fv, unstructured

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ELSEVIER

A natural extension of the conventional finite volume method into polygonal unstructured meshes for CFD application

P. Chow, M. Cross, and K. Pericleous

Centre for Numerical Modelling and Process Analysis, Science, University o f Greenwich, London, U.K.

School o f Computing and Mathematical

A new general cell-centered solution procedure based upon the conventional control or finite volume (CV or FV) approach has been developed for numerical heat transfer and fluid flow which encompasses both structured and unstructured meshes for any kind of mixed polygon cell. Unlike conventional FV methods for structured and block structured meshes and both FV and FE methods for unstructured meshes, tile irregular control volume (ICV) method does not require the shape of the element or cell to be predefined because it simply exploits the concept of fluxes across cell faces. That is, the ICV method enables meshes employing mixtures of triangular, quadrilateral, and any other higher order polygonal cells to be exploited using a single solution procedure. The ICV approach otherwise preserves all the desirable features of conventional FV procedures for a structured mesh; in the current implementation, collocation of variables at cell centers is used with a Rhie and Chow interpolation (to suppress pressure oscillation in the flow field) in the context of the SIMPLE pressure correction solution procedure. In fact all other FV structured mesh-based methods may be perceived as a subset of the ICV formulation. The new ICV formulation is benchmarked using two standard computational fluid dynamics (CFD) problems, i.e., the moving lid cavity and the natural convection driven cavity. Both cases were solved with a variety of structured and unstructured meshes, the latter exploiting mixed polygonal cell meshes. The polygonal mesh experiments show a higher degree of accuracy for equivalent meshes (in nodal density terms) using triangular or quadrilateral cells; these results may be interpreted in a manner similar to the CUPID scheme used in structured meshes for reducing numerical diffusion for flows with changing direction.

Keywords: finite volume, unstructured mesh, computational fluid dynamics

I. Introduct ion

In recent years, the increasing need for solving numerical heat transfer and fluid flow problems in complex geometries has prompted a move toward a finite volume-unstructured mesh (FV-UM) approach. Until fairly recently, the unstructured mesh methodology has been most commonly used by the finite element (FE) method. The conventional finite or control volume (FV or CV) method has the desired conservation properties and compact highly coupled solution procedures necessary for efficient flow calculation but it lacks the unstructured facility needed for treating complex geometries. One such method that com-

Address reprint requests to Dr. P. Chow at the School of Computing and Mathematical Sciences, University of Greenwich, Wellington Street, Woolwich, London SE18 6PF, U.K.

Received 25 July 1995; accepted 11 October 1995.

Appl. Math. Modelling 1996, Vol. 20, February © 1996 by Elsevier Science Inc. 655 Avenue of the Americas, New York, NY 10010

bines the best of both worlds and fits into the FV-UM framework is the control volume based-finite element mesh (CV-FE) method. 1-5 This combines the unstructured mesh aspect of FE with the conservation properties of the FV method. It has been used successfully in the field of numerical heat transfer and fluid flow from aerodynamic flows to solidification analysis. Workers such as Baliga and Patankar, 2 Schneider and Raw, 3 and Lonsdale and Webster 4 have used the method for general fluid flow and heat transfer analysis. Commercial CFD codes such as ASTEC from AEA Technology (Harwell, Oxon, U.K.) that are based upon the CV-FE technique are beginning to emerge. In the aerospace context, Jameson et al., 6-8 Mor- gan et al., 9 Batina, 10 and Barth 11 have used the method for aerodynamic flows, whilest the approach has also been used in solidification by Chow and Cross 5 and in stress- strain analysis by Fryer et al. 12 and Bailey et al. 13 The method adopted by these workers is categorized as the vertex-centered approach because the control volume is formed around the vertices of the element/cell .

0307-904X/96/$15.00 SSDI 0307-904X(95)00156-5

Conventional finite volume method for CFD application: P. Chow et al.

Instead of moving totally to an unstructured framework of FV-UM, a multiblock structured approach has been used successfully in solving some geometrically complex CFD problems by dividing an irregularly shaped geometry into regular blocks and with curvilinear coordinate trans- formation to map each physical block (which can itself be fairly complex) to a solution domain that has a regular mesh structure. Therefore, if the problems can be parti- tioned into regular blocks then it is possible to solve it by the body fitted coordinate (BFC) technique 14-16 or a similar one 17. Commercial CFD codes such as PHOEN- ICS (CHAM Ltd., U.K.) and HARWELL-FLOW3D (AEA Technology) use this approach in their solution procedures. The former uses a staggered grid to avoid pressure oscillation in the flow field, while the latter employs a Rhie and Chow 18 interpolation to suppress the oscillation. The BFC and the classical FV methods may be categorized as the cell-centered approach and have the added simplifying benefit of the element/cell coinciding with the CV.

Another new trend in the FV-UM community is toward mixing the above two approaches. In aerodynamic flows, Weatherill, 19 Peace and Shaw, 20 and Childs et al. 21 have successfully mixed block structured and unstructured meshes for aerospace simulation. In metal casting, Bailey et al. 13 have successfully employed two classes of unstructured mesh in the same solution procedure; heat transfer and solidification is solved using a cell-centered approach with the ICV, while the solid mechanics is solved using the vertex-centered approach. In both applications, they have shown there are significant advantages in mixing different classes of mesh.

In recent years, the aerospace sector has focused a great deal of effort on the FV-UM method using both cell and vertex centered approaches 6-11.22-30 for the numberical study of wing and airfoil configuration and complete aircraft bodies. Most approaches have assumed an inviscid compressible flow, though recently some viscous solutions for the prediction of shocks and surface pressure distribu- tion have been reported. For resolving shock waves accu- rately and efficiently, adaptive meshing is commonly used, and it has been suggested that unstructured meshes provide a natural setting for adaptive and dynamic mesh process- ing. 26,27,30 m typical unstructured mesh is commonly made up of triangular or tetrahedral elements, with Struijs et al. 30 demonstrating the benefit of using polygonal cells for adaptive griding where triangle and quadrilateral elements can easily be combined to allow structured layers, such as boundary layers near solid walls. Recent work by Batina 26 highlights where the block structured approach will have considerable difficulty when aeroelastic deformation of the aircraft is considered but reported how an unstructured approach will have a distinct advantage over a structured mesh in that they can easily treat complex geometric configurations as well as complicated flow physics.

In the work presented here, a new general cell-centered FV-UM method is proposed for treating both structured and unstructured meshes using any mixture of polygonal cell shape with a single solution procedure. The discretization of the new method is cell-centered, i.e., the element is the control volume and the formulation is all based around

the surface of the element for any polygonal cell shape and leads to the name irregular control volume (ICV) method. Since the formulation is for any polygonal shape, it in- cludes all structured and block structured meshes as a subset (see for example Demirdzic and Peric). 17 This has the advantage of a simpler formulation compared with the CV-FE and has been found to reduce computation time significantly in heat transfer analysis. 5 Also, the unstructured framework has the flexibility for treating various features, such as local mesh adaption, refinement, and independent mesh motion. 26.30 The new method has recently been demonstrated to be effective for solidification by conduction only problems by Chow and Cross, 5 and in this paper the extension to coupled fluid flow and heat transfer will be presented together with a study of unstructured meshes using polygonal elements in a cell-centered context.

2. Governing equations

With reference to the cartesian coordinate system (x, y), transient, two-dimensional elliptic fluid flow and heat transfer problems are governed by the following differential equations 3a.

Momentum equations:

o(ou) - - + v . ( o V u ) = v . - - -

Ot

o(o ) Ot

- - + v . ( o w ) = v . - - -

@ Ox + S,

(1)

Op +S,,

Oy (2)

Continity equation:

oo - - + v . ( o v ) = 0 ( 3 ) Ot

Other transported variables are governed by a generic conservation equation of the form

V(P6) - - + V' (pV~b) = V-(F6Vth) +Se~ (4)

Ot

In equations (1)-(4), /x is the dynamic viscosity, p is the density of the fluid, p is the pressure, V is the face resultant velocity, S, and S,, are the sources for the x and y direction, respectively, and u and v are the cartesian velocity components in the respective direction. The symbol th in equation (4) can be used to represent any scalar-dependent variable, such as temperature, enthalpy, turbulence-kinetic energy, etc. The terms F~ and S~, are the diffusion coefficient and the source term, respectively, and are specific to a particular meaning of Oh. Both the momentum and continuity equations can also be repre- sented by the general equation. The four terms in the general differential equation (4) are, from left to right, the transient term, the convection term, the diffusion term, and the source term.

Appl. Math. Modelling, 1996, Vol. 20, February 171


3. Proposed method

3.1. ICV control volume definition

For a given arbitrary element or control volume, all the face contributions that surround it need to be accounted for in a way that it is naturally conservative. By first constructing an outward normal surface vector at each face and then summing all the face contributions, a conservation system results for the control volume. Figure 1 shows the triangle, quadrilateral, and polygonal control volumes with their outward normal surface vectors.

It is in the assembly of the face contributions that the ICV and classical FV or CV methods differ. The classical FV assumes that a cell has a predefined shape, taking advantage of the structured mesh topology that is implicitly imbedded in the formulation and restricts it to elements with four faces in two dimensions and six faces in three dimensions. The ICV has no such presetting of shapes in its formulation and, therefore, the same discretization procedure can be applied to any element/cel l type. Hence, both structured and unstructured meshes can be treated by the same method. In fact, it should be clear that the two methods are identical when the present ICV method is given a structured (rectangular) mesh to process.

3.2. Discretization of the general equation

Integrating the general differential equation (4) over an arbitrary irregular control volume gives

flap6 a05 3,, at dv + f pV05ds i = f F~lds, J~ an i +ffs , dv

(5)

The s i represents the components of the outward normal area vector, with ds 1 = dy and ds 2 = - d x in counter- clockwise traversal of the control volume boundary n i is the coordinate direction in which n I = x and n 2 = y. The terms in equation (5) are evaluated as follows:

Transient term:

f l a p 0 5 (005) - (p05); . I at dv = At Vp (6)

The superscript o denotes the old time-step value, Vp is the volume of the irregular control volume P, At the time step, and 05p denotes value of 05 at the centroid of cell P.

Figure 1. Cells wi th normal surface vectors.

Source term:

f f f , dv = S, Vp (7)

The more general form of the source term S, is: 31

= Sc + Sp05p (8) If a source is nonlinear in 05 it can be appropriately linearized, 31 and cast into the format of equation (8) where the values of S c and Sp are to prevail over the irregular control volume.

Diffusion term:

a05 fr --ds, = Js ani

E (05A - 05P)/~ ax2 + 6Y 2 A = I

X[I~ck(n2Ay--~I~)AX)]A +Cdiff (9)

Here, N s is the total number of control faces and A represents the adjacent control volumes that share a common face with the P control volume. The symbol h is the unit normal to the cell face, where Ax and Ay are the face surface area vector components and tSx and 3y are the distance vector components between the nodes A and P in cartesian coordinates. The convention 0a means the variables inside the brackets are to be evaluated using A and P control volumes.

Cdiee is the cross-diffusion term 17 for the common cell face. This term disappears when the nodal distance vector N is orthogonal (perpendicular) to the surface vector S (see below for N and S) as in the classical case, and it is small compared with the main term if the nonorthogonality is not severe. In the current study, this term will be zero or near zero by using orthogonal meshes both in structured and unstructured cases. Work is underway in addressing highly nonorthogonal meshes in which the cross-diffusion will be significant. 32

Convection term:

Us fPV05dsi= E [R05(uAy - - vAX)]A (10) "s A = I

The face resultant velocity vector V is of the form V = u[ + v~ and the 05 value at the cell face is calculated using the upwind differencing scheme. To express the total convective-diffusive flux across a face, the same format as the standard CV method is employed, i.e.,

a a = O A -I- m a x [ 0 , - CA] (11)

S . N CA=pV'S DA=F6[NI2

The D a and C A are the diffusive and convective parts, respectively. The S^is the outward normal surface vector with S = Aft--Axj~, and N is the nodal distance vector with N = $xi + 6yj. The generalized convection-diffusion formulation using first-order differencing scheme given by Patankar 31 can now be added to equation (11) to give

aa = DAF(I PA I) + m a x [ 0 , - CA] (12)

172 Appl. Math. Modelling, 1996, Vol. 20, February


S~tmetcy

Figure 2.

Hot

STmletz7

Cold

Problem specification.

where PA is the Peclet number, given by CA/D A, and F(] PA l) is the generic function for the various differencing schemes that can be employed. Summing all the adjacent contributions in equation (12) for an irregular control volume P and substituting it with equations (6) and (7) into equation (4) yields a set of equations of the form

Us a t, qbt, -= ~ a A q~a Jr b6e (13)

A = I

where

N, ( p v ) t , at '= E aA Jr - - SpVp

A = 1 At

( b6e - At + ScV P (14)

The dependent variable & in equation (13) can be solved with any suitable linear equation solver. Note, for a structured mesh the system of equations [A]~b = b ~ is identical with that produced by standard FV formulations.

3.3. Pressure-correction equation

The pressure-correction equation is derived from the continuity equation by substituting all the velocity components with the velocity correction formulas. These are derived from the following equations:

p =p* + p ' (15) U = U* J r U ~

v = v* + v' (16)

For a full detailed explanation of the velocity correction, see Patankar. 31 The velocity correction formulas are ex- pressed for a face i as follows:

( A Y ) ' u i=u* + - (PP--P'A) aU i

z~=c ,* - ( a' 1~ (p 'P -pA) (17)

where p' is known as the pressure-correction variable, the u* and u* are the "starred" velocities (i.e., the guessed velocities at the end of the previous iteration), and a~' and a' i' are the respective u and v coefficients. These starred velocities and the u and v coefficients are calculated using the Rhie and Chow 18 interpolation for the collocated grid arrangement employed here. This was undertaken solely because staggered grid arrangements cannot be readily implemented on an unstructured mesh framework. How- ever, if a staggered grid is feasible, then the staggered values would be automatically substituted into the equation.

From equation (11) the convective mass flux for a given face i is

pV. S= [( p u ) i t + (pv) i f ] . (Ay ,~-- z~xif)

= ( p u A y ) i - ( pvAx) i (18)

Substituting the expression given in equation (17) and rearranging in terms of p' gives

p V . S = ( pu*Ay)i-- ( pv*Ax)i

pAy pAx 2 ) ' + a" + a v (Pe--P'A) (19)

i

Therefore, for an irregular control volume P, equation (19) can be written as

Us app' e = ~_~ aAp A Jr bp (20)

A = I

0.8

u) 0.6

>

"~ 0.4

E

8 0 . 2

Buoyancy vs Pressure Gradient forward / backward

I 0 Pressure Gradien~ - - Buoyancy /

0.02 0.04 0.06 0.08 Vertical Distance

0.8

Figure 3.

Buoyancy vs Pressure Gradient Bernoulli

0 Pressure Gradient~

m ~ 0.6 £ ~ m > "~ 0.4

(3 o.2

o o.1 o.o2 0.o4 o.oe 0.o8 o.1

Vertical Distance

(a) Forward/backward differencing (b) with Bernoulli.



where

( pAy2 a A = a,

Us

a p = E aA + - - A = I

Us be=

A = I

pAx2) ] (21) - - + a v

i A

(or), At (22)

[( pu*Ay) i - ( D u * A x ) i ] A -1- _ _ ( P°V)e

At

(23)

The pressure correction, equation (20), can be solved by any suitable solver and then used to update the variables in equations (15) and (16).

3.4. Boundary conditions

For control volumes that have a face coincident with the domain boundary, information is introduced into the equations to complete the formulation and enable it to be solved. For fluid flow and heat transfer, these are the ones appropriate for an inlet, outlet, wall, and symmetry boundary. These are usually specified in terms of external velocity components, u and v, temperature, T, and pressure, p, and they are no different from the standard CV method implementation. Because of the collocated grid arrangement employed here, pressure at the boundary needs to be interpolated.

Interpolating boundary pressure. In the momentum equations (1) and (2), the pressure gradient term requires a pressure at both inlet and wall boundaries. How this pressure is estimated can have a profound influence on the overall behavior of the solution. The straightforward forward/backward differencing can result in a large error in the pressure gradient term when buoyancy plays a major role in the calculations. This can be highlighted with a simple cavity problem that is buoyancy driven, with the top wall hot, the bottom wall cold, and a symmetry condition on both side walls, as illustrated in Figure 2. Of course in this problem the velocity is zero everywhere, and the pressure gradient in the vertical direction equals the gravity force term. Figure 3a shows a plot of the buoyancy and pressure gradient in the vertical direction. The pressure gradient obtained using forwar/backward differencing is under/overpredicted at both the boundaries. Figure 3b shows the same variables being plotted with the boundary pressure estimated using the Bernoulli equation. Again the pressure gradient is under/overpredicted at both the boundaries, but it is a significant improvement over the forward/backward differencing. Both the methods of estimating a pressure value at the boundary will improve with grid refinement. There will always be a finite error though in the pressure gradient for control volumes that coincide with the boundary, owing to the interpolation of the boundary pressure. The staggered grid arrangement has no such problem; no boundary pressure estimation is required.

Pressure correction gradient. With the interpolating of the boundary pressure in the pressure gradient term, there is a corresponding need to interpolate a pressure correction value at the boundary. This is to be used in the pressure correction gradient for updating the velocity components. Forward or backward differencing can be used to interpolate a pressure correction value when the Bernoulli equation is used to interpolate the pressure. A better and more consistent way of evaluating the pressure correction is to use the same basic principle that was used to derive the pressure correction, equation (20). The Bernoulli equation for estimating a boundary pressure is

P pB=pe + -~(V 2 - V 2) + pgAh (24)

With a guessed pressure field p* and starred velocity V*, the guessed boundary value becomes

P p~ =p~ + -~(V . 2 - V 2) + pgAh (25)

With the known boundary velocity V B, subtracting equation (25) from (24) gives

, _ , ) PB - P e + ~ - V2 2 (26)

with V = V* + V' and V' =d A p ' . The pressure correction gradient, dAp', is treated like the pressure gradient term in equations (1) and (2), with d = 1/a e. By just considering the u velocity component case, where the boundary is on the west face of a cell that is regular, the pressure correction gradient can be evaluated as

Us @ ' = ~ ( p ' d y ) a = ( p ' A y ) e + ( p ' A y ) , (27)

A = I

By substituting the boundary pressure correction of equation (26) into equation (27), we have

Ap' = ( p'Ay)e

+ ( p ' e + P ( 2 v ' d V p ' + ( d V p ' ) Z ) ) A y n

(28)

and by regrouping terms we have

( P'~Y)e + P'pAyB = (1 - pV*AyBd ) Vp'

P i 2 -~ Ay(dVpp) (29)

which can be solved for directly or iteratively for use in the velocity corrections. This method works well but it can be expensive in computations. A less expensive route is to take p~ = p~,. This is possible since at convergence V e = V*, thereby making the last term in equation (26) zero.

3.5. Solution procedure

The algorithm used for solving the discretized equations of fluid flow and heat transfer is based upon the semi-implicit



method for pressure-linked equations (SIMPLE) procedure by Patankar and Spalding. 33 The sequence of operations for the SIMPLE algorithm are as follows:

1. Set initial/current values to old time values and guess the pressure field p*;

2. Solve the discretized momentum equations (1) and (2) to obtain u* and v*;

3. Solve the discretized pressure-correction equation (20); 4. Calculate p from p = p* + p ' ; 5. Calculate u and v from their starred values using the

formulas in equation (17); 6. Solve the discretized energy equation (4); 7. Treat the corrected pressure p as new guessed pressure

p*, return to step 2. Repeat the whole procedure until a converged solution is obtained before advancing a time step.

Other variants of this procedure such as SIMPLER, SIM- PLEST, PISO, etc. can all be easily incorporated into the above context.

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: : : : : : : : : : : : : : : : : : : : : : : : : : : : : :

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<? Figure 4. Regular mesh results for Re= 400: (1) velocity vector, (2) pressure, (3) u velocity, (4) v velocity.

4. Solution of fluid flow problems

In this section the proposed method is applied to two well-known benchmark problems. These problems have been solved using both finite volume and finite element techniques by numerous investigators, and these are duly regarded as standard benchmarking for CFD codes and algorithms in fluid flow and heat transfer. The problems are flow in a cavity with a moving lid 34,35 and the natural convection-driven cavity. 36,37 Since the problems are well documented in the literature, their detailed description may be found in Ref. 35 and 36 and will not be reproduced here. Below the results of using the ICV procedure on these problems using both regular and polygonal meshes are compared with the benchmark solutions published by others.

4.1. The moving lid cavity problem

Results with regular elements - rectangular mesh. Re- sults for Reynolds numbers of 100, 400, and 1,000 were obtained and compared with those of Ghia and Ghia 35 which are regarded as the standard benchmark results. Figure 4 shows contour plots of pressure, u and v velocity components, and a velocity vector plot for one of these results with a Reynolds number of 400. Figure 5 shows the result of the u velocity component in the vertical direction and horizontal for v in their respective middle of the cavity compared those of Ghia and Ghia. In this investigation into unstructured meshes, a uniform mesh of 33 X 33 (1,089 cells) and 200 iterations for the solution procedure were found to be sufficient. Solutions were obtained for several first-order convection schemes, upwind, hybrid, exponential, and power law. All the findings

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App l . Ma th . M o d e l l i n g , 1996, Vol . 20, Februa ry 175

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x O 0 o trim lit i i i 4- W i l l

Figure 10. Triangular meshh results for Re = 100.

Ro-100 ( U - v e l o c i t y ) w i t h hoxo, gons l i d l t r l ~ t u l

1 .4

4

l i e

I I i

n ;

gaUl

o . e

0 . 4

O , I

?: : :

• g l . I g . 4 I I . g g . I 'O

le r~up

0 0 I o ~ ~ •1 4. H l ~ l d

Figure 11.

Re-lOB C v - v e l o c l t y Z ) w i t h hexngons i m t l - l t l m i =

n |

g . l l

= q

n w

.,.2 - a I

- o - a

- o , 2

- a l l

- 0 • i l I I I i s 4 o i I I 1

Hexagon-based mesh results for Re = 100.

Re- lO0 ( : U - v e l o c l t y ) with octo, gons MlO i t w . t ~

1 .4

4

O m

n i l

n ?

g i

i ; . l l

O,,I

0 .$

n a

n 4

-.~ - u •

- o . $ . . . . . '.. , .~. , ,.. , ;

Re--t00 C v - v e l O c l t y Z ) with octagons n = = M r I L I = = =

o,~e

0 4

o m

g

- o . m

- o 4

- o 1 5

- I p J

i a l l i i i i i • n 4 o i l l i B 4

Figure 12. Octagon-based mesh results for Re = 100.

the contouring and is not present in the numerical results. (Unfortunately, since we had no software available that can accept data from polygonal meshes, simple programs were specially written for the plotting.) To see how all the results compared with those of Ghia and Ghia, compara- tive plots of the u and v velocity profiles were produced.

For the case where Re = 100, Figures 10 to 12 show comparisons with Ghia and Ghia for the triangle, hexagon, and octagon results, respectively. All three meshes agree well with the benchmark. The Re = 400 comparisons are

shown in Figures 13 to 15. Here both triangle and hexagon meshes under predict the peaks and troughs a little, with the hexagon mesh being slightly better than the triangular one. The octagonal mesh, however, is slightly over the peaks and troughs, but still it is by far the best result for both regular and irregular meshes at the node density specified. The results for the Re = 1,000 case are shown in Figures 16 to 18. Again, both triangular and hexagonal meshes underpredict the peaks and troughs, with the hexagonal doing slightly better than the triangular mesh.

A p p l . M a t h . M o d e l l i n g , 1 9 9 6 , V o l . 20 , F e b r u a r y 1 7 7


However, although the octagonal mesh is slighly underpre- dicting the peaks and troughs, it is by far the best solution in both regular and irregular meshes.

The key feature to note here though is that while Ghia and Ghia used a very fine uniform mesh, 152 × 152 (23,104) cells in obtaining their solution, here only 1,097 cells were used! With these encouraging results other first-order schemes, namely upwind and power law, were used on the octagonal mesh. In both cases the results obtained from the two first-order schemes show a significant improvement over their regular mesh equivalents.

Figures 19 and 20 show the upwind and power law results for Re = 1,000. None of the results obtained with the various schemes on the octagonal mesh were poorer than the best from the regular case.

The reason for the significant improvement with the octagonal mesh lies in the average number of connections with neighboring nodes in a control volume. As experienced with high-order schemes Used in regular meshes, the numberical diffusion for flows with changing direction is significantly reduced by taking more surrounding nodes into account. In the triangular and regular mesh cases, the

R e ' - ~ ' O O C U - v e l o c l ' t . y Z ) w I ' t . h C F l ~ n g l e 5

4,4 4

l |

IhO o ~

all 0.11

o , z

• 4

. m •

.1111

- • . 4

o

o

I I , I l . t I ,S l , I I

Figure 13.

R e - 4 0 0 C V - v e l o c f t y ) w i t h t r l ~ n g l e ~

u i ~ 0 ~ n 4

o • o m . . 0 , 4

• o n

m •

- O . 8

- O , 4

0 i

- o i l • 0 • 0 4 I I e l i i

o . . . , ' . ~ . . . .

Triangular-based mesh results for Re = 400.

R e - . ' 4 0 0 C U - - v e l o c l t . y ) w r t . h hex,',gons ' l t r m t N o

.,I.,I

n i

II,o

la'/

O,|

!.i

-g , ,0

a

• • , • o,,o ii , • •, • 1

Figure 14.

04

ilSi

• a

0,4

z • o n

- o •

- 0 , 8

I J 4 ° ,

oO g o

R e - 4 O O C V - v e l o c l t . y ' ) w l t h h e x n g o n 8 ann I t .Q '~L l~

a i

o • o 4 a • e l l 1

x ~

o ~ 1 i t el • ~ l l

Hexagon-based mesh results for Re = 400.

R o - 4 O O C U - v e l o c l t . y ) w l t . h o c t a g o n s

1,4

' f o •

D.O

o7

g,g ::I :':

- o r

o n l

i I i n -o,~ 0 o , I D.e o , | a • I

Figure 15.

R e - 4 O O C v - v e l o c l t . y ' ) w l t . h o c t a g o n n un

g 4

I 1

e ;

~ - D , I

. m •

- I |

i

D B

- n • I OH o 4 O I i n .i

Octagon-based mesh results for No = 400.

178 Appl . Math . M o d e l l i n g , 1996, Vol . 20, February


number of connections per cell is 3 and 4, respectively. For the hexagonal mesh, the average connection number per cell is 3.97. This low value is due to the large number of triangles in the mesh, thereby reducing the overall connection value. This may also explain why there is so little difference between the results in the hexagonal and the regular cases. In the octagonal case, the average connection value is 5.85, i.e., almost two connections more than that of the regular and hexagonal case and could well

indicate why octagonal solutions are so good. Similar improvements in accuracy have also been suggested recently by Hammond and Barth e8 with triangular elements and Liu and Jameson 7 with quadrilateral elements for aerodynamic flows indicating a preference for the vertex- centered over the cell-centered approach. With the vertex method constructing its control volume around the vertices of the element, it naturally forms a polygonal control volume that is not too dissimilar to the polygon elements

Re-lO00 CU-voloclty:) w l t h otto•on• u it11¢~ 1,1

"I

o •

0 1 g.7

0 , |

• • o

o o

D . |

• 4

-o ,~ o o

-0.2

-o • o

- o . • • o , I I , o ° , l t Ii, ll i 'IL

)fL~O [] ~ r l L t i i + 14e41rld

Figure 16.

R e - 1 0 0 O C V - v e l o c l t y 3 w i t h o c t a g o n s = =

r D I

P

~ - D . ' I

- o •

ii . ,

. : ° un U 4 n l i n .i

,, . . , . . , o , , . . . . ,

Triangular-based mesh results for Re = 1,000.

Re-'lO00 ( U - v e l o c l t y : ) wlth h~gons u l u r ~

I , ' 1 I j . o |

o l 0,7 D.O

• l

:.: 0 , 1

o 4

- n o []

- o , • • g . l l , t • . I l . l 1

o Ob~ ~ Y I ~ O + N~rld

Figure 17.

o'1

o

*D.1

- n •

- o •

~-qO00 C V - v e l o c l t y ~ w i t h h e x s g o ~ = =

0

o

a O l 0 4 on e l 1

0 11111 ~x l lC lO+ I~ I r I o

Hexagon-based mesh results for Re = 1,000.

Re-t000 C U - v o l o c l t y : ) wlth oct••on•

1 . 1

E ° 0

o n

O.•

O •

0 4 • . l

O,|

0 4

- O . I

- o ~

. . . . o . . . . ',o , j . , .,.. , ,

. . . . . . ' 2 . . . , .

Figure 18.

R e - q D n g C V - v e l o c l t y : : ) w i t h o c t a g o n s

i n It4rl~ I ~ l I n q

r n l

n l

u

• I - o •

- 0 . 4

. a • o

- o | ° l 114 n o e l .i

° . . . : . ~ . . . .

Octagon-based mesh results for Re = 1,000.

Appl. Math. Model l ing, 1996, Vol. 20, February 179


~I000 CU-ve locTty ) w i t h octagons

t.-I

o | one

o.7 / 9.0 n |

:.: °

D.t ~ n.I

0

-O,a

i i ° a a . I I,,P o , i ; l;.@ 1

° . . . ; r °. ~ . , . ,

Figure 19.

0 4

o . a

°n •

-o •

°D,4 [

-D •

-o •

Re-t000 CV-veloclty~ wi th octagons a n 0 t e ~ t 1 ~

o

a al o a e l e | t

x c n o

o IPeril i I : an + LlO~lnd

Octagon-based mesh with upwind scheme for Re = 1,000.

Fk~-lO00 CU-velocTty~) wlt.h oct.ogons u t t = - i ~ ~

1, '1

4

o n

n i l

0 , 7

O,e

a l e

D.8

n4

-D,| -e •

ue - . • IN.• p .~ Q . • Ip, i I

o l l ~ a eL o l ÷

Figure 20.

~-IOOO CV-ve loc l t y~ with octngo~

114

u P , a -

D ie

o l

X

-0.I

-o o

a N i l U 4 oau m e

x ~

D ~0uw ~ i n ÷

Octagon-based mesh with power law scheme for Re = 1,000.

Table 1.

U r n a x

A compar ison between the nonuni form mesh and the de Vahl Davis solution

Ra = 103 Ra = 104 Ra = 105 3.638 (3.649) 16.151 (16.178) 34.861 (34.73)

Ra = 106 65.173 (64.63)

y 0.825 Vrnax 3.696 x 0.175 NU o 1.114 Numa x 1.507 x 0.09 NUmi n 0.692 y 0.99

Values in brackets

(0.813) 0.825 (0.823) 0.855 (0.855) 0.855 (0.85) (3.697) 19.7 (19.617) 68.416 (68.59) 220.82 (219.36) (0.178) 0.115 (0.119) 0.07 (0.066) 0.03 (0.0379) (1.117) 2.213 (2.238) 4.54 (4.509) 9.345 (8.817) (1.505) 3.558 (3.528) 7.998 (7.717) 19.778 (17.925) (0.092) 0.145 (0.143) 0.07 (0.081) 0.03 (0.0378) (0.692) 0.586 (0.586) 0.726 (0.729) 1.028 (0.989) (1) 0.99 (1) 0.99 (1) 0.99 (1)

() are from de Vahl Davis.

Table 2.

U m a x

A compar ison between the uniform mesh and the de Vahl Davis solution

Ra = 103 Ra = 104 Ra = 105 3.63 (3.649) 16.127 (16.178) 34.822 (34.73)

ea 65.4

= 106 (64.63)

y 0.803 (0.813) 0.833 (0.823) 0.864 (0.855) 0.864 Vma x 3.677 (3.697) 19.533 (19.617) 67.295 (68.59) 208.8 x 0.167 (0.178) 0.106 (0.119) 0.076 (0.066) 0.045 Nu o 1.118 (1.117) 2.213 (2.238) 4.662 (4.509) 9.629 Numa x 1.51 (1.505) 3.6 (3.528) 8.34 (7.717) 21.133 x 0.076 (0.092) 0.136 (0.143) 0.076 (0.081) 0.152 NUmin 0.69 (0.692) 0.582 (0.586) 0.715 (0.729) 0.979 y 0.985 (1) 0.985 (1) 0.985 (1) 0.985

Values in brackets 0 are f rom de Vahl Davis

(0.85) (219.36)

(0.0379) (8.817)

(17.925) (0.0378) (0.989) (1)



• . . . . . . . . . . . . . . . . . . . . . . . . . .

i[i:~!!:!!!!!!?!?ii)iii!ii ~

!iiiiiiiiiiiiiiiiiii i,iii]

Figure 21. Nonun i fo rm mesh results for Ra = 10s: (1) ve loc i ty vector, (2) temperature , (3) u veloci ty , (4) v veloci ty.

used in this study, providing better solution accuracy than the cell-centered approach with trianglar and quadrilateral elements.

4.2. The natural convection-driven cavity problem

Results with regular e lements - rectangular mesh. Re- sults for Rayleigh (Ra) numbers of 103, 104, 105, and 106 were obtained with the hybrid differencing scheme and compared with those of de Vahl Davis. 37 The solutions were obtained following similar practices used by numerous investigators, such as those of Markatos and Peri- cleous, 39 giving 200 iterations for the solution procedure, using results at one Rayleigh number as initial values for the next higher Rayleigh number and a Prandtl number of 0.71. Here a mesh size of 33 × 33 cells was found suitable. Tables 1 and 2 show the results for both nonuniform and uniform structured meshes compared against the solution

Table 3.

Umax

A compar ison between the octagon and the de Vahl Davis so lut ion

Ra = 103 Ra = 104 Ra = 105 3.697 (3.649) 16.394 (16.178) 35.023 (34.73)

Ra = 106 67.011 (64.63)

y 0.813 (0.813) 0.813 (0.823) 0.844 (0.855) 0.844 (0.85) Vrnax 3.705 (3.697) 19.839 (19.617) 69.923 (68.59) 216.54 (219.36) x 0.186 (0.178) 0.125 (0.119) 0.0625 (0.066) 0.0313 (0.0379) NU o 1.144 (1.117) 2.289 (2.238) 4.431 (4.509) 6.789 (8.817) NUma x 1.548 (1.505) 3.613 (3.528) 7.363 (7.717) 11.919 (17.925) x 0.094 (0.092) 0.156 (0.143) 0.0938 (0.081) 0.0625 (0.0378) Ngmi n 0.703 (0.692) 0.589 (0.586) 0.717 (0.729) 0.958 (0.989) y 0.99 ( 1 ) 0.99 ( 1 ) 0.99 ( 1 ) 0.99 ( 1 )

Values in brackets (1) are f rom de Vahl Davis

Figure 22.

:i iii!! ii!iiiiiiiiii!!!iiiiii' i:!ii!i!!!iii!i!iiiiiiiii!: ! i i ! ! i ! ! ! : : : : : ! ! ! ! ! : : : : : ! i !

/

Octagon-based mesh results for Ra = 106: (1) ve loc i ty vector, (2) temperature, (3) u veloci ty, (4) v veloci ty.


Conventional finite volume method for CI-u application: P. Chow et al.

of de Vahl Davis. For the nonuniform mesh, there is generally a good agreement with that of de Vahl Davis for all Rayleigh numbers. For Ra = 1 0 6, the difference is slightly higher than in the rest because of the thin boundary layers developed at high Rayleigh numbers in the two conducting walls. For the uniform mesh, the results con- form with the expectation, generally good at low Rayleigh numbers but with differences appearing in the high numbers. Figure 21 shows a nonuniform mesh result for Ra = 1 0 6.

Results with polygonal e lements - unstructured mesh. In the natural convection-driven cavity problem, it is well understood that thin boundary layers will develop in the two conducting walls for high Rayleigh numbers. The nonuniform structured mesh has been adapted to pickup the thin boundary layers near the surface of the walls with the mesh remaining orthogonal. This mesh adaption cannot be easily applied to polygonal meshes; it is not a straightforward process even in a square cavity, and the solution for a low-order polygon would not be any better than the regular case. However, even without any adaptation to the mesh, the results of polygonal cells would still be of interest for the study undertaken here. From the work carried out with the moving lid cavity, the octagon mesh proved to give the best results, and consequently the same mesh is used for this investigation with the hybrid scheme.

Table 3 shows the comparison of the octagon results against those of de Vahl Davis. The result is obviously not as good as the nonuniform structured mesh case but it is similar to those of the uniform mesh. The comparison is good at low Rayleigh numbers but differs at high values. This is highlighted extremely well by the u and v velocity contour plot at Ra = 1 0 6 (Figure 22). For the uniform structured mesh, the usual features for a contour plot of the u and c velocity are all present. In the octagonal mesh case, the u velocity plot is identifying the two recirculation zones that are in the de Vahl Davis u velocity plot. In the same plot for the nonuniform structured mesh case, the recirculation is only just visible and it is not as well defined as the octagon. For the v velocity plot, there is a significant difference between the de Vahl Davis and the octagon result. The reason for this is the thin boundary layers at the two conducting walls where both heat and mass transfer is convection-dominated. In this case, the rectangular mesh is better suited than the high-order polygonal ones. Since the flow is aligned with the cell this makes the neighboring diagonal contribution insignificant. This has also been experienced by Patel et al. 40 in a structured mesh context, who employed a CUPID scheme that uses more neighboring nodes in the approximation where convection-dominated flow yields results that are essentially first-order. For recirculating flows, the polygonal cells are able to identify the flow details where the rectangular mesh cannot, as shown in the u velocity plot. This highlights the similarity in characteristics between the high-order polygon cells and the CUPID scheme where both breat more of the surrounding nodes in the discretization.

5. Conclus ions

A new general cell-centered solution procedure based on the classical CV or FV approach has been developed for numerical heat transfer and fluid flow for both structured and mixed polygonal cell unstructured meshes. The ICV method is a natural extension of the classical FV method to facilitate the treatment of geometrically complex problems. The limitation of having to predefine element/cell shapes, implicitly embedded in the conventional FV approach, has been addressed and overcome in the new formulation. Elements/cells such as triangles, quadrilater- als, polygons, and the mixing of different element types may now all be treated in the same solution procedure so that the ICV and convectional FV method are identical given a mesh that is rectangular.

The method was applied to two standard well known CFD benchmark cases: the moving lid cavity and the natural convection-driven cavity to benchmark the new approach with the standard method. In both the cases they were solved first with regular (rectangle) and then irregular element meshes. In the regular case, the results obtained are virtually identical with the findings of other authors using the classical FV method. This is expected, of course, since both the ICV and FV methods reduce to the same set of equations when the mesh has a rectangular structure. However, the polygonal element meshes with first-order schemes yield a significant improvement in numerical accuracy over the rectangular mesh with the same or a similar number of nodes. This improvement in solution accuracy has also been experienced recently by other workers 7,28 using the vertex-centered approach over the cell-centered one with triangular and quadrilateral elements for aerodynamic flows. In these cases, the control volume is constructred around the vertices of the element which naturally forms a polygon cell that is not too dissimilar to the polygon elements used here. However, the vertex-bases scheme is more complicated with respect to its formulation and programmability; it is also more com- putationally expensive. 5 The above improvements in accuracy are simply due to each cell having a greater connec- tivity to its neighbors, thereby reducing numerical diffusion for flows with changing direction. Similar experience has been observed by the authors with the CUPID scheme 40 in a structured mesh context, where once again more surrounding nodes are treated in the formulation giving an improvement in the solution. Therefore, one may view the polygon element/cell in the control volume context as a high-ordered cell which utilizes more neighboring node information in the formulation. Work is on- going to address highly nonorthogonal meshes using the present ICV method in three dimensions.

Nomenc la ture

/x The dynamic velocity p The density of the fluid V The face resultant velocity S u Source for the x direction S~ Source for the y direction



U~ U

r, S , si

n i

At

Sc

A h Ax, Ay 6x, 6y

Cdiff

N S V PA Re Ra

Cartesian velocity components diffusion coefficient Source term represents the components of the outward normal vector (ds t = dy and ds 2 = - d x ) Coordinate direction in which n 1 = x and n 2 = y Volume of the irregular control volume P The time step The value of q~ at the centroid of cell P

Total number of control faces Adjacent control volume(s) The unit normal to the cell face Face surface area vector components Distance vector components between the nodes A and P in cartesian coordinates The cross-diffusion term for the common cell face Nodal distance vector The surface vector The resultant velocity vector The Peclet number Raynolds number The Rayleigh number

References

1. Winslow, A. M, Numerical solution of quasilinear Poisson equation in nonuniform triangle mesh, J. Comput. Phys. 1966, 1, 149-172

2. Baliga, B. R. and Patankar, S.V, A new finite element formulation for convection-diffusion problems, Num. Heat Transfer 1980, 3, 393-409

3. Schneider, G. E. and Raw, M. J, A skewed positive influence coefficient upwinding procedure for control-volume-based finite-element convection-diffusion computation, Num. Heat Transfer 1986, 9, 1-26

4. Lonsdale, R. D. and Webster, R, The application of finite element methods for modelling three dimensional incompressible flow on an unstructured mesh, Numerical Methods in Laminar and Turbulent Flow, eds. C. Tayler et al, Pineridge Press 1989, pp. 1615

5. Chow, P. and Cross, M, An enthalpy control-volume-unstructured- mesh (CV-UM) algorithm for solidification by conduction only, Int. J. Num. Meth. Eng. 1992, 35, 1849-1870

6. Jameson, A, Baker, T. J. and Weatherill, N. P, Calculation of inviscid transonic flow over a complete aircraft, AIAA paper 86-0103, 1986

7. Liu, F. and Jameson, A, Multigrid Navier-Stokes calculations for three-dimensional cascades, A/AA J. 1993, 31, 1785-1791

8. Jameson, A. and Mavriplis, D, Finite volume solution of the two-dimensional Euler equations on regular triangular mesh, A/AA J. 1986, 24, 611-618

9. Morgan, K., Peraire, J. and Peiro, J, The computation of three dimensional flows using unstructured grid, Comp. Meth. Appl. Eng. 1991, 87, 335-352

10. Batina, J, Implicit flux-split Euler schemes for unsteady aerodynamic analysis involving unstructured dynamic meshes, A/AA J. 1991, 29, 1836-1843

11. Barth, T. J, On unstructured grids and solvers, Proceedings of the VK1 Lecture Series 1990-03 on CFD, March 1990

12. Fryer, Y. D., Bailey, C., Cross, C. and Lai, C. -H, A control volume procedure for solving the elastic stress-strain equations on an unstructured mesh, Appl. Math. Modelling 1991, 15, 639-645

13. Bailey, C., Fryer, Y. D., Cross, M. and Chow, P, Predicting the deformation of castings in moulds using a control volume approach on unstructured meshes, Mathematical Modelling for Material Pro- cessing eds. M. Cross et al, Oxford, 1993, pp. 259-272

14. Gordon, W. J. and Hall, C. A, Construction of curvilinear coordinate systems and applications to mesh generation, Int. J. Num. meth. Eng. 1973, 1, 461-477

15. Thompson, J. F, Thames, F. C. and Mastin, C. W, Automatic numerical generation of body-fitted curvi-linear coordinate system for fields containing any number of arbitrary 2D bodies, J. Comput. Phys. 1974, 15, 291-319

16. Thompson, J. F, Warsi, Z. U. A. and Mastin, C. W, Boundary-fitted coordinate system for numerical solution of partial differential equations - - A review, J. Comput. Phys. 1982, 47, 108

17. Demirdzic, I. and Peric, M, Finite volume methods for predication of fluid flow in arbitrarily shaped domains with moving boundaries, Int. J. Num. Meth. Fluids 1990, 10, 771-790

18. Rhie, C. M. and Chow, W. L, A numerical study of the turbulent flow past an isolated airfoil with trailing edge separation, A/AA J. 1982, 21, 1525-1532

19. Weatherill, N. P, Mixed structured-unstructured meshes for aerodynamic flow simulation, Aeronaut. J. 1990

20. Peace, A. J. and Shaw, J. A, The modelling of aerodynamic flows by solution of the Euler equation on mixed polyhedral grids, Int. J. Num. Meth. Eng. 1992, 35, 2003-2229

21. Childs, P. N., Shaw, J. A., Peace, A. J. and Georgala, J. M, SAUNA: A system for grid generation and flow simulation using hybrid structured/unstructured grids, First European Computational Fluid Dynamics Conference, Brussels, September 1992

22. Thareja, R. R., Stewart, J. R., Hassan, O., Morgan, K. and Peraire, J, A point implicit unstructured grid solver for the Euler and Navier- Stokes equations, Int. J. Numt, er. Metho. Fluids 1989, 9, 405-425

23. Mavriplis, D.J. and Jameson, A, Multigrid solution of the Navier- Stokes equations on triangular meshes, AIAA J. 1990, 28, 1415-1425

24. Pelz, R. B. and Jameson, A, Transonic flow calculations using triangular finite elements, A/AA J. 1985, 23, 569-576

25. Batina, J, Accuracy of an unstructured-grid upwind-Euler algorithm for the ONERA M6 wing, J. Aircraft 1991,

26. Batina, J, Unsteady Euler algorithm with unstructured dynamic mesh for complex-aircrafi aerodynamic analysis, AIAA J. 1991, 29, 327- 333

27. Batina, J, Unsteady Euler airfoil solutions using unstructured dynamic meshes, A/AA J. 1990, 28, 1381-1388

28. Hammond, S. W. and Barth, T. J, Efficient massively parallel Euler solver for two-dimensional unstructured grids, AIAA J. 1992, 30, 947-952

29. Strawn, R. C. and Barth, T. J, A finite-volume Euler solver for computing rotary-wing aerodynamics on unstructured meshes, J. Am. Helicopter Soc, 1993, 38, 61-67

30. Struijs, R., Vankeirsbilck, P. and Deconinck, H, An adaptive grid polygonal finite volume method for the compressible flow equations, AIAA paper 89-1957-CP, 1989

31. Patankar, S. V, Numerical Heat Transfer and Fluid Flow, Hemi- sphere, Washington, DC, 1980

32. Croft, N., Cross, M., Pericleous, K. and Chow, P, In preparation. 33. Patankar, S. V. and Spalding, D. B, A calculation procedure for heat,

mass and momentum transfer in three-dimensional parabolic flows, Int. J. Num. Heat Mass Transfer 1972, 15, 1787-1806

34. Burgraff, O. R, Analytical and numberical studies of the study of steady separated flows, J. Fluid Mech. 1966, 24, 113

35. Ghia, K. N. and Ghia, U, Elliptic systems: Finite-difference method III, Handbook of Numerical Heat Transfer, eds. W. J. Minkowycz, et al, Wiley, 1988

36. Jones, I. P, A comparison problem for numberical methods in fluid dynamics: The "double-glazing" problem, Numerical Method in Thermal Problem eds. R. W. Lewis and K. Morgan, Pineridge Press, Swansea, U.K, 1979

37. de Vahl Davis, G, Natural convection in a square cavity: A bench mark numerical solution. Int. J. Num. Meth. Fluids 1983, 3, 249-264

38. Chen, J, The numerical solution of complex fluid flow pehnomena, Ph.D. Thesis, University of Leeds, U. K, 1991

39. Markatos, N. C. and Pericleous, K. A, Laminar and turbulent natural convection in an enclosed cavity, Int. J. Heat Mass Transfer 1984, 27, 755-772

40. Patel, M. K., Cross, M. and Markatos, N. C, An assessment of flow oriented schemes for reducing false diffusion, Int. J. Num. Meth. Eng. 1988, 26, 2279-2304


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