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Computational Fluid Dynamics (Video)
Modules / Lectures
Module I: Introduction
Motivation for CFD and Introduction to the CFD approach
Illustration of the CFD approach through a worked out example
Module II: Governing equations
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.
Module III: Template for the numerical solution of the generic scalar transport equation
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 ap
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
Module IV: Solution of Navier-Stokes equations
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
Module V: Solution of linear algebraic equations
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
Module VI: Dealing with complexity of physics of flow
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
Module VII: Dealing with complexity of geometry of the flow domain
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.
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