1. CHAPTER 12 Finite-Volume (control-Volume) Method-Introduction

2. 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.

3. 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).

4. 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.

5. 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.

6. 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.

7. 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.

8. 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.

9. 12-1 Introduction(8)  Conservation form of the governing equations of fluid flow

10. 12-1 Introduction(9)  One-way and two-way coordinates : 1.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.

11. 12-1 Introduction(10) 2.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. 12-1 Introduction(11) 3.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.

13. 12-1 Introduction(12) 4.Parabolic, elliptic, hyperbolic: a)The term parabolic indicates a one way behavior, while elliptic signifies the two-way concept. b)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

14. 12-1 Introduction(13) c)A hyperbolic problem has a kind of one-way behavior, which is, however, not along coordinate directions but along special-lines called characteristics. d)A situation is parabolic if there exists at least one one-way coordinate: otherwise, it is elliptic. e)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.

15. 12-1 Introduction(14) 5.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.

16. 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

17. 12-2 An Illustrative Example(2) 1.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.

18. 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 W PE (δ x) w (δ x) e △ x Fig. 1 we

19. 12-2 An Illustrative Example(4) 2.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 x Fig. 2 WE we δx w δx e Δp

20. 12-2 An Illustrative Example(5) 3.The discrerization equation: If we evaluate the derivatives dT/dx in eq.(4) from the piecewise-linear profile, the resulting equation will be

21. 12-2 An Illustrative Example(6)

22. 12-2 An Illustrative Example(7) 4.Comments: a)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. b)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.

23. 12-2 An Illustrative Example(8) c)Further, it is important to understand that we need not use the same profile for all quantities. d)Even for given variable, the same profile assumption need not be used for all terms in the equation.

24. 12-2 An Illustrative Example(9) 5.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

25. 12-2 An Illustrative Example(10) Where S c stands for the constant part of S, while S p is the coefficient of T p. With the linearized source expression, the discretization equation will become

26. 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 (a p and neighbor coefficients a nb ) must always be positive.

27. 12-3 The Four Basic Rules(2)  Rule 3:Negative-slope linearization of the source term -When the source term is linearized as S=S C +S P T P, the coefficient S P must always be less than or equal to zero.  Rule 4:Sum of the neighbor coefficients -We require

28. CHAPTER 13 The Finite Volume Method for Diffusion Problems

29. 13-1 Steady One-dimensional Condition(1)  The Basic Equation  The Discretization Equation

30. 13-1 Steady One-dimensional Condition(2)  The Grid Spacing 1.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.

31. 13-1 Steady One-dimensional Condition(3) 2.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.

32. 13-1 Steady One-dimensional Condition(4) 3.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.

33. 13-1 Steady One-dimensional Condition(5)  The Interface Conductivity 1.The most straightforward procedure for obtaining the interface conductivity k e is to assume a linear variation of k between points P and E x EPe (δx) e (δx) e+ (δx) e-

34. 13-1 Steady One-dimensional Condition(6) If the interface e were midway between grid points, f e would be 0.5, and k e would be he arithmetic mean of k p and k E.

35. 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 q e at the interface via

36. 13-1 Steady One-dimensional Condition(8) For the composite slab between points P and E, a steady one-dimensional analysis (without sources) lead to 3.Our main objective is to obtain a good representation for the heat flux q e at the interface via

37. 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 f e =0.5; then Eq. (9) show that k e is the harmonic mean of k p and k E, rather than the arithmetic mean.

38. 13-1 Steady One-dimensional Condition(10) 4. A similar expression can be written for a W.

39. 13-1 Steady One-dimensional Condition(11) 5.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.

40. 13-1 Steady One-dimensional Condition(12)  Iteration 1.Start with a guess or estimate for the values of T at all grid points. 2.From these guessed T´s, calculate tentative values of the coefficients in the discretization equation. 3.Solve the nominally set of algebraic equations to get new values of T. 4.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.

41. 13-1 Steady One-dimensional Condition(13)  Source-Term Linearization T p *: the guess value or the previous-iteration value of T p Example 1: Given S=5-4T -S c =5, S p =-4—recommended -S c =5-4T p *, S p =0 —not impractical -S c =5+7T p *,S p =-11 —a steeper S~T relationship, will slow down the convergence

42. 13-1 Steady One-dimensional Condition(14)  Example 2: Given S=3+7T 1.S c =3,S p =7 —this is not acceptable, as it makes S p positive. The presence of a positive S p many cause divergence. 2.S c =3+7T p *, S p =0 —this is the practice one should follow. 3.S c =3+9T p *, S p =-2 —this is an artificial creative S p. It will, in general, slow down the convergence.

43. 13-1 Steady One-dimensional Condition(15)  Example 3: Given S=4-5T 3 1.S c =4-5T p * 3, S p =0 —this is the lazy-person approach. 2.S c =4, S p =-5T p * 2 —this given S~T curve is steeper than this implies.

44. 13-1 Steady One-dimensional Condition(16) This linearization represents the tangent to the S~T curve at T p * 3.Recommended method:

45. 13-1 Steady One-dimensional Condition(17) 4.S c= 4+20T p *, S p =-25T p * 2 —This givens a steeper S~T curve, which would slow down convergence S T (1) (2) (3) (4)

46. 13-1 Steady One-dimensional Condition(18)  Boundary Conditions: 1.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.

47. 13-1 Steady One-dimensional Condition(19) 2.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 T B. This is done by integrating the differential equation over the “half” control volume shown adjacent to the boundary in the following Figure.

48. 13-1 Steady One-dimensional Condition(20) BIWPE Typical C.V. “Half” C.V. B I (δx) i i Δx qBqB Fig 1 Fig 2

49. 13-1 Steady One-dimensional Condition(21) 3.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

50. 13-1 Steady One-dimensional Condition(22) 4.If q B is specified in terms of a heat transfer coefficient h and a surrounding-fluid temperature T f such that q B =h(T f -T B ) Then, the equation for T B becomes

51. 13-1 Steady One-dimensional Condition(23)  Solution of the Linear Algebra Equation (TDMA) 1.The discretization equations can be written as

52. 13-2 Unsteady One-Dimensional Condition(1)  The general Discretization Equation 1.Unsteady one-dimensional heat-conduction equation

53. 13-2 Unsteady One-Dimensional Condition(2) 2.The discretization equation WPE we

54. 13-2 Unsteady One-Dimensional Condition(3) where f is a weighting factor between 0 and 1 from Eqs.(4) and (5), we can get

55. 13-2 Unsteady One-Dimensional Condition(4)

56. 13-2 Unsteady One-Dimensional Condition(5) 1. Example: Crank-Nicolson, and Fully Implicit Schemes

57. 13-2 Unsteady One-Dimensional Condition(6) 2.Variation of temperature with time for three different schemes Explicit Fully implicit Crank-Nicolson ttt+Δt

58. 13-2 Unsteady One-Dimensional Condition(7) 3.Why would we prefer the fully implicit scheme? a)For f=0 (explicit) scheme → a p T p = a E T E 0 + a w T w 0 +( a p 0 -a E -a W ) T P 0 This means that T P is not related to other unknows such as T E or T W, but is explicitly obtainable in terms of known temperature T P 0, T E 0, T W 0.

59. 13-2 Unsteady One-Dimensional Condition(8) b)For f=0.5, the coefficient of T P 0 in eq(6) becomes a P 0 -(a E + a W )/2. For uniform conductivity and uniform grid spacing, this coefficient can be seen to be ρc(∆x/∆t)-k/ ∆x. Whenever the time step is not sufficiently small, this coefficient could become negative, with its potential for physically unrealistic results. c)For f=1, the coefficient of T P 0 in eq(6) must never become negative. It is for this reason that we shall adopt the fully implicit scheme in this book.

60. 13-2 Unsteady One-Dimensional Condition(9)  The Fully Implicit Discretitation Equation 1.a P T P =a E T E +a W T W +b Where 2. It can be seen that, as ∆t → ∞, this equation reduces to our steady-state discretization equation.

61. 13-3 Two-And Three- Dimensional Situations(1)  Discretization Equation for Two Dimensions

62. 13-3 Two-And Three- Dimensional Situations(2) y x N EW S ∆x ∆y s ew n p

63. 13-3 Two-And Three- Dimensional Situations(3) →a P T P =a E T E +a W T W +a N T N +b where

64. 13-4 Overrelaxation and Underrelaxation(1)  Overrelaxation is often used in conjunction with the Gauss-Seidel method, the resulting scheme being known as successive Over-Relaxation (SOR). With the line-by-line method, the use of overrelaxation is less common. Underrelaxation is a very useful device for nonlinear problem. It is often employed to avoid divergence in the iterative solution of strongly nonlinear equations.

65. 13-4 Overrelation and Underrelaxation(2)  The general discretization equation of the form is

66. 13-4 Overrelation and Underrelaxation(3)  At first, it should be noted that, when the iterations, that is, T P becomes equal to T P *. Eq. (4a) implies that the converged values of T do satisfy the original Eq.(1). Any relaxation scheme, of course must possess this property; the final converged solution, although obtained through the use of arbitrary relaxation factors or similar devices, must still satisfy the original discretization equation.

67. 13-4 Overrelation and Underrelaxation(4)  There are no general rules for choosing the best value of α. The optimum value depends upon a number of factors, such as the nature of the problem, the number of grid points, the grid spacing, and the iterative procedure used. Usually, a suitable value of α can be found by experience.

68. CHAPTER 14 Convection and Diffusion

69. 14-1 Convection-Diffusion Term  The convection term has an inseparable connection with the diffusion term, and therefore, the two terms need to be handled as one unit.  Governing Equations

70. 14-2 Steady One-Dimensional Convection and Diffusion(1)  Governing Equations

71. 14-2 Steady One-Dimensional Convection and Diffusion(2)  A Preliminary Derivation 1.Integration of Eq (1) over the C.V. shown in Fig.1 gives WPE we (δx) w (δx) e C.V.

72. 14-2 Steady One-Dimensional Convection and Diffusion(3) 2.Diffusion term: The same way of chapter 9. 3.Convection term: The factor ½ arises from the assumption of the interfaces being midway. Now, Eq (1) can be written as

73. 14-2 Steady One-Dimensional Convection and Diffusion(4) 4. a) b)Both have the same dimensions; F indicates the strength of the convection (or the flow), while D is the diffusion conductance. c)D always remains positive, F can take either positive or negative values depending on the direction of the fluid flow.

74. 14-2 Steady One-Dimensional Convection and Diffusion(5) 5.Discretization Equation:

75. 14-2 Steady One-Dimensional Convection and Diffusion(6) 6.Discussion: a)Since by continuity Fe=Fw, we do get the property a P = a E + a W b)Eq (3) is also known as the central-difference scheme and is the natural outcome of a Taylor- series formulation.

76. 14-2 Steady One-Dimensional Convection and Diffusion(7) c)Example: D e =D w =1 and F e =F w =4 Consider two sets of values: i.If Φ E =200 and Φ W =100, then Φ P =50! ii.If Φ E =100 and Φ W =200, then Φ P =250! Since, in reality, cannot fall outside the range of 100~200 established by its neighbors, these results are clearly unrealistic.

77. 14-2 Steady One-Dimensional Convection and Diffusion(8) d)Eqs (3a)-(3c) indicate that the coefficients could, at times, become negative. When |F| exceeds 2D, then, depending on whether F is positive or negative, where is a possibility of a E or a W becoming negative. This will be a violation of one of the basic rules.

78. 14-2 Steady One-Dimensional Convection and Diffusion(9)  The Upwind Scheme 1.The upwind scheme recognizes that the weak point in the preliminary formulation is the assumption that the convected property Φ e at the interface is the average of Φ E and Φ P, and it proposes a better prescription. The formulation of the diffusion term is left unchanged, but the convection term is calculated from the following assumption:

79. 14-2 Steady One-Dimensional Convection and Diffusion(10) The value of Φ at an interface is equal to the value of Φ at the grid point on the upwind side of the face. thus, Φ e = Φ p if Fe>0—(4a) and, Φ e = Φ E if Fe<0—(4b) The value of Φ w can be defined similarly. WPE we

80. 14-2 Steady One-Dimensional Convection and Diffusion(11) 2.We shall define 【 A,B 】 to denote the greater of A and B. Then, the upwind scheme implies

81. 14-2 Steady One-Dimensional Convection and Diffusion(12) 3.The discretization equation becomes

82. 14-2 Steady One-Dimensional Convection and Diffusion(13) 4.Discussion: It is evident from Eqs.(6) that no negative coefficients would arise, thus, the solutions will always be physically realistic.

83. 14-2 Steady One-Dimensional Convection and Diffusion(14)  The Exact Solution can be solved exactly if Γ is taken to be constant.(ρu is constant)

84. 14-2 Steady One-Dimensional Convection and Diffusion(15) If a domain 0 ≦ x ≦ L is used, with the boundary conditions

85. 14-2 Steady One-Dimensional Convection and Diffusion(16) 2.The nature of the exact solution: фLфL ф0ф0 0L ф x P>>1 P=1 P=0 P= -1 -P>>1 Fig.2 Exact solution for the one-dimensional convection- diffusion problem

86. 14-2 Steady One-Dimensional Convection and Diffusion(17) a)In the limit of zero Peclet number, we get the pure-diffusion (or conduction) problem, and the ф~x variation is linear. b)When the flow is in the positive x direction (i.e., for positive values of P), the values of ф in the domain seem to be more influenced by the upstream value ф 0. c)For a large positive value of P, the value of ф remains very close to the upstream value ф 0 over much of the domain.

87. 14-2 Steady One-Dimensional Convection and Diffusion(18) d)When the fluid flows in the negative x direction, Φ L becomes the upstream value, which dominates the values of Φ in the domain. e)When a large negative P, the value Φ of over most of the region is very nearly equal to Φ L.

88. 14-2 Steady One-Dimensional Convection and Diffusion(19) 3.Implications: a)It is easy to see why our preliminary derivation failed to give a satisfactory formulation. The Φ ~x profile is far from being linear except for small values of |P|. b)Where |P| is large, the value of Φ at x=L/2 (the interface is nearly equal to the value of Φ at the upwind boundary. This is precisely the assumption made in the upwind scheme; but there it is used for all values of |P|, not just for large value.

89. 14-2 Steady One-Dimensional Convection and Diffusion(20) c)Where |P| is large, dΦ/dx is nearly zero at x=L/2. Thus the diffusion is almost absent. The upwind scheme always calculates the diffusion term from a linear Φ~x profile and thus overestimates diffusion at large value of |P|.

90. 14-2 Steady One-Dimensional Convection and Diffusion(21)  The exponential scheme 1.It is useful to consider a total flux J that is made of the convection flux ρu  and the diffusion flux -Γd  /dx. Thus, with this definition eq (1) becomes which, when integrated over the C.V. shown in fig.1 gives

91. 14-2 Steady One-Dimensional Convection and Diffusion(22) 2.The substitution of eq. (9) into eq. (8) would give the expression for J e :

92. 14-2 Steady One-Dimensional Convection and Diffusion(23) 3.Finally, substitution of eq. (11) and a similar expression for J w into eq. (10) leads to

93. 14-2 Steady One-Dimensional Convection and Diffusion(24) 4.Discussions: a)These coefficients expressions define the exponential scheme. When used for the steady one-dimensional problem, this scheme is guaranteed to produce the exact solution for any value of the Peclet number and for any number of grid points.

94. 14-2 Steady One-Dimensional Convection and Diffusion(25) b)Despite its highly desirable behavior, it is not widely used because i.Exponentials are expensive to compute. ii.Since the scheme is not for two- or three-dimensional situations, nonzero sources, etc., the extra expense of computing the exponentials does not seem to be justified.

95. 14-2 Steady One-Dimensional Convection and Diffusion(26)  The Hybrid scheme 1.To appreciate the connection between the exponential scheme and the hybrid scheme, we shall plot a E /D e vs. p e as follows:

96. 14-2 Steady One-Dimensional Convection and Diffusion(27) 5 4 3 2 Exact Fig. 3 12345 pepe

97. 14-2 Steady One-Dimensional Convection and Diffusion(28) 2.From Fig 3, we can get

98. 14-2 Steady One-Dimensional Convection and Diffusion(29) 3.The hybrid scheme is made up of these three straight lines of Fig.3, so that

99. 14-2 Steady One-Dimensional Convection and Diffusion(30) 4.These expressions can be combined into a compact form as follows:

100. 14-2 Steady One-Dimensional Convection and Diffusion(31) 5.The significance of the hybrid scheme can be understood by observing that a)It is identical with the central-difference scheme for –2 < p e < 2 b)Outside this range it reduces to the upwind scheme in which the diffusion has been set equal to zero. c)The name hybrid is indicative of a combination of the central-difference and upwind scheme, but it is best to consider the hybrid scheme as the three- line approximation to the exact curve, shown in Fig.3

101. 14-2 Steady One-Dimensional Convection and Diffusion(32) 6.The convection-diffusion discretization for the hybrid scheme can now be written as

102. 14-2 Steady One-Dimensional Convection and Diffusion(33)  The Power-Law scheme 1.It can be seen from Fig.3, that the departure of the hybrid scheme from the exact curve is rather large. A better approximation to the exact curve is given by the Power-law scheme.

103. 14-2 Steady One-Dimensional Convection and Diffusion(34) 2.The Power-law expressions for a E can be written as

104. 14-2 Steady One-Dimensional Convection and Diffusion(35)  A Generalized Formulation 1.The general convection-diffusion formulation can be written as

105. 14-2 Steady One-Dimensional Convection and Diffusion(36) 2.The function A(|p|) for different schemes

106. 14-3 Discretization Equation for Two Dimensions(1)  Discretization Equation of 2-D where J x and J y are the total (convection plus diffusion) fluxes defined by

107. 14-3 Discretization Equation for Two Dimensions(2)  The integration of eq(1) over the c.v. shown in Fig.1 would give y x N EW S ∆x ∆y ew n p JnJn JeJe JsJs JwJw Fig.2 s

108. 14-3 Discretization Equation for Two Dimensions(3)

109. 14-3 Discretization Equation for Two Dimensions(4)  In a similar manner, we can integrate the continuity eq over the c.v. and obtain

110. 14-3 Discretization Equation for Two Dimensions(5) 

111. 14-3 Discretization Equation for Two Dimensions(6)  The final discretization equation

112. 14-3 Discretization Equation for Two Dimensions(7) where

113. 14-4 One-Way Space Coordinate(1)  Time is a one-way coordinate. The convection- diffusion formulation reveals that a space coordinate can also become a one-way.  What makes a space coordinate One-Way? When the Peclet number exceeds10, the Power- law scheme will set the downstream-neighbor coefficient equal to zero. (the hybrid scheme does this for a Peclet number greater than 2.)

114. 14-4 One-Way Space Coordinate(2)  Suppose that, in the 2-D situation shown in Fig.2, there is a high flow rate in the positive x direction. Then, for all the grid points along a y-direction line, the coefficient a E will be zero. In other words,  p will be dependent on  W,  N, and  S, but not on a E. Thus the x coordinate will become a one-way coordinate since the  value at any point will be uninfluenced by any of the downstream values.

115. 14-4 One-Way Space Coordinate(3) w N p S E Fig.2

116. 14-4 One-Way Space Coordinate(4)  The outflow Boundary Conditions At the outflow boundary shown in Fig.3, for example, one may not know the temperature or the heat flux. How can we then solve the problem? The answer is surprisingly simple: No boundary- condition information is needed at an outflow boundary. For all grid points p next to the outflow boundary, the coefficient a E will be zero, and hence no boundary values will be needed. In other words, the region near the outflow boundary exhibits, for large Peclet number, local one-way behavior.

117. 14-4 One-Way Space Coordinate(5) Inflow boundary outflow boundary Fig 3 N P S E W

118. 14-4 One-Way Space Coordinate(6)  A particularly bad choice of an outflow- boundary location is the one in which there is an “inflow” over a part of it. An example of this shown in Fig.4. For such a bad choice of the boundary, no meaningful solution can be obtained.

119. 14-4 One-Way Space Coordinate(7) BadGood Fig. 4

120. CHAPTER 15 Calculation of The Flow Field

121. 15-1 Need for a Special Procedure  The real difficulty in the calculation of the velocity field lies in the unknown pressure field. The pressure gradient forms a part of source term for a momentum equation.

122. 15-2 Some Related Difficulties(1)  Representation of the Pressure-Gradient Term 1.To integrate –dp/dx over the control volume shown in Fig.1, we can get p w -p e. WPE we c.v. Fig. 1 x

123. 15-2 Some Related Difficulties(2) 2.To express p w -p e in terms of the grid-point pressure, we may assume a piecewise-linear profile for pressure. Therefore, we can get This means that the momentum equation will contain the pressure difference between two alternate grid points, and not between adjacent ones.

124. 15-2 Some Related Difficulties(3) 3.There is another implication that is far more serious. It can be seen from Fig.2, where a pressure field is proposed in terms of the grid- point values of pressure. p=100500100500100500 Fig. 2 such a zig-zag field cannot be regarded as realistic; but for any grid point p, the corresponding p W -p E can be seen to zero, since the alternate pressure values are everywhere equal.

125. 15-2 Some Related Difficulties(4)  Representation of the continuity Equation If we integrate the continuity equation over the c.v. shown in Fig1, we have u e -u w =0

126. 15-2 Some Related Difficulties(5)  Once again, the use of a piecewise-linear profile for u and of the midway locations of the control-volume faces leads to Thus, the discretized continuity equation demands the quality of velocities at alternate grid points and not at adjacent ones.

127. 15-3 A Remedy : The staggered Grid(1)  The difficulties described so far can be resolved by recognizing that we do not have to calculate all the variables for the same grid points. We can, if we wish, employ a different grid for each dependent variable.

128. 15-3 A Remedy : The staggered Grid(2) Staggered grid: 1.The velocity components are calculated for the points that lie on the face of the control volume, as shown in Fig.3. 2.Other variables are calculated for the grid points(small circles). Fig.3 →=u i,, ↑=v i, =other variables

129. 15-3 A Remedy : The staggered Grid(3) 3.The important advantages are twofold: a)For a typical c.v. shown in Fig.3, it is easy to see that the discretized continuity equation would contain the difference of adjacent velocity components, and this would prevent a wavy velocity field. b)The second important advantage of the staggered grid is that the pressure difference between two adjacent grid points now becomes the natural driving force for the velocity component located between these grid points.

130. 15-4 The Momentum Equations(1)  The discretization equation (2-D) can be written as

131. 15-4 The Momentum Equations(2) The pressure gradient is not included in the source-term S C and S P. e Fig.4a c.v. for u n Fig.4b c.v. for v

132. 15-4 The Momentum Equations(3)  The momentum equation can be solved when the pressure field is given or is somewhat estimated. Unless the correct pressure field is employed, the resulting velocity field will not satisfy the continuity equation. Such an imperfect field based on a guessed pressure field p * will be denoted by u *,v *,w *. This “starred” velocity field will result from the solution of the following equations:

133. 15-5 The Pressure and Velocity Corrections(1)  One aim is to find a way of improving the guessed pressure p * such that the resulting starred velocity field will progressively get closer to satisfying the continuity equation. Let us propose that the correct pressure p is obtained from where p’ will be called the pressure correction.

134. 15-5 The Pressure and Velocity Corrections(2)  Similarly, we can get

135. 15-5 The Pressure and Velocity Corrections(3)  If (1a)-(2a), we have At this point, we shall boldly decide to drop the term ∑a nb u' nb from eq.(5) and the result is

136. 15-5 The Pressure and Velocity Corrections(4)  Eq.(6) will be called the velocity-correction formula, which can also be written as

137. 15-6 The Pressure-Correction Equation(1)  The discretization eq. of continuity eq:

138. 15-6 The Pressure-Correction Equation(2)  Substituting eqs(7) into eq(8), we can obtain

139. 15-6 The Pressure-Correction Equation(3)  If b is zero, it means that the stared velocity, in conjunction with the available value of (ρ p 0 - ρ p ), do satisfy the continuity equation and no pressure correction is needed. The term b this represents a “ mass source”, which the pressure corrections must be annihilate.

140. 15-7 The SIMPLE Algorithm(1)  SIMPLE stands for Semi-Implicit Method for Pressure-Linked Equations.  Sequence of operations: 1.Guess the pressure p *. 2.Solve the momentum eqs, such as (2a)~(2c), to obtain u *,v *,w *. 3.Solve p' eq. 4.Calculate p=p*+p ' 5.Calculate u,v,w from (7a)~(7c) 6.Solve other Φ‘s. 7.Treat the corrected pressure p as a new guessed pressure p *, return to step (2) and repeat the whole procedure until a converged solution is obtain.

141. 15-7 The SIMPLE Algorithm(2)  Discussion of the Pressure-Correction Equation 1.If expressions such as a nb u' nb were retained, they would have to be expressed in terms of the pressure corrections and the velocity corrections at the neighbors of u nb. The omission of the ∑a nb u' nb term enables us to cast the p equation in the same form as the general Φ equation, and to adopt a sequential, one-variable-at-a-time, solution procedure.

142. 15-7 The SIMPLE Algorithm(3) 2.The words semi-implicit in the name SMPLE have been used to acknowledge the omission of the term ∑a nb u' nb. This term represents on indirect or implicit influence of the pressure correction on velocity. Pressure corrections at nearby locations can alter the neighboring velocities and thus a velocity correction at the point under consideration. We do not include this influence and thus work with a scheme that is only partially, and not totally, implicit.

143. 15-7 The SIMPLE Algorithm(4) 3.The omission of any term, would of course, be unacceptable if it meant that the ultimate solution would not be true solution of the discretized forms of the momentum and continuity equation. It also happens that the converged solution given by SIMPLE does not contain any error resulting from the omission of ∑a nb u' nb

144. 15-7 The SIMPLE Algorithm(5) 4.The mass source b serves as a useful indicator of the convergence of the fluid-flow solution. The iterations should be continued until the value of b everywhere becomes sufficiently small. 5.The pressure-correction can be seen to be merely an intermediate algorithm that leads us to the correct pressure field, but has no direct effect on the final solution.

145. 15-7 The SIMPLE Algorithm(6) 6.The pressure-correction equation is prone to divergence unless some under-relaxation is used. A generally successful practice can be described as follows: we under-relax u *,v *,w * while solving the momentum equations (with a relaxation factor α=0.5). Also, we employ It is not implied that these values are the optimum ones or will even produce divergence for all problems. The optimum relaxation factor values are usually problem-dependent.

146. 15-7 The SIMPLE Algorithm(7)  Boundary Conditions for the Pressure- Correction Equation 1.There are two kinds of conditions at a boundary. Either the pressure at the boundary is given (and the velocity is unknown) or the velocity component normal to the boundary is specified. 2.Given pressure at the boundary: If the guessed pressure field p * is arranged such that at a boundary p * =p given, then the value of p' at the boundary will be zero.

147. 15-7 The SIMPLE Algorithm(8) 3.Given normal velocity at the boundary: As shown in Fig.1, the velocity u e is given. It the derivation of the p' equation for the c.v. shown, the flow rate across the boundary face should not expressed in terms of u e * and a correction, but in terms of u e itself u e( (given) p N S E Fig.1 Then, p' E will not appear or a E will be zero in the p' equation. Thus no information about p' E will be needed. *Note:

148. 15-7 The SIMPLE Algorithm(9)  The Relative Nature of Pressure 1.Since no boundary pressure is specified and all the boundary coefficients such as a E will be zero, the p' equation is left without any means of establishing the absolute value of p'. The coefficients of the p' equation are such that a P =∑a nb ; this means that p' and p' +c (c is an arbitrary constant) would both satisfy the p' equation.

149. 15-7 The SIMPLE Algorithm(10) 2.Only difference in the pressure are meaningful (p=p*+p'), and these are not altered by an arbitrary constant to the p' field. Pressure is then a relative variable, not a absolute one.

150. 15-7 The SIMPLE Algorithm(11) 3.In many problems, the value of the absolute pressure is much larger than the local differences in pressure that are encountered. If the absolute values of pressure were for p, round-off errors would arise in calculating differences like p p -p E. It is, therefore, best to set p=0 as a reference value at a suitable point and to calculate all other values of p as pressures relation to start from p'=0 as guess for all point, so that the solution for p' does not acquire a large absolute value.

151. 15-8 A Revised Algorithm: SIMPLER(1)  Motivation 1.SIMPLER stands for SIMPLE Revised. 2.In most cases, it is reasonable to suppose that the pressure-correction equation does a fairly good job of correcting the velocities, but a rather poor job of correcting the pressure.

152. 15-8 A Revised Algorithm: SIMPLER(2)  The Pressure Equation 1.The momentum equation is first written as

153. 15-8 A Revised Algorithm: SIMPLER(3) 2.Although the pressure eq and p' eq are almost identical, there is one major difference: No approximations have been introduced in the derivation of the pressure equation. Thus, if a correct velocity field were used to calculate the pseudo-velocities, the pressure equation would at once give the correct pressure.

154. 15-8 A Revised Algorithm: SIMPLER(4)  The SIMPLER Algorithm The revised algorithm consists of solving the pressure equation to obtain the pressure field and solving the pressure-correction equation only to correct the velocities.

155. 15-8 A Revised Algorithm: SIMPLER(5)  The sequence of operations can be stated as: 1.Start with a guessed velocity field. 2.Calculate pseudo-velocity û,v,ŵ from eqs (1),(2). 3.Solve pressure equation by eq.(4) to obtain the pressure field. 4.Treat this pressure field as p *, solve momentum eq to obtain u *,v *,w * solve p' eq. 5.Solve p' eq 6.Correct the velocity field (u e =u * +d e (p' p -p' e ),etc), but do not correct the pressure. 7.Solve other Φ‘s if necessary. 8.Return to step 2 and repeat until convergence.

156. 15-8 A Revised Algorithm: SIMPLER(6)  Discussion 1.In general, since the pressure-correction equation produces reasonable velocity fields, and the pressure equation works out the direct consequence of a given velocity field, convergence to the final solution should be much faster.

157. 15-8 A Revised Algorithm: SIMPLER(7) 2.In SIMPLE, a guessed pressure field plays an important role. On the other hand, SIMPLER does not use guessed pressures, but extracts a pressure field from a given velocity field. 3.Because of the close similarity between the pressure equation and the pressure-correction equation, the discussion in previous section about boundary conditions for p eq. is also relevant to the pressure equation.

158. 15-8 A Revised Algorithm: SIMPLER(8) 4.Although SIMPLER has been found to give faster convergence than SIMPLE, it should be recognized that one iterations of SIMPLER involves more computational effort. Since SIMPLER requires fewer iterations for convergence, the additional effort per iteration is more than compensated by the overall saving of effort

159. 15-9 The SIMPLEC Algorithm  SIMPLEC stands for SIMPLE-Consistent.  It follows the same steps as the SIMPLE algorithm.  The u-velocity correction equation of SIMPLEC is given by

160. 15-10 Convergence Criterion  a p Φ p =∑a nb Φ nb +b  Residual R=∑a nb Φ nb +b- a p Φ p  Obviously, when the discretization eq is satisfied, R will be zero. A suitable convergence criterion is to require that the largest value of |R| be less than a certain small number.