Draft Notes ME 608 Numerical Methods in Heat, Mass, and Momentum Transfer

Transcription

1 Draft Notes ME 608 Numerial Methods in Heat, Mass, and Momentum Transfer Instrutor: Jayathi Y. Murthy Shool of Mehanial Engineering Purdue University Spring J.Y. Murthy and S.R. Mathur. All Rights Reserved

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3 Contents 1 Mathematial Modeling Conservation Equations Disussion Conservation Form Governing Equations The Energy Equation The Momentum Equation The Speies Equation The General Salar Transport Equation Mathematial Classifiation of Partial Differential Equations Ellipti Partial Differential Equations Paraboli Partial Differential Equations Hyperboli Partial Differential Equations Behavior of the Salar Transport Equation Closure Numerial Methods Overview Mesh Terminology and Types Regular and Body-fitted Meshes Strutured, Blok Strutured, and Unstrutured Meshes Conformal and Non-Conformal Meshes Cell Shapes Node-Based and Cell-Based Shemes Disretization Methods Finite Differene Methods Finite Element Methods Finite Volume Method Solution of Disretization Equations Diret Methods Iterative Methods Auray, Consisteny, Stability and Convergene Auray Consisteny

4 2.5.3 Stability Convergene Closure The Diffusion Equation: A First Look Two-Dimensional Diffusion in Retangular Domain Disretization Disussion Boundary Conditions Dirihlet Boundary Condition Neumann Boundary Condition Mixed Boundary Condition Unsteady Condution The Expliit Sheme The Fully-Impliit Sheme The Crank-Niholson Sheme Diffusion in Polar Geometries Diffusion in Axisymmetri Geometries Finishing Touhes Interpolation of Γ Soure Linearization and Treatment of Non-Linearity Under-Relaxation Disussion Trunation Error Spatial Trunation Error Temporal Trunation Error Stability Analysis Closure The Diffusion Equation: A Closer Look Diffusion on Orthogonal Meshes Non-Orthogonal Meshes Disussion Seondary Gradient Calulation Disrete Equation for Non-Orthogonal Meshes Boundary Conditions Gradient Calulation Strutured Meshes Unstrutured Meshes Influene of Seondary Gradients on Coeffiients Implementation Issues Data Strutures Overall Solution Loop Closure

5 5 Convetion Two-Dimensional Convetion and Diffusion in A Retangular Domain Central Differening Upwind Differening Convetion-Diffusion on Non-Orthogonal Meshes Central Differene Approximation Upwind Differening Approximation Auray of Upwind and Central Differene Shemes An Illustrative Example False Diffusion and Dispersion First-Order Shemes Using Exat Solutions Exponential Sheme Hybrid Sheme Power Law Sheme Unsteady Convetion D Finite Volume Disretization Central Differene Sheme First Order Upwind Sheme Error Analysis Lax-Wendroff Sheme Higher-Order Shemes Seond-Order Upwind Shemes Third-Order Upwind Shemes Implementation Issues Higher-Order Shemes for Unstrutured Meshes Disussion Boundary Conditions Inflow Boundaries Outflow Boundaries Geometri Boundaries Closure Fluid Flow: A First Look Disretization of the Momentum Equation Disretization of the Continuity Equation The Staggered Grid Disussion Solution Methods The SIMPLE Algorithm The Pressure Corretion Equation Overall Algorithm Disussion Boundary Conditions Pressure Level and Inompressibility The SIMPLER Algorithm Overall Algorithm

6 6.7.2 Disussion The SIMPLEC Algorithm Optimal Underrelaxation for SIMPLE Disussion Non-Orthogonal Strutured Meshes Unstrutured Meshes Closure Fluid Flow: A Closer Look Veloity and Pressure Chekerboarding Disretization of Momentum Equation Disretization of Continuity Equation Pressure Chekerboarding Veloity Chekerboarding Co-Loated Formulation The Conept of Added Dissipation Auray of Added Dissipation Sheme Disussion Two-Dimensional Co-Loated Variable Formulation Disretization of Momentum Equations Momentum Interpolation SIMPLE Algorithm for Co-Loated Variables Veloity and Pressure Corretions Pressure Corretion Equation Overall Solution Proedure Disussion Underrelaxation and Time-Step Dependene Co-Loated Formulation for Non-Orthogonal and Unstrutured Meshes Fae Normal Momentum Equation Momentum Interpolation for Fae Veloity The SIMPLE Algorithm for Non-Orthogonal and Unstrutured Meshes Disussion Closure Linear Solvers Diret vs Iterative Methods Storage Strategies Tri-Diagonal Matrix Algorithm Line by line TDMA Jaobi and Gauss Seidel Methods General Iterative Methods Convergene of Jaobi and Gauss Seidel Methods Analysis Of Iterative Methods Multigrid Methods Coarse Grid Corretion Geometri Multigrid

7 8.9.3 Algebrai Multigrid Agglomeration Strategies Cyling Strategies and Implementation Issues Closure

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9 Chapter 1 Mathematial Modeling In order to simulate fluid flow, heat transfer, and other related physial phenomena, it is neessary to desribe the assoiated physis in mathematial terms. Nearly all the physial phenomena of interest to us in this book are governed by priniples of onservation and are expressed in terms of partial differential equations expressing these priniples. For example, the momentum equations express the onservation of linear momentum; the energy equation expresses the onservation of total energy. In this hapter we derive a typial onservation equation and examine its mathematial properties. 1.1 Conservation Equations Typial governing equations desribing the onservation of mass, momentum, energy, or hemial speies are written in terms of speifi quantities - i.e., quantities expressed on a per unit mass basis. For example, the momentum equation expresses the priniple of onservation of linear momentum in terms of the momentum per unit mass, i.e., veloity. The equation for onservation of hemial speies expresses the onservation of the mass of the speies in terms of its mass fration Let us onsider a speifi quantity φ, whih may be momentum per unit mass, or the energy per unit mass, or any other suh quantity. Consider a ontrol volume of size x y z shown in Figure 1.1. We want to express the variation of φ in the ontrol volume over time. Let us assume that φ is governed by a onservation priniple that states Aumulation of φ in the ontrol volume over time t Net influx of φ into ontrol volume Net generation of φ inside ontrol volume (1.1) 9

10 y J x Jx+ x x z Figure 1.1: Control Volume The aumulation of φ in the ontrol volume over time t is given by ρφ t t ρφ t (1.2) Here, ρ is the density of the fluid, is the volume of the ontrol volume ( x y z) and t is time. The net generation of φ inside the ontrol volume over time t is given by S t (1.3) where S is the generation of φ per unit volume. S is also sometimes alled the soure term. Let us onsider the remaining term, the net influx of φ into the ontrol volume. Let J x represent the flux of φ oming into the ontrol volume through fae x, and J x x the flux leaving the fae x x. Similar fluxes exist on the y and z faes respetively. The net influx of φ into the ontrol volume over time t is J x J x x y z t J y J y y x z t J z J z z x y t (1.4) We have not yet said what physial mehanisms ause the influx of φ. For physial phenomena of interest to us, φ is transported by two primary mehanisms: diffusion due to moleular ollision, and onvetion due to the motion of fluid. In many ases, the diffusion flux may be written as J diffusion x Γ φ x (1.5) The onvetive flux may be written as J onvetion x ρuφ (1.6) 10

11 Here, the veloity field is given by the vetor V ui vj wk. Thus the net onvetive and diffusive flux may be written as J x ρuφ Γ φ x x J x x ρuφ Γ φ x x x (1.7) where ρu x is the mass flux through the ontrol volume fae at x. Similar expressions may be written for the y and z diretions respetively. Aumulating terms, and dividing by t Equation 1.1 may be written as ρφ t t t ρφ t Taking the limit x y z t ρφ t It is onvenient to write Equation 1.9 as ρφ t ρuφ x ρvφ y J x J x x x J z J z z z 0, we get J y J y y y S (1.8) J x x J y y J z z S (1.9) ρwφ z x Γ φ x z Γ φ z S y Γ φ y or, in vetor notation ρφ t ρvφ Γ φ S (1.10) Disussion It is worth noting the following about the above derivation: The differential form is derived by onsidering balanes over a finite ontrol volume. Though we have hosen hexahedral ontrol volume on whih to do onservation, we an, in priniple, hoose any shape. We should get the same final governing differential equation regardless of the shape of the volume hosen to do the derivation. 11

12 The onservation equation is written in terms of a speifi quantity φ, whih may be energy per unit mass (J/kg), or momentum per unit mass (m/s) or some similar quantity. The onservation equation is written on a per unit volume per unit time basis. The generation term in Equation 1.10 for example, is the generation of φ per unit volume per unit time. If φ were energy per unit mass, S would be the generation of energy per unit volume per unit time Conservation Form Equation 1.10 represents the onservative or divergene form of the onservation equation. This form is haraterized by the fat that in steady state, in the absene of generation, the divergene of the flux is zero: J 0 (1.11) where J = J x i J y j J z k. By using the ontinuity equation, we may write the nononservative form of Equation 1.10 ρφ t ρv φ Γ φ Γ φ S (1.12) The divergene of J represents the net efflux per unit volume of J. Thus, the onservative form is a diret statement about the onservation of φ in terms of the physial fluxes (onvetion and diffusion). The non-onservative form does not have a diret interpretation of this sort. Numerial methods that are developed with the divergene form as a starting point an be made to reflet the onservation property exatly if are is taken. Those that start from Equation 1.12 an be made to approximate onservation in some limiting sense, but not exatly. 1.2 Governing Equations The governing equations for fluid flow, heat and mass transfer, as well as other transport equations, may be represented by the onservative form, Equation Let us now onsider some speifi ases of the onservation equation for φ The Energy Equation The general form of the energy equation is quite elaborate, though it an also be ast into the general form of Equation For simpliity, let us assume low-speed flow and negligible visous dissipation. For this ase, the energy equation may be written in terms of the speifi enthalpy h as ρh ρvh k T S t h (1.13) 12

13 where k is the thermal ondutivity and T is the temperature. For ideal gases and inompressible substanes, dh C p dt (1.14) so that Equation 1.13 may be written as ρh t ρvh k C p h S h (1.15) Comparing Equation 1.15 with Equation 1.10 shows that the energy equation an be ast into the form of the general onservation equation, with φ h, Γ k C p and S S h The Momentum Equation The momentum equation for a Newtonian fluid in the diretion x may be written as ρu t ρvu µ u p x S u (1.16) Here, S u ontains those parts of the stress tensor not appearing diretly in the diffusion term, and p x is pressure gradient. We see that Equation 1.16 has the same form as the general onservation equation 1.10, with φ u, Γ µ and S p x S u The Speies Equation Consider the transport of a mixture of hemial speies. The equation for the onservation of mass for a hemial speie i may be written in terms of its mass fration, Y i, where Y i is defined as the mass of speies i per mass of mixture. If Fik s law is assumed valid, the governing onservation equation is ρy i t ρvy i Γ i Y i R i (1.17) Γ i is the diffusion oeffiient for Y i in the mixture and R i is the rate of formation of Y i through hemial reations. Again we see that Equation 1.17 has the same form as the general onservation equation 1.10, with φ Y i, Γ Γ i, and S R i. 1.3 The General Salar Transport Equation We have seen that the equations governing fluid flow, heat and mass transfer an be ast into a single general form whih we shall all the general salar transport equation: ρφ t ρvφ Γ φ S (1.18) If numerial methods an be devised to solve this equation, we will have a framework within whih to solve the equations for flow, heat, and mass transfer. 13

14 1.4 Mathematial Classifiation of Partial Differential Equations The general salar transport equation is a seond-order partial differential equation (PDE) governing the spatial and temporal variation of φ. If the properties ρ and Γ, or the generation term S φ are funtions of φ, it is non-linear. Ignoring non-linearities for the moment, we examine the behavior of this equation. It is instrutive to onsider a general seond-order PDE given by aφ xx bφ xy φ yy dφ x eφ y f φ g 0 (1.19) The oeffiients a,b,,d,e, f and g are funtions of the oordinates (x,y), but not of φ itself. The behavior of Equation 1.19 may be lassified aording to the sign on the disriminant! b 2 4a (1.20)!" If 0 the PDE is alled ellipti. If 0, the PDE is alled paraboli. If 0 the PDE is alled hyperboli. Let us onsider typial examples of eah type of equation Ellipti Partial Differential Equations Let us onsider steady heat ondution in a one-dimensional slab, as shown in Figure 1.2. The governing equation and boundary onditions are given by x k T x 0 (1.21) with For onstant k, the solution is given by T 0 # T 0 T L $ T L (1.22) T x T 0 T L T 0 L x (1.23) This simple problem illustrates important properties of ellipti PDEs. These are 1. The temperature at any point x in the domain is influened by the temperatures on both boundaries. 2. In the absene of soure terms, T x is bounded by the temperatures on the boundaries. It annot be either higher or lower than the boundary temperatures. It is desirable when devising numerial shemes that these basi properties be refleted in the harateristis of the sheme. 14

15 x L Figure 1.2: Condution in a One-Dimensional Slab 15

16 1.4.2 Paraboli Partial Differential Equations Consider unsteady ondution in the slab in Figure 1.2. If k, ρ and C p are onstant, Equation 1.13 may be written in terms of the temperature T as where α k given by T t α 2 T x 2 (1.24) ρc p is the thermal diffusivity. The initial and boundary onditions are T x 0 # T i x T 0 t % T 0 T L t # T 0 (1.25) Using a separation of variables tehnique, we may write the solution to this problem as where B n T x t & T 0 2 L ) 0 L T i n' 1 x T 0 sin We note the following about the solution: B n sin nπx L e( αn 2 π 2 L 2 t (1.26) nπx L dx n *,*,* (1.27) 1. The boundary temperature T 0 influenes the temperature T(x,t) at every point in the domain, just as with ellipti PDE s. 2. Only initial onditions are required (i.e., onditions at t 0). No final onditions are required, for example onditions at t. We do not need to know the future to solve this problem! 3. The initial onditions only affet future temperatures, not past temperatures. 4. The initial onditions influene the temperature at every point in the domain for all future times. The amount of influene dereases with time, and may affet different spatial points to different degrees. 5. A steady state is reahed for t. Here, the solution beomes independent of T i x 0. It also reovers its ellipti spatial behavior. 6. The temperature is bounded by its initial and boundary onditions in the absene of soure terms. It is lear from this problem that the variable t behaves very differently from the variable x. The variation in t admits only one-way influenes, whereas the variable x admits two-way influenes. t is sometimes referred to as the marhing or paraboli diretion. Spatial variables may also behave in this way, for example, the axial diretion in a pipe flow. 16

17 " Hyperboli Partial Differential Equations Let us onsider the one-dimensional flow of a fluid in a hannel, as shown in Figure 1.3. The veloity of the fluid, U, is a onstant; also U 0. For t - 0, the fluid upstream of the hannel entrane is held at temperature T 0. The properties ρ and C p are onstant and k 0. The governing equations and boundary onditions are given by: with t ρc p T x ρc p UT 0 (1.28) T x 0 # T i T x. 0 t # T 0 (1.29) You an onvine yourself that Equation 1.28 is hyperboli by differentiating it one with respet to either t or x and finding the disriminant. The solution to this problem is T x t & T + x Ut / 0 (1.30) or to put it another way T x t # T i for t T 0 for t - x U x U (1.31) The solution is essentially a step in T traveling in the positive x diretion with a veloity U, as shown in Figure 1.4. We should note the following about the solution: 1. The upstream boundary ondition (x 0) affets the solution in the domain. Conditions downstream of the domain do not affet the solution in the domain. 2. The inlet boundary ondition propagates with a finite speed, U. 3. The inlet boundary ondition is not felt at point x until t x U Behavior of the Salar Transport Equation The general salar transport equation we derived earlier (Equation 1.10) has muh in ommon with the partial differential equations we have seen here. The ellipti diffusion equation is reovered if we assume steady state and there is no flow. The same problem solved for unsteady state exhibits paraboli behavior. The onvetion side of the salar transport equation exhibits hyperboli behavior. In most engineering situations, the equation exhibits mixed behavior, with the diffusion terms tending to bring in ellipti influenes, and the unsteady and onvetion terms bringing in paraboli or hyperboli influenes. It is sometimes useful to onsider partiular oordinates to be ellipti or paraboli. For example, it is useful in paraboli problems to think about time as the paraboli oordinate and to think of spae as the ellipti oordinate. 17

18 T 0 T T i x U Figure 1.3: Convetion of a Step Profile T 0 T t = 0 T i t = x /U 1 x 1 t = x /U 2 x x 2 Figure 1.4: Temperature Variation with Time 18

19 Though it is possible to devise numerial methods whih exploit the partiular nature of the general salar transport equation in ertain limiting ases, we will not do that here. We will onentrate on developing numerial methods whih are general enough to handle the mixed behavior of the general transport equation. When we study fluid flow in greater detail, we will have to deal with oupled sets of equations, as opposed to a single salar transport equation. These sets an also be analyzed in terms similar to the disussion above. 1.5 Closure In this hapter, we have seen that many physial phenomena of interest to us are governed by onservation equations. These onservation equations are derived by writing balanes over finite ontrol volumes. We have seen that the onservation equations governing the transport of momentum, heat and other speifi quantities have a ommon form embodied in the general salar transport equation. This equation has unsteady, onvetion, diffusion and soure terms. By studying the behavior of anonial ellipti, paraboli and hyperboli equations, we begin to understand the behavior of these different terms in determining the behavior of the omputed solution. The ideal numerial sheme should be able to reprodue these influenes orretly. 19

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21 Chapter 2 Numerial Methods In the previous hapter, we saw that physial phenomena of interest to us ould be desribed by a general salar transport equation. In this hapter, we examine numerial methods for solving this type of equation, and identify the main omponents of the solution method. We also examine ways of haraterizing our numerial methods in terms of auray, onsisteny, stability and onvergene. 2.1 Overview Our objetive here is to develop a numerial method for solving the general salar transport equation. Fundamental to the development of a numerial method is the idea of disretization. An analytial solution to a partial differential equation gives us the value of φ as a funtion of the independent variables (x y z t). The numerial solution, on the other hand, aims to provide us with values of φ at a disrete number of points in the domain. These points are alled grid points, though we may also see them referred to as nodes or ell entroids, depending on the method. The proess of onverting our governing transport equation into a set of equations for the disrete values of φ is alled the disretization proess and the speifi methods employed to bring about this onversion are alled disretization methods. The disrete values of φ are typially desribed by algebrai equations relating the values at grid points to eah other. The development of numerial methods fouses on both the derivation of the disrete set of algebrai equations, as well as a method for their solution. In arriving at these disrete equations for φ we will be required to assume how φ varies between grid points i.e., to make profile assumptions. Most widely used methods for disretization require loal profile assumptions. That is, we presribe how φ varies in the loal neighborhood surrounding a grid point, but not over the entire domain. The onversion of a differential equation into a set of disrete algebrai equations requires the disretization of spae. This is aomplished by means of mesh generation. A typial mesh is shown in Figure 2.1. Mesh generation divides the domain of interest into elements or ells, and assoiates with eah element or ell one or more disrete 21

22 Cell Vertex Figure 2.1: An Example of a Mesh values of φ. It is these values of φ we wish to ompute. We should also distinguish between the disretized equations and the methods employed to solve them. For our purposes, let us say that the auray of the numerial solution, i.e., its loseness to the exat solution, depends only on the disretization proess, and not on the methods employed to solve the disrete set (i.e., the path to solution). The path to solution determines whether we are suessful in obtaining a solution, and how muh time and effort it will ost us. But it does not determine the final answer. (For some non-linear problems, the path to solution an determine whih of several possible solutions is obtained. For simpliity, we shall not pursue this line of investigation here.) Sine we wish to get an answer to the original differential equation, it is appropriate to ask whether our algebrai equation set really gives us this. When the number of grid points is small, the departure of the disrete solution from the exat solution is expeted to be large. A well-behaved numerial sheme will tend to the exat solution as the number of grid points is inreased. The rate at whih it tends to the exat solution depends on the type of profile assumptions made in obtaining the disretization. No matter what disretization method is employed, all well-behaved disretization methods should tend to the exat solution when a large enough number of grid points is employed. 22

23 Node (Vertex) Cell Cell Centroid Fae Figure 2.2: Mesh Terminology 2.2 Mesh Terminology and Types The physial domain is disretized by meshing or gridding it.(we shall use the terms mesh and grid interhangeably in this book). We shall use the terminology shown in Figure 2.2 in desribing our meshes. The fundamental unit of the mesh is the ell (sometimes alled the element). Assoiated with eah ell is the ell entroid. A ell is surrounded by faes, whih meet at nodes or verties. In three dimensions, the fae is a surfae surrounded by edges. In two dimensions, faes and edges are the same. A variety of mesh types are enountered in pratie. These are desribed below Regular and Body-fitted Meshes In many ases, our interest lies in analyzing domains whih are regular in shape: retangles, ubes, ylinders, spheres. These shapes an be meshed by regular grids, as shown in Figure 2.3(a). The grid lines are orthogonal to eah other, and onform to the boundaries of the domain. These meshes are also sometimes alled orthogonal meshes. For many pratial problems, however, the domains of interest are irregularly shaped and regular meshes may not suffie. An example is shown in Figure 2.3(b). Here, grid lines are not neessarily orthogonal to eah other, and urve to onform to the irregular geometry. If regular grids are used in these geometries, stair stepping ours at domain boundaries, as shown in Figure 2.4. When the physis at the boundary are important in determining the solution, e.g., in flows dominated by wall shear, suh an approximation of the boundary may not be aeptable Strutured, Blok Strutured, and Unstrutured Meshes The meshes shown in Figure 2.3 are examples of strutured meshes. Here, every interior vertex in the domain is onneted to the same number of neighbor verties. Figure 2.5 shows a blok-strutured mesh. Here, the mesh is divided into bloks, and the 23

24 (a) (b) Figure 2.3: Regular and Body-Fitted Meshes Figure 2.4: Stair-Stepped Mesh 24

25 Blok Figure 2.5: Blok-Strutured Mesh mesh within eah blok is strutured. However, the arrangement of the bloks themselves is not neessarily strutured. Figure 2.6 shows an unstrutured mesh. Here, eah vertex is onneted to an arbitrary number of neighbor verties. Unstrutured meshes impose fewer topologial restritions on the user, and as a result, make it easier to mesh very omplex geometries Conformal and Non-Conformal Meshes An example of a non-onformal mesh is shown in Figure 2.7. Here, the verties of a ell or element may fall on the faes of neighboring ells or elements. In ontrast, the meshes in Figures 2.3,2.5 and 2.6 are onformal meshes Cell Shapes Meshes may be onstruted using a variety of ell shapes. The most widely used are quadrilaterals and hexahedra. Methods for generating good-quality strutured meshes for quadrilaterals and hexahedra have existed for some time now. Though mesh struture imposes restritions, strutured quadrilaterals and hexahedra are well-suited for flows with a dominant diretion, suh as boundary-layer flows. More reently, as omputational fluid dynamis is beoming more widely used for analyzing industrial flows, unstrutured meshes are beoming neessary to handle omplex geometries. Here, triangles and tetrahedra are inreasingly being used, and mesh generation tehniques for their generation are rapidly reahing maturity. As of this writing, there are no general 25

26 Cell Vertex Figure 2.6: Unstrutured Mesh 26

27 Figure 2.7: Non-Conformal Mesh 27

28 (a) (b) () (d) (e) (f) Figure 2.8: Cell Shapes: (a) Triangle, (b) Tetrahedron, () Quadrilateral, (d) Hexahedron, (e) Prism, and (f) Pyramid purpose tehniques for generating unstrutured hexahedra. Another reent trend is the use of hybrid meshes. For example, prisms are used in boundary layers, transitioning to tetrahedra in the free-stream. In this book, we will develop numerial methods apable of using all these ell shapes Node-Based and Cell-Based Shemes Numerial methods whih store their primary unknowns at the node or vertex loations are alled node-based or vertex-based shemes. Those whih store them at the ell entroid, or assoiate them with the ell, are alled ell-based shemes. Finite element methods are typially node-based shemes, and many finite volume methods are ell-based. For strutured and blok-strutured meshes omposed of quadrilaterals or hexahedra, the number of ells is approximately equal to the number of nodes, and the spatial resolution of both storage shemes is similar for the same mesh. For other ell shapes, there may be quite a big differene in the number of nodes and ells in the mesh. For triangles, for example, there are twie as many ells as nodes, on average. This fat must be taken into aount in deiding whether a given mesh provides adequate resolution for a given problem. From the point of view of developing numerial methods, both shemes have advantages and disadvantages, and the hoie will depend 28

29 " Flow Triangles Boundary Layer Quadrilaterals Figure 2.9: Hybrid Mesh in Boundary Layer on what we wish to ahieve. 2.3 Disretization Methods So far, we have alluded to the disretization method, but have not said speifially what method we will use to onvert our general transport equation to a set of disrete algebrai equations. A number of popular methods are available for doing this Finite Differene Methods Finite differene methods approximate the derivatives in the governing differential equation using trunated Taylor series expansions. Consider a one-dimensional salar transport equation with a onstant diffusion oeffiient and no unsteady or onvetive terms: Γ d2 φ dx 2 S 0 (2.1) We wish to disretize the diffusion term. Referring to the one-dimensional mesh shown in Figure 2.10, we write φ 1 φ 2 x dφ dx 2 x 2 2 d 2 φ dx 2 O x 3 (2.2) 2 and φ 3 φ 2 x dφ dx 2 x 2 2 d 2 φ dx 2 O x 3 (2.3) 2 The term O + x 3 indiates that the terms that follow have a dependene on x n where n 3. Subtrating Equations 2.2 from Equation 2.3 gives dφ dx 2 φ 3 φ 1 2 x O 0 x 2 (2.4) By adding the two equations together, we an write d 2 φ dx 2 2 φ 1 φ 3 2φ 2 x 2 O + x 2 (2.5) 29

30 1 2 3 x x Figure 2.10: One-Dimensional Mesh By inluding the diffusion oeffiient and dropping terms of O + x 2 or smaller, we an write The soure term S is evaluated at the point 2 using Γ d 2 φ dx 2 Γ φ 1 φ 3 2φ 2 2 x 2 (2.6) S 2 S φ 2 (2.7) Substituting Equations 2.6 and 2.7 into Equation 2.1 gives the equation 2Γ x 2 φ 2 Γ x 2 φ 1 Γ x 2 φ 3 S 2 (2.8) This is the disrete form of Equation 2.1. By obtaining an equation like this for every point in the mesh, we obtain an set of algebrai equations in the disrete values of φ. This equation set may be solved by a variety of methods whih we will disuss later in the book. Finite differene methods do not expliitly exploit the onservation priniple in deriving disrete equations. Though they yield disrete equations that look similar to other methods for simple ases, they are not guaranteed to do so in more ompliated ases, for example on unstrutured meshes Finite Element Methods We onsider again the one-dimensional diffusion equation, Equation 2.1. There are different kinds of finite element methods. Let us look at a popular variant, the Galerkin finite element method. Let φ be an approximation to φ. Sine φ is only an approximation, it does not satisfy Equation 2.1 exatly, so that there is a residual R: We wish to find a φ suh that d 2 φ dx 2 S R (2.9) W Rdx ) 0 (2.10) domain W is a weight funtion, and Equation 2.10 requires that the residual R beome zero in a weighted sense. In order to generate a set of disrete equations we use a family of 30

31 ) ) ) weight funtions W i, i 1 2+*,*,* N, where N is the number of grid points, rather than a single weight funtion. Thus, we require W ) i Rdx 0 i 1 2+*,*1* N (2.11) domain The weight funtions W i are typially loal in that they are non-zero over element i, but are zero everywhere else in the domain. Further, we assume a shape funtion for φ, i.e., assume how φ varies between nodes. Typially this variation is also loal. For example we may assume that φ assumes a piee-wise linear profile between points 1 and 2 and between points 2 and 3 in Figure The Galerkin finite element method requires that the weight and shape funtions be the same. Performing the integration in Equation 2.11 results in a set of algebrai equations in the nodal values of φ whih may be solved by a variety of methods. We should note here that beause the Galerkin finite element method only requires the residual to be zero in some weighted sense, it does not enfore the onservation priniple in its original form. We now turn to a method whih employs onservation as a tool for developing disrete equations Finite Volume Method The finite volume method (sometimes alled the ontrol volume method) divides the domain in to a finite number of non-overlapping ells or ontrol volumes over whih onservation of φ is enfored in a disrete sense. It is possible to start the disretization proess with a diret statement of onservation on the ontrol volume, as in Equation 1.9 in the previous hapter. Alternatively we may start with the differential equation and integrate it over the ontrol volume. Let us examine the disretization proess by looking at one-dimensional diffusion with a soure term: d dx Γdφ dx S 0 (2.12) Consider a one-dimensional mesh, with ells as shown in Figure Let us store disrete values of φ at ell entroids, denoted by W, P and E. The ell faes are denoted by w and e. Let us assume the fae areas to be unity. We fous on the ell assoiated with P. We start by integrating Equation 2.12 over the ell P. This yields w e d dx Γdφ dx dx w e Sdx 0 (2.13) so that Γ dφ dx e Γ dφ dx w w e Sdx 0 (2.14) We note that this equation an also be obtained by writing a heat balane over the ell P from first priniples. Thus far, we have made no approximation. 31

32 x W P E e w δ x w δ x e Figure 2.11: Arrangement of Control Volumes We now make a profile assumption, i.e., we make an assumption about how φ varies between ell entroids. If we assume that φ varies linearly between ell entroids, we may write Γ e φ E φ P Γ w φ P φ W δx e δx w S x 0 (2.15) Here S is the average value of S in the ontrol volume. We note that the above equation is no longer exat beause of the approximation in assuming that φ varies in a pieewise linear fashion between grid points. Colleting terms, we obtain where a P φ P a E φ E a W φ W b (2.16) a E Γ e a W Γ w δx e δx w a P a E a W b S x (2.17) Equations similar to Equation 2.16 may be derived for all ells in the domain, yielding a set of algebrai equations, as before; these may be solved using a variety of diret or iterative methods. We note the following about the disretization proess. 1. The proess starts with the statement of onservation over the ell. We then find ell values of φ whih satisfy this onservation statement. Thus onservation is guaranteed for eah ell, regardless of mesh size. 32

33 2. Conservation does not guarantee auray, however. The solution for φ may be inaurate, but onservative. 3. The quantity Γdφ dx e is diffusion flux on the e fae. The ell balane is written in terms of fae fluxes. The gradient of φ must therefore be evaluated at the faes of the ell. 4. The profile assumptions for φ and S need not be the same. We will examine additional properties of this disretization in the next hapter. 2.4 Solution of Disretization Equations All the disretization methods desribed here result in a set of disrete algebrai equations whih must be solved to obtain the disrete values of φ. These equations may be linear (i.e. the oeffiients are independent of φ) or they may be non-linear (i.e. the oeffiients are funtions of φ). The solution tehniques are independent of the disretization method, and represent the path to solution. For the linear algebrai sets we will enounter in this book, we are guaranteed that there is only one solution, and if our solution method gives us a solution, it is the solution we want. All solution methods (i.e. all paths to solution) whih arrive at a solution will give us the same solution for the same set of disrete equations. For non-linear problems, we do not have this guarantee, and the answer we get may depend on fators like the initial guess, and the atual path to solution. Though this is an important issue in omputing fluid flows, we will not address it here. Solution methods may be broadly lassified as diret or iterative. We onsider eah briefly below Diret Methods Using one of the disretization methods desribed previously, we may write the resulting system of algebrai equations as Aφ B (2.18) where A is the oeffiient matrix, φ 32 φ 1 φ 2 +*,*,* 4 T is a vetor onsisting of the disrete values of φ, and B is the vetor resulting from the soure terms. Diret methods solve the equation set 2.18 using the methods of linear algebra. The simplest diret method is inversion, whereby φ is omputed from φ A5 1 B (2.19) A solution for φ is guaranteed if A5 1 an be found. However, the operation ount for the inversion of an N N matrix is O N 2. Consequently, inversion is almost never employed in pratial problems. More effiient methods for linear systems are available. For the disretization methods of interest here, A is sparse, and for strutured meshes it is banded. For ertain types of equations, for example, for pure diffusion, the 33

34 matrix is symmetri. Matrix manipulation an take into aount the speial struture of A in devising effiient solution tehniques for Equation We will study one suh method, the tri-diagonal matrix algorithm (TDMA), in a later hapter. Diret methods are not widely used in omputational fluid dynamis beause of large omputational and storage requirements. Most industrial CFD problems today involve hundreds of thousands of ells, with 5-10 unknowns per ell even for simple problems. Thus the matrix A is usually very large, and most diret methods beome impratial for these large problems. Furthermore, the matrix A is usually non-linear, so that the diret method must be embedded within an iterative loop to update nonlinearities in A. Thus, the diret method is applied over and over again, making it all the more time-onsuming Iterative Methods Iterative methods are the most widely used solution methods in omputational fluid dynamis. These methods employ a guess-and-orret philosophy whih progressively improves the guessed solution by repeated appliation of the disrete equations. Let us onsider an extremely simple iterative method, the Gauss-Seidel method. The overall solution loop for the Gauss-Seidel method may be written as follows: 1. Guess the disrete values of φ at all grid points in the domain. 2. Visit eah grid point in turn. Update φ using φ P a E φ E a W φ W b a P (2.20) The neighbor values, φ E and φ W are required for the update of φ P. These are assumed known at prevailing values. Thus, points whih have already been visited will have reently updated values of φ and those that have not will have old values. 3. Sweep the domain until all grid points are overed. This ompletes one iteration. 4. Chek if an appropriate onvergene riterion is met. We may, for example, require that the maximum hange in the grid-point values of φ be less than 0* 1 %. If the riterion is met, stop. Else, go to step 2. The iteration proedure desribed here is not guaranteed to onverge to a solution for arbitrary ombinations of a P, a E and a W. Convergene of the proess is guaranteed for linear problems if the Sarborough riterion is satisfied. The Sarborough riterion requires that ae aw ap. 1 for all grid points 1 for at least one grid point (2.21) Matries whih satisfy the Sarborough riterion have diagonal dominane. We note that diret methods do not require the Sarborough riterion to be satisfied to obtain a 34

35 solution; we an always obtain a solution to our linear set of equations as long as our oeffiient matrix is not singular. The Gauss-Seidel sheme an be implemented with very little storage. All that is required is storage for the disrete values of φ at the grid points. The oeffiients a P, a E, a W and b an be omputed on the fly if desired, sine the entire oeffiient matrix for the domain is not required when updating the value of φ at any grid point. Also, the iterative nature of the sheme makes it partiularly suitable for non-linear problems. If the oeffiients depend on φ, they may be updated using prevailing values of φ as the iterations proeed. Nevertheless, the Gauss-Seidel sheme is rarely used in pratie for solving the systems enountered in CFD. The rate of onvergene of the sheme dereases to unaeptably low levels if the system of equations is large. In a later hapter, we will use a multigrid method to aelerate the rate of onvergene of this sheme and make it usable as a pratial tool. 2.5 Auray, Consisteny, Stability and Convergene In this setion, we turn to ertain important properties of numerial methods Auray Auray refers to the orretness of a numerial solution when ompared to an exat solution. In most ases, we do not know the exat solution. It is therefore more useful to talk of the trunation error of a disretization method. The trunation error assoiated with the diffusion term using the finite differene method is O + x 2, as shown by Equation 2.5.This simply says that if d 2 φ dx 2 is represented by the first term in Equation 2.5, the terms that are negleted are of O + x 2. Thus, if we refine the mesh, we expet the trunation error to derease as x 2. If we double the x-diretion mesh, we expet the trunation error to derease by a fator of four. The trunation error of a disretization sheme is the largest trunation error of eah of the individual terms in the equation being disretized. The order of a disretization method is n if its trunation error is O 0 x n. It is important to understand that the trunation error tells us how fast the error will derease with mesh refinement, but is not an indiator of how high the error is on the urrent mesh. Thus, even methods of very high order may yield inaurate results on a given mesh. However, we are guaranteed that the error will derease more rapidly with mesh refinement than with a disretization method of lower order Consisteny A onsistent numerial method is one for whih the trunation error tends to vanish as the mesh beomes finer and finer. (For unsteady problems, both spatial and temporal trunation errors must be onsidered). We are guaranteed this if the trunation error is some power of the mesh spaing x (or t). Sometimes we may ome aross shemes 35

36 where the trunation error of the method is O x t. Here, onsisteny is not guaranteed unless x is dereased faster than t. Consisteny is a very important property. Without it, we have no guarantee that mesh refinement will improve our solution Stability The previous two properties refer to the behavior of the disretization method. Stability is a property of the path to solution. For steady state problems, for example, we obtain a disretized set of algebrai equations whih must be solved. We may hoose to solve this set using an iterative method. Depending on the properties of the method, solution errors may be amplified or damped. An iterative solution method is unstable or divergent if it fails to yield a solution to the disrete set. It is also possible to speak of the stability of time-marhing shemes. When solving unsteady problems, we will use numerial methods whih ompute the solution at disrete instants of time, using the solution at one or more previous time steps as initial onditions. Stability analysis allow us to determine whether errors in the solution remain bounded as time marhing proeeds. An unstable time-marhing sheme would not be able to reah steady state in an unsteady heat ondution problem, for example (assuming that a steady state exists). It is possible to analyze iterative and time marhing methods using stability analysis. However, this is most onvenient for linear problems, and is usually too diffiult for most realisti problems. Here, non-linearities in the governing equations, boundary onditions, and properties, as well as oupling between multiple governing equations, make a formal analysis diffiult. In reality the pratitioner of CFD must rely on experiene and intuition in devising stable solution methods Convergene We distinguish between two popular usages of the term onvergene. We may say that an iterative method has onverged to a solution, or that we have obtained onvergene using a partiular method. By this we mean that our iterative method has suessfully obtained a solution to our disrete algebrai equation set. We may also speak of onvergene to mesh independene. By this, we mean the proess of mesh refinement, and its use in obtaining solutions that are essentially invariant with further refinement. We shall use the term in both senses in this book. 2.6 Closure In this hapter, we have presented a broad overview of disretization and introdued terminology assoiated with numerial methods. We have learned that there are a number of different philosophies for disretizing the salar transport equation. Of these, only the finite volume method enfores onservation on eah ell, and thus ensures that both loal and global onservation are guaranteed no matter how oarse the mesh. In the next hapter, we onsider the finite volume method in more detail, and study the properties of the disretizations it produes when applied to diffusion problems. 36

37 Chapter 3 The Diffusion Equation: A First Look In this hapter we turn our attention to an important physial proess, namely diffusion. Diffusion operators are ommon in heat, mass and momentum transfer and an also be used to model eletrostatis, radiation, and other physis. We onsider the disretization and solution of the salar transport equation for both steady and unsteady diffusion problems. We will attempt to relate the properties of our disrete equations with the behavior of the anonial partial differential equations we studied previously. The methodology we develop in this hapter will allow us to examine more ompliated mesh types and physis in later hapters. 3.1 Two-Dimensional Diffusion in Retangular Domain Let us onsider the steady two-dimensional diffusion of a salar φ in a retangular domain. From Equation 1.10, the governing salar transport equation may be written as J S (3.1) where J J x i J y j is the diffusion flux vetor and is given by In Cartesian geometries, the gradient operator is given by J Γ φ (3.2) x i y j (3.3) We note that Equation 3.1 is written in onservative or divergene form. When Γ is onstant and S is zero, the equation defaults to the familiar Laplae equation. When Γ is onstant and S is non-zero, the Poisson equation is obtained. 37

38 N A n δ y n W A w P A e E y A s δ y s S δ x w δ x e x Figure 3.1: Two-Dimensional Control Volume 38

39 3.1.1 Disretization The arrangement of ells under onsideration is shown in Figure 3.1. As in the previous hapter, we fous on ell P and its neighbors, the ells E, W, N and S. Disrete values of φ are stored at ell entroids. We also store the diffusion oeffiient Γ at ell entroids. The faes e,w, n and s are assoiated with area vetors A e, A w, A n and A s. The vetors are positive pointing outwards from the ell P. The volume of the ell P is 7 x y. We begin the proess of disretization by integrating Equation 3.1 over the ell P: ) 8 Jd 9 Next, we apply the divergene theorem to yield J ) da A ) 8 ) 8 Sd (3.4) Sd (3.5) The first integral represents the integral over the ontrol surfae A of the ell. We have made no approximations thus far. We now make a profile assumption about the flux vetor J. We assume that J varies linearly over eah fae of the ell P, so that it may be represented by its value at the fae entroid. We also assume that the mean value of the soure term S over the ontrol volume is S. Thus, or, more ompatly J A e J A w The fae areas A e and A w are given by J A n J A s S (3.6) J f A f S (3.7) f' e w n s A e y i The other area vetors may be written analogously. Further A w y i (3.8) J e A e Γ e y J w A w Γ w y φ x e φ x w (3.9) The transport in the other diretions may be written analogously. In order to omplete the disretization proess, we make one more round of profile assumptions. We assume that φ varies linearly between ell entroids. Thus, Equation 3.9 may be written as J e A e Γ e y φ E φ P δx e J w A w Γ w y φ P φ W δx w 39 (3.10)