Molchanov Alexander M. NUMERICAL METHODS FOR SOLVING THE NAVIER-STOKES EQUATIONS Moscow, 2018 1 Annotation This book is intended for beginners in the field of Computational Fluid Dynamics (CFD), studying in English. If you have never studied CFD before, if you have never worked in the area, and if you have no real idea as to what the discipline is all about, then this book is for you. Although the material has been developed from first principles wherever possible, the book will be of greatest benefit to those who are familiar with the ideas of calculus, elementary vector and matrix algebra and basic numerical methods. The main purpose in writing this book is to provide a simple, satisfying, and motivational approach toward presenting the subject to the reader who is learning about CFD for the first time. In the workplace, CFD is today a mathematically sophisticated discipline. The book is focused on the problems related to aviation and aerospace topics. However, the proposed methods can be easily applied to a wider sphere of science. 2 Preface ................................................................................................................... 5 Navier-Stokes Equations ................................................................................... 5 Application of Computational Fluid Dynamics ................................................ 7 Advantages of a Theoretical Calculation .......................................................... 8 Disadvantages of a Theoretical Calculation ...................................................... 9 1. FLUID DYNAMICS ....................................................................................... 11 1.1. Some useful formulas. .............................................................................. 11 1.2. Fundamental Equations ............................................................................ 13 1.3. Continuity Equation ................................................................................. 14 1.4. Momentum Equation ................................................................................ 16 1.5. Energy Equation ....................................................................................... 21 1.6. Equation of State ...................................................................................... 24 1.7. Vector Form of Equations ........................................................................ 26 1.8. Orthogonal Curvilinear Coordinates ........................................................ 27 1.9. General transport equation ....................................................................... 32 1.10. One-Way and Two-Way Coordinates .................................................... 38 Problems .......................................................................................................... 41 2. DISCRETIZATION METHODS ................................................................... 41 2.1.The Task .................................................................................................... 41 2.2. Taylor-Series Formulation ....................................................................... 45 2.3. Control Volume Approach ....................................................................... 47 2.4. The basic rules derived from a physical sense of control volume method ......................................................................................................................... 51 2.5. Convergence ............................................................................................. 54 2.6.Approximation .......................................................................................... 55 2.7. Stability of a discretization scheme. ....................................................... 58 2.8. Convergence as a consequence of approximation and stability .............. 59 2.9. Spectral analysis of the difference problem ............................................. 60 2.10. Explicit, Crank-Nicolson, and Fully Implicit Schemes ......................... 63 3 2.11. TriDiagonal-Matrix Algorithm .............................................................. 65 2.12. Time-development method for steady state problems ........................... 68 Problems .......................................................................................................... 69 3. NUMERICAL SOLUTION OF THE NAVIER-STOKES EQUATIONS .... 69 3.1. Desirable Numerical Properties ............................................................... 69 3.2. MacCormack Explicit Method ................................................................. 72 3.3. The finite volume approximation of Navier-Stockes equations .............. 74 3.4. Splitting of inviscid fluxes ...................................................................... 76 3.5. Jacobian matrices, eigenvalues, eigenvectors .......................................... 81 3.6. Explicit and implicit Finite Volume Schemes ....................................... 83 3.7. Viscous fluxes .......................................................................................... 85 3.8. Ways to improve the numerical method .................................................. 87 Problems .......................................................................................................... 88 4. SOLUTION OF SYSTEMS OF LINEAR EQUATIONS WITH BLOCK COEFFICIENTS ................................................................................................. 89 4.1. Finite approximating difference/volume equation ................................... 89 4.2. Gauss-Seidel iteration method ................................................................. 91 4.3. Approximate Factorization (AF) .............................................................. 92 4.4. Modified Approximate Factorization (MAF) .......................................... 93 4.5. Block TriDiagonal-Matrix Algorithm ...................................................... 95 5. BOUNDARY CONDITIONS ......................................................................... 96 5.1. Characteristics. Riemann's invariants. ..................................................... 97 5.2. Types of boundary conditions ................................................................ 105 5.2.1. INLET ............................................................................................. 107 5.2.2. OUTLET ......................................................................................... 109 5.2.3. FREE STREAM BOUNDARY ...................................................... 110 5.2.4. WALL ............................................................................................. 113 5.2.5. PLANE (LINE) of SYMMETRY ................................................... 113 5.3. Ghost cells .............................................................................................. 114 5.3.1. Boundary conditions for inviscid fluxes ......................................... 116 5.3.2. Viscous Boundary Condition .......................................................... 119 4 6. COMPUTATIONAL RESULTS .................................................................. 123 6.1. Supersonic jet at high exit static pressure ratio ...................................... 123 6.2. Supersonic oxygen jet in high-temperature ambient. ........................... 125 6.3. Cold under-expanded and over-expanded air jets ................................ 127 6.4. Highly Under-Expanded Chemically Reacting Jets .............................. 132 6.5. Afterburning of exhaust plume. ............................................................. 137 REFERENCES .................................................................................................. 139 Preface Navier-Stokes Equations The Navier-Stokes Equations are the basic governing equations for a viscous, heat conducting fluid. It is a vector equation obtained by applying Second Newton's Law of Motion to a fluid element and is also called the momentum equation. It is supplemented by the mass conservation equation, also called continuity equation and the energy equation. Usually, the term Navier-Stokes equations is used to refer to all of these equations. Navier-Stokes Equations are the governing equations of Computational Fluid Dynamics (CFD). Computational Fluid Dynamics is the simulation of fluids engineering systems using modeling (mathematical physical problem formulation) and numerical methods (discretization methods, solvers, numerical parameters, and grid generations, etc.). The process is as figure 0.1. 5 Figure 0.1. Process of Computational Fluid Dynamics Firstly, we have a fluid problem. To solve this problem, we should know the physical properties of fluid by using Fluid Mechanics. Then we can use mathematical equations to describe these physical properties. This is Navier- Stokes Equations. As the Navier-Stokes Equations are analytical, human can understand it and solve them on a piece of paper. But if we want to solve these equations with computer, we have to translate it to the discretized form. The translators are numerical discretization methods, such as Finite Difference, Finite Element, Finite Volume methods. Consequently, we also need to divide our whole problem domain into many small parts because our discretization is based on them. Then, we can write programs to solve them. The typical languages are Fortran and C. Normally the programs are run on workstations or supercomputers. In the end, we can get our simulation