Lectures on computational fluid dynamics and heat transfer with applications to human thermodynamics Prof. Dr.-Ing. habil. Nikolai Kornev Prof. Dr.-Ing. habil. Irina Cherunova Rostock 2013 Preface The present book is used for lecture courses Computational heat and mass transfer, Mathematical models of turbulence and Design of special cloth given by the authors at the University of Rostock, Germany and Don State Tech- nical University, Russia. Each of lecture courses contains about 14 lectures. The lecture course Compuational heat and mass transfer was written pro- ceeding from the idea to present the complex material as easy as possible. We considered derivation of numerical methods, particularly of the finite vol- ume method, in details up to final expressions which can be programmed. Turbulence is a big and a very complicated topic which is difficult to cover within 14 lectures. We selected the material combining the main physi- cal concepts of the turbulence with basic mathematical models necessary to solve practical engineering problems. The course Design of special cloth uses the material of two parts of this book partially. The material for the third part was gathered from research projects done by the authors of this book within some industrial projects and research works supported by different foundations. We express our gratitude to Andreas Gross, Gunnar Jacobi and Stefan Knochenhauer who carried out CFD calculations for the third part of this book. 2 Contents I Introduction into computational methods for so- lution of transport equations 17 1 Main equations of the Computational Heat and Mass Trans- fer 19 1.1 Fluid mechanics equations . 19 1.1.1 Continuity equation . 19 1.1.2 Classification of forces acting in a fluid . 20 1.1.2.1 Body forces . 20 1.1.2.2 Surface forces . 21 1.1.2.3 Properties of surface forces . 21 1.1.3 Navier Stokes Equations . 23 1.2 Heat conduction equation . 26 2 Finite difference method 29 2.1 One dimensional case . 29 2.2 Two dimensional case . 32 2.3 Time derivatives. Explicit versus implicit . 33 2.4 Exercises . 34 3 Use of boundary conditions in Finite Difference Method 35 3.1 Zero boundary conditions . 35 3.2 Extension to arbitrary non zero boundary conditions . 36 4 Stability and artificial viscosity of numerical methods 39 4.1 Artificial viscosity . 39 4.2 Stability. Courant Friedrich Levy criterion (CFL) . 41 4.3 Exercise . 43 5 Simple explicit time advance scheme for solution of the Navier Stokes Equation 45 5.1 Theory . 45 3 5.2 Mixed schemes . 46 5.3 Staggered grid . 47 δunun 5.4 Approximation of − i j ..................... 50 δxj @uxux @uxuy @ux @ux 5.4.1 Approximation of − @x − @y = −ux @x − uy @y .. 51 @uxuy @uyuy @uy @uy 5.4.2 Approximation of − @x − @y = −ux @x − uy @y .. 52 δun 5.5 Approximation of δ i ..................... 53 δxj δxj 5.6 Calculation of the r.h.s. for the Poisson equation (5.6) . 53 5.7 Solution of the Poisson equation (5.6) . 53 5.8 Update the velocity field . 53 5.9 Boundary conditions for the velocities . 54 5.10 Calculation of the vorticity . 54 6 Splitting schemes for solution of multidimensional problems 55 6.1 Splitting in spatial directions. Alternating direction implicit (ADI) approach . 55 6.2 Splitting according to physical processes. Fractional step methods . 56 6.3 Increase of the accuracy of time derivatives approximation us- ing the Lax-Wendroff scheme . 59 7 Finite Volume Method 61 7.1 Transformation of the Navier-Stokes Equations in the Finite Volume Method . 61 7.2 Sample . 62 7.2.1 Pressure and unsteady terms . 62 7.2.2 Convection term of the x-equation . 63 7.2.3 Convection term of the y-equation . 63 7.2.4 X-equation approximation . 64 7.2.5 Y-equation approximation . 65 7.3 Explicit scheme . 65 7.4 Implicit scheme . 66 7.5 Iterative procedure for implicit scheme . 67 7.6 Pressure correction method . 70 7.7 SIMPLE method . 70 7.7.1 Pressure correction equation . 71 7.7.2 Summary of the SIMPLE algorithm . 73 8 Overview of pressure correction methods 75 8.1 SIMPLE algorithm . 75 8.2 SIMPLE algorithm in OpenFOAM . 76 4 8.3 PISO algorithm . 76 8.3.1 First iteration . 76 8.3.2 Second iteration . 76 8.3.3 Correction . 77 8.3.4 Summary . 77 8.3.5 PISO algorithm in OpenFOAM . 78 8.4 SIMPLEC algorithm . 78 9 Computational grids 81 9.1 Grid types . 81 9.2 Overset or Chimera grids . 82 9.3 Morphing grids . 82 II Mathematical modelling of turbulent flows 85 10 Physics of turbulence 87 10.1 Definition of the turbulence . 87 10.2 Vortex dynamics . 87 10.2.1 Vorticity transport equation . 87 10.2.2 Vorticity and vortices . 88 10.2.3 Vortex amplification as an important mechanism of the turbulence generation . 89 10.2.4 Vortex reconnection . 92 10.2.5 Richardson poem (1922) . 93 10.2.6 Summary . 94 10.3 Experimental observations . 94 10.3.1 Laminar - turbulent transition in pipe. Experiment of Reynolds . 95 10.3.2 Laminar - turbulent transition and turbulence in jets . 97 10.3.3 Laminar - turbulent transition in wall bounded flows . 100 10.3.4 Distribution of the averaged velocity in the turbulent boundary layer . 103 11 Basic definitions of the statistical theory of turbulence 109 11.1 Reynolds averaging . 109 11.2 Isotropic and homogeneous turbulence . 110 11.3 Correlation function. Integral length . 110 11.3.1 Some relations in isotropic turbulence . 113 11.3.2 Taylor microscale λ ................... 115 11.3.3 Correlation functions in the Fourier space . 116 5 11.3.4 Spectral density of the turbulent kinetic energy . 117 11.4 Structure functions . 118 11.4.1 Probability density function . 118 11.4.2 Structure function . 118 12 Kolmogorov theory K41 123 12.1 Physical background . 123 12.2 Dissipation rate . 125 12.3 Kolmogorov hypotheses . 126 12.4 Three different scale ranges of turbulent flow . 128 12.5 Classification of methods for calculation of turbulent flows. 131 12.6 Limitation of K-41. Kolmogorov theory K-62 . 131 12.6.1 Exercises . 133 13 Reynolds Averaged Navier Stokes Equation (RANS) 137 14 Reynolds Stress Model (RSM) 143 14.1 Derivation of the RSM Equations . 143 14.1.1 Step 1 . 143 14.1.2 Step 2 . 144 14.1.3 Step 3 . 144 14.1.4 Analysis of terms . 146 15 Two equations RANS models 149 15.1 Derivation of the k-Equation . 149 15.2 Derivation of the "-Equation . 152 15.3 The k − " model . 154 15.4 Low Reynolds k − " model . 156 15.5 The k − ! model . 156 15.6 The k − ! SST model . 157 15.7 Method of wall functions . 160 15.7.1 Equilibrium wall functions . 161 15.7.2 Non- equilibrium wall functions . 163 16 Large Eddy Simulation (LES) 165 16.1 LES filtering . 165 16.1.1 Properties of filtering . 166 16.2 LES equations . 167 16.3 Smagorinsky model . 168 16.4 Model of Germano ( Dynamic Smagorinsky Model) . 170 16.5 Scale similarity models . 173 6 16.6 Mixed similarity models . 174 16.7 A-posteriori and a-priori tests . 176 17 Hybrid URANS-LES methods 179 17.1 Introduction . 179 17.2 Detached Eddy Simulation (DES) . 180 17.3 Hybrid model based on integral length as parameter switching between LES and URANS . 183 17.4 Estimations of the resolution necessary for a pure LES on the example of ship flow . 186 III CFD applications to human thermodynamics 189 18 Mathematical model of the ice protection of a human body at high temperatures of surrounding medium 191 18.1 Designations . 192 18.1.1 List of symbols . 192 18.1.2 Subscripts . 193 18.1.3 Superscripts . 193 18.2 Introduction . 193 18.3 Human body and ice protection models . 194 18.4 Mathematical model . 195 18.5 Results . 199 18.5.1 Design of the protection clothes . 199 18.5.2 Experimental proof of numerical prediction . 200 18.6 Discussion . 204 19 CFD Design of cloth for protection of divers at low temper- atures under current conditions 207 20 CFD application for design of cloth for protection from low temperatures under wind conditions. Influence of the wind on the cloth deformation and heat transfer from the body. 211 20.1 Wind tunnel measurements of pressure distribution . 211 20.2 Numerical simulations of pressure distribution and comparison with measurements . ..