1. © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Fluid Flow: Application of Numerical Methods

2. © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Objectives  Understand the application of numerical methods.  Learn about domain discretization.  Investigate discretization of equations.  Compare different numerical methods.  Understand the process of numerical analysis.  Become familiar with the use of CFD software, such as Autodesk Simulation Multiphysics. Section 5 – Fluid Flow Module 2: Numerical Methods Page 2

3. © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Understanding Numerical Methods  Numerical Methods are used when an approximate analysis can be deemed accurate enough.  Due to the nonlinear and complex nature of flow equations, exact solutions are possible for only a handful of cases.  Even when using numerical methods, simplifications have to be made in the problem being analyzed to yield an answer.  Three discretization schemes used in numerical methods are: Finite Element Method (FEM) Finite Difference Method (FDM) Finite Volume Method (FVM) Section 5 – Fluid Flow Module 2: Numerical Methods Page 3

4. © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Domain Discretization: Part I  In Numerical Fluid flow analysis, a continuous domain is replaced by a discrete domain using a grid.  In a continuous domain, a result (e.g., velocity) can be found at any point in the domain.  In a discrete domain, results are calculated only at the grid points (nodes) or at the centers of control volumes (CVs) defined by those grid points.  Values for other positions are extrapolated from grid point results.  Continuous DomainDiscrete Domain Section 5 – Fluid Flow Module 2: Numerical Methods Page 4

5. © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Domain Discretization: Part II  Similarly, when solving fluid flow in a CFD software application, the flow domain must be discretized into a number of nodes.  These elements can be quadrilateral or triangular. Cell Nodes Quadrilateral MeshTriangular Mesh Section 5 – Fluid Flow Module 2: Numerical Methods Page 5

6. © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Discretization of Equations- Techniques  Numerical discretization techniques used in commercially popular applications are:  Finite Element Method (FEM) – popular in structural mechanics  Finite Volume Method (FVM) – popular in CFD  Finite Difference Method (FDM) – popular in optimization and flow studies requiring less accuracy  The differences lie in how the equations are discretized, or converted into discrete form over a number of points. FEM is mainly popular for structural analysis (left) but can also be applied for CFD analysis (right) Section 5 – Fluid Flow Module 2: Numerical Methods Page 6

7. © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Discretization of Equations Taylor Series: Part I  Once the domain is discretized, the equation also needs to be discretized, or converted into discrete form over a number of points.  The Taylor Series is given below: from this series: Section 5 – Fluid Flow Module 2: Numerical Methods Page 7

8. © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Discretization of Equations Taylor Series: Part II  Replace the continuum with discrete points:  Approximate derivatives:  Central  Backward  Forward Section 5 – Fluid Flow Module 2: Numerical Methods Page 8

9. © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Discretization of Equations, Taylor Series III  Example of partial differential equation (PDE) with spatial and temporal derivatives:  For Space discretization  Index “i” is used with Backward differencing scheme.  For Time discretization  Index “n” is used with Forward differencing scheme.  Notice that the PDE has been reduced to an algebraic equation. Section 5 – Fluid Flow Module 2: Numerical Methods Page 9

10. © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Discretization of Equations Finite Difference Method: Part I  A system of flow governed by the following equation:  Can be discretized using Taylor series algebraic equations: 124 Section 5 – Fluid Flow Module 2: Numerical Methods Page 10

11. © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Discretization of Equations Finite Difference Method: Part II  If there is a boundary condition (B.C) of u 1 =0, then:  This matrix can be solved using a direct or iterative matrix method.  More nodes = more equations to solve.  The higher the accuracy of a Taylor Series, the more terms in the equation.  A computer can greatly help to solve the complex system of equations resulting from a large, finely meshed domain. Section 5 – Fluid Flow Module 2: Numerical Methods Page 11

12. © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community FDM vs “FEM” and “FVM”  FDM is an easy to implement, easy to understand and easy to program scheme.  FDM does not show good results for unstructured meshes.  Compared to FEM and FVM, FDM is very a crude scheme.  In-house CFD codes based on FDM do exist, but most commercial software for CFD are based on either FEM or FVM.  In the next slide, differences between FEM and FVM are explored. Section 5 – Fluid Flow Module 2: Numerical Methods Page 12

13. © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Comparison between FEM and FVM for CFD FEMFVM Solves both structural mechanics and flow/thermal problems Is used only for flow/thermal problems Equations are discretized over a number of points Governing equations are solved over discrete control volumes (CV) More stable compared to FVMLess stable, convergence can sometimes require manipulation Requires high amount of memory, limits solution of large flow domains Requires less memory, a mesh with up to 5 million CVs can be solved on a PC Is capable of solving cases involving Fluid–Solid Interaction (FSI) Schemes for FVM based FSI have been devised, but are difficult to implement Solves non-Newtonian fluid flow (e.g., plastic flow in molds) much better than FVM Can solve non-Newtonian fluids, but not as effective as FEM Discretizes conservative form of equations Recasts and discretizes integral form of equations Section 5 – Fluid Flow Module 2: Numerical Methods Page 13

14. © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Process of Numerical Analysis  To solve a problem numerically, the following steps are required: (First simplify geometry if possible) Establishing problem boundaries and flow assumptions (e.g., inlet/outlet, walls, density constant) Discretization of the domain Generation of equations for each nodal point (by using FDM, FEA) Solving those equations (using direct or iterative matrix scheme) Actual GeometrySimplified Geometry Geometry simplification often involves elimination of unnecessary curves and details that may have negligible or no influence on the flow. This helps mesh creation or domain discretization by reducing complexity. Section 5 – Fluid Flow Module 2: Numerical Methods Page 14

15. © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Flow Process Numerical Analysis Simplification FVM FDM FEM Initial / Boundary Conditions Discretization Solving Convergence Results Section 5 – Fluid Flow Module 2: Numerical Methods Page 15

16. © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Questions for establishing workflow  The first step of the analysis process is to formulate the flow problem by seeking answers to the following questions: What is the objective of the analysis? What is the easiest way to obtain that objective? What geometry should be included? What are the freestream and/or operating conditions? What dimensionality of the spatial model is required? (1D, 2D, axisymmetric, 3D) What should the flow domain look like? What temporal modelling is appropriate? (is flow steady or unsteady) What is the nature of the viscous flow? (inviscid, laminar, turbulent) How should the fluid be modelled? (compressible or incompressible) Section 5 – Fluid Flow Module 2: Numerical Methods Page 16

17. © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Using CFD software  A CFD software application breaks down the analysis using the following steps: Pre-Processing Post-Processing Solving (number crunching) Section 5 – Fluid Flow Module 2: Numerical Methods Page 17

18. © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Video: Application of Numerical Methods  The video for this module on application of numerical methods covers:  Domain discretization  Discretization of equations  The concept of numerical analysis  How computers have helped  Types of discretization and their applications Measuring the circumference of a circle Section 5 – Fluid Flow Module 2: Numerical Methods Page 18

19. © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Summary  Navier–Stokes is a complex equation and can be highly nonlinear for many flow cases.  There are relatively few cases where an exact solution to this equation can be found, and they involve a great amount of assumptions and simplification.  We replace these equations with small linear equations which are applicable at very small intervals.  This is called domain discretization and discretization of equations.  The result is a large number of simultaneous equations. Section 5 – Fluid Flow Module 2: Numerical Methods Page 19

20. © 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity Education Community Summary  To solve these equations, computers are used.  Because of advancements in computer technology, large flow domains can now be solved.  FEA, FDM and FVM are different types of discretizing schemes that have found applications in different areas.  For instance, FVM is widely popular for CFD.  FEA is used largely in structural analyses and also in complex CFD problems Section 5 – Fluid Flow Module 2: Numerical Methods Page 20