layton w j introduction to the numerical analysis of incompressible viscous flows (siam, 2008)(isbn 9780898716573)(233s) mnssinhvienzone com

Si nh V ie nZ on e C om Introduction to the Numerical Analysis of Incompressible Viscous Flows SinhVienZone.com https://fb.com/sinhvienzonevn C O M P U TAT I O N A L S C I E N C E & E N G I N E E R I N G Computational Science and Engineering (CS&E) is widely accepted, along with theory and experiment, as a crucial third mode of scientific investigation and engineering design This series publishes research monographs, advanced undergraduate- and graduate-level textbooks, and other volumes of interest to a wide segment of the community of computational scientists and engineers The series also includes volumes addressed to users of CS&E methods by targeting specific groups of professionals whose work relies extensively on computational science and engineering Editor-in-Chief Omar Ghattas University of Texas at Austin Rolf Jeltsch ETH Zurich Ted Belytschko Northwestern University Chris Johnson University of Utah Clint Dawson University of Texas at Austin Laxmikant Kale University of Illinois Lori Freitag Diachin Lawrence Livermore National Laboratory Efthimios Kaxiras Harvard University C e on Jelena Kovacevic Carnegie Mellon University Charbel Farhat Stanford University nZ James Glimm Stony Brook University Habib Najm Sandia National Laboratory Alex Pothen Old Dominion University nh V ie Teresa Head-Gordon University of California–Berkeley and Lawrence Berkeley National Laboratory Series Volumes om Editorial Board David Keyes, Associate Editor Columbia University Layton, William, Introduction to the Numerical Analysis of Incompressible Viscous Flows Ascher, Uri M., Numerical Methods for Evolutionary Differential Equations Zohdi, T I., An Introduction to Modeling and Simulation of Particulate Flows Si Biegler, Lorenz T., Omar Ghattas, Matthias Heinkenschloss, David Keyes, and Bart van Bloemen Waanders, Editors, Real-Time PDE-Constrained Optimization Chen, Zhangxin, Guanren Huan, and Yuanle Ma, Computational Methods for Multiphase Flows in Porous Media Shapira, Yair, Solving PDEs in C++: Numerical Methods in a Unified Object-Oriented Approach SinhVienZone.com https://fb.com/sinhvienzonevn William Layton University of Pittsburgh Pittsburgh, Pennsylvania Si nh V ie nZ on e C om Introduction to the Numerical Analysis of Incompressible Viscous Flows Society for Industrial and Applied Mathematics Philadelphia SinhVienZone.com https://fb.com/sinhvienzonevn om C Copyright © 2008 by the Society for Industrial and Applied Mathematics e 10 nZ on All rights reserved Printed in the United States of America No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher For information, write to the Society for Industrial and Applied Mathematics, 3600 Market Street, 6th Floor, Philadelphia, PA 19104-2688 USA Trademarked names may be used in this book without the inclusion of a trademark symbol These names are used in an editorial context only; no infringement of trademark is intended ie FEMLAB is a registered trademark of COMSOL AB Si nh V The cover was produced from images created by and used with permission of the Scientific Computing and Imaging (SCI) Institute, University of Utah; J Bielak, D O’Hallaron, L Ramirez-Guzman, and T Tu, Carnegie Mellon University; O Ghattas, University of Texas at Austin; K Ma and H Yu, University of California, Davis; and Mark R Petersen, Los Alamos National Laboratory More information about the images is available at http://www.siam.org/books/series/csecover.php Library of Congress Cataloging-in-Publication Data Layton, W J (William J.) Introduction to the numerical analysis of incompressible viscous flows / William Layton p cm — (Computational science and engineering ; 6) Includes bibliographical references and index ISBN 978-0-898716-57-3 Viscous flow—Mathematical models Numerical analysis Fluid mechanics I Title QA929.L39 2008 532’.053301518—dc22 2008018448 is a registered trademark SinhVienZone.com https://fb.com/sinhvienzonevn om Contents Preface xiii Zo Mathematical Preliminaries: Energy and Stress 1.1 Finite Kinetic Energy: The Hilbert Space L2 (2 ) 1.1.1 Other norms 1.2 Finite Stress: The Hilbert Space X := H01 () 1.2.1 Weak derivatives and some useful inequalities 1.3 Some Snapshots in the History of the Equations of Fluid Motion 1.4 Remarks on Chapter 1.5 Exercises 3 10 12 15 15 Approximating Scalars 2.1 Introduction to Finite Element Spaces 2.2 An Elliptic Boundary Value Problem 2.3 The Galerkin–Finite Element Method 2.4 Remarks on Chapter 2.5 Exercises 17 17 26 30 33 34 37 37 39 43 48 49 Si nh Vi en Mathematical Foundations xi ne Foreword I ix C List of Figures Vector and Tensor Analysis 3.1 Scalars, Vectors, and Tensors 3.2 Vector and Tensor Calculus 3.3 Conservation Laws 3.4 Remarks on Chapter 3.5 Exercises v SinhVienZone.com https://fb.com/sinhvienzonevn vi 51 Approximating Vector Functions 4.1 Introduction to Mixed Methods for Creeping Flow 4.2 Variational Formulation of the Stokes Problem 4.3 The Galerkin Approximation 4.4 More About the Discrete Inf-Sup Condition 4.4.1 Other div-stable elements 4.5 Remarks on Chapter 4.6 Exercises 53 53 56 59 63 66 66 68 The Equations of Fluid Motion 5.1 Conservation of Mass and Momentum 5.2 Stress and Strain in a Newtonian Fluid 5.2.1 More about internal forces 5.2.2 More about V 5.3 Boundary Conditions 5.4 The Reynolds Number 5.5 Boundary Layers 5.6 An Example of Fluid Motion: The Taylor Experiment 5.7 Remarks on Chapter 5.8 Exercises 71 71 74 75 76 78 83 87 91 92 95 The Steady Navier–Stokes Equations 6.1 The Steady Navier–Stokes Equations 6.2 Uniqueness for Small Data 6.2.1 The Oseen problem 6.3 Existence of Steady Solutions 6.4 The Structure of Steady Solutions 6.5 Remarks on Chapter 6.6 Exercises 99 99 106 108 110 114 117 117 Approximating Steady Flows 7.1 Formulation and Stability of the Approximation 7.2 A Simple Example 7.3 Errors in Approximations of Steady Flows 7.4 More on the Global Uniqueness Conditions 7.5 Remarks on Chapter 7.6 Exercises 121 121 124 125 131 132 133 en Si nh Vi Zo ne C Steady Fluid Flow Phenomena om II Contents III Time-Dependent Fluid Flow Phenomena 137 The Time-Dependent Navier–Stokes Equations 139 8.1 Introduction 139 8.2 Weak Solution of the NSE 141 SinhVienZone.com https://fb.com/sinhvienzonevn Contents vii 8.3 8.4 8.5 Kinetic Energy and Energy Dissipation 145 Remarks on Chapter 147 Exercises 148 Approximating Time-Dependent Flows 9.1 Introduction 9.2 Stability and Convergence of the Semidiscrete Approximations 9.3 Rates of Convergence 9.4 Time-Stepping Schemes 9.5 Convergence Analysis of the Trapezoid Rule 9.5.1 Notation for the discrete time method 9.5.2 Error analysis of the trapezoid rule 9.6 Remarks on Chapter 9.7 Exercises 151 151 154 158 161 165 165 168 175 176 179 179 181 182 183 186 189 190 192 194 195 197 197 197 198 198 199 200 200 C om Models of Turbulent Flow 10.1 Introduction to Turbulence 10.2 The K41 Theory of Homogeneous, Isotropic Turbulence 10.2.1 Fourier series 10.2.2 The inertial range 10.3 Models in Large Eddy Simulation 10.3.1 A first choice of T 10.4 The Smagorinsky Model for T 10.5 Near Wall Models: Boundary Conditions for the Large Eddies 10.6 Remarks on Chapter 10 10.7 Exercises Vi Nomenclature Vectors and Tensors Fluid Variables Basic Function Spaces and Norms A.3.1 Other norms A.4 Velocity and Pressure Spaces and Norms A.5 Finite Element Notation A.6 Turbulence Si nh Appendix A.1 A.2 A.3 en Zo ne 10 Bibliography 203 Index 211 SinhVienZone.com https://fb.com/sinhvienzonevn om C ne Zo en Vi nh Si SinhVienZone.com https://fb.com/sinhvienzonevn om List of Figures This flow is exciting but far beyond what is reliably computable! 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 A curve and its piecewise linear interpolant A typical basis function Mesh with two “bad” triangles (upper left and lower right) Mesh following a curve Mesh for flow around cylinder at Re = 40 Different mesh density for Re = 200 The finite element space X h A sketch of the basis function j (x, y) The correspondence between basis functions and nodes A mesh resolving a circular transition region One of the three linear basis functions The cubic bubble function An inhomogeneous medium A typical adaptive FEM mesh 3.1 Geometry of a simple shear flow 41 4.1 4.2 4.3 4.4 4.5 4.6 Typical experimental realization of creeping flow Velocity vectors for Stokes flow Streamlines for Stokes flow Linear-constant pair violates stability The MINI element Another element satisfying the discrete inf-sup condition 54 55 55 60 65 69 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 Stress-deformation relation is nonlinear Verifying no-slip at low stresses Driven cavity domain and boundary conditions An example of flow in the driven cavity Discontinuous boundary velocities induce infinite stress A fluid flows across a surface A typical flow over a step Exploring the Cauchy stress vector 77 79 80 80 81 82 82 83 Si nh Vi en Zo ne C 1.1 ix SinhVienZone.com https://fb.com/sinhvienzonevn 18 19 20 20 21 21 22 22 23 24 25 25 27 33 x List of Figures Geometry of the forward-backward step Flow in pipes at increasing Reynolds numbers A boundary layer flow Depiction of a boundary layer Setup of counter rotating cylinders Three Taylor cells Schematic of flow between rotating cylinders 83 84 88 89 92 93 94 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 Velocity vectors for Re = Streamlines for Re = Velocity vectors for Re = 40 Streamlines for Re = 40 Velocity vectors Re = 200 Vorticity contours Re = 200: the von Karman vortex street Vorticity contours at Re = 1000 Voticity contours at Re = 2000 FEM mesh for Re = 40 FEM mesh for Re = 200 Shear flow between parallel plates The map T(N(u)) Behavior of flow between rotating cylinders 100 100 100 100 101 101 102 102 103 103 111 113 115 8.1 Depiction of k and epsilon 147 9.1 Streamtubes in a “simple” three-dimensional flow 152 10.1 10.2 10.3 10.4 10.5 A Gaussian filter (heavy) and rescaled (thin) A curve, its mean (heavy line) and fluctuation (dashed) Eddies are shed and roll down channel Smagorinsky model predicts flow reaches equilibrium quickly u¯ does not vanish on ∂2 Si nh Vi en Zo ne C om 5.9 5.10 5.11 5.12 5.13 5.14 5.15 SinhVienZone.com https://fb.com/sinhvienzonevn 187 188 191 192 193 ||6 (u − uh )|| ≤ C(3, f, β h ) inf ||6 (u − v h )|| + inf ||p − q h || v7 X h SinhVienZone.com q7 Qh https://fb.com/sinhvienzonevn 130 Chapter Approximating Steady Flows Proof The first inequality is proved already For the second, pick (v h , q h ) to be the Stokes projection of (u, q) in (Xh , Qh ) and use Lemma of Chapter For specific choices of div-stable elements, rates of convergence can be determined Corollary (rates of convergence for the MINI element) Suppose that the global uniqueness condition (7.3.1) holds and let X h , Qh be the MINI element Then, ||6 (u − uh )|| ≤ C(f, ν)h{||66 u|| + ||6 p||} ||6 (u − uh )|| ≤ C(f, ν)h om If additionally is a convex, planar polygon, and f L2 (), then ne C Proof The first inequality follows by using interpolation theory for linear elements from Chapter The second inequality needs one additional ingredient: regularity theory for the Stokes (and Navier–Stokes) equation In the indicated setting, it is known that solutions of the Stokes problem satisfy ||66 u|| + ||6 p|| ≤ C()||f || Zo It is possible to prove the same about the steady NSE using a bootstrap argument as follows Rewrite the NSE as a Stokes problem with the right-hand side f − u · u The above regularity result implies en ||66 u|| + ||6 p|| ≤ C(, ν)||f − u · u|| ≤ C(||f || + ||u · u||) Vi Now, by the Cauchy–Schwarz and Ladyzhenskaya inequalities, 2 ||u · u||2 = |u · u|2 dx ≤ |u|2 |6 u|2 dx 2 nh ≤ |u| dx 2  12 2 |6 u| dx  12 = ||u||2L4 () ||6 u||2L4 () Si Thus, by Theorem of Chapter 1, 1 ||u · u|| ≤ C||u||||6 u|| ||66 u|| , and using Exercise of Chapter with p = and q = 4/3 gives, for any ε > 0, ε ε ||u · u|| ≤ ||66 u||2 + C(ε)(||u||||6 u|| ) ≤ ||66 u||2 + C(ε, 2) ||6 u||2 4 Thus, we have for the NSE solution ||66 u|| + ||6 p|| ≤ C(, ν)||f − u · u|| ≤ C(||f || + ||u · u||) ε ≤ C||f || + ||66 u||2 + C(ε, 2) ||6 u||2 , and it follows that ||66 u|| + ||6 p|| ≤ C(, ν)||f ||, which completes the proof SinhVienZone.com https://fb.com/sinhvienzonevn 7.4 More on the Global Uniqueness Conditions 131 Corollary (rates of convergence for the Hood–Taylor element) Suppose that the global uniqueness condition (7.3.1) holds and let X,h Qh be the Hood–Taylor element Then, ||6 (u − uh )|| ≤ C(f, ν)h2 {||666 u|| + ||66 p||} Proof This follows like the previous corollary Under the discrete inf-sup condition optimal error estimates for the pressure also hold (Exercise 94) The proof is almost identical to the Stokes problem More on the Global Uniqueness Conditions om 7.4 ne C For nonlinear equations, such as the Navier–Stokes equations, it is known that a regular solution for the nonstationary problem need not exist for all times t ≥ 0, it need not converge towards the solution of the stationary problem as t → , when the boundary conditions and the forces converge towards a stationary solution O A Ladyzhenskaya, in [62] Zo There is scarcely any question in dynamics more important for Natural Philosophy than the stability of motion W Thompson and P G Tait, in Treatise on Natural Philosophy, Oxford, 1867, p 346 nh Vi en The error analysis in this chapter was based on a global uniqueness assumption on the data and (through that) the Reynolds number Since fluid flow is inherently time-dependent, it is important to understand what this condition means for the time-dependent NSE To this end, let (u3 (x), p (x)) be a solution of the (nondimensionalized) steady NSE: −Re−1 u3 + u3 · u3 + p = f (x) in , ∇ · u3 = in , u3 = on ∂2 and (p, 1) = (7.4.1) Si Under the small data condition for global uniqueness, Re2 N ||f ||3 < 1, (7.4.2) the solution of (7.4.1) is unique and finite element approximations converge to (u3 , p3 ) as h It is worthwhile to try to understand the meaning of the above global uniqueness condition in the time-dependent problem Accordingly, let (u(x, t), p(x, t)) be the solution of the time-dependent problem with the same f (x): ut − Re−1 u + u · u + p = f (x) in , ∇ · u = and u(x, 0) = u0 (x) in (7.4.3) u = on ∂2 and (p, 1) = Let w(x, t) = u(x, t) − u3 (x) Subtracting the variational formulations of (7.3.2) and (7.4.3) gives that w : [0, ) V satisfies (wt , v) + (u · u, v) − (u3 · u3 , v) + Re−1 (6 w, v) = ∀ v V , SinhVienZone.com https://fb.com/sinhvienzonevn 132 Chapter Approximating Steady Flows w(x, 0) = u(x, 0) − u3 (x) Adding and subtracting terms and setting v = w gives d ||w||2 + Re−1 ||6 w||2 = −(u · w, w) − (w · u3 , w) dt = − (w · u3 , w) ≤ N ||6 w|| ||6 u3 || ||6 w|| ≤ using the bound on u3 ≤ N Re||f ||3 ||6 w||2 om Thus, C d ||w||2 + Re−1 (1 − N Re2 ||f ||3 )||6 w||2 ≤ dt Now the small data condition implies N Re2 ||f ||3 ≤ α < or − N Re2 ||f ||3 ≥ − α > ne Thus, by the Poincaré–Friedrichs inequality Zo d ||w||2 ≤ −2 Re−1 (1 − α)||6 w||2 ≤ −δ||w||2 dt where δ = Re−1 (1 − α) CP F > Integration gives en ||w||2 (t) ≤ e−δt ||w(0)|| and as t → ||u(x, t) − u3 (x)|| Remarks on Chapter Si 7.5 nh Vi exponentially fast as t → Thus, the global uniqueness condition for the steady problem is really a condition for global, exponential, asymptotic stability of the (unique) steady solution in the timedependent problem Our presentation of the error analysis in the uniqueness case is a simplification of that in the fundamental book of Girault and Raviart [44] The last section, treating physical stability of the steady solution of the time-dependent problem, hints at the vast literature on stability of fluid motion See Galdi [41] and Straughan [94] and the references therein for more of this beautiful area Fluid flow typically is three-dimensional and time-dependent When working on simplified models (e.g., steady NSE, boundary layer equations) or settings (e.g., two dimensions), the meaning of every condition imposed needs to be understood in the context of the three-dimensional, time-dependent Navier–Stokes equations When the Reynolds number is larger, the global uniqueness condition no longer holds It is known, however, that solutions are generically nonsingular A convergence theory for finite element methods is also in place for this case as well The error analysis in the nonuniqueness case is the fundamental next step beyond the presentation of this chapter It is SinhVienZone.com https://fb.com/sinhvienzonevn 7.6 Exercises 133 Exercises C 7.6 om explained lucidly in [46], and the full technical details of two complementary approaches are given in [44] and [45] A posteriori error analysis of nonlinear problems can be performed in the small (that is, beginning with the assumption that the approximate solution is close enough to the true solution), e.g., Verfürth [100] This assumption is reasonable once priori error estimates are known Ideally, complete knowledge precedes computations; practically, for nonlinear problems it does not There are many problems for which insight into the behavior of the numerical method can be obtained from one type of estimate, and other types of estimates are unavailable until numerical calculations give more insight into the fluid velocity For nonlinear problems, analytical insight and computational experience must grow together or both will stagnate ne Exercise 84 (another useful skew-symmetrization) Define b∗3 (u, v, w) := (u · v, w) + ((∇ · u)v, w) Show that b∗3 (·, ·, ·) is skew symmetric on X and prove continuity of ∗3 b (u, v, w) N h := Zo Exercise 85 Suppose (LBBh ) holds Define sup uh ,v h ,w h V h (uh · v h , wh ) ||6 uh || ||6 v h || ||6 w h || en Look up and present the proof in [44] that N h dense in (X, Q) as h N as h 0, provided (Xh , Qh ) becomes Vi Exercise 86 Let x be the fixed point of a contractive map: x = T (x ) in X Show that its Galerkin approximation in Xh ⊂ X exists and is unique Si nh Exercise 87 (a reaction-diffusion problem) (a) Let f (x) L2 (), > 0, and consider the nonlinear BVP: find u(x) satisfying Let g : R − u = f (x) + g(u) in , u = on ∂2 (7.6.1) R be globally Lipschitz: |g(u) − g(v)| ≤ L|u − v| ∀ u, v R Let X := {u L2 () : u L2 () and u = on ∂2 } Find the variational formulation of (7.6.1) in X (b) Let T (f ) = w be the solution of: w X satisfies (6 w, v) = (f, v) ∀ v X Show (7.6.1) is equivalent to the fixed point problem: find u X with u = F(u) := T (f + g(u)) SinhVienZone.com https://fb.com/sinhvienzonevn 134 Chapter Approximating Steady Flows Reconsider (7.6.1) Show F(·) is a contraction for small enough using the equivalent inner product and norm (u, v)X := (6 u, v)L2 () and ||u||X = ||6 u|| (c) Let P h : X Xh be the orthogonal projection with respect to (·, ·)X := (6 u, v)L2 () uh = P h F(uh ) Prove an error estimate for u − uh for small (d) Reconsider om Show that the Galerkin approximation uh Xh to (7.6.1) satisfies C − u = f (x) + g(u) in , u = on ∂2, ne for small Prove an error estimate for u − uh by direct assault Mimic one of the proofs of Céa’s lemma and use the Lipschitz condition and small to hide the extra terms in the right-hand side arising from the nonlinearity Zo Exercise 88 Prove an error bound on ||p − p h || assuming (LBBh ) holds Exercise 89 (lagging the nonlinearity) Consider the problem of solving the nonlinear system for (uh , ph ) Show that the fixed point iteration en ν(6 uhn+1 , v h ) + b3 (uhn , uhn , v h ) = (f, v h ) ∀ v h V h Vi converges locally under a (very) small data condition and that each step requires the solution of a discrete Stokes problem nh Exercise 90 (a (better) fixed point iteration) Analyze convergence of the iteration ν(6 uhn+1 , v h ) + b3 (uhn , uhn+1 , v h ) = (f, v h ) ∀ v h V h Si Find the small data condition sufficient for convergence and compare it with the condition arising in the previous exercise Exercise 91 (another skew-symmetrization) Let (uh , ph ) be the finite element approximation to the steady NSEs using b∗3 (·, ·, ·) (defined in Exercise 84) instead of b3 (·, ·, ·) Prove a stability bound for the method Under a small data condition, prove convergence Exercise 92 Write down Newton’s method in V h for calculating uh V h From this, find its equivalent formulation in (X h , Qh ) Exercise 93 Consider calculating u3 by solving u(x, t) and time-stepping to t = via the following: given t > and u0 calculate (un+1 V ) via  n+1  u − un , v + Re−1 (6 un+1 , v) + b3 (un , un+1 , v) = (f, v) ∀v V t SinhVienZone.com https://fb.com/sinhvienzonevn 7.6 Exercises 135 Prove that un+1 exists for any n Show ||un − u3 || data condition Is a condition on t needed? as n → under the small Si nh Vi en Zo ne C om Exercise 94 (pressure errors) Prove an error estimate for the error in the pressure under the discrete inf-sup condition SinhVienZone.com https://fb.com/sinhvienzonevn om C ne Zo en Vi nh Si SinhVienZone.com https://fb.com/sinhvienzonevn ... http://www.siam.org/books/series/csecover.php Library of Congress Cataloging-in-Publication Data Layton, W J (William J. ) Introduction to the numerical analysis of incompressible viscous flows / William Layton p cm — (Computational... the jet and the water where the two mix and in which the jet slows down and the water is dragged along by the jet This mixing layer also spreads out as the jet flows along, and, farther from the. .. build upon those of incompressible viscous flows, such as viscoelasticity, plasmas, compressible flows, coating flows, flows of mixtures of fluids, and bubbly flows The world of fluid motion is