CHAPTER 12 Finite-Volume (control-Volume) Method-Introduction
12-1 Introduction (1) • In developing what has become known as the finite-volume method, the conservation principles are applied to a fixed region in space known as a control volume, are somewhat interchangeably in the literature.
12-1 Introduction(2) • In the finite-volume approach, a point of view is taken that is distinctly different from finite-difference method(or Taylar-series method ). In the Taylar-series method, we accepted the PDE as the correct and appropriate from of the conservation principle(physical law) governing our problem and merely turned to mathematical tools to develop algebraic approximations to derivatives. We never again considered the physical law represented by the PDE. In the finite-volume method, the conservation statement is applied in a form applicable to a region in space (control volume).
12-1 Introduction(3) • This integral form of the conservation statement is usually well known from the first principles, or it can in most cases, be developed from the PDE form of the conservation from.
12-1 Introduction(4) • The feature of the FV method is shared in common with the finite-element methods. The FV procedure can, in fact, be considered as a variant of the finite-element method, although it is, from another point of view, just a particular type of finite-difference method.
12-1 Introduction(5) • As an example, consider unsteady 2-D heat conduction in a rectangular-shaped solid. The problem domain is divided up into control volume with associated points. We can establish the control volumes first and place grid points in the centers of the volumes (cell-centered method) or establish the grid first and then fix the boundaries of the control volumes (cell-vertex method) by, for example placing the boundaries halfway between grid points.
12-1 Introduction(6) • The General Differential Equation The differential equation obeying the generalized conservation principle can be written by the general differential equation as :dependent variable, such as velocity components (u,v,w), h or T, k, ε concentration, etc.
12-1 Introduction(7) : diffusion coefficients S: source term The four terms of eq.(1) are the unsteady term, the convection term, the diffusion term and the source term. *Note: The “conservation form” of the PDE is also referred to as “conservation law form” or “divergence form”, i.e., all spatial derivatives appear purely as divergences.
12-1 Introduction(8) • Conservation form of the governing equations of fluid flow
12-1 Introduction(9) • One-way and two-way coordinates : • Definitions: a two-way coordinate is such that the conditions at a given location in that coordinate are influenced by changes in conditions on either side of that location. A one-way coordinate is such that the conditions at a given location in that coordinate are influenced by changes in the conditions on only one side of that location.
12-1 Introduction(10) • Examples: one-dimensional steady heat conduction in a rod provides one example of a two-way coordinate. The temperature of any given point in the rod can be influenced by changing the temperature of either end. Normally, space coordinates are two-way coordinates. Time, on the other hand, is always a one-way coordinate. During the unsteady cooling of a solid, the temperature at a given instant can be influenced by changing only these conditions that prevailed before that instant.
12-1 Introduction(11) • Space as a one-way coordinate: If there is a strong unidirectional flow in the coordinate direction, then significant influences travel only from upstream. The conditions at a given point are then affected largely by the upstream conditions, and very little by the downstream ones. It is true that convection is a one-way process, but diffusion (which is always present) has two-way influences. However., then the flow rate is large, condition overpowers diffusion and thus make the space coordinate nearly one-way.
12-1 Introduction(12) • Parabolic, elliptic, hyperbolic: • The term parabolic indicates a one way behavior, while elliptic signifies the two-way concept. • It would be more meaningful if situations were described as being parabolic or elliptic in a given coordinate. Thus, the unsteady heat condition problem, which is normally called parabolic, is actually parabolic in time and elliptic in all coordinate. A two-dimension boundary layer is parabolic in the stream wise coordinate and elliptic in the cross-stream coordinate
12-1 Introduction(13) • A hyperbolic problem has a kind of one-way behavior, which is, however, not along coordinate directions but along special-lines called characteristics. • A situation is parabolic if there exists at least one one-way coordinate: otherwise, it is elliptic. • A flow with one one-way space coordinate is sometimes called a boundary-layer-type flow, while a flow with all two-way coordinate is referred to as a recirculating flow.
12-1 Introduction(14) • Computational implications: The motivation for the foregoing discussion about one-way and two-way coordinates is that, it a one-way coordinate can be identified in a given situation, substantial economy of computer storage and computer time is possible.
12-2 An Illustrative Example(1) • The FV method used the integral form of the conservation equation(eq.1) as the starting point: • Let us consider steady one-dimensional heat conduction governed by
12-2 An Illustrative Example(2) • Preparation: To derive the discrerization equation, we shall employ the grid-point cluster shown in Fig.1. We focus attention on the grid point P, which has the grid points E and W as its neighbors.(E denotes the east side, while W stands for the west side). The dashed lines show the faces of the control volume. The letters e and w denote these faces.
12-2 An Illustrative Example(3) For one-dimensional problem under consideration, we shall assume a unit thickness in the y and z directions. Thus, the volume of the cv shown is △x ×1 ×1. If we integrate eq(3) over the cv, we get (δ x)w (δ x)e w e P E W Fig. 1 △ x
12-2 An Illustrative Example(4) • Profile assumption: To make further further progress, we need a profile assumption or an interpolation formula. Here, linear interpolation functions are used between the grid points, as shown in Fig 2. T Fig. 2 x w e W Δp E δxw δxe
12-2 An Illustrative Example(5) • The discrerization equation: If we evaluate the derivatives dT/dx in eq.(4) from the piecewise-linear profile, the resulting equation will be
12-2 An Illustrative Example(7) • Comments: • In general, it is convenient to extend eq.(6) into multidimensional form as where nb denotes a neighbor, and the summation is to be taken over all the neighbors. • In deriving eq(6), we have used the simplest profile assumption that enabled us to evaluate dT/dx. Of course, many other interpolation functions would have been possible.
12-2 An Illustrative Example(8) • Further, it is important to understand that we need not use the same profile for all quantities. • Even for given variable, the same profile assumption need not be used for all terms in the equation.
12-2 An Illustrative Example(9) • Treatment of source term: The discretization equations will be solved by the techniques for linear algebraic equations. The procedure for “linearizing” a given S~T relationship is necessary. Here, it is sufficient to express the overage value S as
12-2 An Illustrative Example(10) Where Sc stands for the constant part of S, while Sp is the coefficient of Tp. With the linearized source expression, the discretization equation will become
12-3 The Four Basic Rules(1) • Rule 1:Consistency at a control-volume face -When a face is common to two adjacent control volumes, the flux across it must be represented by the same expression in the discretization equations for the two control volumes • Rule 2:Positive coefficients -All coefficients (ap and neighbor coefficients anb) must always be positive.
12-3 The Four Basic Rules(2) • Rule 3:Negative-slope linearization of the source term -When the source term is linearized as S=SC+SPTP, the coefficient SPmust always be less than or equal to zero. • Rule 4:Sum of the neighbor coefficients -We require
CHAPTER 13 The Finite Volume Method for Diffusion Problems
13-1 Steady One-dimensional Condition(1) • The Basic Equation • The Discretization Equation
13-1 Steady One-dimensional Condition(2) • The Grid Spacing • For the grid points shown in 8.4, it it not necessary that the distances (δx)e and (δx)w be equal. Indeed, the use of non-uniform grid spacing is often desirable, for it enables us to deploy computing power effectively. In general, we shall obtain an accurate solution only when the grid is sufficiently fine, but there is no need to employ a fine grid in regions where the dependent variable T changes rather slowly with x. On the other hand, a fine grid is required where the T~x variation is steep.
13-1 Steady One-dimensional Condition(3) • A misconception seems prevail that non-uniform grid lead to less accuracy than do uniform grids. There is no sound basis for such an assertion. Also there are no universal rules about what maximum (or minimum) ratio the adjacent grid intervals should maintain.
13-1 Steady One-dimensional Condition(4) • Since the T~x distribution is not known before the problem is solved, how can we design an appropriate non-uniform grid? First: One normally has some qualitative expectations about the solution, from which some guidance can be obtained. second: preliminary coarse-grid solutions can be used to find the pattern of the T~x variation; then a suitable non- uniform grid can be constructed.
13-1 Steady One-dimensional Condition(5) • The Interface Conductivity • The most straightforward procedure for obtaining the interface conductivity ke is to assume a linear variation of k between points P and E (δx)e (δx)e- (δx)e+ x P e E
13-1 Steady One-dimensional Condition(6) If the interface e were midway between grid points, fe would be 0.5, and ke would be he arithmetic mean of kp and kE.
13-1 Steady One-dimensional Condition(7) 2.We shall shortly show that this simple-minded approach leads to rather incorrect implications in some cases and cannot accurately handle the abrupt changes of conductivity that may occur in composite materials. Fortunately, a much better alternative is available. 3.Our main objective is to obtain a good representation for the heat flux qe at the interface via
13-1 Steady One-dimensional Condition(8) For the composite slab between pointsPand E, a steady one-dimensional analysis (without sources) lead to 3.Our main objective is to obtain a good representation for the heat flux qe at the interface via
13-1 Steady One-dimensional Condition(9) Combination of Eqs.(4) —— (6) yields When the interface e is placed midway between p and E, we have fe=0.5; then Eq. (9) show that ke is the harmonic mean of kp and kE, rather than the arithmetic mean.
13-1 Steady One-dimensional Condition(10) A similar expression can be written for aW.
13-1 Steady One-dimensional Condition(11) • The recommended interface conductivity formula (7) is based on the steady, no-source, one-dimensional situation in which the conductivity varies in a stepwise fashion from one control volume to the next. Even in situations with nonzero sources or with continuous variation of conductivity, it performs much better then the arithmetic-mean formula.
13-1 Steady One-dimensional Condition(12) • Iteration • Start with a guess or estimate for the values of T at all grid points. • From these guessed T´s, calculate tentative values of the coefficients in the discretization equation. • Solve the nominally set of algebraic equations to get new values of T. • With these T´s as better guesses, return to step 2 and repeat the process until further repetitions cease to produce significant changes in the values of T.
13-1 Steady One-dimensional Condition(13) • Source-Term Linearization Tp*: the guess value or the previous-iteration value of Tp Example 1: Given S=5-4T -Sc=5, Sp=-4—recommended -Sc=5-4Tp *, Sp=0 —not impractical -Sc=5+7Tp *,Sp=-11 —a steeper S~T relationship, will slow down the convergence
13-1 Steady One-dimensional Condition(14) • Example 2: Given S=3+7T • Sc=3,Sp=7 —this is not acceptable, as it makes Sp positive. The presence of a positive Sp many cause divergence. • Sc=3+7Tp*, Sp=0 —this is the practice one should follow. • Sc=3+9Tp*, Sp=-2 —this is an artificial creative Sp. It will, in general, slow down the convergence.
13-1 Steady One-dimensional Condition(15) • Example 3: Given S=4-5T3 • Sc=4-5Tp*3, Sp=0—this is the lazy-person approach. • Sc=4, Sp=-5Tp*2—this given S~T curve is steeper than this implies.
13-1 Steady One-dimensional Condition(16) This linearization represents the tangent to the S~T curve at Tp* • Recommended method:
13-1 Steady One-dimensional Condition(17) • Sc=4+20Tp*, Sp=-25Tp*2—This givens a steeper S~T curve, which would slow down convergence S (1) (2) (3) (4) T
13-1 Steady One-dimensional Condition(18) • Boundary Conditions: • Typically, three kinds of boundary conditions are encountered in heat condition. These are -Given boundary temperature. -Given boundary heat flux -Boundary heat flux specified via a heat transfer coefficient and the temperature of the surrounding fluid.
13-1 Steady One-dimensional Condition(19) • If the boundary temperature is given, no particular difficulty arises, and no additional equations are required. When the boundary temperature is not given, we need to construct an additional equation for TB. This is done by integrating the differential equation over the “half” control volume shown adjacent to the boundary in the following Figure.
13-1 Steady One-dimensional Condition(20) “Half” C.V. B I W P E Fig 1 Typical C.V. (δx)i qB B i I Δx Fig 2
13-1 Steady One-dimensional Condition(21) • Apply the principles of energy conservation • Applying the principles of energy conservation over C.V. of Fig.2 and noting that the heat flux q stands for -k(dT/dx), we get
13-1 Steady One-dimensional Condition(22) • If qB is specified in terms of a heat transfer coefficient h and a surrounding-fluid temperature Tf such that qB =h(Tf-TB) Then, the equation for TB becomes