1. © Fluent Inc. 9/5/2015L1 Fluids Review TRN-98-004 Solution Methods

2. © Fluent Inc. 9/5/2015L2 Fluids Review TRN-98-004 Overview u Properties of Numerical Solution Methods u FVM and FEM solution methods u Characteristics of solution algorithms u Equations solvers u Underrelaxation u Convergence

3. © Fluent Inc. 9/5/2015L3 Fluids Review TRN-98-004 Numerical Solution Methods (1) u The important components of a numerical solution method are: 1.Mathematical model of flow n e.g. equations of motion- unsteady and steady, compressible and incompressible, 2D and 3D, turbulence, etc. 2. Discretization Method n Approximation of the differential equations by a system of algebraic equations s Finite Difference Method (FDM) s Finite Volume Method (FVM) s Finite Element Method (FEM) 3. Coordinate system n cartesian or cylindrical, curvilinear orthogonal and non-orthogonal coordinate systems

4. © Fluent Inc. 9/5/2015L4 Fluids Review TRN-98-004 Numerical Solution Methods (2) 4. Numerical Grid n The solution domain is subdivided by the grid. The algebraic conservation equations for the variables are computed on a finite number of control volumes or elements in the domain. n Types of Grids s Structured grids s Multi-block-structured grids s Unstructured grids 5. Finite Approximations n Discretizing the solution domain gives rise to errors from the approximation of the continuous differential functions n FDM - approximate the derivatives through the Taylor series expansion n FVM - approximate the surface and volume integrals n FEM - choose weighting functions Unstructured surface grid for vehicle aerodynamic analysis.

5. © Fluent Inc. 9/5/2015L5 Fluids Review TRN-98-004 Numerical Solution Methods (3) 6. Solution Criteria and Convergence Criteria n This is the topic of this lecture n Methods of solving the system of algebraic equations n The nonlinear nature of the governing equations requires an iterative solution method. Convergence criteria determine when to terminate the iterative process. Accuracy and efficiency are considered.

6. © Fluent Inc. 9/5/2015L6 Fluids Review TRN-98-004 Properties of Solution Methods l Consistency l Stability l Convergence l Conservation l Boundedness l Realizability l Accuracy

7. © Fluent Inc. 9/5/2015L7 Fluids Review TRN-98-004 FVM - Solution Algorithms u The discretized form of the governing conservation equations can be written as: n where nb denotes the cell neighbors of cell P u In a 2D structured grid, the face P has four neighbors (E,W,N,S). In a 3D grid, a cell has six neighbors. u In an unstructured grid, the number of neighbors depends on the cell shape and mesh topology. u The above algebraic equation is written for each transport variable, that is, velocity, temperature, species concentration and turbulence quantities. n P E W N S j i e s w

8. © Fluent Inc. 9/5/2015L8 Fluids Review TRN-98-004 FVM - Solution Algorithms u The solution of the Navier-Stokes equations is complicated by the lack of an independent equation for pressure. Pressure is linked to all three momentum equations u The pressure-velocity coupling algorithm SIMPLE (Semi-Implicit Pressure Linked Equations), and it’s variants, are used. u Concept: l the momentum equations are used to compute velocity l a pressure equation is derived from the continuity equation l a discrete pressure correction equation is derived from the discrete forms of the pressure and momentum equations l the pressure correction equation is updated with pressure and a mass flux balance through a mass correction

9. © Fluent Inc. 9/5/2015L9 Fluids Review TRN-98-004 Finite Volume Solution Methods u The Finite Volume Solution method can either use a “segregated” or a “coupled” solution procedure. u The solution procedure of each method is the same.

10. © Fluent Inc. 9/5/2015L10 Fluids Review TRN-98-004 Segregated Solution Procedure Update properties. Solve momentum equations (u, v, w velocity). Solve pressure-correction (continuity) equation. Update pressure, face mass flow rate. Solve energy, species, turbulence, and other scalar equations. Converged? Stop No Yes

11. © Fluent Inc. 9/5/2015L11 Fluids Review TRN-98-004 Coupled Solution Procedure Solve continuity, momentum, energy, and species equations simultaneously. Converged? Stop NoYes Solve turbulence and other scalar equations. Update properties.

12. © Fluent Inc. 9/5/2015L12 Fluids Review TRN-98-004 Unsteady Solution Procedure u Same procedure for segregated and coupled solvers: Execute segregated or coupled procedure, iterating to convergence Take a time step Requested time steps completed? NoYes Stop Update solution values with converged values at current time

13. © Fluent Inc. 9/5/2015L13 Fluids Review TRN-98-004 FVM - Linear Equation Solvers  Consider the system of algebraic equations for variable  l The above system of equations is arranged in a matrix and solved iteratively. u For a structured grid, the coefficient matrix is banded. Special line-by-line iterative techniques such as the Line Gauss-Seidel (LGS) method may be used. l LGS method involves solving the equations in a “line” simultaneously. l The equations are set-up in a tri-diagonal matrix solved via Gaussian elimination u For an unstructured grid, no line structure exists. Point-iterative methods are used, e.g., the Point Gauss-Seidel (PGS) technique. u LGS/PGS locally reduce errors but can miss long-wavelength errors. Multigrid acceleration will speed up the LGS/PGS convergence.

14. © Fluent Inc. 9/5/2015L14 Fluids Review TRN-98-004 FVM - Line Gauss-Seidel (LGS) Method u The LGS method is used on structured grids and involves the following steps: l simultaneously solve the equations in the sweep direction l march to next row or column Line to be solved Values from previous sweep Values from previous iteration Flow Marching direction sweeping direction

15. © Fluent Inc. 9/5/2015L15 Fluids Review TRN-98-004 FVM - The Multigrid Solver u The LGS and PGS solvers both transmit the influence of near-neighbors effectively and are less effective at transmitting the influence of far away grid points and boundaries, thereby, slowing convergence. u “Multigrid” solver accelerates convergence for: l Large number of cells l Large cell aspect ratios  x/  y > 20 l Large differences in thermal conductivity l Such as in conjugate heat transfer u General concept of multigrid is the same for structured and unstructured grids, although the implementation is different.

16. © Fluent Inc. 9/5/2015L16 Fluids Review TRN-98-004 The Multigrid Concept (1) u Multigrid solver uses a sequence of grids going from fine to coarse. u Influence of boundaries and far-away points more easily transmitted to interior on coarse meshes than on fine meshes. l In coarse meshes, grid points are closer together in the computational space and have fewer computational cells between any two spatial locations. u Fine meshes give more accurate solutions.

17. © Fluent Inc. 9/5/2015L17 Fluids Review TRN-98-004 The Multigrid Concept (2) u The solutions on the coarser meshes is used as a starting point for solutions on the finer meshes. l Coarse-mesh solution contains influence of boundaries and far neighbors. l These effects felt more easily on coarse mesh. l Accelerates convergence on fine mesh. u Final solution obtained for original (fine) mesh. l Coarse mesh calculations: l only accelerates convergence l do not change final answer fine mesh corrections summed equations (or volume-averaged solution) coarse mesh

18. © Fluent Inc. 9/5/2015L18 Fluids Review TRN-98-004  For stability the change in a variable  p value from iteration to iteration is reduced by an “under-relaxation” factor,  : For example, an under-relaxation of 0.2 restricts the change in  P to 20% of the computed change of  for one iteration. FVM - Under-relaxation u Equation set being solved is non-linear. u Equation for one variable may depend on other variables, e.g., l Temperature l Mass fraction

19. © Fluent Inc. 9/5/2015L19 Fluids Review TRN-98-004 u Residual at point P is defined as: u An overall measure of the residual in the domain is: u Residuals can be scaled relative to the starting residual FVM - Residuals and Convergence u At convergence: l All discrete conservation equations (momentum, energy, etc.) are obeyed in all cells to a specified tolerance. l The solution no longer changes with additional iterations. l Mass, momentum, energy and scalar balances are obtained. u “Residuals” measure imbalance (or error) in conservation equations.

20. © Fluent Inc. 9/5/2015L20 Fluids Review TRN-98-004 Finite Element Solution Methods u We seek a solution to the equation of the form: K(u) u = F u A solution method is made up of two parts l Algorithm: solution organization scheme l Equation solver: solves linear system of equations u We shall consider two algorithms and two equation solvers

21. © Fluent Inc. 9/5/2015L21 Fluids Review TRN-98-004 FEM Algorithms and Equation Solvers u Algorithms: l fully-coupled l segregated u Equation solvers: l Gaussian elimination l Iterative methods: n non-symmetric equation systems n symmetric equation systems (pressure eqns.)

22. © Fluent Inc. 9/5/2015L22 Fluids Review TRN-98-004 Fully-Coupled Algorithm (1) u The most common solution scheme is the so-called Newton-Raphson iteration, or Newton’s method for short u First, re-write the equation as: R(u) = K(u) u - F u Using a Taylor series expansion and some further manipulations, we arrive at:

23. © Fluent Inc. 9/5/2015L23 Fluids Review TRN-98-004 Fully-Coupled Algorithm (2) u Advantages: l converges very rapidly u Disadvantages: l requires good initial guess l calculation of J -1 (u i ) is expensive u Alternatives: l Modified Newton-Raphson: evaluate J -1 (u i ) only once l Quasi-Newton: update J -1 (u i ) in a simple manner graphic representation of Newton’s method

24. © Fluent Inc. 9/5/2015L24 Fluids Review TRN-98-004 Segregated Algorithm (1) u K(u) u = F is never formed u Rather, it is decomposed into a set of decoupled equations: l K u u - C x p = f u u momentum equation l K v v - C y p = f v v momentum equation l C x T u + C y T v = 0continuity equation l K T T = f T energy (scalar) equation u No explicit equation for pressure! l Replace continuity equation with Poisson-type pressure matrix equation (derived from manipulating discretized momentum and continuity eqn’s) u The pressure can be calculation in a number of ways

25. © Fluent Inc. 9/5/2015L25 Fluids Review TRN-98-004 Segregated Algorithm (2) u Pressure projection method l given the current values of u, v and T, obtain an approximate pressure bo solving a discrete pressure equation l relax the pressure, i.e.: l using pnew, solve the momentum equations and energy equation using the newly computed velocities, solve for the pressure correction,  p adjust the velocity field (so that it obeys the incompressibility constraint) using  p u Advantage: less memory use u Disadvantage: more iterations Each equation set can be solved iteratively (inner iteration) or simulaneously (Gaussian elimination) uvpTuvpT outer iteration

26. © Fluent Inc. 9/5/2015L26 Fluids Review TRN-98-004 Equation Solvers u Iterative l Non-symmetric equation systems: n Conjugate gradient squared n GMRES l Symmetric equation systems (pressure): n Conjugate gradient n Conjugate residual u Gaussian elimination

27. © Fluent Inc. 9/5/2015L27 Fluids Review TRN-98-004 Underrelaxation u Two forms are used l explicit (similar to FVM approach) n carries some “history” forward n used with fully-coupled method n also used for pressure in segregated method l implicit n alters the weighting term for matrix diagonal n used for other equations (not pressure) with segregated method

28. © Fluent Inc. 9/5/2015L28 Fluids Review TRN-98-004 Convergence u Various quantities can be used to judge convergence of an FEM solution u The more commonly used are: l Relative change in solution between iterations ||U i - U i-1 || / ||U i || < tolerance l Relative numerical accuracy (R is residual vector) ||R i || / ||R 0 || < tolerance