Vortex Methods: Theory and Practice

Vortex Methods: Theory and Practice GEORGES-HENRI COTTET Université Joseph Fourier in Grenoble PETROS D. KOUMOUTSAKOS ETH-Zürich and CTR, NASA Ames/Stanford University iii PUBLISHED BY THE PRESS SYNDICATE OF THE UNIVERSITY OF CAMBRIDGE The Pitt Building, Trumpington Street, Cambridge, United Kingdom CAMBRIDGE UNIVERSITY PRESS The Edinburgh Building, Cambridge CB2 2RU, UK http://www.cup.cam.ac.uk 40 West 20th Street, New York, NY 10011-4211, USA http://www.cup.org 10 Stamford Road, Oakleigh, Melbourne 3166, Australia Ruiz de Alarcón 13, 28014 Madrid, Spain c Cambridge University Press 2000 ° This book is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2000 Printed in the United States of America Typeface in Times Roman 10/13 pt. System LATEX 2ε [TB] A catalog record for this book is available from the British Library. Library of Congress Cataloging in Publication Data Cottet, G.-H. (Georges-Henri). 1956– Vortex methods: theory and practice / Georges-Henri Cottet. Petros D. Koumoutsakos. p. cm. ISBN 0-521-62186-0 1. Navier–Stokes equations – Numerical solutions. 2. Vortex-motion. I. Koumoutsakos, Petros D. II. Title. QA925.C68 1999 532′ .0533′ 01515353 – dc21 99-12277 CIP ISBN 0 521 62186 0 hardback iv Contents Preface 1 1.1 1.2 1.3 page ix Definitions and Governing Equations Kinematics of Vorticity Dynamics of Vorticity Helmholtz’s and Kelvin’s Laws for Vorticity Dynamics 2 Vortex Methods for Two-Dimensional Flows 2.1 An Introduction to Two-Dimensional Vortex Methods: Vortex Sheet Computations 2.2 General Definition 2.3 Cutoff Examples and Construction of Mollified Kernels 2.4 Particle Initializations 2.5 The Case of No-Through-Flow or Periodic Boundary Conditions 2.6 Convergence and Conservation Properties 3 3.1 3.2 3.3 3.4 Three-Dimensional Vortex Methods for Inviscid Flows Vortex Particle Methods Vortex Filament Methods Convergence Results The Problem of the Vorticity Divergence for Vortex Particle Methods v 1 2 5 7 10 10 18 22 26 31 34 55 56 61 75 84 vi 4 4.1 4.2 4.3 4.4 4.5 5 5.1 5.2 5.3 5.4 5.5 5.6 Contents Inviscid Boundary Conditions Kinematic Boundary Conditions Kinematics I : The Helmholtz Decomposition Kinematics II: The Ψ–ω and the u–ω Formulations Discretization of the Integral Equations Accuracy Issues Related to the Regularization near the Boundary 90 92 92 96 110 Viscous Vortex Methods Viscous Splitting of the Navier–Stokes Equations Random-Walk Methods Resampling Methods The Method of Particle Strength Exchange Other Redistribution Schemes Subgrid-scale Modeling in Vortex Methods 121 124 130 141 145 159 164 Vorticity Boundary Conditions for the Navier–Stokes Equations 6.1 The No-Slip Boundary Condition 6.2 Vorticity Boundary Conditions for the Continuous Problem 6.3 Viscous Splitting Algorithms 114 6 172 174 179 181 7 Lagrangian Grid Distortions: Problems and Solutions 7.1 Circulation Processing Schemes 7.2 Location Processing Techniques 206 208 219 8 8.1 8.2 8.3 237 238 244 251 Hybrid Methods Assignment and Interpolation Schemes Vortex-In-Cell Methods Eulerian–Lagrangian Domain Decomposition Appendix A Mathematical Tools for the Numerical Analysis of Vortex Methods A.1 Data Particle Approximation A.2 Classical and Measure Solutions to Linear Advection Equations A.3 Mathematical Facts about the Flow Equations 261 262 267 278 Contents B.1 B.2 B.3 B.4 Appendix B Fast Multipole Methods for Three-Dimensional N-Body Problems Multipole Expansions The Poisson Integral Method Computational Cost Tree Data Structures vii 284 286 291 293 294 Bibliography 301 Index 311 1 Definitions and Governing Equations Vorticity plays an important role in fluid dynamics analysis, and in many cases it is advantageous to describe dynamic events in a flow in terms of the evolution of the vorticity field. The vorticity field (ω) is related to the velocity field (u) of a flow as ω = ∇ × u. (1.0.1) It follows from this definition that vorticity is a solenoidal field: ∇ · ω = 0. (1.0.2) In a Cartesian coordinate system (x, y, z) this relation yields the following relationships between the velocity componenets (u x , u y , u z ) and the vorticity components (ωx , ω y , ωz ): ωx = ∂u y ∂u z ∂u x ∂u x ∂u y ∂u z − , ωy = − , ωz = − . ∂y ∂z ∂z ∂x ∂x ∂y (1.0.3) In two dimensions the vorticity field has only one nonzero component (ωz ) orthogonal to the (x, y) plane, thus automatically satisfying solenoidal condition (1.0.2). The circulation Ŵ of the vorticity field around a closed curve L, surrounding a surface S with unit normal n is defined by Z Z ω · n d S, (1.0.4) u · dr = Ŵ = S L where dr denotes an element of the curve. There are several physical interpretations of the definition of vorticity. We will adopt the point of view that vorticity is a solid-body-like rotation that can 1 2 1. Definitions and Governing Equations be imparted to the elements because of a stress distribution in the fluid. Hence when we consider a vorticity-carrying fluid element, the increment of angular velocity (dÄ) across an infinitesimal distance (dr) over the element is given by dÄ = 1 ω × dr. 2 (1.0.5) When we can track the translation and deformation of vorticity-carrying fluid elements, because of the kinematics and dynamics of the flow field we are able to obtain a complete description of the flow field. Considering the vorticitycarrying fluid elements as computational elements is the basis of the vortex methods that we analyze in this book. The close link of numerics and physics is the essense of vortex methods, and it is a point of view that will be emphasized throughout this book. In this introductory chapter we present fundamental definitions and equations relating to the kinematics and the dynamics of the vorticity field. In Section 1.1 we introduce the description of flow phenomena in terms of Eulerian and Lagrangian points of view. Using these two descriptions, we present in Section 1.2 the dynamic laws governing the evolution of the vorticity field in a viscous, incompressible flow field. In Section 1.3 we present Helmholtz’s and Kelvin’s laws governing the motion of the vorticity field. 1.1. Kinematics of Vorticity There are two different ways of expressing the behavior of the fluid that may be classified as the Lagrangian and the Eulerian point of view. Their difference lies in the choice of coordinates we wish to use to describe flow phenomena. 1.1.1. Lagrangian Description When the fluid is viewed as a collection of fluid elements that are freely translating, rotating, and deforming, then we may identify the dependent quantities of the flow field (such as the velocity, temperature, etc.) with these individual fluid elements. In that sense the Lagrangian viewpoint is a natural extension of particle mechanics. To obtain a full description of the flow we need to identify the initial location of the fluid elements and the initial value of the dependent variable. The independent variables are then the initial location of a point (x0p ) and time (T ). By following the trajectories of the collection of fluid elements, we are able to sample at every location in space and instant in time the quantity of interest. The primary flow quantity in this description is the velocity of the individual fluid elements. The velocity of a fluid element that is residing in an inertial 1.1. Kinematics of Vorticity 3 frame of reference at X p is expressed as up = ∂X p . ∂T (1.1.1) The acceleration of a fluid particle in a Lagrangian frame is expressed as ap = ∂u p . ∂T (1.1.2) The Lagrangian description is ideally suited to describing phenomena in terms of the vorticity of the flow field. 1.1.2. Eulerian Description In this description of the flow, our observation point is fixed at a certain location x of the flow field. The flow quantities as they are changing with time t are considered as functions of x. Unlike in Lagrangian methods the location of our observation point remains unchanged by time, and it is the change of the values of the dependent variables at the observation point that describes the flow field. The Eulerian and the Lagrangian quantities of the flow are related as x = X(x0 , T ), (1.1.3) t = T. (1.1.4) The Eulerian description of the flow is the most commonly used method to describe flow phenomena in the fluid mechanics literature. In this description, individual fluid elements and their history are not tracked explicitly, but rather it is the global picture of the field that is changing with time that provides us with the description of the flow. 1.1.3. The Material Derivative The material derivative allows us to relate the Eulerian and the Lagrangian time derivatives of a dependent variable. Let Q be a quantity of the flow expressed in a Lagrangian frame as Q(x0 , T ) and let q be the same quantity expressed in an Eulerian frame, that is, q(x, t). Then we would have that Q(x0 , T ) = q[x = X(x0 , T ), t]. (1.1.5) 4 1. Definitions and Governing Equations So the rate of change of Q with time T may be related to the rate change of q with time t with the chain rule for differentiation as ∂q ∂x ∂q ∂t ∂Q = · + , ∂T ∂x ∂ T ∂t ∂ T (1.1.6) and since we have for the velocity of a fluid particle that u = ∂x/∂ T then ∂Q ∂q ∂q = + u· . ∂T ∂t ∂x (1.1.7) The first term is the local rate of change of a variable, and the second term is the convective change of the dependent variable. The substantial derivative (i.e., the rate of change of quantity in a Lagrangian frame) is a convenient way of understanding several phenomena in fluid mechanics, and Stokes has given it a special symbol: ∂( ) D( ) = + (u · ∇)( ). Dt ∂t (1.1.8) From the definition of the substantial derivative we may easily see then that Dx = u. Dt (1.1.9) We may also determine the rate of change of a material line element (dr) by using the definition of the substantial derivative as D(dr) = du = ∂ j udr j = dr · ∇u. Dt (1.1.10) 1.1.4. Reynold’s Transport Theorem As an illustrative example of the Lagrangian and the Eulerian descriptions of the flow, we may consider the rate of change of the volume integral of the quantity Q in a material volume [V (t)] with surface [S(t)] having normal n and velocity u, i.e., Z d Q d V. (1.1.11) dt V (t) Contributions for this rate of change are given by the local rate R R of change of Q, V (t) ∂ Q/∂t d V , as well as from the motion of the boundary S(t) Q(u·n) d S 1.2. Dynamics of Vorticity 5 [note that for small times dt we may write d V = d S(u · n) dt] so that we have Z Z Z ∂Q d Q(u · n) d S, (1.1.12) Q dV = dV + dt V (t) S(t) V (t) ∂t By using vector calculus we may write Z Z Z ∂Q d ∇ · (Qu) d V, Q dV = dV + dt V (t) V (t) ∂t V (t) (1.1.13) or by using the expression for the substantial derivative we may write that Z Z Z DQ d Q∇ · u d V. (1.1.14) Q dV = dV + dt V (t) V (t) V (t) Dt which is known as Reynold’s transport theorem for the quantity Q. 1.2. Dynamics of Vorticity The motion of an incompressible Newtonian fluid is governed by the following equations that express the conservation of mass and momentum of fluid in Eulerian and Lagrangian frames [160]. In the Eulerian description we consider the development of the flow field as it is observed at a fixed point P of the domain, while in the Lagrangian description we consider the equations from the point of view of a material fluid element that moves with the local velocity of the flow. The conservation of mass can be expressed as Eulerian Description: ∂ρ ∂t + Rate of accumulation of mass per unit volume at P = 0. ∇ · (ρu) (1.2.1) Net flow rate of mass out of P per unit volume Lagrangian Description: Dρ Dt Rate of change of the density of a fluid element = −ρ Mass per unit volume ∇ · u. (1.2.2) Particle-volume expansion rate The conservation of momentum can be expressed in terms of the velocity (u) and the pressure P of the flow field as 6 1. Definitions and Governing Equations Eulerian Description: ρ ∂u ∂t ρu · ∇ u + Rate of increase of momentum at P −∇ P = Net flow rate of momentum carried in P by ρu ν1u, + Net pressure force (1.2.3) Net viscous force where ν denotes the kinematic viscosity of the fluid. Lagrangian Description: ρ Du Dt −∇ P = Acceleration of a fluid particle ν1u. + Net pressure force (1.2.4) Net viscous force With definition of vorticity (1.0.1) the momentum equations for an incompressible, Newtonian fluid of uniform density can be expressed in Lagrangian and Eulerian forms as Eulerian Description: ρ ∂ω ∂t ρu · ∇ ω + Rate of increase of vorticity = Net flow rate of vorticity ρω · ∇u + Vortex stretching ν1ω. (1.2.5) Viscous diffusion Lagrangian Description: ρ Dω Dt Rate of change of particle vorticity = ρω · ∇u + Rate of deforming vortex lines ν1 · ω. (1.2.6) Net rate of viscous diffusion Note that in the velocity–vorticity formulation the pressure of the flow can be recovered from the equation 1 1P = −∇ · ρ µ ¶ 1 2 |u| − u × ω . 2 (1.2.7) In the case of a viscous, Newtonian flow of a fluid with nonuniform density, rotation can be imparted to the fluid elements because of the baroclinic generation 1.3. Helmholtz’s and Kelvin’s Laws for Vorticity Dynamics of vorticity. In this case the equation for the vorticity field is µ ¶ D(ω/ρ) 1 1 1 = ω · ∇ u + ν1ω + ∇ P × ∇ . Dt ρ ρ ρ 7 (1.2.8) 1.3. Helmholtz’s and Kelvin’s Laws for Vorticity Dynamics In order to characterize the kinematic evolution of the vorticity field it is useful to introduce some geometrical concepts. We consider the vector of the vorticity field and we identify the lines that are tangential to this vector as vortex lines. In turn, a collection of these lines can form vortex surfaces or vector tubes. The motions of fluid elements carrying vorticity obey certain laws that were first outlined by Helmholtz for the inviscid evolution of the vorticity and further extended by Kelvin to include the effects of viscosity. From the solenoidal condition for the vorticity field, integrating over a volume of fluid with nonzero vorticity, and using the Gauss theorem, we obtain that Z Z ω · n d S = 0, (1.3.1) ∇ · ω dV = V S where V denotes the volume of the fluid encompassed by the surface S. When we consider a vortex tube, Eq. (1.3.1) dictates that the strength of the vortex tube is the same at all cross sections. This is Helmholtz’s first theorem. When Eq. (1.3.1) is applied to a vorticity tube with cross sections A1 and A2 with respective uniform normal vorticity components ω1 = ω · n1 and ω2 = ω · n2 (Fig. 1.1) we obtain that |ω1 |A1 = |ω2 |A2 = |Ŵ| (1.3.2) independently of the behavior of the vorticity field between the two crosssections of the vortex tube. Equation (1.3.2) defines the circulation (Ŵ) of the vortex tube. When we consider the Lagrangian description of the inviscid evolution of the vorticity field in an incompressible flow (with ρ = 1), Eq. (1.2.6) can be expressed as Dω = ω · ∇u. Dt (1.3.3) Comparing Eqs. (1.3.3) and (1.1.10) for the evolution of material lines, Ddr = dr · ∇u, Dt (1.3.4) 8 1. Definitions and Governing Equations Figure 1.1. Sketch of vortex lines and vortex tube. we observe that in a circulation-preserving motion the vortex lines are material lines. This is Helmholtz’s second theorem for the motion of vorticity elements. As a result of this law, fluid elements that at any time belong to one vortex line, however they may be translated, remain on the vortex line. A result of the first and the second laws is the property of vortex lines and tubes: that no matter how they evolve, they must always form closed curves or they must have their ends in the bounding surface of the fluid. Kelvin extended the laws of Helmholtz in order to account for the effects of viscosity and at the same time provide a different physical interpretation for the motion of vorticity-carrying fluid elements in terms of the circulation around a closed curve. From the definition of circulation for a line around a cross section of a vortex tube we obtain that Ŵ = Z u · dr. (1.3.5) L Now by using the Lagrangian form of the velocity–pressure formulation for the 1.3. Helmholtz’s and Kelvin’s Laws for Vorticity Dynamics acceleration of the material particles we obtain Z DŴ D u · dr = Dt Dt L Z Z Ddr Du · dr + · du. = Dt L Dt L As we are tracking material lines we obtain that Z Z Ddr · du = u · du = 0. L Dt L 9 (1.3.6) (1.3.7) (1.3.8) Using Eq. (1.3.8) and momentum equation (Eq. 1.2.4), we can express Eq. (1.3.7) as Z Du DŴ = · dr (1.3.9) Dt L Dt Z Z = − 1u · dr. (1.3.10) ∇ P · dr + ν L L Noting that the pressure term integrates to zero, we obtain that Z DŴ (1u ) · dr. = ν Dt L (1.3.11) In the case of an inviscid flow, the right-hand side of Eq. (1.3.11) is zero and the circulation of material elements is conserved. This is Kelvin’s theorem for the modification of circulation of fluid elements. In the case of baroclinic flow the circulation around a material line can be modified because of the baroclinic generation of vorticity, and Kelvin’s theorem is modified as Z Z 1 DŴ (1u ) · dr + ∇ρ × ∇ P · n d S. (1.3.12) = ν 2 Dt ρ L Note that the second term on the right-hand side is an integral over the area encompassed by the material curve. Equation (1.3.12) is known as Bjerken’s theorem.