Introduction to CFD BasicsRajesh BhaskaranLance CollinsThis is a quick-and-dirty introduction to the basic concepts underlying CFD. The con-cepts are illustrated by applying them to simple 1D model problems. We’ll invoke theseconcepts while performing “case studies” in FLUENT. Happily for us, these model-problemconcepts extend to the more general situations in the case studies in most instances. Sincewe’ll keep returning to these concepts while performing the FLUENT case studies, it’s worthyour time to understand and digest these concepts.We discuss the following topics briefly. These topics are the minimum necessary toperform and validate the FLUENT calculations to come later.1. The Need for CFD2. Applications of CFD3. The Strategy of CFD4. Discretization Using the Finite-Difference Method5. Discretization Using The Finite-Volume Method6. Assembly of Discrete System and Application of Boundary Conditions7. Solution of Discrete System8. Grid Convergence9. Dealing with Nonlinearity10. Direct and Iterative Solvers11. Iterative Convergence12. Numerical Stability13. Turbulence modeling1The Need for CFDApplying the fundamental laws of mechanics to a fluid gives the governing equations for afluid. The conservation of mass equation is∂ρ∂t+ ∇ · (ρ~V ) = 0and the conservation of momentum equation isρ∂~V∂t+ ρ(~V · ∇)~V = −∇p + ρ~g + ∇ · τijThese equations along with the conservation of energy equation form a set of coupled, non-linear partial differential equations. It is not possible to solve these equations analyticallyfor most engineering problems.However, it is possible to obtain approximate computer-based solutions to the governingequations for a variety of engineering problems. This is the subject matter of ComputationalFluid Dynamics (CFD).Applications of CFDCFD is useful in a wide variety of applications and here we note a few to give you an idea ofits use in industry. The simulations shown below have been performed using the FLUENTsoftware.CFD can be used to simulate the flow over a vehicle. For instance, it can be used to studythe interaction of propellers or rotors with the aircraft fuselage The following figure showsthe prediction of the pressure field induced by the interaction of the rotor with a helicopterfuselage in forward flight. Rotors and prop ellers can be represented with models of varyingcomplexity.The temperature distribution obtained from a CFD analysis of a mixing manifold is shownbelow. This mixing manifold is part of the passenger cabin ventilation system on the Boeing767. The CFD analysis showed the effectiveness of a simpler manifold design without theneed for field testing.2Bio-medical engineering is a rapidly growing field and uses CFD to study the circulatory andrespiratory systems. The following figure shows pressure contours and a cutaway view thatreveals velocity vectors in a blood pump that assumes the role of heart in open-heart surgery.CFD is attractive to industry since it is more cost-effective than physical testing. However,one must note that complex flow simulations are challenging and error-prone and it takes alot of engineering expertise to obtain validated solutions.The Strategy of CFDBroadly, the strategy of CFD is to replace the continuous problem domain with a discretedomain using a grid. In the continuous domain, each flow variable is defined at every pointin the domain. For instance, the pressure p in the continuous 1D domain shown in the figurebelow would be given asp = p(x), 0 < x < 1In the discrete domain, each flow variable is defined only at the grid points. So, in thediscrete domain shown below, the pressure would be defined only at the N grid points.pi= p(xi), i = 1, 2, . . . , NContinuous Domain Discrete Domain x=0 x=1 x1 xi xN 0 ≤ x ≤ 1 x = x1, x2, …,xN Grid point Coupled PDEs + boundary conditions in continuous variablesCoupled algebraic eqs. indiscrete variables In a CFD solution, one would directly solve for the relevant flow variables only at the gridpoints. The values at other locations are determined by interpolating the values at the gridpoints.The governing partial differential equations and boundary conditions are defined in termsof the continuous variables p,~V etc. One can approximate these in the discrete domain interms of the discrete variables pi,~Vietc. The discrete system is a large set of coupled,algebraic equations in the discrete variables. Setting up the discrete system and solving it(which is a matrix inversion problem) involves a very large number of repetitive calculations,a task we humans palm over to the digital computer.3This idea can be extended to any general problem domain. The following figure showsthe grid used for solving the flow over an airfoil. We’ll take a closer look at this airfoil gridsoon while discussing the finite-volume method.Discretization Using the Finite-Difference MethodTo keep the details simple, we will illustrate the fundamental ideas underlying CFD byapplying them to the following simple 1D equation:dudx+ um= 0; 0 ≤ x ≤ 1; u(0) = 1 (1)We’ll first consider the case where m = 1 when the equation is linear. We’ll later considerthe m = 2 case when the equation is nonlinear.We’ll derive a discrete representation of the above equation with m = 1 on the followinggrid:x1=0 x2=1/3 x3=2/3 x4=1 ∆x=1/3 This grid has four equally-spaced grid points with ∆x being the spacing between successivepoints. Since the governing equation is valid at any grid point, we haveÃdudx!i+ ui= 0 (2)where the subscript i represents the value at grid point xi. In order to get an expression for(du/dx)iin terms of u at the grid points, we expand ui−1in a Taylor’s series:ui−1= ui− ∆xÃdudx!i+ O(∆x2)Rearranging givesÃdudx!i=ui− ui−1∆x+ O(∆x) (3)4The error in (du/dx )idue to the neglected terms in the Taylor’s series is called the truncationerror. Since the truncation error above is O(∆x), this discrete representation is termed first-order accurate.Using (3) in (2) and excluding higher-order terms in the Taylor’s series, we get thefollowing discrete equation:ui− ui−1∆x+ ui= 0 (4)Note that we have gone from a differential equation to an algebraic equation!This method of deriving the discrete equation using Taylor’s series expansions is calledthe finite-difference method. However, most commercial CFD codes use the finite-volume orfinite-element methods which are better suited for modeling flow past complex