behavior of ocean waves at site is an essential prerequisite for almost all activities in the ocean including planning, design, construction and operation related to harbor, coastal and structures. The waves of major concern to a harbor engineer are generated by the action of wind. The wind creates a disturbance in the sea which is restored to its calm equilibrium position by the action of gravity and hence resulting waves are called wind generated gravity waves.
    [Show full text]
  • 2.016 Hydrodynamics Free Surface Water Waves
    2.016 Hydrodynamics Reading #7 2.016 Hydrodynamics Prof. A.H. Techet Fall 2005 Free Surface Water Waves I. Problem setup 1. Free surface water wave problem. In order to determine an exact equation for the problem of free surface gravity waves we will assume potential theory (ideal flow) and ignore the effects of viscosity. Waves in the ocean are not typically uni-directional, but often approach structures from many directions. This complicates the problem of free surface wave analysis, but can be overcome through a series of assumptions. To setup the exact solution to the free surface gravity wave problem we first specify our unknowns: • Velocity Field: V (x, y, z,t ) = ∇φ(x, y, z,t ) • Free surface elevation: η(x, yt, ) • Pressure field: p(xy,,z , t ) version 3.0 updated 8/30/2005 -1- ©2005 A. Techet 2.016 Hydrodynamics Reading #7 Next we need to set up the equations and conditions that govern the problem: • Continuity (Conservation of Mass): ∇=2φ 0 for z <η (Laplace’s Equation) (7.1) • Bernoulli’s Equation (given some φ ): 2 ∂φ 1 pp− a ∂t +∇2 φ + ρ + gz = 0 for z <η (7.2) • No disturbance far away: ∂φ ∂t , ∇→φ 0 and p = pa − ρ gz (7.3) Finally we need to dictate the boundary conditions at the free surface, seafloor and on any body in the water: (1) Pressure is constant across the free surface interface: p = patm on z =η . ⎧∂φ 1 2 ⎫ p =−ρ ⎨ − V − gz⎬ + c ()t= patm . (7.4) ⎩ ∂t 2 ⎭ Choosing a suitable integration constant, ct( ) = patm , the boundary condition on z =η becomes ∂φ 1 ρ{ + V 2 + gη} = 0.
    [Show full text]
  • THERMODYNAMICS, HEAT TRANSFER, and FLUID FLOW, Module 3 Fluid Flow Blank Fluid Flow TABLE of CONTENTS
    Department of Energy Fundamentals Handbook THERMODYNAMICS, HEAT TRANSFER, AND FLUID FLOW, Module 3 Fluid Flow blank Fluid Flow TABLE OF CONTENTS TABLE OF CONTENTS LIST OF FIGURES .................................................. iv LIST OF TABLES ................................................... v REFERENCES ..................................................... vi OBJECTIVES ..................................................... vii CONTINUITY EQUATION ............................................ 1 Introduction .................................................. 1 Properties of Fluids ............................................. 2 Buoyancy .................................................... 2 Compressibility ................................................ 3 Relationship Between Depth and Pressure ............................. 3 Pascal’s Law .................................................. 7 Control Volume ............................................... 8 Volumetric Flow Rate ........................................... 9 Mass Flow Rate ............................................... 9 Conservation of Mass ........................................... 10 Steady-State Flow ............................................. 10 Continuity Equation ............................................ 11 Summary ................................................... 16 LAMINAR AND TURBULENT FLOW ................................... 17 Flow Regimes ................................................ 17 Laminar Flow ...............................................
    [Show full text]
  • PRESSURE 68 Kg 136 Kg Pressure: a Normal Force Exerted by a Fluid Per Unit Area
    CLASS Second Unit PRESSURE 68 kg 136 kg Pressure: A normal force exerted by a fluid per unit area 2 Afeet=300cm 0.23 kgf/cm2 0.46 kgf/cm2 P=68/300=0.23 kgf/cm2 The normal stress (or “pressure”) on the feet of a chubby person is much greater than on the feet of a slim person. Some basic pressure 2 gages. • Absolute pressure: The actual pressure at a given position. It is measured relative to absolute vacuum (i.e., absolute zero pressure). • Gage pressure: The difference between the absolute pressure and the local atmospheric pressure. Most pressure-measuring devices are calibrated to read zero in the atmosphere, and so they indicate gage pressure. • Vacuum pressures: Pressures below atmospheric pressure. Throughout this text, the pressure P will denote absolute pressure unless specified otherwise. 3 Other Pressure Measurement Devices • Bourdon tube: Consists of a hollow metal tube bent like a hook whose end is closed and connected to a dial indicator needle. • Pressure transducers: Use various techniques to convert the pressure effect to an electrical effect such as a change in voltage, resistance, or capacitance. • Pressure transducers are smaller and faster, and they can be more sensitive, reliable, and precise than their mechanical counterparts. • Strain-gage pressure transducers: Work by having a diaphragm deflect between two chambers open to the pressure inputs. • Piezoelectric transducers: Also called solid- state pressure transducers, work on the principle that an electric potential is generated in a crystalline substance when it is subjected to mechanical pressure. Various types of Bourdon tubes used to measure pressure.
    [Show full text]
  • On the Generation of Vorticity at a Free-Surface
    Center for TurbulenceResearch Annual Research Briefs On the generation of vorticity at a freesurface By T Lundgren AND P Koumoutsakos Motivations and ob jectives In free surface ows there are many situations where vorticityenters a owin the form of a shear layer This o ccurs at regions of high surface curvature and sup ercially resembles separation of a b oundary layer at a solid b oundary corner but in the free surface ow there is very little b oundary layer vorticity upstream of the corner and the vorticity whichenters the owisentirely created at the corner Ro o d has asso ciated the ux of vorticityinto the ow with the deceleration ofalayer of uid near the surface These eects are quite clearly seen in spilling breaker ows studied by Duncan Philomin Lin Ro ckwell and Dabiri Gharib In this pap er we prop ose a description of free surface viscous ows in a vortex dynamics formulation In the vortex dynamics approach to uid dynamics the emphasis is on the vorticityvector which is treated as the primary variable the velo city is expressed as a functional of the vorticity through the BiotSavart integral In free surface viscous ows the surface app ears as a source or sink of vorticityand a suitable pro cedure is required to handle this as a vorticity b oundary condition As a conceptually attractivebypro duct of this study we nd that vorticityis conserved if one considers the vortex sheet at the free surface to contain surface vorticity Vorticity which uxes out of the uid and app ears to b e lost is really gained by the vortex sheet
    [Show full text]
  • Pressure and Fluid Statics
    cen72367_ch03.qxd 10/29/04 2:21 PM Page 65 CHAPTER PRESSURE AND 3 FLUID STATICS his chapter deals with forces applied by fluids at rest or in rigid-body motion. The fluid property responsible for those forces is pressure, OBJECTIVES Twhich is a normal force exerted by a fluid per unit area. We start this When you finish reading this chapter, you chapter with a detailed discussion of pressure, including absolute and gage should be able to pressures, the pressure at a point, the variation of pressure with depth in a I Determine the variation of gravitational field, the manometer, the barometer, and pressure measure- pressure in a fluid at rest ment devices. This is followed by a discussion of the hydrostatic forces I Calculate the forces exerted by a applied on submerged bodies with plane or curved surfaces. We then con- fluid at rest on plane or curved submerged surfaces sider the buoyant force applied by fluids on submerged or floating bodies, and discuss the stability of such bodies. Finally, we apply Newton’s second I Analyze the rigid-body motion of fluids in containers during linear law of motion to a body of fluid in motion that acts as a rigid body and ana- acceleration or rotation lyze the variation of pressure in fluids that undergo linear acceleration and in rotating containers. This chapter makes extensive use of force balances for bodies in static equilibrium, and it will be helpful if the relevant topics from statics are first reviewed. 65 cen72367_ch03.qxd 10/29/04 2:21 PM Page 66 66 FLUID MECHANICS 3–1 I PRESSURE Pressure is defined as a normal force exerted by a fluid per unit area.
    [Show full text]