This is page i Printer: Opaque this Computational Turbulent Incompressible Flow Applied Mathematics: Body & Soul Vol 4 Johan Hoffman and Claes Johnson 24th February 2006 ii This is page iii Printer: Opaque this Contents I Overview 4 1 Main Objective 5 2 Mysteries and Secrets 7 2.1 Mysteries . 7 2.2 Secrets . 8 3 Turbulent flow and History of Aviation 13 3.1 Leonardo da Vinci, Newton and d'Alembert . 13 3.2 Cayley and Lilienthal . 14 3.3 Kutta, Zhukovsky and the Wright Brothers . 14 4 The Navier{Stokes and Euler Equations 19 4.1 The Navier{Stokes Equations . 19 4.2 What is Viscosity? . 20 4.3 The Euler Equations . 22 4.4 Friction Boundary Condition . 22 4.5 Euler Equations as Einstein's Ideal Model . 22 4.6 Euler and NS as Dynamical Systems . 23 5 Triumph and Failure of Mathematics 25 5.1 Triumph: Celestial Mechanics . 25 iv Contents 5.2 Failure: Potential Flow . 26 6 Laminar and Turbulent Flow 27 6.1 Reynolds . 27 6.2 Applications and Reynolds Numbers . 29 7 Computational Turbulence 33 7.1 Are Turbulent Flows Computable? . 33 7.2 Typical Outputs: Drag and Lift . 35 7.3 Approximate Weak Solutions: G2 . 35 7.4 G2 Error Control and Stability . 36 7.5 What about Mathematics of NS and Euler? . 36 7.6 When is a Flow Turbulent? . 37 7.7 G2 vs Physics . 37 7.8 Computability and Predictability . 38 7.9 G2 in Dolfin in FEniCS . 39 8 A First Study of Stability 41 8.1 The linearized Euler Equations . 41 8.2 Flow in a Corner or at Separation . 42 8.3 Couette Flow . 45 8.4 Reflections on Stability and Perspectives . 46 9 d'Alembert's Paradox and Bernoulli's Law 49 9.1 Introduction . 49 9.2 Bernoulli, Euler, Ideal Fluids and Potential Solutions . 50 9.3 d'Alembert's Paradox . 51 9.4 A Vector Calculus Identity . 52 9.5 Bernoulli's Law . 52 9.6 Potential Flow around a Circular Cylinder . 53 9.7 Zero Drag/Lift of Potential Flow . 53 9.8 Ideal Fluids and Vorticity . 54 9.9 d'Alembert's Computation of Zero Drag/Lift . 55 9.10 A reformulation of the momentum equation . 56 10 Prandtl's Resolution of d'Alembert's Paradox 57 10.1 Quotation from a Standard Source . 57 10.2 Quotation from Prandtl's 1904 report . 58 10.3 Discussion of Prandtl's Resolution . 59 11 New Resolution of d'Alembert's Paradox 63 11.1 Drag of a Circular Cylinder . 63 12 Turbulence and Chaos 69 12.1 Weather as Deterministic Chaos . 69 12.2 Predicting the Temperature in M˚alilla . 71 Contents v 12.3 Chaotic Dynamical System . 71 12.4 The Harmonic Oscillator as a Chaotic System . 74 12.5 Randomness and Foundations of Probability . 75 12.6 NS chaotic rather than random . 78 12.7 Observability vs Computability . 79 12.8 Lorenz system . 80 12.9 Lorenz, Newton and Free Will . 81 12.10Algorithmic Information Theory . 82 12.11Statistical Mechanics and Roulette . 83 13 A $1 Million Prize Problem 85 13.1 The Clay Institute Impossible $1 Million Prize . 85 13.2 Towards a Possible Formulation . 87 13.3 Well-Posedness According to Hadamard . 88 13.4 -Weak Solutions . 88 13.5 Existence of -Weak Solutions by Regularization . 90 13.6 Output Sensitivity and the Dual Problem . 91 13.7 Reformulation of the Prize Problem . 93 13.8 The standard approach to uniqueness . 94 14 Weak Uniqueness by Computation 97 14.1 Introduction . 97 14.2 Uniqueness of cD and cL . 98 14.3 Non-Uniqueness of D(t) . 99 14.4 Stability of the dual solution with respect to time sampling 99 14.5 Conclusion . 104 15 Existence of -Weak Solutions by G2 105 15.1 Introduction . 105 15.2 The Basic Energy Estimate for the Navier{Stokes Equations 106 15.3 Existence by G2 . 107 15.4 A Posteriori Output Error Estimate for G2 . 109 16 Stability Aspects of Turbulence in Model Problems 111 16.1 The Linearized Dual Problem . 111 16.2 Rotating Flow . 114 16.3 A Model Dual Problem for Rotating Flow . 114 16.4 A Model Dual Problem for Oscillating Reaction . 116 16.5 Model Dual Problem Summary . 116 16.6 The Dual Solution for Bluff Body Drag . 117 16.7 Duality for a Model Problem . 117 16.8 Ensemble Averages and Input Variance . 118 17 A Convection-Diffusion Model Problem 121 17.1 Introduction . 121 vi Contents 17.2 Pointwise vs Mean Value Residuals . 121 17.3 Artificial viscosity from least squares stabilization . 123 18 Reynolds Stresses In and Out 125 18.1 Introducing Reynolds Stresses . 125 18.2 Removing Reynolds Stresses . 126 19 Smagorinsky Viscosity In and Out 127 19.1 Introducing Smagorinsky Viscosity . 127 19.2 Removing Smagorinsky Viscosity . 129 20 G2 for Euler 131 20.1 Introduction . 131 20.2 Euler/G2 as a Model of the World . 133 20.3 Solution of the Euler Equations by G2 . 134 20.4 Drag of a Square Cylinder . 135 20.5 Instability of the pointwise potential solution . 137 20.6 Temperature . 137 20.7 G2 as Dissipative Weak Solutions . 147 20.8 Entropy, G2 and Physics . 151 20.9 Analysis of Instability of the Potential Solution . 152 20.10Proof that Euler/G2 is a dissipative weak solution . 153 21 Resolution of Loschmidt's Mystery 155 21.1 Irreversibility in Reversible Systems . 155 21.2 Euler/G2 as a Model of Thermodynamics . 156 21.3 Euler/G2 vs Physics . 156 21.4 The World as a Clock with Finite Precision . 157 21.5 Direction of Time . 157 21.6 Finite Precision Computation . 158 21.7 Dissipation . 160 21.8 Coupling to Particle Systems . 161 21.9 Imperfect Nature and Mathematics? . 163 21.10A New Paradigm? . 164 21.11The Prize Problem Again . 164 22 Secrets of Ball Sports 167 22.1 Introduction . 167 22.2 Dimples of a Golf Ball: Drag Crisis . 168 22.3 Topspin in Tennis: Magnus Effect . 168 22.4 Roberto Carlos: Reverse Magnus Effect? . 170 22.5 Pitch in Baseball . 171 23 Secrets of Flight 173 23.1 Generation of Lift . 173 Contents vii 23.2 Generation of Drag . 173 24 Summary so far 177 24.1 Outputs of -weak solutions . 177 24.2 Chaos and Turbulence . 178 24.3 Computational Turbulence . 180 24.4 Irreversibility . 180 II Computational Method: G2 181 25 G2 for Navier-Stokes Equations 183 25.1 Introduction . 183 25.2 Development of G2 . 184 25.3 The Incompressible Navier-Stokes Equations . 185 25.4 G2 as Eulerian cG(p)dG(q) . 186 25.5 Neumann Boundary Conditions . 187 25.6 No Slip and Slip Boundary Conditions . 187 25.7 Outflow Boundary Conditions . 187 25.8 Shock Capturing . 188 25.9 Basic Energy Estimate for cG(p)dG(q) . 188 25.10G2 as Eulerian cG(1)dG(0) . 189 25.11Eulerian cG(1)cG(1) . 190 25.12Basic Energy Estimate for cG(1)cG(1) . 190 25.13Slip with Friction Boundary Conditions . 191 26 Discrete solvers 193 26.1 Fixed point iteration using multigrid/GMRES . 193 27 G2 as Adaptive DNS/LES 195 27.1 An a posteriori error estimate . 195 27.2 Proof of the a posteriori error estimate . 197 27.3 Interpolation error estimates . 198 27.4 G2 as Adaptive DNS/LES . 199 27.5 Computation of multiple output . 200 27.6 Mesh refinement . 201 28 Implementation of G2 with FEniCS 203 29 Moving Meshes and ALE Methods 205 29.1 Introduction . 205 29.2 G2 formulation . 205 29.3 Free boundary . 207 29.4 Laplacian Mesh Smoothing . 208 29.5 Mesh Smoothing by Local Optimization . 208 29.6 Object in a box . 212 viii Contents 29.7 Sliding mesh . 213 III Flow Fundamentals 217 30 Bluff Body Flow 219 30.1 Introduction . 219 30.2 Drag and Lift . 220 30.3 An Alternative Formula for Drag and Lift . ..