DOCSLIB.ORG

Stream Function and Velocity Potential)

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Stream Function and Velocity Potential) NPTEL – Mechanical – Principle of Fluid Dynamics Module 3 : Lecture 1 INVISCID INCOMPRESSIBLE FLOW (Fundamental Aspects) In general, fluids have a well-known tendency to move or flow. The slight change in shear stress or appropriate imbalance in normal stresses will cause fluid motion. Fluid kinematics deals with various aspects of fluid motion without concerning the actual force that causes the fluid motion. In this particular section, we shall consider the ‘field’ concept to define velocity/ acceleration of fluid by virtue of its motion. In the later part, some ‘visualization’ concepts are introduced to define the motion of the fluid qualitatively as well as quantitatively. There are two general approaches in analyzing the fluid motion. In the first method (Lagrangian approach), the individual fluid particles are considered and their properties are studied as a function of time. In the second method (Eulerian approach), the ‘field’ concept is introduced and the properties are completely prescribed as the functions of space and time. In other words, the attention is focused at fixed points in space as the fluid passes those points. Velocity and Acceleration Field Since the ‘continuum’ assumption holds well for fluids, the description of any fluid property (such as density, pressure, velocity, acceleration etc.) can be expressed as a function of location. These representation as a function of spatial coordinates is called as “field representation” of the flow. One of the most important fluid variables is the velocity field. It is a vector function of position and time with components uv, and w as scalar variables i.e. V=++ u( xyzt,,,) iˆˆ v( xyzt ,,,) j w( xyzt ,,,) kˆ (3.1.1) The magnitude of the velocity vector i.e. V=( uvw22 ++ 2) , is the speed of fluid. The total time derivative of the velocity vector is the acceleration vector field (a) of the flow which is given as, dV d{ u( xyzt,,,)} d{ v( xyzt ,,,)} d{ w( xyzt,,,)} a==++ ijkˆˆ ˆ (3.1.2) dt dt dt dt Joint initiative of IITs and IISc – Funded by MHRD Page 1 of 56 NPTEL – Mechanical – Principle of Fluid Dynamics For instance, the scalar time derivative of u is expressed as, d{ u( xyzt,,,)} ∂∂u u dx ∂ u dy ∂ u dz =+++ dt∂∂ t t dt ∂ t dt ∂ t dt ∂∂∂uuu ∂ u =+++uvw (3.1.3) ∂∂∂ttt ∂ t ∂u = +∇(Vu. ) ∂t When u is replaced with vwand in the above equation, then the corresponding expressions would be, d{ v( xyzt,,,)} ∂v =+∇(Vv. ) dt∂ t (3.1.4) d{ w( xyzt,,,)} ∂w = +∇(Vw. ) dt∂ t Now, summing them into a vector quantity, one may write Eq. (3.1.2) in compact form as, dV∂ V ∂∂ V V ∂∂ V V a= = + u + v + w = +∇( VV. ) dt∂ t ∂∂ x y ∂∂ z t (3.1.5) ∂∂ ∂ ∂∂∂ where, Vu= + v + w and ∇=iˆˆ + j + kˆ ∂∂xy ∂ z ∂∂ xy ∂ z ∂V In the above equation, is called as “local acceleration” and the second part i.e. ∂t ∂∂VV ∂ V uvw++ is called a “convective acceleration”. The total time derivative ∂∂xy ∂ z is called as “substantial/material” derivative. This field concept can be applied to any variable (vector or scalar). For example, one may write the total derivative for pressure and temperature field as, dp∂ p ∂∂ p p ∂∂ p p = +u + v + w = +∇( Vp. ) dt∂ t ∂∂ x y ∂∂ z t (3.1.6) dT∂ T ∂∂ T T ∂∂ T T = +u + v + w = +∇( VT. ) dt∂ t ∂∂ x y ∂∂ z t Joint initiative of IITs and IISc – Funded by MHRD Page 2 of 56 NPTEL – Mechanical – Principle of Fluid Dynamics Visualization of Fluid Flow The quantitative and qualitative information of fluid flow can be obtained through sketches, photographs, graphical representation and mathematical analysis. However, the visual representation of flow fields is very important in modeling the flow phenomena. In general, there are four basic types of line patterns used to visualize the flow such as timeline, pathline, streakline and streamlines. Regardless of how the results are obtained, (i.e. analytically/experimentally/computationally) it is necessary to plot the data to get the feel of flow parameters that vary in time and/or shape (such as profile plots, vector plots and contour plots). (a) Timeline: A ‘timeline’ is a set of fluid particles that form a line at a given instant (Fig. 3.1.1-a). Thus, it is marked at same instant of time. Subsequent observations of the line provide the information of the flow field. They are particularly useful in situations where uniformity of flow is to be examined. (b) Pathline: It is the actual path traversed by a given fluid particle as it flows from one point to another (Fig. 3.1.1-b). Thus, the pathline is a Lagrangian concept that can be produced in the laboratory by marking the fluid particle and taking time exposure photograph of its motion. Pathlines can be calculated numerically for a known velocity field (V ) i.e. tend = + x xstart ∫ V dt (3.1.7) tstart (c) Streakline: A streakline consists of all particles in a flow that has previously passed through a common point (Fig. 3.1.1-c). Here, the attention is focused to a fixed point in space (i.e. Eulerian approach) and identifying all fluid particles passing through that point. These lines are laboratory tool rather than analytical tool. They are obtained by taking instantaneous photographs of selected particles that have passed through a given location in the flow field. Joint initiative of IITs and IISc – Funded by MHRD Page 3 of 56 NPTEL – Mechanical – Principle of Fluid Dynamics (d) Streamline: These are the lines drawn in the flow field so that at a given instant, they are tangent to the direction of flow at every point in the flow field (Fig. 3.1.1-d). Since the streamlines are tangent to the velocity vector at every point in the flow field, there can be no flow across a streamline. Mathematically, these lines are obtained analytically by integrating the equations defining lines tangent to the velocity field. In a two dimensional flow field as shown in the figure, the slope of the streamline is equal to the tangent of the angle that velocity vector makes with x-axis i.e. dy v = (3.1.8) dx u This equation can be integrated to obtain the equation of streamlines. When bundles of streamlines are considered in a flow field, it constitutes a ‘stream tube’ (Fig. 3.1.1-e). Since streamlines are everywhere parallel to the local velocity, fluid cannot cross a streamline, so fluids within a stream tube remain there and cannot cross the boundary at stream tube. Fig. 3.1.1: Basic line patterns in fluid flow: (a) Timelines; (b) Pathline; (c) Streakline; (d) Streamline; (e) Streamtube. Joint initiative of IITs and IISc – Funded by MHRD Page 4 of 56 NPTEL – Mechanical – Principle of Fluid Dynamics The following observations can be made about the fundamental line patterns; 1. Mathematically, it is convenient to calculate a streamline while other three are easier to generate experimentally. 2. The streamlines and timelines are instantaneous lines while pathlines and streakline are generated by passage of time. 3. In a steady flow, all the four basic line patterns are identical. Since, the velocity at each point in the flow field remains constant with time, consequently streamline shapes do not vary. It implies that the particle located on a given streamline will always move along the same streamline. Further, the consecutive particles passing through a fixed point in space will be on the same streamline. Hence, all the lines are identical in a steady flow. They do not coincide for unsteady flows. (f) Graphical data analysis techniques: Profile plots, vector plots and contour plots are few important techniques in which fluid flow properties can be analyzed. The profile plot (Fig. 3.1.2-a) indicates the variation of any scalar property (such as pressure, temperature and density) along some desired direction in a flow field. Using this plot, it is possible to examine the relative behavior of all variables in a multivariate data set. Fig. 3.1.2: Graphical representation of data analysis technique: (a) profile plot, (b) vector plot, (c) contour plot. Joint initiative of IITs and IISc – Funded by MHRD Page 5 of 56 NPTEL – Mechanical – Principle of Fluid Dynamics A vector plot (Fig. 3.1.2-b) is an array of arrows indicating the magnitude and direction of a vector property at an instant of time. Although, streamlines indicate the direction of instantaneous velocity field, but does not directly indicate the magnitude of velocity. A useful flow pattern for both experimental and computational fluid flow is the ‘vector plot’ that indicates the magnitude and direction of instantaneous vector property. A contour plot (Fig. 3.1.2-c) is a two-dimensional plot of a three-dimensional surface showing lines where the surface intersects planes of constant elevation. Thus, they are curves with constant values of scalar property (or magnitude of vector property) at an instant of time. They can be filled in with either colors or sheds of gray representing the magnitude of the property. Joint initiative of IITs and IISc – Funded by MHRD Page 6 of 56 NPTEL – Mechanical – Principle of Fluid Dynamics Module 3 : Lecture 2 INVISCID INCOMPRESSIBLE FLOW (Kinematic Description of Fluid Flow) In order to illustrate this concept, we consider a typical fluid element of certain volume at any arbitrary time as shown in Fig. 3.2.1. After certain time interval, it has moved and changed its shape as well as orientation drastically. However, when we limit our attention to an infinitesimal particle of volume dV(= δδδ x.
Load more

Recommended publications
  • Variational Formulation of Fluid and Geophysical Fluid Dynamics
    Advances in Geophysical and Environmental Mechanics and Mathematics Gualtiero Badin Fulvio Crisciani Variational Formulation of Fluid and Geophysical Fluid Dynamics Mechanics, Symmetries and Conservation Laws Advances in Geophysical and Environmental Mechanics and Mathematics Series editor Holger Steeb, Institute of Applied Mechanics (CE), University of Stuttgart, Stuttgart, Germany More information about this series at http://www.springer.com/series/7540 Gualtiero Badin • Fulvio Crisciani Variational Formulation of Fluid and Geophysical Fluid Dynamics Mechanics, Symmetries and Conservation Laws 123 Gualtiero Badin Fulvio Crisciani Universität Hamburg University of Trieste Hamburg Trieste Germany Italy ISSN 1866-8348 ISSN 1866-8356 (electronic) Advances in Geophysical and Environmental Mechanics and Mathematics ISBN 978-3-319-59694-5 ISBN 978-3-319-59695-2 (eBook) DOI 10.1007/978-3-319-59695-2 Library of Congress Control Number: 2017949166 © Springer International Publishing AG 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication.
    [Show full text]
  • General Meteorology
    Dynamic Meteorology 2 Lecture 9 Sahraei Physics Department Razi University http://www.razi.ac.ir/sahraei Stream Function Incompressible fluid uv 0 xy u ; v Definition of Stream Function y x Substituting these in the irrotationality condition, we have vu 0 xy 22 0 xy22 Velocity Potential Irrotational flow vu Since the, V 0 V 0 0 the flow field is irrotational. xy u ; v Definition of Velocity Potential x y Velocity potential is a powerful tool in analysing irrotational flows. Continuity Equation 22 uv 2 0 0 0 xy xy22 As with stream functions we can have lines along which potential is constant. These are called Equipotential Lines of the flow. Thus along a potential line c Flow along a line l Consider a fluid particle moving along a line l . dx For each small displacement dl dy dl idxˆˆ jdy Where iˆ and ˆj are unit vectors in the x and y directions, respectively. Since dl is parallel to V , then the cross product must be zero. V iuˆˆ jv V dl iuˆ ˆjv idx ˆ ˆjdy udy vdx kˆ 0 dx dy uv Stream Function dx dy l Since must be satisfied along a line , such a line is called a uv dl streamline or flow line. A mathematical construct called a stream function can describe flow associated with these lines. The Stream Function xy, is defined as the function which is constant along a streamline, much as a potential function is constant along an equipotential line. Since xy, is constant along a flow line, then for any , d dx dy 0 along the streamline xy Stream Functions d dx dy 0 dx dy xy xy dx dy vdx ud y uv uv , from which we can see that yx So that if one can find the stream function, one can get the discharge by differentiation.
    [Show full text]
  • Future Directions of Computational Fluid Dynamics
    Future Directions of Computational Fluid Dynamics F. D. Witherden∗ and A. Jameson† Stanford University, Stanford, CA, 94305 For the past fifteen years computational fluid dynamics (CFD) has been on a plateau. Due to the inability of current-generation approaches to accurately predict the dynamics of turbulent separated flows, reliable use of CFD has been restricted to a small regionof the operating design space. In this paper we make the case for large eddy simulations as a means of expanding the envelope of CFD. As part of this we outline several key challenges which must be overcome in order to enable its adoption within industry. Specific issues that we will address include the impact of heterogeneous massively parallel computing hardware, the need for new and novel numerical algorithms, and the increasingly complex nature of methods and their respective implementations. I. Introduction Computational fluid dynamics (CFD) is a relatively young discipline, having emerged during the last50 years. During this period advances in CFD have been paced by advances in the available computational hardware, which have enabled its application to progressively more complex engineering and scientific prob- lems. At this point CFD has revolutionized the design process in the aerospace industry, and its use is pervasive in many other fields of engineering ranging from automobiles to ships to wind energy. Itisalso a key tool for scientific investigation of the physics of fluid motion, and in other branches of sciencesuch as astrophysics. Additionally, throughout its history CFD has been an important incubator for the formu- lation and development of numerical algorithms which have been seminal to advances in other branches of computational physics.
    [Show full text]
  • Potential Flow Theory
    2.016 Hydrodynamics Reading #4 2.016 Hydrodynamics Prof. A.H. Techet Potential Flow Theory “When a flow is both frictionless and irrotational, pleasant things happen.” –F.M. White, Fluid Mechanics 4th ed. We can treat external flows around bodies as invicid (i.e. frictionless) and irrotational (i.e. the fluid particles are not rotating). This is because the viscous effects are limited to a thin layer next to the body called the boundary layer. In graduate classes like 2.25, you’ll learn how to solve for the invicid flow and then correct this within the boundary layer by considering viscosity. For now, let’s just learn how to solve for the invicid flow. We can define a potential function,!(x, z,t) , as a continuous function that satisfies the basic laws of fluid mechanics: conservation of mass and momentum, assuming incompressible, inviscid and irrotational flow. There is a vector identity (prove it for yourself!) that states for any scalar, ", " # "$ = 0 By definition, for irrotational flow, r ! " #V = 0 Therefore ! r V = "# ! where ! = !(x, y, z,t) is the velocity potential function. Such that the components of velocity in Cartesian coordinates, as functions of space and time, are ! "! "! "! u = , v = and w = (4.1) dx dy dz version 1.0 updated 9/22/2005 -1- ©2005 A. Techet 2.016 Hydrodynamics Reading #4 Laplace Equation The velocity must still satisfy the conservation of mass equation. We can substitute in the relationship between potential and velocity and arrive at the Laplace Equation, which we will revisit in our discussion on linear waves.
    [Show full text]
  • Stream Function for Incompressible 2D Fluid
    26/3/2020 Navier–Stokes equations - Wikipedia Continuity equation for incompressible fluid Regardless of the flow assumptions, a statement of the conservation of mass is generally necessary. This is achieved through the mass continuity equation, given in its most general form as: or, using the substantive derivative: For incompressible fluid, density along the line of flow remains constant over time, therefore divergence of velocity is null all the time Stream function for incompressible 2D fluid Taking the curl of the incompressible Navier–Stokes equation results in the elimination of pressure. This is especially easy to see if 2D Cartesian flow is assumed (like in the degenerate 3D case with uz = 0 and no dependence of anything on z), where the equations reduce to: Differentiating the first with respect to y, the second with respect to x and subtracting the resulting equations will eliminate pressure and any conservative force. For incompressible flow, defining the stream function ψ through results in mass continuity being unconditionally satisfied (given the stream function is continuous), and then incompressible Newtonian 2D momentum and mass conservation condense into one equation: 4 μ where ∇ is the 2D biharmonic operator and ν is the kinematic viscosity, ν = ρ. We can also express this compactly using the Jacobian determinant: https://en.wikipedia.org/wiki/Navier–Stokes_equations 1/2 26/3/2020 Navier–Stokes equations - Wikipedia This single equation together with appropriate boundary conditions describes 2D fluid flow, taking only kinematic viscosity as a parameter. Note that the equation for creeping flow results when the left side is assumed zero. In axisymmetric flow another stream function formulation, called the Stokes stream function, can be used to describe the velocity components of an incompressible flow with one scalar function.
    [Show full text]
  • A Dual-Potential Formulation of the Navier-Stokes Equations Steven Gerard Gegg Iowa State University
    Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1989 A dual-potential formulation of the Navier-Stokes equations Steven Gerard Gegg Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Aerospace Engineering Commons, and the Mechanical Engineering Commons Recommended Citation Gegg, Steven Gerard, "A dual-potential formulation of the Navier-Stokes equations " (1989). Retrospective Theses and Dissertations. 9040. https://lib.dr.iastate.edu/rtd/9040 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS The most advanced technology has been used to photo­ graph and reproduce this manuscript from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are re­ produced by sectioning the original, beginning at the upper left-hand corner and continuing from left to right in equal sections with small overlaps.
    [Show full text]
  • Div-Curl Problems and H1-Regular Stream Functions in 3D Lipschitz Domains
    Received 21 May 2020; Revised 01 February 2021; Accepted: 00 Month 0000 DOI: xxx/xxxx PREPRINT Div-Curl Problems and H1-regular Stream Functions in 3D Lipschitz Domains Matthias Kirchhart1 | Erick Schulz2 1Applied and Computational Mathematics, RWTH Aachen, Germany Abstract 2Seminar in Applied Mathematics, We consider the problem of recovering the divergence-free velocity field U Ë L2(Ω) ETH Zürich, Switzerland of a given vorticity F = curl U on a bounded Lipschitz domain Ω Ï R3. To that end, Correspondence we solve the “div-curl problem” for a given F Ë H*1(Ω). The solution is expressed in Matthias Kirchhart, Email: 1 [email protected] terms of a vector potential (or stream function) A Ë H (Ω) such that U = curl A. After discussing existence and uniqueness of solutions and associated vector potentials, Present Address Applied and Computational Mathematics we propose a well-posed construction for the stream function. A numerical method RWTH Aachen based on this construction is presented, and experiments confirm that the resulting Schinkelstraße 2 52062 Aachen approximations display higher regularity than those of another common approach. Germany KEYWORDS: div-curl system, stream function, vector potential, regularity, vorticity 1 INTRODUCTION Let Ω Ï R3 be a bounded Lipschitz domain. Given a vorticity field F .x/ Ë R3 defined over Ω, we are interested in solving the problem of velocity recovery: T curl U = F in Ω. (1) div U = 0 This problem naturally arises in fluid mechanics when studying the vorticity formulation of the incompressible Navier–Stokes equations. Vortex methods, for example, are based on the vorticity formulation and require a solution of problem (1) at every time-step.
    [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]
  • Useful Identities and Theorems from Vector Calculus
    Appendix A Useful Identities and Theorems from Vector Calculus A.1 Vector Identities A · (B × C) = C · (A × B) = B · (C × A) A × (B × C) = B(A · C) − C(A · B) (A × B) × C = B(A · C) − A(B · C) ∇×∇f = 0 ∇·(∇×A) = 0 ∇·( f A) = (∇ f ) · A + f (∇·A) ∇×( f A) = (∇ f ) × A + f (∇×A) ∇·(A × B) = B · (∇×A) − A · (∇×B) ∇(A · B) = (B ·∇)A + (A ·∇)B + B × (∇×A) + A × (∇×B) ∇·(AB) = (A ·∇)B + B(∇·A) ∇×(A × B) = (B ·∇)A − (A ·∇)B − B(∇·A) + A(∇·B) ∇×(∇×A) =∇(∇·A) −∇2 A A.2 The Gradient Theorem For two points a, b in a space where a scalar function f with spatial derivatives everywhere well-defined up to first order, b (∇ f ) · d = f (b) − f (a), a independently of the integration path between a and b. P. Charbonneau, Solar and Stellar Dynamos, Saas-Fee Advanced Course 39, 215 DOI: 10.1007/978-3-642-32093-4, © Springer-Verlag Berlin Heidelberg 2013 216 Appendix A: Useful Identities and Theorems from Vector Calculus A.3 The Divergence Theorem For any vector field A with spatial derivatives of all its scalar components everywhere well-defined up to first order, (∇·A)dV = A · nˆ dS , V S where the surface S encloses the volume V . A.4 Stokes’ Theorem For any vector field A with spatial derivatives of all its scalar components everywhere well-defined up to first order, (∇×A) · nˆ dS = A · d , S γ where the contour γ delimits the surface S, and the orientation of the unit nor- mal vector nˆ and direction of contour integration are mutually linked by the right-hand rule.
    [Show full text]
  • Potential Flow
    1.0 POTENTIAL FLOW One of the most important applications of potential flow theory is to aerodynamics and marine hydrodynamics. Key assumption. 1. Incompressibility – The density and specific weight are to be taken as constant. 2. Irrotationality – This implies a nonviscous fluid where particles are initially moving without rotation. 3. Steady flow – All properties and flow parameters are independent of time. (a) (b) Fig. 1.1 Examples of complicated immersed flows: (a) flow near a solid boundary; (b) flow around an automobile. In this section we will be concerned with the mathematical description of the motion of fluid elements moving in a flow field. A small fluid element in the shape of a cube which is initially in one position will move to another position during a short time interval as illustrated in Fig.1.1. Fig. 1.2 1.1 Continuity Equation v u = velocity component x direction y v y y y v = velocity component y direction u u x u x x x v x Continuity Equation Flow inwards = Flow outwards u v uy vx u xy v yx x y u v 0 - 2D x y u v w 0 - 3D x y z 1.2 Stream Function, (psi) y B Stream Line B A A u x -v The stream is continuity d vdx udy if (x, y) d dx dy x y u and v x y Integrated the equations dx dy C x y vdx udy C Continuity equation in 2 2 y x 0 or x y xy yx if 0 not continuity Vorticity equation, (rotational flow) 2 1 v u ; angular velocity (rad/s) 2 x y v u x y or substitute with 2 2 x 2 y 2 Irrotational flow, 0 Rotational flow.
    [Show full text]
  • Introduction to Computational Fluid Dynamics by the Finite Volume Method
    Introduction to Computational Fluid Dynamics by the Finite Volume Method Ali Ramezani, Goran Stipcich and Imanol Garcia BCAM - Basque Center for Applied Mathematics April 12–15, 2016 Overview on Computational Fluid Dynamics (CFD) 1. Overview on Computational Fluid Dynamics (CFD) 2 / 110 Overview on Computational Fluid Dynamics (CFD) What is CFD? I Fluids: mainly liquids and gases I The governing equations are known, but not their analytical solution: thus, we approximate it I By CFD we typically denote the set of numerical techniques used for the approximate solution (prevision) of the motion of fluids and the associated phenomena (heat exchange, combustion, fluid-structure interaction . ) I The solution of the governing differential (or integro-differential) equations is approximated by a discretization of space and time I From the continuum we move to the discrete level I The CFD is deeply connected to the improvement of computers in the last decades 3 / 110 Overview on Computational Fluid Dynamics (CFD) Applications of CFD I Any field where the fluid motion plays a relevant role: Industry Physics Medicine Meteorology Architecture Environment 4 / 110 Overview on Computational Fluid Dynamics (CFD) Applications of CFD II Moreover, the research front is particularly active: Basic research on fluid New numerical methods mechanics (e.g. transition to turbulence) Application oriented (e.g. renewable energy, competition . ) 5 / 110 Overview on Computational Fluid Dynamics (CFD) CFD: limits and potential I Method Advantages Disadvantages Experimental 1. More realistic 1. Need for instrumentation 2. Allows “complex” problems 2. Scale effects 3. Difficulty in measurements & perturbations 4. Operational costs Theoretical 1. Simple information 1.
    [Show full text]