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Fluid Flow: Application of Numerical Methods © 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 Section 5 – Fluid Flow Objectives Module 2: Numerical Methods Page 2 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. © 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 Section 5 – Fluid Flow Understanding Numerical Methods Module 2: Numerical Methods Page 3 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) © 2011 Autodesk Finite Difference Method (FDM) 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. Finite Volume Method (FVM) www.autodesk.com/edcommunity Education Community Section 5 – Fluid Flow Domain Discretization: Part I Module 2: Numerical Methods Page 4 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. Continuous 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. Discrete Domain Values for other positions are extrapolated from grid point results. © 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 Section 5 – Fluid Flow Domain Discretization: Part II Module 2: Numerical Methods Page 5 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 Quadrilateral Mesh Triangular Mesh Nodes © 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 Section 5 – Fluid Flow Discretization of Equations- Techniques Module 2: Numerical Methods Page 6 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) © 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 Section 5 – Fluid Flow Module 2: Numerical Methods Page 7 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: df d 2 f (x) d n f (x) n f ( x x) f ( x) (x) 2 ......... n dx dx 2 dx n! from this series: df { f ( x x) f ( x)} dx x © 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 Section 5 – Fluid Flow Discretization of Equations Taylor Series: Part II Module 2: Numerical Methods Page 8 Replace the continuum with discrete points: x ....,xi 1 , xi , xi 1 ,.... u ....,ui 1 , ui , ui 1 ,.... Approximate derivatives: Central Backward Forward © 2011 Autodesk ui 1 ui 1 u 2 x x i ui ui 1 u x x i ui 1 ui u x x i 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 Section 5 – Fluid Flow Discretization of Equations, Taylor Series III Module 2: Numerical Methods Page 9 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. © 2011 Autodesk u u a 0 t x ui ui 1 u a x t i n 1 i u u u a t x a x n n 1 n n ui ui (ui ui 1 ) t u 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. n i n i www.autodesk.com/edcommunity n i 1 Education Community Section 5 – Fluid Flow Discretization of Equations Finite Difference Method: Part I Module 2: Numerical Methods Page 10 A system of flow governed by the following equation: u ui 0 x i x 1 2 4 Can be discretized using Taylor series algebraic equations: u1 (1 x) u2 0 u2 (1 x) u3 0 u3 (1 x) u4 0 © 2011 Autodesk i 2 i 3 i 4 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 Section 5 – Fluid Flow Module 2: Numerical Methods Page 11 If there is a boundary condition (B.C) of u1=0, then: 0 0 0 u1 1 1 1 1 x 0 0 u2 0 0 1 1 x 0 u 3 0 0 1 1 x u4 0 0 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. © 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 Section 5 – Fluid Flow FDM vs “FEM” and “FVM” Module 2: Numerical Methods Page 12 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. © 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 Section 5 – Fluid Flow Comparison between FEM and FVM for CFD Module 2: Numerical Methods Page 13 FEM Solves both structural mechanics and flow/thermal problems Equations are discretized over a number of points More stable compared to FVM Requires high amount of memory, limits solution of large flow domains Is capable of solving cases involving Fluid–Solid Interaction (FSI) Solves non-Newtonian fluid flow (e.g., plastic flow in molds) much better than FVM Discretizes conservative form of equations © 2011 Autodesk FVM Is used only for flow/thermal problems Governing equations are solved over discrete control volumes (CV) Less stable, convergence can sometimes require manipulation Requires less memory, a mesh with up to 5 million CVs can be solved on a PC Schemes for FVM based FSI have been devised, but are difficult to implement Can solve non-Newtonian fluids, but not as effective as FEM Recasts and discretizes integral form of equations 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 Section 5 – Fluid Flow Process of Numerical Analysis Module 2: Numerical Methods Page 14 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 Geometry © 2011 Autodesk Simplified Geometry 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. 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. www.autodesk.com/edcommunity Education Community Section 5 – Fluid Flow Flow Process Module 2: Numerical Methods Page 15 FVM Numerical Analysis Simplification FDM Initial / Boundary Conditions FEM Discretization Solving Convergence Results © 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 Section 5 – Fluid Flow Questions for establishing workflow Module 2: Numerical Methods Page 16 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) © 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 Section 5 – Fluid Flow Using CFD software Module 2: Numerical Methods Page 17 A CFD software application breaks down the analysis using the following steps: Pre-Processing Solving (number crunching) Post-Processing © 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 Section 5 – Fluid Flow Video: Application of Numerical Methods Module 2: Numerical Methods Page 18 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 © 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 Section 5 – Fluid Flow Summary Module 2: Numerical Methods Page 19 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. © 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 Section 5 – Fluid Flow Summary Module 2: Numerical Methods Page 20 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 © 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