Computational Fluid Dynamics

Lectures

Module I: Introduction

1. Motivation for CFD and Introduction to the CFD approach Video
2. Illustration of the CFD approach through a worked out example Video

Module II: Governing equations

3. Eulerian approach, Conservation Equation, Derivation of Mass Conservation Equation and Statement of the momentum conservation equation Video
4. Forces acting on a control volume; Stress tensor; Derivation of the momentum conservation equation ; Closure problem; Deformation of a fluid element in fluid flow Video
5. Kinematics of deformation in fluid flow; Stress vs strain rate relation; Derivation of the Navier-Stokes equations Video
6. Equations governing flow of incompressible flow; Initial and boundary conditions; Wellposedness of a fluid flow problem Video
7. Equations for some simple cases; Generic scalar transport equation form of the governing equations; Outline of the approach to the solution of the N -S equations. Video

Module III: Template for the numerical solution of the generic scalar transport equation

8. cut out the first 30s; Spatial discretization of a simple flow domain; Taylor’s series expansion and the basis of finite difference approximation of a derivative; Central and one-sided difference approximations; Order of accuracy of… Video
9. Finite difference approximation of pth order of accuracy for qth order derivative; cross -derivatives; Examples of high order accurate formulae for several derivatives Video
10. One -sided high order accurate approximations; Explicit and implicit formulations for the time derivatives Video
11. Numerical solution of the unsteady advection equation using different finite difference approximations Video
12. Need for analysis of a discretization scheme; Concepts of consistency, stability and convergence and the equivalence theorem of Lax ; Analysis for consistency Video
13. Statement of the stability problem; von Neumann stability analysis of the first order wave equation Video
14. Consistency and stability analysis of the unsteady diffusion equation; Analysis for two- and three -dimensional cases; Stability of implicit schemes Video
15. Interpretation of the stability condition; Stability analysis of the generic scalar equation and the concept of upwinding ; Diffusive and dissipative errors in numerical solution; Introduction to the concept of TVD schemes Video

Module IV: Solution of Navier-Stokes equations

16. Template for the generic scalar transport equation and its extension to the solution of Navier-Stokes equa tions for a compressible flow. Video
17. Illustration of application of the template using the MacCormack scheme for a three-dimensional compressible flow Video
18. Stability limits of MacCormack scheme; Limitations in extending compressible flow schemes to incompre ssible flows ; Difficulty of evaluation of pressure in incompressible flows and listing of various approaches Video
19. Artificial compressibility method and the streamfunction-vorticity method for the solution of NS equations and their limitations Video
20. Pressur e equation method for the solution of NS equations Video
21. Pressure-correction approach to the solution of NS equations on a staggered grid; SIMPLE and its family of methods Video

Module V: Solution of linear algebraic equations

22. Need for effici ent solution of linear algebraic equations; Classification of approaches for the solution of linear algebraic equations. Video
23. Direct methods for linear algebraic equations; Gaussian elimination method Video
24. Gauss-Jordan method; LU decomposition method; TDMA and Thomas algorithm Video
25. Basic iterative methods for linear algebraic equations: Description of point -Jacobi, Gauss-Seidel and SOR methods Video
26. Convergence analysis of basic iterative schemes; Diagonal dominance condition for convergence; Influence of source terms on the diagonal dominance condition; Rate of convergence Video
27. Application to the Laplace equation Video
28. Advanced iterative methods: Alternating Direction Implicit Method; Operator splitting Video
29. Advanced iterative methods; Strongly Implicit Proc edure; Conjugate gradient method; Multigrid method Video
30. Illustration of the Multigrid method for the Laplace equation Video

Module VI: Dealing with complexity of physics of flow

31. Overview of the approach of numerical solution of NS equations for simple domains; Introduction to complexity arising from physics and geometry Video
32. Derivation of the energy conservation equation Video
33. Derivation of the species conservation equation; dealing with chemical reactions Video
34. Turbulence; Characteri stics of turbulent flow; Dealing with fluctuations and the concept of time-averaging Video
35. Derivation of the Reynolds -averaged Navier -Stokes equations; identification of the closure problem of turbulence; Boussinesq hypothesis and eddy viscosity Video
36. Reynol ds stresses in turbulent flow; Time and length scales of turbulence; Energy cascade; Mixing length model for eddy viscosity Video
37. One-equation model for turbulent flow Video
38. Two -equation model for turbulent flow; Numerical calculation of turbulent reacting flows Video
39. Calculation of near-wall region in turbulent flow; wall function approach; near-wall turbulence models Video

Module VII: Dealing with complexity of geometry of the flow domain

40. Need for special methods for dealing with irregular fl ow geometry; Outline of the Body-fitted grid approach ; Coordinate transformation to a general, 3-D curvilinear system Video
41. Transformation of the governing equations; Illustration for the Laplace equation; Appearance and significance of cross -derivative terms; Concepts of structured and unstructured grids. Video
42. Finite vol ume method for complicated flow domain; Illustration for the case of flow through a duct of triangular cross -section. Video
43. Finite volume method for the general case Video
44. Generation of a structured grid for irregular flow domain; Algebraic methods; Elliptic grid generation method Video
45. Unstructured grid generation; Domain nodalization; Advancing front method for triangulation Video
46. Delaunay triangulation method for unstructured grid generation Video
47. Co -located grid approach for irregular geometries; Pressure correction equation for a co -located structured grid; Pressure correction equation for a co-located unstructured grid. Video