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Ram Ramanan 8/10/2015 FD and FV 1 ME 5337/7337 Notes-2005-001 Introduction to Computational Fluid Dynamics Lecture 5: Discretization, Finite Volume Methods

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Ram Ramanan 8/10/2015 FD and FV 2 ME 5337/7337 Notes-2005-001 Transport Equations Mass conservation The integral form of mass conservation equation is where is the density in domain , v the velocity of the fluid and n the unit normal to the boundary, S.

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Ram Ramanan 8/10/2015 FD and FV 3 ME 5337/7337 Notes-2005-001 Transport Equations Momentum Conservation T = Stress tensor, n = normal to the boundary b = body force (gravity, centrifugal, Coriolis, Lorentz etc..)

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Ram Ramanan 8/10/2015 FD and FV 4 ME 5337/7337 Notes-2005-001 Transport Equations Energy transport T = temperature, k = thermal conductivity, c = specific heat at constant pressure, Q = heat flux (Species transport is similar no specific heat term)

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Ram Ramanan 8/10/2015 FD and FV 5 ME 5337/7337 Notes-2005-001 Finite Volume Methods See class slides for finite volume methods

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Ram Ramanan 8/10/2015 FD and FV 6 ME 5337/7337 Notes-2005-001 Discretization Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 7 ME 5337/7337 Notes-2005-001 Overview The Task Why discretization? Discretization Methods Dealing with Convection and Diffusion Discretization Errors Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 8 ME 5337/7337 Notes-2005-001 u 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 (PDEs) u 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 PDEs 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 small The Task Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 9 ME 5337/7337 Notes-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 PDEs is replaced with discrete values What is discretization? Pipe discretized into cells Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 10 ME 5337/7337 Notes-2005-001 Discretizing 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 domain Illustration of the cells Continuous domain Discretized domain Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 11 ME 5337/7337 Notes-2005-001 Solving the PDEs The are a number of methods for the solution of the governing PDEs 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.

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Ram Ramanan 8/10/2015 FD and FV 12 ME 5337/7337 Notes-2005-001 Finite Difference Method - Introduction Oldest method for the numerical solution of PDEs 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 its neighboring grid points Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 13 ME 5337/7337 Notes-2005-001 Finite Difference Method - Concept The finite difference method is based on the Taylor series expansion about a point, x xx xx u i-1 u i+1 uiui Subtracting the two eqns above givesAdding the two eqns above gives Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 14 ME 5337/7337 Notes-2005-001 Finite Difference Method - Application Consider the steady 1-dimensional convection/diffusion equation: From the Taylor series expansion, get Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 15 ME 5337/7337 Notes-2005-001 Finite Difference Method - Algebraic form of PDE Substitute the discrete forms of the differentials to get: Algebraic form of PDE Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 16 ME 5337/7337 Notes-2005-001 Finite Difference Method - Summary Discretized the one-dimensional convection/diffusion equation The derivatives were determined from a Taylor series expansion Advantages of FDM: simple and effective on structured grids Disadvantages of FDM: conservation is not enforced unless with special treatment, restricted to simple geometries Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 17 ME 5337/7337 Notes-2005-001 Finite Volume Method - Introduction Using Finite Volume Method, the solution domain is subdivided into a finite number of small control volumes by a grid The grid defines to boundaries of the control volumes while the computational node lies at the center of the control volume The advantage of FVM is that the integral conservation is satisfied exactly over the control volume Control volume Computational node Boundary node Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 18 ME 5337/7337 Notes-2005-001 Finite Volume Method - Typical Control Volume The net flux through the control volume boundary is the sum of integrals over the four control volume faces (six in 3D). The control volumes do not overlap The value of the integrand is not available at the control volume faces and is determined by interpolation P E W N S SW SE NE NW j i n e s w xx yy xwxw xexe Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 19 ME 5337/7337 Notes-2005-001 Finite Volume Method - Application Consider the one-dimensional convection/diffusion equation The finite volume method (FVM) uses the integral form of the conservation equations over the control volume: Integrating the above equation in the x-direction across faces e and w of the control volume and leaving out the source term gives The values of at the faces e and w are needed P E W N S j i n e s w Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 20 ME 5337/7337 Notes-2005-001 Finite Volume Method - Interpolation Using a piecewise-linear interpolation between control volume centers gives where } n linear interpolation between nodes n face is midway between nodes n equivalent to Central Difference Scheme (CDS) n Under assumption of continuity, discrete form of PDE from FVM is identical to FDM Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 21 ME 5337/7337 Notes-2005-001 Finite Volume Method - Exact Solution Exact solution with boundary conditions: = o at x = 0, = L at x = L Peclet number, Pe, is the ratio of the strengths of convection and diffusion When the Peclet number is high (positive or negative), the profile is highly non-linear The Central-Difference Scheme relies on a linear interpolation which will fail to capture the gradient changes in the variable LL oo L -Pe >> 1 Pe = -1 Pe = 1 Pe >> 1 Pe = 0 Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 22 ME 5337/7337 Notes-2005-001 Finite Volume Method - Interpolation The piecewise-linear or CDS interpolation may give rise to numerical errors (oscillatory or checkerboard solutions). CDS was used only as an example of discretization and is inappropriate for most convection/diffusion flows A large number of interpolation techniques in FLUENT software that are improvements on the CDS. Some of these, in increasing level of accuracy, are: First-Order Upwind Scheme Power Law Scheme Second-Order Upwind Scheme Higher Order Blended Second-Order Upwind/Central Difference Quadratic Upwind Interpolation (QUICK) Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 23 ME 5337/7337 Notes-2005-001 Sources of Numerical Errors - FDM & FVM Discretization Errors from inexact interpolation of nonlinear profile (FVM) Truncation Errors due to exclusion of Higher Order Terms (FDM) Domain discretization not well resolved to capture flow physics Artificial or False Diffusion due to interpolation method and grid Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 24 ME 5337/7337 Notes-2005-001 False Diffusion (1) False diffusion is numerically introduced diffusion and arises in convection dominant flows, i.e., high Pe number flows Consider the problem below: Hot fluid Cold fluid T = 100C T = 0C u If there is no false diffusion, the temperature along the diagonal will be 100C u False diffusion will occur due to the oblique flow direction and non-zero gradient of temperature in the direction normal to the flow u Grid refinement coupled with a higher-order interpolation scheme will minimize the false diffusion Diffusion set to zero k =0 Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 25 ME 5337/7337 Notes-2005-001 False Diffusion (2) 8 x 8 64 x 64 First-order Upwind Second-order Upwind Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 26 ME 5337/7337 Notes-2005-001 Finite Volume Method - Summary The FVM uses the integral conservation equation applied to control volumes which subdivide the solution domain, and to the entire solution domain The variable values at the faces of the control volume are determined by interpolation. False diffusion can arise depending on the choice of interpolation scheme The grid must be refined to reduce smearing of the solution as shown in the last example Advantages of FVM: Integral conservation is exactly satisfied, Not limited to grid type (structured or unstructured, cartesian or body- fitted) Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 27 ME 5337/7337 Notes-2005-001 Finite Element Method - Introduction Using Finite Element Method, the solution domain is subdivided into a finite number of small elements by a grid The grid defines to boundaries of the elements and location of nodes for higher-order elements, there can be mid-side nodes also FEM uses multi-dimensional shape functions which afford geometric flexibility and limit false diffusion Finite element Computational node Mid-side node Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 28 ME 5337/7337 Notes-2005-001 Finite Element Method - Typical Element 9-noded quadrilateral Within each element, the velocity and pressure fields are approximated by: where u i, v i, p i are the nodal point unknowns and i and i are interpolation functions Quadratic approximation for velocity, linear approximation for pressure required to avoid spurious pressure modes nodes with u, v, p nodes with u, v Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 29 ME 5337/7337 Notes-2005-001 Finite Element Method - Interpolation The solution on an element is represented as: node ii i+1 where N are the basis functions. We choose basis functions that are 1 at one node of the element and 0 at all other the nodes. Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 30 ME 5337/7337 Notes-2005-001 Finite Element Method - Application Recall the one-dimensional convection/diffusion equation Most often, the finite element method (FEM) uses the Method of Weighted Residuals to discretize the equation Multiply governing equation by weight function W i and integrate over the element How do we choose the W i ? For Galerkin FEM, replace W i by N i, the shape or basis functions Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 31 ME 5337/7337 Notes-2005-001 Finite Element Method - Weak form Use integration by parts to obtain the weak formulation involves first derivatives rather than second derivatives We can now substitute the interpolation function for and evaluate the required integrals to produce the discrete equation: Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 32 ME 5337/7337 Notes-2005-001 Finite Element Method - Wiggles (1) Wiggles occur in FEM when linear weighting functions are used Typical cures are: Petrov-Galerkin method: where the weighting function is different from the shape function. The weighting function is asymmetric, being skewed in the direction of the upwind element ii+1i-1 u WiWi NiNi Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 33 ME 5337/7337 Notes-2005-001 Finite Element Method - Wiggles (2) Artificial tensor viscosity: the viscosity in the streamline direction is augmented this is equivalent to skewing the weighting of the convection terms towards the upstream direction flow U C D shapes of Galerkin and Streamline Upwind weighting functions for nodal point C Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 34 ME 5337/7337 Notes-2005-001 Finite Element Method - Summary FEM solves the weak form of the governing equations weak form requires continuity of lower order operators only very similar to using the divergence theorem in FVM The technique is conservative in a weighted sense The weight functions an easily be made multi-dimensional this limits false diffusion Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 35 ME 5337/7337 Notes-2005-001 Summary The concept of discretization was introduced. The three main methods of discretization, FDM, FVM, and FEM were detailed. The one-dimensional convection/diffusion was used to illustrate the different methods of discretizing the partial differential equations into algebraic equations. The process of discretization was shown to introduce errors (though minimal in some cases) through truncation of higher order terms (FDM), approximation of the integrals (FVM). The correct interpolation of the convective fluxes across the cell faces minimizes errors in both discretization methods. Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 36 ME 5337/7337 Notes-2005-001 Designing Grids for CFD Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 37 ME 5337/7337 Notes-2005-001 Outline Why is a grid needed? Element types Grid types Grid design guidelines Geometry Solution adaption Grid import Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 38 ME 5337/7337 Notes-2005-001 Why is a grid needed? The grid: designates the cells or elements on which the flow is solved is a discrete representation of the geometry of the problem has cells grouped into boundary zones where b.c.s are applied The grid has a significant impact on: rate of convergence (or even lack of convergence) solution accuracy CPU time required Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 39 ME 5337/7337 Notes-2005-001 Element Types Many different cell/element and grid types are available choice depends on the problem and the solver capabilities Cell or element types 2D: 3D: triangle (tri) 2D prism (quadrilateral or quad) tetrahedron (tet) pyramid prism with quadrilateral base (hexahedron or hex) prism with triangular base (wedge) Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 40 ME 5337/7337 Notes-2005-001 Grid Types (1) Single-block, structured grid i,j,k indexing to locate neighboring cells grid lines must pass all through domain Obviously cant be used for very complicated geometries Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 41 ME 5337/7337 Notes-2005-001 Grid Types (2) Multi-block, structured grid uses i,j,k indexing with each block of mesh grid can be made up of (somewhat) arbitrarily-connected blocks More flexible than single block, but still limited cross-section: Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 42 ME 5337/7337 Notes-2005-001 Grid Types (3) Unstructured grid tri or tet cells arranged in arbitrary fashion no grid index, no constraints on cell layout There is some memory/CPU overhead for unstructured referencing CFD stands for Cow Fluid Dynamics! Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 43 ME 5337/7337 Notes-2005-001 Grid Types (4) Hybrid grid use the most appropriate cell type in any combination triangles and quadrilaterals in 2D tetrahedra, prisms and pyramids in 3D can be non-conformal: grids lines dont need to match at block boundaries triangular surface mesh on car body is quick and easy to create prism layer efficiently resolves boundary layer tetrahedral volume mesh is generated automatically non-conformal interface Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 44 ME 5337/7337 Notes-2005-001 Grid Design Guidelines: Quality (1) Quality: cells/elements are not highly skewed Two methods for determining skewness: 1. Based on the equilateral volume: Skewness = Applies only to triangles and tetrahedra optimal (equilateral) cell actual cell circumcircle Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 45 ME 5337/7337 Notes-2005-001 Grid Design Guidelines: Quality (1) 2. Based on the deviation from a normalized equilateral angle: Skewness (for a quad) = Applies to all cell and face shapes High skewness values inaccurate solutions & slow convergence Keep maximum skewness of volume mesh < 0.95 Possible classification based on skewness: Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 46 ME 5337/7337 Notes-2005-001 Grid Design Guidelines: Resolution Pertinent flow features should be adequately resolved Cell aspect ratio (width/height) should be near 1 where flow is multi- dimensional Quad/hex cells can be stretched where flow is fully-developed and essentially one-dimensional inadequate better flow OK! Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 47 ME 5337/7337 Notes-2005-001 Grid Design Guidelines: Smoothness Change in cell/element size should be gradual (smooth) Ideally, the maximum change in grid spacing should be 20%: smooth change in cell size sudden change in cell size AVOID! xixi x i+1 Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 48 ME 5337/7337 Notes-2005-001 Grid Design Guidelines: Total Cell Count More cells can give higher accuracy downside is increased memory and CPU time To keep cell count down: use a non-uniform grid to cluster cells only where theyre needed use solution adaption to further refine only selected areas Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 49 ME 5337/7337 Notes-2005-001 Geometry The starting point for all problems is a geometry The geometry describes the shape of the problem to be analyzed Can consist of volumes, faces (surfaces), edges (curves) and vertices (points). Geometry can be very simple... or more complex geometry for a cube Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 50 ME 5337/7337 Notes-2005-001 Geometry Creation A good preprocessor provides tools for creating and modifying geometry. Geometry can also be imported from other CAD programs. Various file types exist: IGES ACIS STL STEP DXF various proprietary (Universal files, etc.) Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 51 ME 5337/7337 Notes-2005-001 Solution Adaption Q: how do you ensure adequate grid resolution, when you dont necessarily know the flow features? A: solution-based grid adaption! The grid can be refined or coarsened by the solver based on the developing flow: solution values gradients along a boundary inside a certain region Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 52 ME 5337/7337 Notes-2005-001 Grid Import Grids can also be created by many CAD programs (I-DEAS, Patran, ANSYS, etc.). These can be imported into the solver. Be sure to check grid quality! A grid acceptable for stress analysis may not be good enough for CFD. Repair/improve if necessary. Courtesy: Fluent, Inc.

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Ram Ramanan 8/10/2015 FD and FV 53 ME 5337/7337 Notes-2005-001 Summary Design and construction of a quality grid is crucial to the success of the CFD analysis. Appropriate choice of grid type depends on: geometric complexity flow field cell/element types supported by solver Hybrid meshing offers the greatest flexibility. Take advantage of solution adaption. Courtesy: Fluent, Inc.