Unformatted text preview:

Overview of CFD Solution MethodologiesOutlineIngredientsSurvey of MethodsFinite Difference Method (FDM)Finite Volume Method (FVM)Finite Element Method (FEM)Finite Difference: Basic MethodologyFinite Difference: Pro’s and Con’sFinite Volume: Basic MethodologyFinite Volume: Pro’s and Con’sFinite Element: Basic MethodologyFinite Element: Pro’s and Con’sSummary© Fluent Inc. 01/14/19J1Fluids ReviewTRN-1998-004Overview of CFD Solution Methodologies© Fluent Inc. 01/14/19J2Fluids ReviewTRN-1998-004OutlineIngredientsOverview of Solution MethodologiesFinite DifferenceFinite Volume (i.e., Control Volume)Finite Element Strengths and Weaknesses© Fluent Inc. 01/14/19J3Fluids ReviewTRN-1998-004IngredientsCFD solution methodequation discretizationdomain discretization (grid)solution of algebraic equationstreatment of convection and source terms© Fluent Inc. 01/14/19J4Fluids ReviewTRN-1998-004Survey of MethodsMany CFD techniques existThe most common are:Finite DifferenceFinite Volume or Control VolumeFinite ElementThe focus of this talk is to introduce these threeThere are certainly many other approaches, including:control volume/finite elementspectralspectral elementboundary elementlattice gasand more!© Fluent Inc. 01/14/19J5Fluids ReviewTRN-1998-004Finite Difference Method (FDM)Historically, the oldest of the threeTechniques published as early as 1910 by L. F. RichardsonSeminal paper by Courant, Fredrichson and Lewy (1928) derived stability criteria for explicit time steppingFirst ever numerical solution: flow over a circular cylinder by Thom (1933)Scientific American article by Harlow and Fromm (1965) clearly and publicly the idea of “computer experiments” for the first time — CFD is born!!© Fluent Inc. 01/14/19J6Fluids ReviewTRN-1998-004Finite Volume Method (FVM)Has its roots in the Finite Difference MethodFirst well-documented use was by Evans and Harlow (1957) at Los Alamos and Gentry, Martin and Daley (1966)Was attractive because:while variables may not be continuously differentiable across shocks and other discontinuities,mass, momentum and energy would always be conservedLate 70’s, early 80’s saw development of body-fitted gridsBy early 90’s, unstructured grid methods had appearedflowcompressible flow over a wedgecontours of density© Fluent Inc. 01/14/19J7Fluids ReviewTRN-1998-004Finite Element Method (FEM)Earliest use was by Courant (1943) for solving St. Venant torsion problemClough (1960) gave the method its nameMethod was refined greatly in the 60’s and 70’s, mostly for analyzing structural mechanics problemFEM analysis of fluid flow was developed in the mid- to late 70’s coextrusionmetal insertcontours of velocity magnitude© Fluent Inc. 01/14/19J8Fluids ReviewTRN-1998-004Finite Difference: Basic MethodologyThe domain is discretized into a series of grid pointsa “structured” (ijk) mesh is requiredThe governing equations are discretized (converted to algebraic form)first and second derivatives are approximated by truncated Taylor series expansionsThe resulting set of linear algebraic equations is solved iteratively or simultaneouslyijij© Fluent Inc. 01/14/19J9Fluids ReviewTRN-1998-004Finite Difference: Pro’s and Con’sAdvantages:simple derivation, implementationDisadvantages:relatively simple gridsmass, momentum, energy not conserved on coarse grids© Fluent Inc. 01/14/19J10Fluids ReviewTRN-1998-004Finite Volume: Basic MethodologyDivide the domain into control volumes (c.v.’s)Integrate the differential equation over the control volume and apply the divergence theorem.To evaluate derivative terms, values at the control volume faces are needed: have to make an assumption about how the value varies.Result is a set of linear algebraic equations; one for each c.v.Solve iteratively or simultaneously.•••••••••••••••••••••© Fluent Inc. 01/14/19J11Fluids ReviewTRN-1998-004Finite Volume: Pro’s and Con’sAdvantages:basic FV control volume balance does not limit cell shapemass, momentum, energy conserved even on coarse gridsefficient, iterative solvers well developedDisadvantages:Simplest implementation uses 1-D assumptions during differencing of convection/diffusion terms — leads to false diffusion (multi-dimensional approaches now available)© Fluent Inc. 01/14/19J12Fluids ReviewTRN-1998-004Finite Element: Basic MethodologyDomain is divided into elements.Most FEM methods use some variant of the Method of Weighted Residuals.we seek an approximate solution to the governing equationstherefore, we seek to minimize the residual (or error) in some weighted sense over the domainChoose a shape function which is used to interpolate values between node points.Multiply the governing equations by a weight function and integrate to obtain the “weak” formulation (contains first derivatives, not second).Solve algebraic equations iteratively or simultaneously.© Fluent Inc. 01/14/19J13Fluids ReviewTRN-1998-004Finite Element: Pro’s and Con’sAdvantages:multi-dimensional shape functions give geometric flexibility and limit false diffusion“natural” imposition of flux boundary conditionsmass is conserved locally and globally, momentum is conserved globally and nearly locallyfor same solution accuracy, generally requires fewest grid pointsDisadvantages:traditionally used simultaneously solution of all equations (memory and CPU intensive) — iterative solves now availableperhaps more complicated to develop, debug and maintain© Fluent Inc. 01/14/19J14Fluids ReviewTRN-1998-004SummaryFinite volume method and finite element method are now the most popular and successful methodsEach has its own advantages:FVM seems to enjoy an advantage in memory use and speed for very large problems, higher speed flows and source term dominated flows (like combustion)FEM solutions can be very accurate using generally smaller gridsBeing a node based scheme with natural imposition of traction boundary conditions, FEM seems better suited for deforming mesh free surface


View Full Document

SMU ME 7337 - Overview of CFD Solution Methodologies

Download Overview of CFD Solution Methodologies
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Overview of CFD Solution Methodologies and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Overview of CFD Solution Methodologies 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?