Learn · Computational Fluid Dynamics
Lecture 28 of 47Module V: Solution of linear algebraic equations
Advanced iterative methods: Alternating Direction Implicit Method; Operator splitting
Prof. Sreenivas Jayanti · IIT Madras
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All the lectures
- Motivation for CFD and Introduction to the CFD approach
- Illustration of the CFD approach through a worked out example
- Eulerian approach, Conservation Equation, Derivation of Mass Conservation Equation and Statement of the momentum conservation equation
- Forces acting on a control volume; Stress tensor; Derivation of the momentum conservation equation ; Closure problem; Deformation of a fluid element in fluid flow
- Kinematics of deformation in fluid flow; Stress vs strain rate relation; Derivation of the Navier-Stokes equations
- Equations governing flow of incompressible flow; Initial and boundary conditions; Wellposedness of a fluid flow problem
- 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.
- 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…
- Finite difference approximation of pth order of accuracy for qth order derivative; cross -derivatives; Examples of high order accurate formulae for several derivatives
- One -sided high order accurate approximations; Explicit and implicit formulations for the time derivatives
- Numerical solution of the unsteady advection equation using different finite difference approximations
- Need for analysis of a discretization scheme; Concepts of consistency, stability and convergence and the equivalence theorem of Lax ; Analysis for consistency
- Statement of the stability problem; von Neumann stability analysis of the first order wave equation
- Consistency and stability analysis of the unsteady diffusion equation; Analysis for two- and three -dimensional cases; Stability of implicit schemes
- 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
- Template for the generic scalar transport equation and its extension to the solution of Navier-Stokes equa tions for a compressible flow.
- Illustration of application of the template using the MacCormack scheme for a three-dimensional compressible flow
- 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
- Artificial compressibility method and the streamfunction-vorticity method for the solution of NS equations and their limitations
- Pressur e equation method for the solution of NS equations
- Pressure-correction approach to the solution of NS equations on a staggered grid; SIMPLE and its family of methods
- Need for effici ent solution of linear algebraic equations; Classification of approaches for the solution of linear algebraic equations.
- Direct methods for linear algebraic equations; Gaussian elimination method
- Gauss-Jordan method; LU decomposition method; TDMA and Thomas algorithm
- Basic iterative methods for linear algebraic equations: Description of point -Jacobi, Gauss-Seidel and SOR methods
- Convergence analysis of basic iterative schemes; Diagonal dominance condition for convergence; Influence of source terms on the diagonal dominance condition; Rate of convergence
- Application to the Laplace equation
- Advanced iterative methods: Alternating Direction Implicit Method; Operator splitting
- Advanced iterative methods; Strongly Implicit Proc edure; Conjugate gradient method; Multigrid method
- Illustration of the Multigrid method for the Laplace equation
- Overview of the approach of numerical solution of NS equations for simple domains; Introduction to complexity arising from physics and geometry
- Derivation of the energy conservation equation
- Derivation of the species conservation equation; dealing with chemical reactions
- Turbulence; Characteri stics of turbulent flow; Dealing with fluctuations and the concept of time-averaging
- Derivation of the Reynolds -averaged Navier -Stokes equations; identification of the closure problem of turbulence; Boussinesq hypothesis and eddy viscosity
- Reynol ds stresses in turbulent flow; Time and length scales of turbulence; Energy cascade; Mixing length model for eddy viscosity
- One-equation model for turbulent flow
- Two -equation model for turbulent flow; Numerical calculation of turbulent reacting flows
- Calculation of near-wall region in turbulent flow; wall function approach; near-wall turbulence models
- 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
- Transformation of the governing equations; Illustration for the Laplace equation; Appearance and significance of cross -derivative terms; Concepts of structured and unstructured grids.
- Finite vol ume method for complicated flow domain; Illustration for the case of flow through a duct of triangular cross -section.
- Finite volume method for the general case
- Generation of a structured grid for irregular flow domain; Algebraic methods; Elliptic grid generation method
- Unstructured grid generation; Domain nodalization; Advancing front method for triangulation
- Delaunay triangulation method for unstructured grid generation
- 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.