Calculation of flows using three-dimensional overlapping grids and multigrid methods

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, VOL. 38,259-282 (1995)

CALCULATION OF FLOWS USING THREE-DIMENSIONAL OVERLAPPING GRIDS AND MULTIGRID METHODS

J. Y. TU AND L. FUCHS’

Department of Mechanics, The Royal Institute of Technology, S-100 44, Stockholm, Sweden

SUMMARY A computational methodology combining overlapping grid techniques with multigrid methods has been developed for three-dimensional flow calculations in or around complex geometries. The computational accuracy, efficiency and capability of the present approach are investigated in this paper. The incompressible Navier-Stokes equations are discretized using a finite volume method on a semi-staggered grid. The discrete problem is solved by a multigrid algorithm. Some numerical examples are chosen for evaluating numerical accuracy: (a) a straight pipe for which the exact solution is known; (b) curved pipes where previous experimental and numerical data are available; (c) an axisymmetric sudden expansion. The performance of the multigrid method on overlapping grids is assessed. Several cases of flows in stationary and time- dependent complex geometries are given to demonstrate the capability and the potential of the methods that we employ.

KEY WORDS Complex geometries Numerical accuracy and efficiency 3D overlapping grids Multigrid methods Finite-volume Engine flows

1. INTRODUCTION

Computational Fluid Dynamics (CFD) is becoming an essential tool in the understanding of fluid physics and in engineering design. There are three major challenges for CFD: (i) the physics of the flow itself (e.g. turbulence); (ii) geometrical complexity of realistic configurations; and (iii) main- taining required numerical accuracy with good computational economy. This paper addresses the latter two aspects, namely, suggesting a computational methodology in which complex three-dimensional (3-D) problems can be handled easily, with rather good accuracy and computational efficiency.

In many industrial applications such as flow simulations in Internal Combustion (IC) engines, it is extremely difficult to generate a single, structured, grid that covers the entire flow field. Such grids have often highly skewed cells and poor cell distribution. Decomposition of the physical domain into topologically simpler sub-domains, or zones, is an effective solution to many grid generation difficulties. The main advantages of domain decomposition are that (a) it can reduce the topological complexity of a complicated geometry, permitting each component to be more easily fitted with an appropriated grid; (b) the relative motion between the physical objects can be treated by means of movement of zones; and (c) it provides a convenient mechanism for splitting a problem into smaller portions for parallel computation and also facilitates the use of different models/approximations to the physics in the different regions.

In general, domain decomposition techniques can belong to one of two classes: ‘patched grids’ or ‘overlapping grids’. The grid patching method requires that the different grids are joined

Present address: Department of Heat- and Power-Engineering, Lund Institute of Technology, S-22100, Lund, Sweden

CCC 0029-5981/95/020259-24 0 1995 by John Wiley & Sons, Ltd.

Received 27 July 1994

260 J. Y. TU AND L. FUCHS

together along common boundaries. The level of continuity of grid lines across the patches may vary, however, depending on the problem under consideration. The Overlapping Grid (OG) technique is more flexible in handling multicomponent configurations since the boundaries between neighbouring zones can be arbitrary. The OG-approach requires an interpolation procedure to transfer information among the different grids. A number of publications concern- ing OG-techniques and their applications have recently appeared in literature. However, most of these publications are limited to two-dimensional situations.' - 5 Fewer 3-D cases have been considered.6 - * Large-scale demonstration of the capability, accuracy and efficiency of using 3-D overlapping grid techniques is still limited. The present study addresses these questions by computing several cases for which experimental and numerical data are available. In order to address these questions, the basic two-dimensional overlapping grid version developed by Fu~hs ' ,~ has been extended to three dimensions with enhanced flexibility. Any number of overlapping grids can be generated and they may be moved arbitrarily relative to each other. Each zone is generated independently as a local body-fitted mesh so that geometrically non- similar multicomponent configurations can be handled easily. The incompressible Navier-Stokes equations are discretized on the present OG-system by using a finite volume (FV) method. The multigrid (MG) method is integrated into the present OG-technique to accelerate the convergence of the iterative solver. It has been found that the MG-scheme is mandatory for the overlapping grid system to compensate for the additional increase in computational speed due to the information exchange among the sub-grids. A flux correction schemeg at the inter-zonal information exchange is introduced to ensure global mass balance in each zone on all levels of the MG-procedure. This scheme is required to maintain the convergence rate of the MG-technique on an OG-system.

An alternative approach to OG is to use unstructured meshes. Such an alternative offers a similar flexibility of handling multicomponent geometries," as an OG-system. Futhermore, unstructured grids may be used in a straightforward manner for adaptive refinement. Adaptive grid refinements can be carried out also with the OG-system, by introducing locally refined sub- domains.'' The main advantages of the OG-technique as compared to unstructured grids are twofolded: the OG-system is most applicable to situations where one has solid multiple objects, that may move relative to each other during the computations or a design process. Using the OG-technique one avoids the need to generate completely new grids for each situation, since the grids in each zone are moved relative to each other. (This requires the addition of acceleration terms to the governing equations.) The other advantage of OG is that the overhead is smaller, since the general interfaces are among few number of grid systems. The local regularity (stride) of the data leads to a reduction in computational time, which is experienced to be of a factor of 2-3. Furthermore, one may use, to a larger extent, sub-grids that are orthogonal (or even Cartesian) uniformly spaced grids. This implies that the number of computational operations for expressing the discrete approximations to the differential equations, decreases substantially. In the limiting case, however, as the number of zones (grids) equals the number of cells, the OG-system (patched) will reduce to an unstructured mesh.

With the main emphasis being on studying the accuracy, efficiency and potential of the numerical method, our attention is focused here on the calculation of laminar flows in different flow domains. One case for turbulent flow is also carried out using a k--E turbulence model. The computational accuracy is investigated by using the currently developed code to compute a number of cases for which either exact solutions or experimental and/or numerical data are available. The computational efficiency of the present numerical procedure using multigrid method is evaluated by comparing with global single-grid calculations. Finally, a number of cases with rather complex geometries are chosen to illustrate the ability and the potential of the present method in predicting flows in such configurations.

OVERLAPPING GRIDS AND MULTIGRID METHODS 26 1

2. FORMULATION OF THE PHYSICAL PROBLEM

We assume that the fluid is incompressible and the flow may be either steady or time-dependent. The variables are non-dimensionalized using a length scale D and a velocity scale Uo. The governing equations are written in the following form:

Continuity: v-u = 0

Momentum: au 1 - + v-uu = - V P + -v2u at R e

where U = (u, v, w) is the velocity vector, with Cartesian components in x-, y- and z-direction, respectively. P represents the pressure Re is the Reynolds number based on the reference velocity Uo, the reference length D and the kinematic viscosity v.

No-slip condition is applied at solid walls and the velocity vector is assumed to be given at inlets, whereas a zero-gradient condition is applied at the outflow boundaries.

For the turbulent flow case, a k--E turbulence model is used for turbulence closure. Further details are given in References 12 and 13:

- + V . ( U k ) = V . ak at (3)

where G is the rate of generation of turbulent kinetic energy. The effective viscosity is defined by

where v( is the laminar viscosity. The turbulent viscosity is then given by

k2 V( = c, -

&

This model has five constants that assume the following values:

C , = 0.09, C1 = 1.44, C2 = 1.92, ak = 1.0, a, = 1.3 (7)

3. NUMERICAL METHODS

3.1. Overlapping grids

An overlapping grid is constructed to cover the region on which the governing equations are to be solved. The basic idea of the overlapping grid technique, used here, is to employ a separate body-fitted grid for each component in a multicomponent configuration and then to interface the grids in a manner which allows for efficient solution of the governing equations. One of the main advantages of using overlapping grids is that it can reduce the topological complexity of a complicated geometry, permitting each component to be more easily fitted with an appropriate grid. Usually, such appropriate grids are often structured grids (orthogonal or non-orthogonal). Currently, we use grids of a library of nine basic mesh types as shown in Figure 1. The overall grid-system is generated by combining the basic grids.

262

A

J. Y. TU AND L. FUCHS

,. 4

ITYPE=l

dlYPE=3

f

2 I

UITYPE-2

ITYE-4

.c ATYPE=6

2 I

Compared to the single-grid approach, the storage of numerical data for the overlapping grid system is more complicated. The data structure that we employ here is an extension of the data structure used previously.' All the dependent variables and the grid parameters are stored in one-dimensional arrays. A pointer system is defined so that each sub-grid can be accessed directly. The data are organized by grid levels and sub-grid numbers. Within each grid level, the variables of the first sub-grid are stored at the beginning, followed by those of the second sub-grid, and so on. The position of the first variable entry of each sub-grid is calculated according to grid

OVERLAPPING GRIDS AND MULTIGRID METHODS 263

levels and the number of sub-grids. The data for internal boundary points in different sub-grids are also stored in additional shorter one-dimensional arrays and are independently managed by an auxiliary pointer system. This type of data structure allows access to each sub-grid independently and it is then easy to deal with the interfaces among the different grids. This grid system allows also addition/deletion of locally refined sub-grids. I

3.2. Finite volume formulation

The finite volume method is used to discretize the Navier-Stokes equations using Cartesian velocity components. By this approach one avoids the need for transformation of co-ordinates and it makes the information exchange procedure among different grids simpler. We define a semi-staggered grid system in which all the velocity components are defined at the cell vertices while all scalar variables (such as the pressure, P, the turbulence kinetic energy, k, and its rate of dissipation, E ) are defined at the cell centre. The main advantage of using the semi-staggered grid system is that it requires a simpler interpolation procedure when compared with the fully staggered grids. On the other hand, one does not have to specify boundary conditions on the pressure as is required in the case of complete collocation of the dependent variables. The three-dimensional finite volume (i.e. main control volume) for the scalar transport equations and the continuity equation is the grid cell itself (i.e. cell-centred finite volume). For the momentum equations, the main control volume is formed by joining the cell centres surrounding the cell vertex (i.e. cell-vertex finite volume). Both the cell-vertex and the cell-centred finite volumes are illustrated in Figure 2. A wedge-shaped control volume and a control volume which consists of multiple wedge-shaped volumes, as shown in Figure 3, are used particularly in cases of grid degeneration (i.e. at the axis of a cylindrical mesh).

Figure 2. Illustration of main and sub-control volumcs in the physical space.


(4 (b)

Figure 3. Illustrations of wedged shape control volumes: (a) for the continuity equation; (b) for the momentum equations

An arbitrary FV-element for the governing equations in the physical space is illustrated in Figure 2. A cell marked by the black solid lines indicates the main control volume which encloses a computational point in the centre of the element. The dashed volume, called a sub-control volume, encloses the central point of one surface of the main control volume. The following notations are used to discretize the first- and second-order derivatives: Sl denotes the projected area of the surface 1 of the main control volume in the e-direction; Sf, denotes the projected area of the surface r of the sub-control volume 1 in the e-direction; 4 and 4, express the function values (assumed to be constants) on the surface 1 of the main control volume and in the surface r of the sub-control volume 1, respectively; V and 6 are the volumes of the main and the sub-control volume I, respectively; 1, r = 1, 2, . . ., 6; e = 1, 2, 3, (x, y, 2).

The first-order derivatives can be expressed as

1 a@ 1 1 6 - lim - -dV= lim - @dS‘w-

a@ _- ax, v+o v ax, v-0 .Is V l = l

and the approximate Laplace operator can be evaluated by 1 3 6 1 6

As can be seen, no co-ordinate or equation transformation is required and only the grid parameters such as areas and volumes of the cells have to be calculated. The central difference like approximation in equation (9) gives rise to high-frequency oscillations in the pressure. The numerical damping used here is defined by a fourth-order difference operator which is similar to the scheme proposed by Jame~on.’~

3.3. Multigrid and inter-grid data exchange

The multigrid procedure is used for the fast solution of discretized equations. More details of MG methods can be found in References 15 and 16. Only those aspects that are directly relevant to the implementation on the overlapping grid system will be described in the following.

Interpolating and exchanging inner-boundary information is of primary importance for the efficiency of the overlapping-multigrid technique. In the ‘classical’ Schwarz iteration, the solution in each domain is computed, and then data are exchanged among the zones. Stuben and Trottenberg,” on the basis of numerical experiments on a single Poisson equation with Dirichlet boundary conditions, indicated that the solution efficiency of a simple combination of the

OVERLAPPING GRIDS AND MULTlGRlD METHODS 265

straightforward combination of the Schwarz iterations with a multigrid technique is limited by the convergence properties of the Schwarz method. They proposed a ‘direct’ smoothing strategy which updates the inner-boundary information right after every smoothing step on each sub-grid. Their results showed that the covergence factors are insensitive to the amount of grid overlap. Fuchs” and Meakin et a1.” conducted intensive numerical experiments on the subject, solving the Navier--Stokes equations on two-dimensional overlapping grid system. They noted that the more frequently the information is exchanged amongst overlapping domains, the less expensive are the corresponding global calculations . It has also been observed that there is a minimum overlapping region, depending on the discrete ‘molecule’ and the number of sweeps on each grid.

Here, the inter-zonal updating procedure is integrated into the MG-cycles. There is some freedom in the choice of the order of relaxation and the order of polynomials used for interpolation. One possibility is to relax all sub-grids before updating the inner boundaries. From experience, however, it seems that a good procedure involves interpolation after each sub-grid is relaxed. In the MG-procedure, the relaxation consists of Symmetrical Line Relaxations (SLR) to smooth all the transport equations and a point relaxation for the continuity equation by updating the velocity vector and pressure at each computational ce11.16 The residuals are transferred from a finer grid to a coarser grid through volume-weighted restrictions. The computed corrections on the coarse grids are prolonged to the next fine grids through trilinear interpolations. It should be noted that the restriction step has to maintain the conservation of mass. On the semi-staggered grid system conservation deficiency occurs at cells which are close to the solid boundary. As illustrated in Figure 4, the velocity U, on a coarse grid point, neighbouring to a solid boundary, is computed by taking the average of the surrounding velocities U, on the finer grid. Here, we consider a finite volume surface for mass flux computation as indicated by thick lines in Figure 4 and assume the value of velocity on the surface to be constant (U, = U,) except the value on the wall boundary (where U, = U, = 0). The surface areas are defined by Si. The mass flux on the coarse, M,, and finer, Mf, grid are

M , = 0.25(Uc + U, + 0 + O ) ( S , + Sz + S3 + Sd) = 05Uc(S1 + Sz + S3 + S,) (10)

/ / Wall bounsary ’ U, = U[= 0

0 -Finite volume surface centre for mass flux computation on coarse grid

0 - Finite volume surface centre for mass flux computation on finer grid

Figure 4. Illustration of the correction scheme on a semi-staggered grid

266

and

I. Y. TU AND L. FUCHS

Mf = 0.25(Uf + Uf + 0 + 0)Si + 0*25(Uf + Uf + 0 + 0)Sz

+ 0'25(Uf + Uf + Uf + Uf)Sj + 0*25(Uf + Uf + Uf + Uf)S4

= Ur(0'5S1 + 0.5s~ + S 3 + S4) (1 1) It can be seen that the contributions from the solid boundary points to mass flux computations on the coarse- and on the fine-grids are different. A simple way to correct this error can be attained by changing u,' (UE is the velocity neighbouring to the solid boundary on a coarse grid before the correction to U:):

u,p = C'Ut (12)

where U: is the corrected velocity, the superscript e denotes x-, y- and z-co-ordinate directions, and C' is a correction factor obtained by ensuring M, = Mf from (10) and (11) and is computed by

C" = (Se, + s; + 2s; + 2%) (Se, + s; + se, + s:)

where Sf(i = 1, . . . ,4) is the projected area in the e-direction. If

(S: + S; + S: + Si) = 0 then C" = 0 (14) The comparison between the convergence results with and without the correction is presented in Section 4.

A simple alternative to this scheme is to use 'slip' boundary conditions on coarse grids such that global continuity is satisfied.

Y

i Grid X 1

Figure 5. A two-dimensional illustration of the interpolation procedure


The interpolation of data from one grid to another is carried out in the transformed space (r, s, t). A two-dimensional illustration of this process is shown in Figure 5. A Newton’s method13 is used to determine the location of the interpolated point in the transformed space. The three-dimensional Lagrange interpolation formula is used for calculating the values of the dependent variables:

where &is the function value at the interpolated point and &,,, are the function values at selected nodes, n is the interpolating order which is taken to be 1, 3, 5, . . ., etc. Equation (15) is implemented as a sequence of three one-dimensional interpolations.

4. RESULTS AND DISCUSSIONS

4.1. Order of interpolation and accuracy

An important question related to overlapping grids is how to choose the order of interpolation so that the overall accuracy will be as good as the accuracy of the discretization formulae. The answer to this question depends on the order of the governing equation and the order of accuracy of the discretization formulae. In the work of Chesshire and Hen~haw,~ a detailed theoretical analysis and numerical experiments in one- and two-dimensional situations were carried out. It was shown that for solving a second-order partial differential equation to second-order accuracy it is necessary to use third-order interpolation. Similar experimental observation has been made by Stuben and Trottenberg.” In order to address this question for three-dimensional cases and the Navier-Stokes equations, numerical experiments have been performed.

4.1.1. Flow through a straight p ipe . In this case (Case l), a straight pipe (the ratio of the diameter to the length of the pipe is 1 : 6) is selected for calculating a fully developed flow by using overlapping grids or a single grid without overlapping. The Reynolds number Re (based on the maximum inlet velocity and the diameter of pipe) is 50. To construct the overlapping grids, the pipe is artificially divided into three zones, as illustrated in Figure 6. In Figure qa), a parabolic velocity profile with no transverse components is set at both inlet and outlet. The exact solution is also a parabolic profile, with a constant pressure gradient along the axis. The error in the solution versus the number of computational points are depicted in Figure 6(a). Here Ni denotes the number of interior grid points along the axis of pipe and (“errorl”) indicates the root mean square of the difference between the computed and exact solution. It is shown that the current numerical scheme used on a global single grid and with a third-order Lagrange interpolation on overlapping grids results in an expected numerical accuracy (close to the second-order accuracy). Futhermore, when first-order Lagrange interpolation is employed, the convergence rate is only slightly affected, due to flow degeneracy (vanishing streamwise velocity gradients).

In order to evaluate the effect of the order of interpolation on solution accuracy, swirl is added to both inlet and outlet, as illustrated in Figure qb). The swirl decreases and disappears in the middle of the pipe due to viscous effect. Since no exact solution is available, a sequence of global grid solutions is extrapolated to estimate the error in the solution computed on overlapping grids. The extrapolated solution is obtained by Richardson’s extrapolation based on two grids of which the finest grid has 17 x 38 x 141 nodes and the coarser contains each other node point. It is clear from the plotted results [Figure qb)], that the solution with third-order Lagrange interpolation maintains the desired accuracy, whereas the first-order Lagrange interpolation is less accurate.

268

log (errorl)


log (error2)

+ 1st order Lapaange's 3rd order Lagrange's

A Single grids

f intp. intp.

-1 / + 1st order Lagrange intp.

3rd order Lagrange intp. I ' I ' -15 . -1 ' ' 5 -354 ' -1.5 ' -1 ' -

-3.5 5

( 4 log(l/NiJ (b) log(l/NJ

Figure 6. Convergence vs. number of grid nodes for laminar flow in straight pipe: (a) without swirl; (b) with swirl

Since we use a symmetric Lagrange interpolation formulation the even-order Lagrange interpolations are not considered. It is concluded that the third-order interpolation formulation is adequate in ensuring a global second-order accuracy of the solution on 3-D overlapping grids. All the results presented in the following have been calculated using third-order Lagrange interpolations at grid interfaces.

4.1.2. Flow through curved pipes. Flow through curved pipes has been studied by many researchers both experimentally and numerically. We consider the flow in a curved pipe for which the geometry is shown in Figure 7(a). For this geometry, the curvature ratio is given by I = a/R, = 3 (a is the radius of the pipe) and the Dean number is K = 2R,(I)"' = 183. At the inlet a uniform velocity profile is imposed and a zero-gradient condition is applied to the outlet. The domain is divided into two zones. Each zone consists of three grid levels of which the finest grid has 17 x 34 x 61 nodes. A comparison with the experimental results of Agarwal et a1." is shown in Figures 7(b)-7(d) at three angles = 15.1", 30" and 60", where the profiles are compared along radial line (6 = OO). Good agreement between the numerical results using overlapping grids and the experimental ones is observed.

The effect of grid refinement on the numerical solution has also been investigated. The grids used in this study are shown in Table I. The axial velocity profiles in the plane 4 = 60" computed on different grids are given in Figure 8. As seen, the solution converges as the grid is being refined. The converged solution is in rather good agreement with the experimental data.

Next, consider a high Reynolds number and turbulent flow through a curved pipe. Data from an experimental study performed by Enayet et al." are used for comparison with our computed results. The details of the geometry are given elsewhere,22 and consists of a 90" pipe bend with a mean radius of curvature of 3.2 times the diameter of the pipe, and inlet and outlet extensions of 2.0 and 3.2 times the diameter, respectively (Figure 9). Here, the whole computational domain is

OVERLAPPING GRIDS AND MULTIGRID METHODS

1.4.

269

1.4

Axial velocity W, i+---

0.5 1 1.5

R

(a) Geometry and co-ordinates for the pipe bend (b)

Table I. The different grids used in the calculations

Distribution of grids in two zones Number of total

No. Level 1 Level 2 Level 3 grid points

1 2 x(5 x l o x 11) 2 x(9 x 18 x 21) 7904 2 2 x ( 3 x 7 x 9 ) 2x(5x12x17) 2x(9x22x33) 32536 3 2x(4x8x13) 2x(7x14x25) 2x(13x26x49) 66044 4 2x(5x lOx16) 2x(9x18x31) 2x(17x34x61) 82160 5 2 x(5 x 10 x 21) 2 x(9 x 18 x41) 2 x(17 x 34x 81) 109020


Axial velocity W.

1.66

0:s i 1 :S

R Figure 8. Effect of grid refinement on axial velocity profile in the 60" plane. Lines 1,2, . . . , 5 correspond to grids in Table

I; (I) experimental measurement from Reference 19

Figure 9. Geometry and co-ordinates for a curved pipe (Case 3)

OVERLAPPlNG GRIDS AND MULTIGRID METHODS 271

divided into three zones. In each of them there are three grid levels for the MG-procedure. The measured velocity profile2' is used as inlet boundary condition and a zero-gradient condition is applied at the outlet. The Reynolds number, based on the maximum inlet velocity and the diameter of the inlet, is 1093 for the laminar flow and 43000 for the turbulent case. The k-& turbulence model, together with 'wall functions', is used in the latter case.

The calculated axial velocity contours at 4 = 30", 60" and at a distance equal to the pipe diameter from the end of the bend, are displayed in Figures 10 and 11, and are compared to the measurements of Enayet et al." and to other available numerical calculations.22 From these figures it is concluded that at such level of Reynolds numbers the agreement between the different data is more or less of qualitative character.

4.1.3. Flow through a symmetric sudden expansion. The last case (Case 4) for evaluating numerical accuracy is the flow through a symmetrical sudden expansion, which has also been investigated experimentally by Durst et aLZ3 The geometrical characteristics and grid distribu- tions are shown at the top of Figure 12. The Reynolds number, based on maximum inlet velocity ( Wi,,,x) and step height h, is 56. The channel expansion ratio is 3 : 1. Fully developed flow profile is given at the inlet and zero-gradient condition is imposed at the outlet. Figure 12 also shows the centreline velocity distribution. Good agreement between the present computation and the experimental measurements is observed.

4.2. Computational eficiency

Case 3 which is described above, has been used for assessing the performance of the MG algorithm on the overlapping grids. The same problem has also been solved on a single grid

Figure 10. Comparison of axial velocity contours, Re = 1093: (a) at 4 = 30"; (b) at 4 = 60"; (c) at x = D from the end of bend

272

1.1.


M 1Od

D.M

Figure 11 . Comparison of axial mean velocity contours, turbulent flow Re = 43 OOO: (a) at 4 = 30"; (b) at 4 = 60"; (c) at x = D from the end of bend

0.9.

0.7.

0.5.

9r2635. C d d l Cdd 3 Crld 1 13r41x33 1 3 X 3 4 X 4 1 13rUx42 h

\ \ - Prctent compuhlion

I Measurement from Ref.

L,r = 0.112m

22

. 011 ' 0;2 ' 013 ' 0:4 ' ( 0.3 5

%.I

Figure 12. Centreline velocity distribution (laminar flow in an axisymmetric sudden expansion, Re = h W.,../v = 56, h/d = 1.0)


system without overlapping. In these computations, the number of node points in the grid without overlapping is 9 x 26 x 85 and the Reynolds number is taken to be 150.

The convergence history of the residual of the continuity equation is shown in Figure 13. The convergence rate is best (as should be) when the global mass is conserved at ail levels in the MG-procedure. The MG-solver is by far faster than the corresponding single grid counterpart.

4.3. Further applications

of the present overlapping, finite volume and multigrid flow solver. The following computational examples are selected to illustrate the flexibility and the potential

4.3.1. Flow through a butterfly valve within an S-type duct. Another problem that has been computed by the present method is the flow through a butterfly valve within an S-type duct. A 3-D overlapping grid system for such problem is shown in Figure 14(a). Two body-fitted grids are generated independently, one for the butter-fly valve and one for the S-type duct, respectively. In this case, those grid points in a grid which lie inside the solid object are flagged as unused points which are excluded from the computation. On the edges of the zones (marked by circles and dark dots), ‘internal boundary’ points are defined. The data for these internal boundary points in the different grids are arranged in an additional one-dimensional array and is independently managed by an auxiliary pointer system. The function values for these internal points are obtained by interpolation from the overlapping region.

The computed laminar flow field is plotted in Figure 14(b), with a magnified view around the valve. Fully developed velocity profiles are imposed at both inlet and outlet. Due to the low

Moss Residual

0*01 9

3E-06 ‘ 0 20 40 60 80 100

WORK UNITS

Figure 13. Convergence histories of the different schemes on overlapping grids (Case 3): (-) MG with coarse. grid mass conservative correction scheme; (- -) MG without coarse grid mass conservative correction scheme; (- - -) Single-grid

scheme on the h e s t grids alone


(b)

Figure 14. A case with a butterfly valve in an S-type duct: (a) the overlapping grids; (b) the computed flow field.

Reynolds number, Re = 10 (based on the diameter of the inlet and the maximum inlet velocity), the flow is not separated.

4.2.2. Flow through a curved-duct with a non-uniform cross-section. Flow through curved passages occurs in many engineering applications. The main feature of the flow in curved ducts is the presence of cross-stream, twin counter-rotating (‘Dean’) vortices. These are generated due to the action of centrifugal forces on the primary flow. The flow through a curved duct with a circular cross-section has been computed and compared for the validation of the current code.


In order to illustrate the general capability of the present numerical scheme, a more complex application has been carried out. The non-uniform cross-sectional curved-duct with a sudden expansion inlet is shown in Figure 15(a). The finest grid has 9 x 26 x 25 nodes in the inlet duct and 13 x 42 x 65 in the main duct. The boundary conditions are similar to those used above. Figures 15(b) and 15(c) show the velocity vectors and pressure contours in the plane of symmetry.

The axial development of the secondary flow is depicted in Figure 16. The non-uniformity of the cross-section seems to have a rather small effect on the shape of the vortices; however, they do increase the strength due to a decrease of the effective cross-sectional area. The figures also show that the centres of the pairs of vortices move from the inside towards the outside, as the turning angle increases. Finally, the vortex motion weakens as it leaves the curved passage. This behaviour is as expected and has also been predicted by previous numerical calculations in the case of circular cross-section.2',23 It is interesting to note, however, that the centre of the pair of vortices has moved towards the 'inner corners' as the flow leaves the curved passage.

4.2.3. Flow in a curved bifurcating vessel. Another application of the present method is for the calculation of blood The human arteries are highly curved and have a large number of branches. In order to illustrate the potential of the present method for this kind of problem, the aortic-arch geometry is constructed and the flow in this configuration is computed. The geometry and the grid used in the present calculation are shown in Figure 17(a). Three local body-fitted grids are used for the three components which consist of the arch and the two branches. Each grid has three grid levels for the MG-procedure and the finest grid has a total of (13 x 42 x 33) and 2 x (1 3 x 34 x 29) node points, respectively.

A Newtonian fluid and laminar conditions are assumed. The fully developed flow is specified at the inlet and three outlets ensuring the global mass balance. The Reynolds number is 30, based on the average velocity at the inlet. A 3-D view of velocity field in terms of velocity vectors is plotted in Figure 17(b). The velocity and pressure contours in the symmetry plane are shown in Figure 18.

Figure 15. A non-uniform cross-sectional curved duct with a sudden expansion, Re = 50"; (a) 3-D view of overlapping grids; (b) and (c) velocity field and pressure contours in the plane of symmetry, respectively

276

Insine Inside


Figure 16. Development of the secondary flow: (a) Re = 5 0 (b) Re = 100

@)

4.2.4. Flow in an IC-engine geometry. The final application chosen is relevant to a time- dependent flow in a non-fixed geometry, such as in an IC-engine. The topological complexity of the IC-engine problem is due to multicomponent configuration which consists of a cylindrical combustion chamber, multiple S-type intake ports and curved-duct exhaust ports, as well as moving parts (such as a moving piston). An overlapping grid system for such an IC-engine geometry is depicted in Figure 19. Five local body-fitted grids are generated for each component and the grids on the cross-section of the inlet and the outlet ports are stretched along the central axis. The moving piston (assumed to be flat) is managed by using a time-independent grid system

OVERLAPPING GRIDS AND MULflGRlD METHODS 277

"T Figure 17. (a) A 3-D view of overlapping grids for a curved bifurcating vessel; (b) velocity field

in which only a cell layer close to the face of the piston is attached, while the rest of the grid system itself remains unchanged. Details about this type of grid system are presented elsewhere* where computed results, in a considerably simpler geometry, have been compared with experimental data. The Reynolds number (based on the inlet characteristic parameters) is taken to be 100. Uniform laminar flow at the inlets/outlets during the intake/exhaust stroke process is specified from the overall mass balance. The initial condition is that the piston is stationary at the Top

278 J. Y . TU AND L. FUCHS

Figure 18. (a) Velocity contours for case in Figure 19, (b) pressure contours

Dead Center (TDC) and the flow everywhere is set to be at rest. The flow inside the chamber is driven by the motion of the piston away from the top dead centre, according to a simple harmonic motion, i.e. the motion of the piston follows a cosine wave.

Figure 20 depicts 3-D velocity vector plots of the intake stroke flow at an equivalent crank angle (CA) of 8 = 90" and of the exhaust stroke flow at 0 = 270". During the first half of the intake


Figure 19. 3-D views of overlapping grids for an IC-engine configuration

stroke, a primary vortex is formed in the volume of chamber due to the impinging of the inlet jet-flow on the piston face, as shown in Figure 2qa). During the exhaust stroke process, the exhaust ports are assumed to be fully opened and the intake ports to be closed. When the piston moves away from the bottom dead centre, most of the flow changes direction towards the head of the chamber, as shown in Figures 20(c)-20(d). These flow patterns are qualitatively similar to the experimental observations of Morse et al.”

280 J. Y. TU AND L. FUCHS'

Figure 20. 3-D views of velocity fields for case in Figure 21: (a) and (b) at crank angle 0 = 90"; (c) and (d) at crank angle e = 2700

5. CONCLUDING REMARKS

A computational methodology which includes a numerical technique that can readily treat complex geometries is described in this paper. The main features of this method are the use of the finite volume discretization on semi-staggered overlapping grids, and a multigrid solver. The current code offers great flexibility and efficiency in treating 3-D flow problems in complex geometries. The greatest advantage of the OG scheme is evident for cases with complex geometries, with solid objects moving relative to each other. Also, when one can use a set of

OVERLAPPING GRIDS AND MULTIGRID METHODS 28 1

structured orthogonal, locally uniform, grids for the discretization of the computational domain, the OG technique offers several advantages. The number of operations to express the discrete approximations to the governing equations is considerably smaller on uniform orthogonal grids as compared to general non-uniform grids. Furthermore, structured grids, with constant stride, can offer improved code performance on computers with vector units or cached memory access. The examples given in this paper demonstrate that one may use a set of orthogonal, uniform grids even for complex geometries, which by itself improves code efficiency and increases the flexibility in solving problems with geometrical perturbations.

The accuracy of the code has been demonstrated by comparing the computed results with some available experimental and numerical data. The performance of the numerical procedure is assessed by comparing with an equivalent calculation on a single global grid. These calculations show that the information exchange procedure does not impair numerical accuracy or numerical efficiency.

ACKNOWLEDGEMENT

This work was supported financially by the Swedish National Board for Technical Development (STU grant no.8902268).

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Calculation of flows using three-dimensional overlapping grids and multigrid methods