Introduction to Computational Fluid Dynamics Lecture 5: Discretization, Finite Volume MethodsTransport EquationsSlide 3Slide 4Finite Volume MethodsDiscretizationOverviewThe TaskWhat is discretization?Discretizing the domainSolving the PDE’sFinite Difference Method - IntroductionFinite Difference Method - ConceptFinite Difference Method - ApplicationFinite Difference Method - Algebraic form of PDEFinite Difference Method - SummaryFinite Volume Method - IntroductionFinite Volume Method - Typical Control VolumeFinite Volume Method - ApplicationFinite Volume Method - InterpolationFinite Volume Method - Exact SolutionSlide 22Sources of Numerical Errors - FDM & FVMFalse Diffusion (1)False Diffusion (2)Finite Volume Method - SummaryFinite Element Method - IntroductionFinite Element Method - Typical ElementFinite Element Method - InterpolationFinite Element Method - ApplicationFinite Element Method - “Weak” formFinite Element Method - Wiggles (1)Finite Element Method - Wiggles (2)Finite Element Method - SummarySummaryDesigning Grids for CFDOutlineWhy is a grid needed?Element TypesGrid Types (1)Grid Types (2)Grid Types (3)Grid Types (4)Grid Design Guidelines: Quality (1)Slide 45Grid Design Guidelines: ResolutionGrid Design Guidelines: SmoothnessGrid Design Guidelines: Total Cell CountGeometryGeometry CreationSolution AdaptionGrid ImportSlide 53Ram Ramanan 01/15/19FD and FV 1ME 5337/7337Notes-2005-001Introduction to Computational Fluid DynamicsLecture 5: Discretization, Finite Volume MethodsRam Ramanan 01/15/19FD and FV 2ME 5337/7337Notes-2005-001Transport EquationsMass conservationThe integral form of mass conservation equation iswhere ρ is the density in domain Ω , v the velocity of the fluid and n the unit normal to the boundary, S. - SdSdt0)( nvRam Ramanan 01/15/19FD and FV 3ME 5337/7337Notes-2005-001Transport EquationsMomentum ConservationT = Stress tensor, n = normal to the boundaryb = body force (gravity, centrifugal, Coriolis, Lorentz etc..)- SdSdtfnvvv )( -SddSn bTf)(Ram Ramanan 01/15/19FD and FV 4ME 5337/7337Notes-2005-001Transport EquationsEnergy transportT = temperature, k = thermal conductivity, c = specific heat at constant pressure, Q = heat flux(Species transport is similar – no specific heat term)  --S SQddSnTkdScTcTdt)()( nvRam Ramanan 01/15/19FD and FV 5ME 5337/7337Notes-2005-001Finite Volume MethodsSee class slides for finite volume methodsRam Ramanan 01/15/19FD and FV 6ME 5337/7337Notes-2005-001Discretization Courtesy: Fluent, Inc.Ram Ramanan 01/15/19FD and FV 7ME 5337/7337Notes-2005-001OverviewThe TaskWhy discretization?Discretization MethodsDealing with Convection and DiffusionDiscretization Errors Courtesy: Fluent, Inc.Ram Ramanan 01/15/19FD and FV 8ME 5337/7337Notes-2005-001The Navier-Stokes equations equations governing the motion of fluid, in this instance, around a vehicle, are highly non-linear, second order partial differential equations (PDE’s)Exact solutions only exist for a small class of simple flows, e.g., laminar flow past a flat plateA numerical solution of a PDE or system of PDE’s consists of a set of numbers from which the distribution of the variable  can be obtained from the setThe variable  is determined at a finite number of locations known as grid points or cells. This number can be large or smallThe TaskCourtesy: Fluent, Inc.Ram Ramanan 01/15/19FD and FV 9ME 5337/7337Notes-2005-001Discretization is the method of approximating the differential equations by a system of algebraic equations for the variables at some set of discrete locations in space and timeThe discrete locations are grid/mesh points or cellsThe continuous information from the exact solution of PDE’s is replaced with discrete values What is discretization?Pipe discretized into cellsCourtesy: Fluent, Inc.Ram Ramanan 01/15/19FD and FV 10ME 5337/7337Notes-2005-001Discretizing the domainTransforming the physical model into a form in which the equations governing the flow physics can be solved can be referred to as discretizing the domainIllustration of the cellsContinuous domainDiscretized domainCourtesy: Fluent, Inc.Ram Ramanan 01/15/19FD and FV 11ME 5337/7337Notes-2005-001Solving the PDE’sThe are a number of methods for the solution of the governing PDE’s on the discretized domain The most important discretization methods are:Finite Difference Method (FDM)Finite Volume Method (FVM)Finite Element Method (FEM)Courtesy: Fluent, Inc.Ram Ramanan 01/15/19FD and FV 12ME 5337/7337Notes-2005-001Finite Difference Method - IntroductionOldest method for the numerical solution of PDE’sProcedure:Start with the conservation equation in differential formSolution domain is covered by gridApproximate the differential equation at each grid point by approximating the partial derivatives from the nodal values of the function giving one algebraic equation per grid pointSolve the resulting algebraic equations for the whole grid. At each grid point you solve for the unknown variable value and the value of it’s neighboring grid pointsCourtesy: Fluent, Inc.Ram Ramanan 01/15/19FD and FV 13ME 5337/7337Notes-2005-001Finite Difference Method - ConceptThe finite difference method is based on the Taylor series expansion about a point, xx xui-1ui+1ui xxuuxxuxxuuuiiiii as defined is whereH.O.T.212221 xxuuxxuxxuuuiiiii as defined is whereH.O.T.212221Subtracting the two eqns above gives 2112xOxuuxuiiiAdding the two eqns above gives 2211222xOxuuuxuiiiiCourtesy: Fluent, Inc.Ram Ramanan 01/15/19FD and FV 14ME 5337/7337Notes-2005-001Finite Difference Method - ApplicationConsider the steady 1-dimensional convection/diffusion equation:From the Taylor series expansion, get xxxuxxxxxxxxiiiiiii1112121