a calculation procedure for solution of incompressible ...

Numerical Heat Transfer, Part B, 38:93^131, 2000 Copyright # 2000 Taylor & Francis 1040-7790/00 $12.00 + .00

A CALCULATION P ROCEDUR E FOR SOLUTION OF INC OM P RESSIBLE NAVIER- STOK ES EQUATIONS ON CURVILINEAR NON STAGGERED GRIDS S. Ray Department of Mechanical Engineering, Jadavpur University, Calcutta 700 032, India

A. W. Date Department of Mechanical Engineering, Indian Institute of Technology, Bombay, Powai, Mumbai 400 076, India A recently proposed pressure-correction algorithm for solution of incompressible Navier-Stokes equations on nonstaggered grids introduced the notion of smoothing pressure correction to overcome the problem of checkerboard prediction of pressure [9]. The algorithm was derived for equations in Cartesian coordinates. In this article, the algorithm is extended to solution of Navier-Stokes equations in general curvilinear coordinates. By way of application, two cavity ýow problems and two internal ýow problems are solved. Comparisons with benchmark solutions or experimental data and (or) previous solutions employing staggered grids are made to validate the calculation procedure.

1

INTROD UCTION

Solution of Navier-Stokes equations on nonstaggered grids offers considerable advantages over that in which the equations are solved on staggered grid. For example, to the uninitiated, discretization of transport equations on nonuniform grids and in general curvilinear coordinates becomes easier on nonstaggered grids, since all scalar and vector quantities are de¢ned (or stored) at the same grid location (or node) and bookkeeping requirements are considerably reduced. Straightforward ¢nite differencing of Navier Stokes equations on nonstaggered grid, however, leads to checkerboard (or zigzag) prediction of pressure [1, 2]. The nature of this zigzag prediction in physical problems is demonstrated by Date [3]. Its notional appearance is shown in [2].1 Such nonphysical prediction Received June 14, 1999; revised January 13, 2000. One of us, Dr. Subhashis Ray, would like to thank the authorities of Jadavpur University for granting leave during this research work. Address correspondence to Dr. Subhashis Ray, Application Software and R&D Laboratory, Department of Mechanical Engineering, Jadavpur University, Calcutta 700 032, India. E-mail: [email protected] 1 It must be noted that Patankar [2] considers the case in which the true pressure distribution is constant. Date [3], however, has shown that in a real physical problem, if the true pressure distribution were to be constant or linear, zigzag prediction should not occur. To that extent, Patankar’s illustration of zigzagness is only notional. 93

94

S. RAY AND A. W. DATE

NOM ENCLATUR E ai A Bj Ci d D

Dh e G J keq L p P Pr qf r Ruj Ra Re S T ui Ui UL Um

cell face area perpendicular to xi direction coef¢cients of f in the discretized equation body force in the jth momentum equation constants (i ˆ m, 1, 2) diameter diffusion coef¢cient in the f conservation equation; also, source term in the discretized conservation equation hydraulic diameter turbulent kinetic energy production of turbulent kinetic energy Jacobian of the coordinate transformation equivalent thermal conductivity length of the side of the inclined cavity pressure Peclet number Prandlt number £ux of f radius residuals in the discretized uj momentum equation Rayleigh number Reynolds number source term in the f conservation equation temperature Cartesian velocity components (i ˆ 1, 2, 3) contravariant velocity in the xi direction velocity of the driven lid mean velocity through the duct

xi Xo a b

d e ev y G m n r s t xi f Superscripts i l m o

Cartesian coordinates (i ˆ 1, 2, 3) length of the axisymmetric constriction underrelaxation parameter geometric coef¢cients; Also, angle of inclination of the inclined side wall height of the axisymmetric constriction dissipation vertical eccentricity angle from the vertical position diffusivity dynamic viscosity kinematic viscosity ( ˆ m/r) density turbulent Prandtl number stress curvilinear coordinates (i ˆ 1, 2, 3) general variable

¡

inner iteration number mean outer correction averaged value

Subscripts eff m nb s W,E,S, etc. w,e,s, etc.

effective mass conserving neighboring points smoothing pertaining to nodes pertaining to cell faces

0

occurs because the determination of discretized velocities through momentum equations at a node becomes unlinked from the pressure at the node. This problem arising out of straightforward control-volume discretization is called the problem of pressure^velocity decoupling. The cure for the decoupling problem was ¢rst proposed by Rhie and Chow [4], in which the velocities at the control-volume faces (henceforth called cell faces) arising out of discretization of convection and mass conservation equations were modeled in such a way that they accounted for the departure from one-dimensional

SOLUTION OF INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

95

linearity in the local variation of pressure [3]. Later, Peric¨ [5] introduced another model expression, different in appearance but reducible to the Rhie and Chow [6] form. Still later, Date [3] introduced a cure in which the cell-face velocities were calculated as arithmetic means of the neighboring nodal velocities (as in the staggered-grid procedure of Patankar and Spalding [1]), but the pressure gradient appearing in the nodal momentum equations was evaluated by interpolation. This interpolation was shown to be fourth-order accurate. Both the above type of cures, while being satisfactory in several £ow situations [7], lead to certain embarrassment when the pressure variations are steep and when grid spacings are coarse.2 Miller and Schmidt [8], for example, showed that in certain regions of a channel £ow with a backward-facing step, the modeled cell-face velocities not only do not equal with those predicted by the staggered grid procedure, they do not even remain bounded between the neighboring nodal velocities [3]. Similarly, for £ow in a square-sectioned duct with a 180¯ sharp bend, it was found3 that the fourth-order-accurate pressure gradient interpolation is inadequate in the bend region and zig-zag pressure distribution is predicted, unless a very ¢ne mesh is used in the bend region. Noting the above type of embarrassment, Date [9] traced the problem of checkerboard pressure prediction to the manner in which the pressure-correction equation is derived. This new exploration lead to the recognition of the need to de¢ne a smoothing pressure correction. Date [9] showed that introduction of the latter obviates the need for complex modeling of cell-face velocity 4 as well as that for higher-order evaluation of the nodal pressure gradient. The importance of curing the checkerboarding problem through smoothing pressure correction lies in the fact that the nodal momentum equations can now be straightforwardly discretized through conventional means, while the application of smoothing pressure correction itself is reduced to a simple additive operation that is easy to implement not only on orthogonal grids, but also on general curvilinear and even on unstructured grids, as will become apparent in the present article. The main purpose of this article is to demonstrate the application of the smoothing pressure correction using general curvilinear coordinates and a nonstaggered grid. Section 2 describes the essential features of discretization of the Navier-Stokes and scalar transport equations along with the derivation of the new pressure-correction equation. Section 3 reports the solutions to two cavity £ow problems and two internal £ow problems. Finally, the conclusions are reported in Section 4.

2

Note that the notion of coarseness here is with respect to pressure variations and not with respect to velocity variations. In many £ows, the regions of steep pressure and velocity variations do not always coincide. Most computational £uid dynamics (CFD) analysts can anticipate regions of steep velocity variations, but not necessarily those of steep pressure variations. 3 unpublished work by Date. 4 It must be noted that the effect of this complex modeling on the ``false’’ numerical diffusion does not appear to have been ascertained in the literature.

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S. RAY AND A. W. DATE

2 2.1

M ATH EM ATICAL FOR M ULATION

D iscretization of Transport Equations

The Navier-Stokes and scalar transport equations in general curvilinear coordinates (xi , i ˆ 1, 2, 3) can be written as J

3 X

t

…rf† ‡

iˆ1

rUi f ¡ Di

xi

f xi

ˆ Sf

…1†

where D i and Sf are interpreted in Table 1. Further de¢nitions are given in Table 2. The tables show that when f ˆ 1, the mass conservation equation is retrieved. The momentum equations are written for Cartesian velocity components (f ˆ uj ). It is important to note that the term Di f xi represents the dummy diffusion normal to xi ˆ constant surface. The true Cartesian diffusion is included in the source terms

Table 1. Meaning of Di and Sf f

Di

Sf

1 uj

0 meff a2i J

T

Geff T a2i J Geff k a2i J Geff a2i J

0 j 3 ¡S3kˆ1 b k p xk ‡ Siˆ1

e e

3 k S kˆ1 bi tjk ¡ Di uj

3 k xi xi Skˆ1 bi qT k ‡ Di T k e 3 3 xi G ¡ re ¡ S iˆ1 Skˆ1 bi qk ‡ Di e xi 3 3 k e …e k†…C1 G ¡ C2 re† ¡ Siˆ1 Skˆ1 bi qk ‡ Di

¡S3iˆ1

xi

xi ‡ Bj

xi

xi

Table 2. Terms in Table 1 Symbol

Meaning

De¢nition

Ui

Contravariant £ow velocity

S3jˆ1 b i uj

j

Stresses

f

…meff J†Skˆ1 bk uj

qk

Flux of f

k ¡…Geff J†Sjˆ1 bj f xj

meff

Effective viscosity

f Geff

m ‡ mt ˆ m ‡ cm re2 e (m/Pr) ‡ cm re2 …esf †

bij

Effective diffusivity for scalar f Geometric coef¢cients

J

Cell volume

ai

Cell-face area

G Bj

Production of T.K.E Body force

ti

j

f

… xj

3

i

3

j

xk ‡ b k ui

xk †… xk xi † ¡ … xj xi †… xk k 3 Skˆ1 bi xk xi , i ˆ 1, 2, or 3 2 1 2 3 j Sjˆ1 bi 3 j tij Skˆ1 bk J … ui xk †

xk ¡ 23 rdij e

xk †, k6ˆi, j

SOLUTION OF INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

97

Figure 1. A typical control volume in curvilinear coordinates.

f through gradients of stresses, tij , and scalar £uxes qk .5 For turbulent £ows, the e^e (k^e) model of turbulence is employed with model constants Cm ˆ 0.09, C1 ˆ 1.44, C2 ˆ 1.92, se ˆ 1.0, se ˆ 1.3, and sT ˆ 0.9 [11, 12]. With respect to the control volume surrounding node P (see Figure 1), integration of Eq. (1) over the control volume yields the following discretized equations:

AP fP ˆ AE fE ‡ AW fW AN fN ‡ AS fS ‡ AT fT ‡ AB fB ‡ Df

5

…2†

This is done to provide for coding £exibility if higher-order turbulence models, such as the nonlinear model of Craft et al. [10], for example, are to be employed.

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S. RAY AND A. W. DATE

where AP ˆ AE ‡ AW ‡ AN ‡ AS ‡ AT ‡ AB ‡ D ˆ J Sf ‡

JroP Dt

roP foP Dt

…3† …4†

Coef¢cients Ai (i ˆ E, W, N, S, T, B) represent the convective and diffusive in£uences. Thus, AE and AW , for example, are given by AE ˆ D1e ¡

…rU 1 †e ¤ ‰A ¡ 1 ‡ Max…¡Pe 0†Š Pe

…5†

…rU1 †w ¤ ‰A ¡ 1 ‡ Max…Pw 0†Š Pw

…6†

AW ˆ D1w ¡

where P ( ˆ rU1 D1 ) represents local Peclet number and A* assumes the following relations in different schemes for convective terms: A¤ ˆ 1 ˆ Max‰0 1 ¡ 0 5jPjŠ

upwind difference scheme hybrid differencing scheme 5

ˆ Max‰0 …1 ¡ 0 1jPj† Š ˆ jPj exp…jPj ¡ 1†

power law scheme exponential differencing scheme

…7a† …7b† …7c†

…7d†

In the present article, all the calculations are performed with a power law scheme. In Eq. (5), contravariant £ow velocity at cell face ``e,’’ for example, is calculated as where

U1e ˆ b11e u1e ‡ b21e u2e ‡ b31e u3e

…8†

U1e ˆ 0 5…u1P ‡ u1E † u2e ˆ 0 5…u2P ‡ u2E † u3e ˆ 0 5…u3P ‡ u3E †

…9a† …9b† …9c†

Evaluations of AN , AS , AT , AB are straightforward extensions of the above. The details of the boundary conditions are given in Appendix A. In order to discretize the source terms, pressure and other scalar variables are required at the cell faces. They are obtained from the nodal values by linear interpolation.

2.2

Pressure- Correction Equation

In the SIMPLE algorithm, Eqs. (2) are solved iteratively in a sequential manner for each f. Underrelaxation of the equations is then often required. Thus, Eq. (2) for velocity component uj reads as ˆ ul‡1 j

a AP uj

nb

Anb ul‡1 j nb ¡

3 kˆ1

b jk

pl‡1 ‡ …1 ¡ a†ulj xk

…10†

SOLUTION OF INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

99

where nb denotes summation over neighbors of node P, and superscript 1 refers to the iteration number. In order to derive the pressure-correction equation, Eqs. (10) are subjected to a mass conservation constraint [that is, Eq. (1) with f ˆ 1]. Then, omitting the transient term for brevity, one obtains 3

xi

iˆ1

3

rUil‡1 ˆ

3

xi

iˆ1

rbji ul‡1 j

ˆ0

jˆ1

…11†

Substituting for ul‡1 from Eq. (10), the above equation yields j 3 iˆ1

3

xi

jˆ1

rb ji ulj ˆ

3

3

xi

iˆ1 3

ˆ

xi

iˆ1

iˆ1

rb ji a AP uj ulj ¡ AP uj

3

Anb ul‡1 j nb ‡

nb

kˆ1

b jk

pl‡1 xk

rUil

…12†

Now, writing ˆ ulj ‡ Uj0 ul‡1 j

…13†

pl‡1 ˆ pl ‡ p0m

…14†

where u0j is the velocity correction and p0m is the mass conserving pressure correction. It can be readily shown that 3 iˆ1

3

xi

jˆ1

rb ji a 3 j p0m b AP uj kˆ1 k xk

3

ˆ

iˆ1

xi

rUil ¡

3

3

iˆ1

xi

jˆ1

rbji a Ru AP uj j

…15†

where Ruj are the residuals in the discretized momentum equations, given by Ruj ˆ AP uj ulj ¡

nb

Anb ulj nb ‡

3 kˆ1

b jk

pl xk

…16†

Equation (15) thus represents the mass-conserving pressure-correction equation on nonstaggered grids. The left-hand side of this equation involves cross-derivatives of p0m . However, to afford a Poisson equation form, only normal derivatives are retained by setting k ˆ i. Since Eq. (15) is only an approximate estimator of p0m , the above violation is not a serious one. Moreover, when convergence is achieved, the neglected terms are expected to be negligible. Thus, Eq. (15) becomes 3 iˆ1

3

xi

jˆ1

2

rb ji a p0m AP uj xi

3

ˆ

iˆ1

xi

rU il ¡

3 iˆ1

3

xi

jˆ1

rb ji a ¤ R AP uj uj

…17†

where R¤uj ˆ AP uj ulj ¡

nb

Anbulj nb ‡ bji

pl x

…18†

When Eq. (17) is discretized, R¤uj are required at the cell-face locations (namely, at e, w, n, s, t, and b). At these locations Anb and therefore AP uj coef¢cients are not

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S. RAY AND A. W. DATE

available when computing on nonstaggered grids. Thus, it is of interest to calculate R¤uj without evaluating Anb coef¢cients at the cell faces. 2.2.1. Evaluation of R¤uj . It is now possible to reinterpret the proposals of Peric¨ [5] and Rhie and Chow [4]. Thus, the former author evaluated R¤uj at cell face ``e,’’ for example, as [6] R¤uj

e

ˆ AP uj e

uj P ‡ uj E 1 ¡ 2 2

nb

Anb ulj nb

P

AP uj P

nb

‡

Anb ulj nb

E

‡

AP uj E

bj1 pl AP uj e x1

e

…19† with 1 1 1 1 ˆ ‡ AP uj e 2 AP uj P AP uj E

…20†

In Eq. (19), Anb terms are now required at grid nodes (P and E in this case), and these are known. Rhie and Chow [4], on the other hand, evaluated R¤uj as R¤uj

e

pl x1

ˆ b j1

pl

where the average pressure gradient interpolation as pl x1

e

ˆ

e

pl x1

1 2

¡

pl x1

…21†

e

x1 is evaluated by one-dimensional

P

‡

pl x1

…22†

E

Two comments are now in order, 1.

If R¤uj P , and R¤uj E are taken as zero (as they must be at full convergence), then Eq. (19) reduces to R¤uj

2.

e

bj1 pl AP uj x1

ˆ AP uj e

e

¡

j 1 b1 2 AP uj

pl x1

P

‡

bj1 AP uj

pl x1

E

…23†

The above equation has remarkable similarity to Eq. (21). In fact, the two equations are identical if bj1 and AP uj at P, E, and e, respectively, are taken to be equal. Substitution of Eq. (19) or (21) in Eq. (17) reduces the p0m equation to the form 3 iˆ1

3

xi

jˆ1

2

rbji a p0m ˆ AP uj xi

3 iˆ1

xi

rUi¤

…24†

SOLUTION OF INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

101

where U i¤ ˆ Ui ¡

abji a ¤ R AP uj uj

…25†

Equation (24) has the form of the pressure-correction equation that can be derived for a staggered grid [13]. When this equation is discretized, Ui¤ are required at the cell faces. Rhie and Chow [4] and Peric¨ [5] treated this arti¢cial velocity Ui¤ as the model of the true cell-face velocity referred to in Section 1. Date [3, 6] showed that this model form is not unique, nor does it represent the momentum-conserving cell-face velocity that would be predicted by a staggered grid procedure. As such, writing of Ui¤ is merely a matter of elegance; its usage in calculation of convective coef¢cients has no theoretical basis. Date [9] therefore adopted an alternative approach. Thus, Eq. (18) for cell face ``e’’, for example, is rewritten as R¤uj

e

ˆ AP uj ulj ¡

X nb

Anb ulj nb ‡ bj1 e

pl x1

e

…26†

where the overbar denotes averaging over all the six immediate nodal neighbors of ``e.’’ The need for this multidimensional averaging (as opposed to one-dimensional averaging adopted by previous authors) will be clear shortly. Further, the averaged net momentum £ux is represented as AP uj ulj ¡

X

Anb ulj nb

nb

e

ˆ ¡bj1

pl x1

e

…27†

The concept of multidimensional averaging and its derivation for the twodimensional case is presented in Appendix B. The extension of the derivation to the three-dimensional case is fairly straightforward and hence is not presented in this article. It is clear that the averaged pressure gradient at ``e’’ can now be written as " # pl 1 pl pl ˆ ‡ …28† x1 x1 2 x1 e

Pe

Ee

where pl x1

Pe

ˆ plE ¡ plP

pl x1

Ee

l ˆ p¡l E ¡ pP

…29†

In Eq. (28), subscript Pe accounts for departure from multidimensional linearity in the variation of pressure at P, and subscript Ee likewise accounts for departure at E. The notion of average pressure for a one-dimensional variation of pressure can be readily understood from Figure 2. In Eq. (29), pP and pE , however, represent multidimensionally averaged pressures for a three-dimensional variation

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S. RAY AND A. W. DATE

Figure 2. The notion of average pressure in one dimension.

of pressure [see Eq. (40) for de¢nition]. Thus it follows that AP uj ulj ¡

X

Anb ulj nb

nb

e

ˆ¡

bj1 ¡ l p ¡ plP ‡ plE ¡ plP 2 E

bj ˆ¡ 1 2

pl x1

p¡l ‡ x1 e

…30†

e

Further substitution in Eq. (26) therefore yields R¤uj

ˆ bj1

p0s x1

e

…31†

p0s ˆ 0 5…p ¡ p†

…32†

e

where

SOLUTION OF INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

103

2.2.2. F urther simpli® cation. Thus, with substitution of equations, similar to Eq. (31), Eq. (17) can now be rewritten as 3 iˆ1

3

xi

jˆ1

2

rb ji a p0 AP uj xi

3

ˆ

iˆ1

xi

rUil

…33†

where p 0 is the total pressure correction and is given by p0 ˆ p0m ‡ p0s

…34†

Note that Eq. (33) can also be derived from the total pressure-correction equation derived by Date [9] in Cartesian coordinates by employing the chain rule of coordinate transformation. Writing of Eq. (33) is permissible since multipliers of p0m xi and p0s xi in Eq. (17) are identical. Further, when computing on a nonstaggered grid, the boundary pressures are linearly extrapolated from their near-boundary values.6 Thus, p0s n

boundary

ˆ0

…35†

The above boundary condition is also valid for p0m for incompressible £ows. Therefore, Eq. (33) can be solved with p0 n

boundary

ˆ0

…36†

The ¢nite difference form of Eq. (33) is same as Eq. (2) with f ˆ p 0 and Df ˆ ¡ rU1l e ¡ rU1l

w

‡ rU2l n ¡ rU2l s ‡ rU3l t ¡ rU3l

b

…37†

and the coef¢cients Ai in Eq. (2) are given by AE ˆ ra

a21 AP uj

e

AS ˆ ra

a22 AP uj

etc s

…38†

where AP uj coef¢cients at the cell faces are evaluated by simple averaging. Thus, AP uj e ˆ 0.5 (AP uj P ‡ AP uj E †, AP uj s ˆ 0.5 …AP uj P ‡ AP uj S †, etc. Note that writing of Ai coef¢cients in the above manner is permissible since AP coef¢cients are identical for all velocities, uj , when computation is carried out on a nonstaggered grid.

2.3

The C omplete Solution Algorithm

The complete solution sequence for one iteration is as follows, 1. 2.

6

Guess pl distribution. Solve Eq. (2) for f ˆ uj with coef¢cients Ai calculated according to Eqs.(5), (6), etc., and using boundary conditions described in Appendix A. This yields the uj distribution.

Note that this is not required for a staggered grid procedure, since velocities are solved at the cell faces.

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S. RAY AND A. W. DATE

3. 4.

Solve Eq. (33) for p0 with cell-face contravariant velocities in Eq. (37) calculated according to Eqs. (8) and (9) and using the boundary conditions in Eq. (36). Recover p0m from Eq. (34) as p0mP ˆ p0P ¡ p0sP ˆ p0P ¡ 0 5 plP ¡ plP

…39†

where plP ˆ

1 l p ‡ plx2 ‡ plx3 3 x1

P

…40†

and plx1 , for example is de¢ned as (see Appendix B for details) plx1 ˆ

‡ l PEl Dr¡ 1 ‡ P W Dr1 ‡ ¡ ‡ Dr1 Dr1 1 2

3

Dr¡ 1

ˆ

iˆ1

2

…xi P ¡ xi W †

5.

ˆ

iˆ1

…42† 1 2

3

Dr‡ 1

…41†

2

…xi E ¡ xi P †

…43†

Correct guessed pressure pl and velocities ulj as 0 l pl‡1 P ˆ pP ‡ ap pm P

l ul‡1 j P ˆ uj P ¡ a

3 iˆ1

bji

p0m xi

…44† AP uj

…45†

Note that the above sequence is same as that required in the original SIMPLE algorithm [2] for a staggered grid except for step 4, where the mass-conserving pressure correction p0m is recovered from the total pressure correction p 0. The above algorithm predicts smooth pressure distribution mainly because of inclusion of p0s [see Eq. (34)]. Hence, p0s is called the smoothing pressure correction. Note that p0s , which represents the difference between the point value and the cell-averaged value of pressure [see Eqs. (32) and (40)], will tend to vanish when the mesh size is re¢ned. The problem of checkerboard prediction of pressure is thus associated with the coarseness of mesh size used with respect to the pressure variation. Further, if the true pressure variation were to be multidimensionally linear or constant, then p0s ˆ 0 and the checkerboarding problem should not arise, as has been demonstrated by Date [3]. It may also be noted that on a nonstaggered grid, r ¢ V is not equal to zero, as the cell face velocities are calculated from the arithmatic mean values [linear

SOLUTION OF INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

105

interpolation, see Eqs. (8) and (9)]. The purpose of smoothing pressure correction is to compensate for this. Therefore, the mass conservation is checked from the evaluation of the left-hand side of Eq. (17) after the mass-conserving pressure correction, p0m , is calculated. At convergence, p0m does go to zero and p0s equals p 0 (which is nonzero in this case).

3

TEST C ASES

Two different £ow situations are considered to verify the present numerical method, one in which the recirculating £ow takes place within a cavity and another where the £ow occurs through ducts. All the present calculations are performed with a power law scheme, in which the convective coef¢cients are calculated from the arithmatic mean velocities [see Eqs. (8) and (9)]. 3.1

C avity F lows

Under this cavity £ows, two problems are considered: 1. 2.

Laminar £ow in a lid-driven cavity with inclined side walls Natural-convection heat transfer in concentric and eccentric horizontal cylindrical annuli.

3.1.1. Laminar ¯ ow in a lid-driven c avity with inclined side walls. The geometry of the problem, with side wall inclination b, is presented in Figure 3. The bottom and side walls are stationary, whereas the top wall moves with a velocity UL . The £ow is assumed to be laminar. Demirdzic¨ et al. [14] have presented benchmark solutions for this problem employing 320£320 control

Figure 3. Geometry of lid-driven cavity with inclined side walls.

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S. RAY AND A. W. DATE

volumesì (or 322£322 nodes). The grid lines used in the present study are parallel to the boundary walls and are generated by algebraic mapping. The study is carried out on increasingly ¢ner grids, starting from a very coarse grid (21£21) to a very ¢ne grid (81£81). Results with b ˆ 45¯ . The centerline Cartesian velocities for Re( ˆ UL L/n) ˆ 100 and 1,000 are presented in Figures 4 and 5, respectively. The ¢gures clearly indicate that grid-independent solutions are obtained with much coarser grids for lower Re. The comparison with the benchmark solutions, particularly at Re ˆ 100, is excellent. Results with b ˆ 30¯ . With the decrease in the inclination angle, the numerical grid becomes increasingly nonorthogonal. Therefore results are obtained with b ˆ 30¯ to check the capability of the present numerical method to handle grid nonorthogonality. Here also the centerline velocities are compared with the predictions of Demirdzic¨ et al. [14] in Figures 6 and 7 for Re ˆ 100 and 1,000, respectively. Comparison of Figures 5 and 7 shows that the vertical velocities along CL2 are suppressed considerably for Re ˆ 1,000. Compared to the earlier case of b ˆ 45¯ , the u1 velocity distribution for Re ˆ 1,000 with an 81£81 grid shows better agreement with the ¢ne-grid solution of Demirdzic¨ [14]. As expected, the results also clearly show that for any value of b, the grid-independent solutions are obtained with a much ¢ner grid for higher Re. Finally, the typical pressure distributions for b ˆ 45¯ and 30¯ , obtained with a 41£41 grid, are shown in Figures 8 and 9, respectively. In the ¢gure, pref is the pressure at the reference point, chosen at x1 ˆ 0.5 and x2 ˆ 0.0. The pressure distributions are observed to be smooth. 3.1.2. Natural-convection heat transf er in concentric and eccentric horizontal cylindrical annuli. The geometry of this test case is presented in Figure 10. Kuehn and Goldstein [15, 16] have presented experimental data for this problem. For a concentric annulus, they have conducted experiments with air and water [15]. The Rayleigh number based on the gap between the cylinders (Ra ˆ gbL3 …Ti ¡ To †/na, L ˆ ro ¡ ri ) has been varied from 2.11£104 (for air) to 9.76£105 (for water). For the eccentric annulus, they have considered two different eccentricities (vertical eccentricities, ev L ˆ 0.652 and ¡0.623) and presented the results for air. The diameter ratio for their study has been kept constant at 2.6, for which the dimensionless radii of the inner and outer cylinders are obtained as 0.625 and 1.625, respectively. In these articles, they have also presented the variation of local equivalent conductivities (keq ) for inner and outer cylinders as a function of angular position (y). These are de¢ned as keq i …y† ˆ

T njri y do di ln 2…Ti ¡ To † di

…46†

keq o …y† ˆ

T njro y do do ln 2…Ti ¡ To † di

…47†

SOLUTION OF INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

Figure 4. Velocity distributions along centerlines for b ˆ 45¯ , Re ˆ 100.

107

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S. RAY AND A. W. DATE

Figure 5. Velocity distributions along centerlines for b ˆ 45 ¯ , Re ˆ 1,000 (legend as given in Figure 4).

SOLUTION OF INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

109

Figure 6. Velocity distributions along centerlines for b ˆ 30¯ , Re ˆ 100 (legend as given in Figure 4).

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S. RAY AND A. W. DATE

Figure 7. Velocity distributions along centerlines for b ˆ 30 ¯ , Re ˆ 1,000 (legend as given in Figure 4).

SOLUTION OF INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

Figure 8. Pressure variation along centerlines for b ˆ 45¯ , 41£41 grid.

111

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S. RAY AND A. W. DATE

Figure 9. Pressure variation along centerlines for b ˆ 30¯ , 41£41 grid.

SOLUTION OF INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

113

Figure 10. Geometry of eccentric annulus.

In order to obtain solutions for this problem, a numerical grid is generated using radial lines emerging from the center of the smaller cylinder as one set of coordinate lines. The other set of coordinate lines is generated by dividing these radial lines in de¢nite proportions. In this analysis, a 49 (circumferential)£31 (radial) numerical grid is employed to discretize the computational domain. The temperature of the inner cylinder is set to unity (Ti ˆ 1), whereas that for the outer cylinder is assumed to be zero (To ˆ 0). For concentric con¢guration (ev L ˆ 0), following Kuehn and Goldstein’s [15] experiment, the Rayleigh number is taken as 4.7£104 . For eccentric con¢gurations two Rayleigh numbers are considered: 4.8£104 and 4.93£104 for ev L ˆ 0.652 and ¡0.623, respectively. For all the cases, air, with Pr ˆ 0.706, is considered as the working £uid. The local equivalent conductivities for different eccentricities are compared with the experimental data of Kuehn and Goldstein [15, 16] in Figure 11. The results show reasonable agreement with the experimental data. For all the cases under consideration, the comparison is better for the outer cylinders. For both the eccentric cases (Figures 11b and 11c), predictions for the inner cylinder, particularly near y ˆ 0, deviate from the experimental data. This deviation is more for ev L ˆ 0.652.

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S. RAY AND A. W. DATE

Figure 11. Local equivalent conductivities for concentric and eccentric annuli.

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115

0.652. Earlier, Karki and Patankar [13], Kobayashi and Pereira [17], Choi et al. [18], and Date [19] solved the same problem, using 32£22, 31£21, and 51£31 numerical grids, respectively. Their results also show similar departures near y ˆ 0 at the inner cylinder. Typical pressure distributions along different radial lines (y ˆ 30¯ , 90¯ , and 150¯ ) as a function of normalized radial distance are shown in Figure 12. Again, they are smooth.

3.2.

F low through ducts

Under £ow through ducts, two problems are considered: 1. 2.

Laminar £ow through a circular duct with an axisymmetric constriction Laminar and turbulent £ow through a 90¯ -bend duct of square cross section

3.2.1. Laminar ¯ ow in a circular duc t with an axisymmetric constriction. The geometry of the axisymmetric constriction is presented in Figure 13. The problem is typical of arterial stenoses. The shape of the stenoses is speci¢ed as a cosine curve and is given as [20] d px1 r ˆ1¡ 1 ‡ cos ro 2ro Xo

for ¡ Xo µ x1 µ Xo

…48†

The numerical grid for this problem is shown in Figure 14 (31 in the radial direction and 71 in the axial direction). The results are obtained for d ro ˆ 2 3 and Xo ro ˆ 4 0 For all cases (Reynolds number, de¢ned as Re ˆ 2Um ro n† the radius of the unconstricted duct is chosen as unity …ro ˆ 1†. A fully developed velocity pro¢le is prescribed at the inlet …x1 ˆ ¡8 0† At the exit …x1 ˆ 50† the £ow is assumed to be fully developed and all the gradients are set equal to zero. A typical streamline plot for Re ˆ 50 is presented in Figure 15. The comparison of separation and reattachment lengths obtained by the present method with the experiments of Young and Tsai [20] is presented in Table 3. The results show that although the present method predicts the separation point with reasonable accuracy, the prediction for the reattachment point differs from the experimental data. The deviation is less at higher Re. The wall pressure distribution (here, pref is the pressure at x1 ˆ 0 and x2 ˆ 0† with the axial distance is presented in Figure 16. Here also, a smooth pressure distributioin is predicted by the present method. 3.2.2. Laminar and Turbulent Flow through a 90¯ bend duct of square cross section. The geometry of the 90¯ -bend curved duct of square cross section is shown in Figure 17, along with the coordinate system. Due to symmetry, only half of the duct cross section is chosen for analysis. All lengths are nondimensionalized with respect to the hydraulic diameter of the duct, Dh (which is also the side of the duct). The velocities are normalized with respect to the average

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S. RAY AND A. W. DATE

Figure 12. Pressure distribution for natural convection in eccentric annulus.

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117

Figure 13. Geometry of axisymmetric constriction.

Figure 14. Typical numerical grid chosen for axisymmetric constriction.

axial (streamwise) velocity, Um The numerical grid is ¢rst generated at the inlet plane by algebraic mapping, with closer spacing near the wall. The cross section is then rotated to generate grid at other cross sections along the bend. Predictions for laminar ¯ ow. Results for laminar £ow are obtained for Re ˆ 792 and are compared with the experiments of Taylor et al. [21], performed on a duct with ri Dh ˆ 1 8 The velocity pro¢le at the inlet is prescribed by ¢tting the experimental data of Taylor et al. [21] at a section 0.5Dh upstream of the bend. This treatment is the same as that adopted by Tamamidis and Assanis [22], who obtained numerical solutions for this problem. The ¢nal numerical grid chosen for the present computation is 17 (in x1 )£31 (in x2 )£73 (in x3 ). In Figure 18, the streamwise velocity pro¢les along the centerline (x1 ˆ 0.0) at different cross sections are compared with the experimental data of Taylor et al. [21]. Tamamidis and Assanis [22] used a 22£22£57 nonstaggered grid and employed Rhie and Chow’s [4] interpolation formula to estimate the cell face velocities in convective coef¢cients as well as in mass £ux calculations. Numerical results are

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S. RAY AND A. W. DATE

Table 3. Comparison of predicted Xs Xo and Xr Xo Separation point Xs Xo

Reattachment point Xr Xo

Re

Experiment [20]

Prediction

Experiment [20]

Prediction

20 40 60 80 100

0.49 0.37 0.34 0.30 0.29

0.496 0.358 0.307 0.288 0.276

1.46 1.87 2.54 3.29 4.15

0.947 1.626 2.291 2.933 3.553

Figure 15. Streamlines for £ow through axisymmetric constriction: (a) Re ˆ 50; (b) Re ˆ 100.

obtained with hybrid (dotted line) and FOUB (¢fth-order upwind scheme, chain line) schemes for convective coef¢cients. In the ¢gure, the radial distance, r, is normalized according to r¤ ˆ …r ¡ ri † …ro ¡ ri †. From the ¢gures, it is observed that as the £uid negotiates the bend, due to centrifugal force on the £uid, the velocity peak shifts toward the outer wall. The effect of curvature becomes more prominent at larger values of y. Even at 0.25D h downstream of the bend, axial velocities are observed to be higher near the outer

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119

Figure 16. Wall pressure variation for £ow through circular duct with axisymmetric constriction.

wall. It is also obvious that, trendwise, the present predictions (solid lines) with the power law scheme compare well with the experimental data. Since no computational data are available in [22] with a power law scheme, no direct comparison can be made at this stage. The wall pressure variation with the distance along the centerline, as presented in Figure 19, shows smooth prediction of pressure. In the ¢gure, the normalized pressure is de¢ned as …p ¡ pref † …1 2Um2 †, where, pref is the pressure at the centerline at the inlet plane. Predictions for turbulent ¯ ow. In the case of turbulent £ow, the Reynolds number is taken as 40,000 (the same as in the experiment of Taylor et al. [21]). A grid of 17£31£89 nodes is used to discretize the computational domain. The inlet boundary condition for the velocity is prescribed by setting a plug £ow pro¢le at 5D h upstream of the bend (this treatment is similar to that of Tamamidis and Assanis [22]). The inlet turbulent intensity is assumed to be 5% of the mean £ow. Therefore, the inlet turbulent kinetic energy is speci¢ed as 0.25% of the mean kinetic energy. The dissipation is obtained from the following relation: eˆ

Cm3 4 e3 L

2

…49†

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S. RAY AND A. W. DATE

Figure 17. Geometry of 90¯ -bend duct of square cross section.

where L is the characteristic length scale, which is assumed to be 1% of the hydraulic diameter to yield mt /m^100. A wall function approach [11, 12] is adopted to provide the boundary condition at a near-wall node. Other boundary conditions are the same as those used for laminar £ow. The comparison of the predicted and experimental [21] streamwise velocity distribution along the centerline at different planes is presented in Figure 20. From the ¢gure it is observed that the velocity pro¢les are much £atter than those predicted for the laminar £ow situation. At lower values of y (y ˆ 30¯ and 60¯ ), the streamwise velocities are higher near the inner wall. The pattern changes signi¢cantly at a plane, 0.25Dh downstream of the bend. It is also clear from the ¢gure that, qualitatively and trendwise, the present predictions show reasonable agreement with the experiment. Finally, the wall pressure distribution is presented in Figure 21 (the normalized

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121

Figure 18. Streamwise velocity distribution along centerline for laminar £ow through 90¯ -bend square duct.

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S. RAY AND A. W. DATE

Figure 19. Pressure variation for laminar £ow through 90¯ -bend duct of square cross section.

pressure is de¢ned in the same fashion as de¢ned in the case of laminar £ow). Here also, a smooth pressure distribution is obtained by the present method.

4

C ONC LUSIONS

In the present study, the complete pressure-correction algorithm of Date [9] is extended to predict various two- and three-dimensional laminar and turbulent £ow problems requiring use of general curvilinear coordinates. Following are the ¢ndings of the present study. 1. 2.

The predicted pressure distributions in all problems are found to be smooth. Thus the employment of smoothing pressure correction in curvilinear coordinates is validated. It is shown that the pressure checkerboarding problem should not arise when the true pressure variation is multidimensionally linear or constant. When nonlinearity is present, checkerboarding will occur only on coarse grids with respect to pressure.

SOLUTION OF INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

123

Figure 20. Streamwise velocity distribution along centerline for turbulent £ow through 90¯ -bend square duct.

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S. RAY AND A. W. DATE

Figure 21. Pressure variation for turbulent £ow through 90¯ -bend duct of square cross section.

3.

The present predictions provide reasonably good agreement with existing experimental and computational results, thus providing validity of the method in curvilinear coordinates.

R EF ER ENCES 1. S. V. Patankar and D. B. Spalding, A Calculation Procedure for Heat, Mass and Momentum Transfer in Three-Dimensional Parabolic Flows, Int. J. Heat Mass Transfer, vol. 15, pp. 1787^1806, 1972. 2. S. V. Patankar, Numerical Heat Transfer and Fluid Flow, Hemisphere, Washington, DC, 1980. 3. A. W. Date, Solution of Navier-Stokes Equations on Non-Staggered Grid, Int. J. Heat Mass Transfer., vol. 36, pp. 1913^1922, 1993. 4. C. M. Rhie and W. L. Chow, Numerical Study of the Turbulent Flow Past an Airfoil with Trailing Edge Separation, AIAA J., vol. 21, pp. 1525^1532, 1983. 5. M. Peric¨, A Finite Volume Method for the Prediction of Three Dimensional Fluid Flow in Complex Ducts, Ph.D. thesis, University of London, 1985. 6. A. W. Date, On Interpolation of Cell-Face Velocities in the Solution of N-S Equations Using Non-staggered Grids, Tech. Rep. SFB 210/T/79, University of Karlsruhe, Germany, 1991.

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125

7. A. W. Date, A Calculation Procedure for Prediction of Heat Momentum and Mass Transfer in Elliptic Flows Using Non-Staggered Grids, in A. R. Balakrishnan and S. Srinivasmurthy (eds.), Proc. 1st ISHMT-ASME Heat and Mass Transfer Conf., Paper HMT-94-010, pp. 95^108, 1994. 8. T. F. Miller and F. W. Schmidt, Use of Pressure-Weighted Interpolation Method for the Solution of the Incompressible Navier Stokes Equations on a Nonstaggered Grid System, Numer. Heat Transfer, vol. 14, pp. 213^233, 1988. 9. A. W. Date, A Complete Pressure Correction Algorithm for Solution of Incompressible Navier-Stokes Equations on Non-Staggered Grid, Numer. Heat Transfer B, vol. 29, pp. 441^458, 1996. 10. T. J. Craft, B. E. Launder, and K. Suga, Development and Application of Cubic Eddy Viscosity Model of Turbulence, Int. J. Heat Fluid Flow, vol. 17, pp. 108^115, 1996. 11. B. E. Launder and D. B. Spalding, Mathematical Models of Turbulence, Academic Press, London, 1972. 12. B. E. Launder and D. B. Spalding, The Numerical Computation of Turbulent Flows, Comput. Meth. Appl. Mech. Eng., vol. 3, pp. 269^289, 1974. 13. K. C. Karki and S. V. Pakankar, Solution of Some Two Dimensional Incompressible Flow Problems Using a Curvilinear Coordinate System Based Calculation Procedure, Numer. Heat Transfer, vol. 14, pp. 309^321, 1988. 14. I. Demirdzic¨, Z. Lilek, and M. Peric¨, Fluid Flow and Heat Transfer Test Problems for Non-orthogonal Grids: Bench Mark Solutions, Int. J. Numer. Meth. Fluids, vol. 15, pp. 329^354, 1992. 15. T. H. Kuehn and R. J. Goldstein, An Experimental Study of Natural Convection in the Annulus between Horizontal Concentric Cylinders, J. Fluid Mech., vol. 74, pp. 695^719, 1976. 16. T. H. Kuehn and R. J. Goldstein, An Experimental Study of National Convection Heat Transfer in Concentric and Eccentric Horizontal Annuli, ASME J. Heat Transfer, vol. 100, pp. 635^640, 1978. 17. M. H. Kobayashi and J. C. F. Pereira, Calculation of Incompressible Laminar Flows on a Nonstaggered Nonorthogonal Grid, Numer. Heat Transfer B, vol. 19, pp. 243^262, 1991. 18. S. K. Choi, H. Y. Nam, and M. Cho, Use of Staggered and Nonstaggered Grid Arrangements for Incompressible Flow Calculations on Nonorthogonal Grids, Numer. Heat Transfer B, vol 25, pp. 193^204, 1994. 19. A. W. Date, Numerical Prediction of Natural Convection Heat Transfer in Horizontal Annulus, Int. J. Heat Mass Transfer, vol. 29, pp. 1457^1464, 1986. 20. D. F. Young and F. Y. Tsai, Flow Characteristics in Models of Arterial StenosesöI. Steady Flow, J. Biomechanics, vol. 6 pp. 395^410, 1973. 21. A. M. K. P. Taylor, J. H. Whitelaw, and M. Yianneskis, Measurement of Laminar and Turbulent Flow in a Curved Duct with Thin Inlet Boundary Layers, NASA CR 3367, 1981. 22. P. Tamamidis and D. N. Assanis, Three Dimensional Incompressible Flow Calculations with Alternative Discretization Schemes, Numer. Heat Transfer B, vol. 24, pp. 57^76, 1993.

AP PEND IX A: IM PLEM ENTATION OF THE B OUND AR Y C ONDITIONS In the present calculation procedure, the boundary nodes are eliminated from the discretized equations for the near-boundary control volumes by applying suitable boundary conditions. During any iteration, the numerical solution is obtained only for the interior points. The values of the boundary points are extracted subsequently

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S. RAY AND A. W. DATE

from the appropriate boundary conditions. Brief description of the boundary conditions and their implementation are given here.

A.1

In¯ ow

At the inlet, all the values (ui ’s, T, etc.) are assumed to be known. However, turbulent kinetic energy and dissipation requires estimation. Typically, ein ˆ …Tu uin †2

where

3

uin ˆ

…50†

u2i

…51†

…5--10%†

…52†

iˆ1

Tu ^ 0 05--0 1 Dissipation is estimated as

1 2

ein ˆ

Cm3 4 ein L

…53†

The length scale, L, is to be chosen suitably.

A.2

Exit and Symmetry

Typically, the normal gradients are prescribed for exit and symmetry boundary conditions. Since the contravariant base vectors (ai ) are normal to the coordinate surfaces of constant xi , the normal derivative of a scalar variable f on such surfaces is given by f n

xi

ai ¢ …rf† jai j

ˆ ˆ

1 3 n 2 nˆ1 …bi †

J

3 nˆ1

bni bni

f ‡ xi

3

bni bnj

nˆ1

f ‡ xj

3

bni bnk

nˆ1

f xk

…54†

If f/ n is zero on the boundary, then f xi

xi

ˆ¡

3 nˆ1

bni bnj … f xj † ‡ 3 nˆ1

3 nˆ1

bni bni

bni bnk … f xk †

…55†

Normally, at the exit and symmetry, the gradients of contravariant velocities (for symmetry plane of constant xi , Ui is set equal to zero) are prescribed. The Cartesian velocity components are extracted from the conditions posed on the contravariant velocities. However, evaluation of scalar variables is fairly straightforward.

SOLUTION OF INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

A.3

127

Wall

A.3.1. Wall boundary condition f or velocity. Consider the lth (for xl direction) momentum equation and apply the same to a near-boundary control volume, where xi is constant. Let us consider both inner and outer boundaries as shown in Figure 22 (for simplicity, a two-dimensional representation is provided). In the ¢gure, P is the near-boundary node for which discretization is to be carried out. The dotted line shows the control volume to which the lth momentum equation is applied. Let us consider the momentum equation as given in Eq. (1) and consider the derivatives only with respect to xi (since these derivatives will involve the boundary point, B). Evaluation of the other derivatives are similar to that carried out for the interior points. Now, the terms containing the boundary points in the expanded xi derivatives can be clubbed into a source term Sxi . For the outer boundary control volume, we obtain Sx‡i ˆ ¡rUi‡ ul

3 X

‡ B

kˆ1

bki tlk

B

…56†

Similarly, at inner boundary of constant xi , the source term is obtained as Sx¡i ˆ rUi¡ ‡ ul

3 X

¡ B

kˆ1

bki tlk

B

Figure 22. Two-dimensional representation of near-boundary control volume.

…57†

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S. RAY AND A. W. DATE

For an impervious wall we obtain Sx‡i ˆ

3 kˆ1

Sx¡i ˆ ¡

bki tlk 3

…58a†

B

bki tlk

…58b†

B

kˆ1

Now, S3kˆ1 bki tlk can be expressed as ttot xl ai , where, ttot xl is the total stress in the xl direction. Therefore, at the outer and inner boundaries, the source terms may be obtained as Sxi ˆ ttot xl ai ˆ meff

Vxt l

¡ Vxt l P ai jDnj

B

…59†

In the above expression, jDnj is the normal distance of point P from the boundary and Vxt l is the xl component of the tangential velocity vector. They are evaluated as follows: 3 n nˆ1 bi …

Dn ˆ

bli ai

Vxt l ˆ ul 1 ¡

x n xi †

…60†

3 n 2 nˆ1 …bi †

2

3

¡

un nˆ1 n6 ˆl

bli ai

bni ai

…61†

In general, source terms are written in the form Sf ˆ Su f ¡ Sp f fP for diagonal dominance. Thus, they are obtained as Su ul

m ai ˆ eff Dn

bl 1¡ i ai

2

B

Sp ul ˆ

3

ul B ¡

bli ai

nˆ1 n6 ˆl

meff ai bl 1¡ i Dn ai

B

bni ai

B

…un B ¡ un P †

…62†

2

…63†

B

For turbulent £ow, meff rut k ˆ ln…Ey‡ † Dn m ˆ jDnj

for

y‡

11 6

otherwise

…64† …65†

For laminar £ows, the second equality holds. ut is calculated as 1 2 ut ˆ Cm1 4 eP

…66†

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129

A.3.2. Wall boundary c ondition f or temperature. Consider the transformed energy equation as given in Eq. (1) and apply the same at a near-boundary control volume, as shown in Figure 22. Here also, the expanded xi derivatives, containing the boundary point, are clubbed together in a source term, Sxi . Therefore, for the outer boundary control volume, the source term is obtained as 3 X

Sx‡i ˆ ¡rUi‡ T

¡ B

bki

kˆ1

qk Cp

B

…67†

Similarly, at the inner boundary control volume, this is obtained as Sx¡i ˆ rUi¡ T

3 X B

‡

kˆ1

bki

qk Cp

B

…68†

For an impervious wall, the source terms in Eqs. (65) and (66) can be combined together and can be expressed as Sxi ˆ ¨

3 X

bki

kˆ1

qk Cp

…69†

B

The total heat £ow across a surface of constant xi can be written as Qw ˆ qw dAi ˆ

3 X

bki qk

kˆ1

…70†

where qw is the heat £ux normal to the surface of constant xi . At the outer and inner boundaries, this heat £ux is considered to be positive if it is added to the control volume. Therefore, at the outer and inner boundaries, Sxi may be written as Sxi ˆ

qw dAi keff …TB ¡ TP † ˆ dAi jDnj Cp Cp

…71†

When qw is speci¢ed, the ¢rst equality is used to calculate Sxi and the second equality is used to recover TB . When TB is speci¢ed, the ¢rst equality is used to recover qw . In laminar £ow, keff ˆ k and k Cp ˆ m/Pr. For turbulent £ow, keff is recovered from the law of the wall for temperature, given by TP‡ ˆ

T B ¡ TP ˆ sT …PF ‡ u‡ P† qw rCp ut

where the function PF is taken as

"

PF ˆ 9 24

Pr sT

…72†

#

0 75

¡1

…73†

Thus, from Eq. (69), keff ˆ

rCp ut jDnj ¡ sT PF ‡ u‡ P

…74†

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S. RAY AND A. W. DATE

A.3.3. Wall boundary condition f or T.K.E. ( ) and dissipation ( e) . In the ``wall-function’’ approach, the source term Se at the near-wall node P is calculated as Vt ¡ reP Se ˆ J…G ¡ re† ˆ J ttot n B " # 3 2 meff X t t ˆJ Vxi B ¡ Vxi P ¡reP …Dn†2 iˆ1 where eP is the average dissipation and is obtained as Z jDnj 1 eP ˆ e dn jDnj 0 u2 ut ¡ ‡ ˆ t ln Ey Dn k

…75†

…76†

Substituting for ln (Ey‡ )/k from Eq. (63) and using the value of ut as given in Eq. (64), the above equation is written as eP ˆ Thus, Su e and Sp e are written as " Su e ˆ J

ru2t 1 2 c eP meff m

3 meff X

…Dn†2

iˆ1

" sp e ˆ J

…77† #

Vxt i B

¡

r2 u2t c1m 2

Vxt i P

2

…78†

#

meff

…79†

For the e equation, the value at P is frozen at u3t …kjDnj† with Su e ˆ 1030‰u3t …kjDnj†Š and Sp e ˆ 1030.

AP PEND IX B : M ULTID IM ENSIONAL AVER AGING Consider a two-dimensional control volume for simplicity. The term p x1 je in the present method is obtained by multidimensional averaging. Considering the locations P, E, ne, and se, as the neighbors of ``e’’ and interpolating on the physical plane, the term can be expressed as (here ri are the physical coordinates, coinciding with xi ) " # ‡ Dr¡ p 1 1 r1 p p 2 e p x1 j ne ‡ Dr2 e p x1 jse ˆ ‡ ‡ …80† ‡ x1 e 2 2 x1 e r1 P r1 E Dr¡ 2 e ‡ Dr2 e

SOLUTION OF INCOMPRESSIBLE NAVIER-STOKES EQUATIONS

131

Using the de¢nition of pxi , as given in Eq. (41), it can be readily shown that r1 x1

p r1

e

P

‡

p r1

E

ˆ

r1 x1

e

pE ¡ pP px1 E ¡ px1 P ‡ Dr‡ Dr‡ 1P 1P

ˆ pE ¡ pP ‡ px1 E ¡ px1 P

…81†

Now, consider the other term in Eq. (80).

‡ Dr¡ Dr¡ Dr¡ 2 e p x1 j ne ‡ Dr2 e p x1 j se ˆ ¡ 2 e ‡ …pnE ¡ pn † ‡ ¡ 2 e ‡ …psE ¡ ps † ‡ ¡ ‡ Dr2 e Dr2 e Dr2 e ‡ Dr2 e Dr2 e ‡ Dr2 e

…82† pnE , pn , psE , and ps are pressures at different cell faces and hence are obtained from arithmatic mean values. Using these values, Eq. (82) can be written as ‡ Dr¡ 1 2 e p x1 j ne ‡ Dr2 e p x1 j se ˆ pE ¡ pP ‡ px2 E ¡ px2 P ‡ ¡ ‡ Dr2 e Dr2 e 2

…83†

Using Eqs. (81) and (83), Eq. (80) can now be written as p x1

e

ˆ

1 1 1 pE ¡ pP ‡ px1 E ¡ px1 P ‡ pE ¡ pP ‡ px2 E ¡ px2 P 2 2 2

…84†

With the help of the de¢nition of cell-averaged pressure, p [see Eq, (40)], the above equation is written as p x1

e

1 1 ˆ …pE ¡ pP ‡ pE ¡ pP † ˆ 2 2

p x1

e

‡

p x1

e

…85†