AML2506 Biomechanics and Flow Simulation Day Introduction to Computational Fluid Dynamics Session Speaker Dr. M. D. Deshpande M.S. Ramaiah School of Advanced Studies - Bangalore 1
Session Objectives At the end of this session the delegate would ldhave understood: The fundamentals of numerical methods as applied to CFD problems Deriving the algebraic analogue equations Solving two simple problems numerically The role of boundary conditions i The connection to the complex 3-D CFD problems M.S. Ramaiah School of Advanced Studies - Bangalore 2
Session Topics Mathematical preliminaries pertaining to numerical methods Solving the Laplace equation numerically Solving the Diffusion equation numerically The importance of the boundary conditions and their numerical treatment Appreciation of complex 3-D CFD problems of relevance to the present study. M.S. Ramaiah School of Advanced Studies - Bangalore 3
Introduction to Computational Fluid Dynamics (1) Introduction to CFD in this session involves: Fluid Mechanics, Mathematics, Numerical Analysis and Actual solving of some simple examples using a calculator. Two examples will be taken and solved in the class as a group activity: Laplace equation in 2-D in Cartesian coordinates. Diffusion equation in 1-D and time. Following gpoints will be kept in mind during this exercise: M.S. Ramaiah School of Advanced Studies - Bangalore 4
Introduction to Computational Fluid Dynamics (2) Physical meaning of these equations Linear and Nonlinear equations Steady state and Evolving solutions Importance of Boundary Conditions Dirichlet B. C. Neumann B. C. Mixed B. C. M.S. Ramaiah School of Advanced Studies - Bangalore 5
Numerical Methods PEMP-AML2506 System of Algebraic equations Direct and Iterative Solutions (Jacobi or Gauss- Seidel Iteration) Steady state and Time evolutionary solutions Derivation of the algebraic analogue equations Truncation and round-off errors Numerical Application of B. C.s Single precision and Double precision arithmetic M.S. Ramaiah School of Advanced Studies - Bangalore 6
Numerical Discretisation of a Simple Equation To see how these three discretisation techniques are used, we will consider the discretisation of the time dependent diffusion equation: U 2 U 2 t x which consists of a first derivative in the time direction t and a second derivative in the space direction x. This is a parabolic partial differential equation that can be used to model the temporal changes in the diffusion of some quantity through a medium. As an aside, there are three classifications of partial differential equations ; elliptic, parabolic and hyperbolic. Equations belonging to each of these classifications behave in different ways both physically and numerically. In particular, the direction along which any changes are transmitted is different for the three types. Depending on the flow, the governing equations of fluid motion can exhibit all three classifications. M.S. Ramaiah School of Advanced Studies - Bangalore 7
The Main Discretisation Methods PEMP-AML2506 Finite-Difference Method Discretise the governing differential equations directly; e.g. 0 u v u x y t1, j u 2x t1, j v t, j1 v 2y t, j1 M.S. Ramaiah School of Advanced Studies - Bangalore 8
Finite-Volume Method PEMP-AML2506 Discretise the governing integral equations directly; e.g. net mass outflow ua e ua w va n va s 0 M.S. Ramaiah School of Advanced Studies - Bangalore 9
10 o C Y Numerical Solution of 2D Laplace Equation on a Rectangle. Laplace eqn satisfies the Potential problem. 40 0 C 5 6 C 3 4 30 0 C 1 2 20 0 C x Exmp: Steady state Heat Conduction. Elliptic-type p.d.e. PEMP-AML2506 B.C. needed on a closed domain (see Fig). Dirichlet-type B.C. is shown (i.e. for T) Can also give Neumann-type (normal T-derivative) or mixedtype. Can solve numerically either directly or iteratively; (Jacobi or Gauss-Seidel iteration). M.S. Ramaiah School of Advanced Studies - Bangalore 10
Time t e I. C.. 5 10 20 15 10 ( o C) Table 2. Grid for solving the Diffusion Equation. M.S. Ramaiah School of Advanced Studies - Bangalore 11 X
Review Mathematical preliminaries pertaining to numerical methods have been discussed. Finite difference analogue algebraic equations are derived. The Laplace equation is solved numerically on a rectangular domain. The Diffusion equation is solved numerically. The role of the boundary conditions and their numerical treatment is discussed at length. An appreciation of complex 3-D CFD problems of relevance to the present study is attempted. M.S. Ramaiah School of Advanced Studies - Bangalore 12
Thank you M.S. Ramaiah School of Advanced Studies - Bangalore 13