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

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"el,
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 of Greenwich, London, U.K. School of 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. Introduction
In recent years, the increasing need for solving numerical heat transfer and fluid flow problems in complex geome- tries has prompted a move toward a finite volume-unstruc- tured 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 cou- pled solution procedures necessary for efficient flow calcu- lation 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 oscilla- tion 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 un- structured mesh in the same solution procedure; heat trans- fer and solidification is solved using a cell-centered ap- proach 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 air- craft 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 ele- ments 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 deforma- tion 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 discretiza- tion 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 unstruc- tured framework has the flexibility for treating various features, such as local mesh adaption, refinement, and independent mesh motion. 26.30 The new method has re- cently 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 unstruc- tured 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 differen- tial equations 3a.
Momentum equations:
o(ou)
-- + v. (oVu) = v. -
Ot
o(o )
Ot
-- + v. (ow) = v. - --
@
Ox +
S,
(1)
Op
+S,, Oy
(2)
Continity equation:
oo
-- + v. (ov) = 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 sym- bol 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
Conventional finite volume method for CFD application: P. Chow et al.
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 construct- ing 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 implic- itly imbedded in the formulation and restricts it to ele- ments 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 dis- cretization procedure can be applied to any element/cell 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
=-dx 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:
flap05
(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 with normal surface vectors.
Source term:
f ff, 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 com- mon 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-
max[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)
+ max[0,-
CA]
(12)
172 Appl. Math. Modelling, 1996, Vol. 20, February
Conventional finite volume method for CFD application: P. Chow et al.
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, (pv)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 struc- tured 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 conti- nuity 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* Jr 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: (AY) '
ui=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' '
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 equa- tion. From equation (11) the convective mass flux for a given face i is pV.
S= [(
pu)it+
(pv)if]
.(Ay,~-- z~xif) = ( puAy)i- ( pvAx)i
(18) Substituting the expression given in equation (17) and rearranging in terms of p' gives
pV. 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 80.2
Buoyancy vs Pressure Gradient
forward
backward
I 0 Pressure
radien~
-- 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.
Appl. Math. Modelling, 1996, Vol. 20, February 173
Conventional finite volume method for CFD application: P. Chow et al.
where (
pAy2 a A = a, Us
ap = E aA + --
A=I
Us be=
A=I
pAx2) ] (21)
--
a v i A
(or),
At (22)
[(
pu*Ay)i- (
Du*Ax 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 equa- tions 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 bound- ary. These are usually specified in terms of external veloc- ity 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 arrange- ment 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 for- ward/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 buoy- ancy and pressure gradient in the vertical direction. The pressure gradient obtained using forwar/backward differ- encing 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 esti- mating 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 interpo- late a pressure correction value when the Bernoulli equa- tion 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 equa- tion (25) from (24) gives
,_,
)
PB -Pe + ~ - V2 2
(26) with V = V* + V' and V' =dAp'. 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 correc- tion gradient can be evaluated as
Us @'= ~ (p'dy)a=(p'Ay)e+(p'Ay),
(27)
A=I
By substituting the boundary pressure correction of equa- tion (26) into equation (27), we have
Ap' = ( p'Ay)e +(p'e+P(2v'dVp'+(dVp')Z))Ayn
(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
174 Appl. Math. Modelling, 1996, Vol. 20, February

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