169 USING THE ANSYS FLUENT FOR SIMULATION OF ... - doiSerbia

APTEFF, 43, 1-342 (2012) DOI: 10.2298/APT1243169M

UDK: 532.54:66.011:004.4 BIBLID: 1450-7188 (2012) 43, 169-178 Original scientific paper

USING THE ANSYS FLUENT FOR SIMULATION OF TWO-SIDED LID-DRIVEN FLOW IN A STAGGERED CAVITY Jelena Đ. Marković*, Nataša Lj. Lukić, Jelena D. Ilić, Branislava G. Nikolovski, Milan N. Sovilj and Ivana M. Šijački University of Novi Sad, Faculty of Technology, Bulevar Cara Lazara 1, 21000 Novi Sad, Serbia

This paper is concerned with numerical study of the two-sided lid-driven fluid flow in a staggered cavity. The ANSYS FLUENT commercial software was used for the simulation, In one of the simulated cases the lids are moving in opposite directions (antiparallel motion) and in the other they move in the same direction (parallel motion). Calculation results for various Re numbers are presented in the form of flow patterns and velocity profiles along the central lines of the cavity. The results are compared with the existing data from the literature. In general, a good agreement is found, especially in the antiparallel motion, while in the parallel motion the same flow pattern is found, but the velocity profiles are slightly different. KEY WORDS: cavity benchmark; fluid flow; two-sided lid driven cavity; parallel motion; antiparallel motion INTRODUCTION In the past decades, flow in a lid-driven cavity has been studied extensively as one of the most popular fluid problems in the computational fluid dynamics (CFD). This classical problem has attracted considerable attention because the flow configuration is relevant to a number of industrial applications. ANSYS FLUENT uses conventional algorithms for calculation of macroscopic variables. Computational advantages of this commercial software are simplicity of the problem setup, parallel computing and higher precision. Two-sided lid-driven staggered cavity appears to be a synthesis of two benchmark problems: a lid-driven cavity and backward facing step. Furthermore, it has all the main features of a complex geometry. Nonrectangular two-sided lid-driven cavities have been recently introduced and investigated as a potential benchmark problem by Zhou et al. (1), Nithiearasu and Liu (2) and Tekic et al. (3). Zhou et al. Presented a solution for the flow in a staggered cavity obtained by using wavelet-based discrete singular convolution. Nithiarasu and Liu solved the same problem using the artificial compressibility-based 

* Corresponding author: Jelena Marković , University of Novi Sad, Faculty of Technology, Bulevar Cara Lazara 1, 21000 Novi Sad, Serbia, e-mail: [email protected]

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APTEFF, 43, 1-342 (2012) DOI: 10.2298/APT1243169M

UDK: 532.54:66.011:004.4 BIBLID: 1450-7188 (2012) 43, 169-178 Original scientific paper

characteristic-based split scheme. Tekic et al. solved this problem by using the latticeBoltzmann method. The aim of this work was to study two-sided lid-driven staggered cavity utilizing the commercial software package FLUENT. Solutions are presented in the parallel and antiparallel motion of the lid and the flow pattern which develops under these conditions.

Figure 1. Schematic diagram of two-sided lid-driven staggered cavity: (a) antiparallel; (b) parallel motion. MATHEMATICAL FORMULATION General Scalar Transport Equation: Discretization and Solution - ANSYS FLUENT uses a control-volume-based technique to convert a general scalar transport equation to an algebraic equation that can be solved numerically. This control volume technique consists of the integration of the transport equation about each control volume, yielding a discrete equation that expresses the conservation law on a control-volume basis. Discretization of the governing equations can be illustrated most easily by considering the unsteady conservation equation for transport of a scalar quantity Φ. This is demonstrated by the following equation written in integral form for an arbitrary control volume V as follows:  [1] V t dV    v  d A      d A  V S dV where ρ is the density, v - velocity vector; A - surface area vector;  - diffusion coefficient for Φ, S source of Φ per unit volume. Equation [1] is applied to each control volume, or cell, in the computational domain. The two-dimensional, triangular cell shown in Figure 1 is an example of such a control volume. Discretization of Equation [1] on a given cell yield N N  V   f v f  f  A f    f  A f  SV [2] t f f

 faces

170

 faces

APTEFF, 43, 1-342 (2012) DOI: 10.2298/APT1243169M

UDK: 532.54:66.011:004.4 BIBLID: 1450-7188 (2012) 43, 169-178 Original scientific paper

where Nfaces represents the number of faces enclosing the cell, Φf is the value of convected through the face f, A f is the area of the face f and V is the cell volume. The equations solved by ANSYS FLUENT take the same general form as the one given above and apply readily to multi-dimensional, unstructured meshes composed of arbitrary polyhedra.

Figure 2. Control volume used to illustrate discretization of a scalar transport equation. For relatively uncomplicated problems (laminar flows with no additional models activated) in which convergence is limited by the pressure-velocity coupling, a converged solution can often be obtained more quickly using SIMPLEC. With SIMPLEC, the pressure-correction under-relaxation factor is generally set to 1.0, which aids in convergence speedup. In the present study, a slightly more conservative under-relaxation value was used, and it is equal to 0.7 .Special practices related to the discretization of the momentum and continuity equations and their solution by means of the pressure-based solver is most easily described by considering the steady-state continuity and momentum equations in the integral form:

 v  d A  0

[3]

  vv  d A   pI  d A    d A   FdV

[4]

V

where I is the identity matrix,  is the stress tensor, and F is the force vector. Discretization of the Momentum Equation - previously described a discretization scheme for a scalar transport equation is also used to discretize the momentum equations. For example, the x-momentum equation can be obtained by setting   u :

aP u 

a nb

nb

u nb 

p

^

f

Ai  S

[5]

If the pressure field and face mass fluxes are known, Equation [5] can be solved in the previously outlined manner, and a velocity field can be obtained. However, the pressure field and face mass fluxes are not known a priori and have to be obtained as a part of the solution. There are important issues with respect to the storage of pressure and the discretization of the pressure gradient term. ANSYS FLUENT uses a co-located scheme, whereby pressure and velocity are both stored at cell centers. However, Equation [5] requires the value of the pressure at the face between cells c0 and c1, shown in Figure 2. 171

APTEFF, 43, 1-342 (2012) DOI: 10.2298/APT1243169M

UDK: 532.54:66.011:004.4 BIBLID: 1450-7188 (2012) 43, 169-178 Original scientific paper

Therefore, an interpolation scheme is required to compute the face values of pressure from the cell values. Discretization of continuity equation- Equation [1] may be integrated over the control volume to yield the following discrete equation N faces

J

f

Af  0

[6]

f

where Jf is the mass flux through the face vn . In order to proceed further, it is necessary to relate the face values of the velocity, vn , to the stored values of velocity at the cell centers. Linear interpolation of cell-centered velocities to the face results in an unphysical checker-boarding of pressure. ANSYS FLUENT uses a procedure similar to that outlined by Rhie and Chow (4) to prevent checkerboarding. The face value of velocity is not averaged linearly; instead, momentum-weighted averaging, using weighting factors based on the aP coefficient from the equation [5], is performed. Using this procedure, the face flux, Jf, may be written as: ^ a p ,c vn ,c  a p ,c vn ,c Jf  f  d f (( pc  (p) c  r0 )  ( pc  (p) c  r1 ))  J f  d f ( pc  pc1 ) [7] a p ,c  a p ,c 0

0

1

0

1

1

0

0

1

1

0

where pc , pc and vn ,c , vn ,c , are the pressures and normal velocities, respectively, 0

1

0

1

^

within the two cells on either side of the face, and J f contains the influence of velocities in these cells (Figure 2). The term d f is a function of a P , the average of the momentum equation of the a P coefficients for the cells on either side of the face f. Spatial Discretization - By default, FLUENT stores discrete values of the scalar  at the cell centers (c0 and c1 in Figure 2). However, the face values  f are required for the convection terms in Equation [2] and they have to be interpolated from the cell center values. This is accomplished using an upwind scheme. Upwinding means that the face value f is derived from quantities in the cell upstream, or „upwind“, relative to the direction of the normal velocity vn in Equation [2]. The diffusion terms are centraldifferenced and are always second-order accurate. When second-order accuracy is desired, the quantities at cell faces are computed using a multidimensional linear reconstruction approach (5,6). In this approach, higherorder accuracy is achieved at cell faces through a Taylor series expansion of the cellcentered solution about the cell centroid. Thus, when second-order upwinding is selected, the face value f is computed using the following expression:

 f ,SOU      r

[8]

where  and  are the cell-centered value and its gradient in the upstream cell, and r is the displacement vector from the upstream cell centroid to the face centroid. This formulation requires the determination of the gradient  in each cell. Finally, the gradient  is limited so that no new maxima or minima are introduced. 172

APTEF FF, 43, 1-342 (2012) DOI: 10.2298/APT1243169 1 M

UDK: 53 32.54:66.011:004.4 BIBLID: 1450-7188 (2012) ( 43, 169-178 Origin nal scientific paper

Siimulation setup - Mesh was createdd with 140x140 number n of elemen nts with grid refinem ment adjacent to thhe walls. Densityy of the fluid was set to 1 kg/m3, and a viscosity to 0.001 1 Pas. Reynolds number n was calculated as Re = uL L/. where ρ repreesents the fluid densiity; μ is dynamic visocity v of the fluuid; L is the characcteristical length of o cavity, and u lid veelocity in the x dirrection. The veloccity of the movingg lid was calculateed based on desired Re number. Bounndary conditions w were set as no-sliip for the left and right wall, and for th he upper and bottoom moving lid as moving walls witth defined velocitty and direction of moving m depending on the case (parrallel or antiparalllel). Starting con nditions for the first-o order upwind scheeme were taken as 0.5 velocity of the t moving lids, and results were used as starting conditiions for the seconnd-order upwind sccheme. RESULTS AND DISCUSSIION Validatioon or results of oone-sided lid –driiven square cavitty In n order to validatee the simulation m mehod, a popular benchmark b probleem of one-sided lid drriven square caviity is simulated ffor different Re numbers n and com mpared with the resulsst in the available literature. Figure 3 shows the u- annd v-velocity proffile, through the geom metric center of thhe cavity. The oobtained results are a in good agreeement with the resultts of Chen et al. (66) and Ghia et al ((7).

Figu ure 3. Velocity proofiles u – and v- aalong the vertical and a horizontal cen nterlines of the sqquare cavity. Antiparalllel motion of the lids l The results for anttiparallel motion oof lids are listed in i Table 1. Stream mfunction contours at various Re nuumbers are presentted in Figure 4, while w the results obtained o for the veloccity u – and v-proofiles through the mid-section of thhe staggered caviity are given in Figurre 5. For comparisson sake, the resullts obtained by Teekic et al. (3) are also a presented. 173

APTEF FF, 43, 1-342 (2012) DOI: 10.2298/APT1243169 1 M

UDK: 53 32.54:66.011:004.4 BIBLID: 1450-7188 (2012) ( 43, 169-178 Origin nal scientific paper

Itt is evident that with w the increase iin the Re numberr, extreme values of the velocity comp ponents also increease in magnitudde. Furthermore, the t inertial forcess are dominant comp pared to the viscoous ones. As a reesult, the gradiennts close to the moving m lids are stronger for higher Re.. As A previously menntioned, three studdies on staggered cavity, (1)-(3) sh howed unsteady behav vior for Re numbbers above 1000. In the present stuudy, symmetric and a asymmetric patterrns are achieved even e at Re numbeers lower than 10000. Multiple vorticces are formed, moree precisely there arre three primary vvortices, although in Table 1 the thiird vortex is referred d to as secondary for easier compaarison of the resultts. Primary vortices are all vertically aligned along thee mid-section of thhe cavity. Opposeed to this, secondaary vortices are locateed in the left and right r bottom corneer of the cavity.

Figure 4. Streamffunction contours at various Re num mbers – antiparallel motion. With W the increase of o Re, the primaryy vortex located inn the left bottom corner c grows at the ex xpense of the prim mary vortex locatted in the upper right r corner. With h the further increasse of the Re numbber, the bottom lefft corner vortex disappears, and theere are two primary y vortices along the long diagonall of the cavity, seecondary vorticess appear in the corneers next to the movving lid.

Figu ure 5. Velocity proofiles u – and v- aalong the vertical and a horizontal cen nterlines of the staggered cavity c –antiparalleel motion (Rea – Tekic T et al. results (3)) Velocity V profiles allong the vertical ccenterline of the cavity c differ for so ome Re values. The most m notable diffference is for Re =100. While the results of Tekic et al. (3) show 174

APTEF FF, 43, 1-342 (2012) DOI: 10.2298/APT1243169 1 M

UDK: 53 32.54:66.011:004.4 BIBLID: 1450-7188 (2012) ( 43, 169-178 Origin nal scientific paper

moree flattened profiless, ANSYS FLUEN NT results show thhe existence of a sine-like s curve, moree similar to the profiles p which T Tekic et al. (3) showed s for higheer Re numbers (Re=1000). The veloccity profiles along the horizontal centerline are almost identical. Conssidering that there is a very good aggreement betweenn the present study y and the work of Gh hia et al. (6) andd Chen et al. (7), and also betweenn results of Tekicc et al. (3) and previiously mentioned authors, the reasoons for disagreements with present study could be found d in different Re number definitionn and different booundary condition ns implementations caused by the diffferent numerical aapproach. To summarize the results, the locaations of the centers of the vorticees are listed in Tablee 1 and comparedd with (1) and (33). It can be noticed that the results for primary vorticces are in good aggreement with the results in the avaiilable literature. Tablle 1. Locations annd secondary vortiices – antiparallel motion, aZhou et al. (1), b Tekic et al.(33), c present study

(xc1,yc1) (0.9781, 1.1600) (0.4219, 0.2518) (0.9637, 1.1551) (0.4494, 0.2543) (1.10383,1.17135) (0.41911,0.343648) (1.0172, 1.1091) (0.3828, 0.2889)

First secondary vortex (xc2,yc2) (1.3556, 0.4405) (0.0444, 0.9595) (1.3484, 0.4476) (0.0460, 0.9543) (1.27995, 0.69013) (0.032408, 0.8355) (1.3556, 0.4486) (0.0444, 0.9514)

100b

(1.0031, 1.1382) (0.4082, 0.2693)

100c

(1.20251, 1.25965) (0.452649,0.360417)

Re 50a 50b 50c 100a

Primary vortex

R Re

Primary vortex (xc1,yc1)

First secondary vortex (xc2,yc2) (1.3500, 0.4656) (0.0500, 0.9344) (1.3522, 0.4607) (0.0554, 0.9506) (1.25479, 0.60078) (0.05631, 0.60016) (1.3250, 0.4844) (0.0750, 0.9063)

Second secondary vortex (xc3,yc1) (0.4703, 1.625) (0.9219, 0.2375) (0.4382, 1.1345) (0.9824, 0.2862)

40 00a

(0.7000, 0.7000)

40 00b

(0.6822, 0.6859)

40 00c

(0.472214,0.445062))

000a 10

(0.7000, 0.7000)

(1.3502, 0.4457) (0.0460, 0.9543)

10 000b

(0.7000, 0.7000)

(1.3250, 0.4844) (0.0750, 0.9063)

(0.8811,0.2167) (0.5301, 1.1962)

(0.90263, 0.89427) (0.04669, 0.84673)

000c 10

(0.6934, 0.6972)

(1.3371, 0.4851) (0.0722, 0.9280)

(0.82157, 0.13123) (0.61360, 0.96692)

(0.42548, 1.07007) (0.7256,0.2000) (0.5339, 1.1907)

Parrallel motion As A expected, paralllel motion of thee opposite lids deevelops a differen nt flow pattern comp pared to the antipparallel motion. F Figure 6 shows the t streamfunctio on contours for differrent Re numbers.

mfunction contouurs at various Re numbers n – parallel motion. Figure 6. Stream Itt can be noticed thhat two primary coounter-rotating voortices are presentt and that a free shearr layer forms betw ween them. Compaared to the previoous studies of flow w inside rectangularr cavities, where the t free shear layeer is formed alongg a horizontal cen nterline (6), (8), 175

APTEF FF, 43, 1-342 (2012) DOI: 10.2298/APT1243169 1 M

UDK: 53 32.54:66.011:004.4 BIBLID: 1450-7188 (2012) ( 43, 169-178 Origin nal scientific paper

in thee staggered cavityy the free shear layyer is formed alonng the shorter diag gonal. The flow is no longer symmetriccal due to the uppper lid moving froom the offset, whille the lower lid movees towards the offfset. At low Re nuumbers, a secondaary vortex is preseent close to the corneer of the right waall and offset. As the Re numberr increases, this vortex v gains in streng gth at the cost of the upper primaryy vortex. At higееr Re numbers, secondary vortex causees splitting of thee primary vortex and formation of a second secon ndary vortex as show wn in Figure 6. Booth primary vorticces have become more m prominent and a larger in size, so o that viscous efffects are confined to the thin bounddary layers close to the walls of the caavity (9). As menntioned by Sahin aand Owens (10), fluid f begins to rottate like a rigid body with a constant angular a velocity aat high Re numbers. Figure 7 show ws the u- and vveloccity profiles alongg mid sections of the cavity. As diiscussed in the prrevious section, with the increase in thhe Re number, exttreme velocity vaalues also increasee in magnitude. Furth her, the free shearr layer formed beetween the two prrimary vortices sh hrinks with the increase in the Re nuumber due to turbbulence. The proffiles confirm asym mmetrical flow aboutt the horizontal ceenterline of the cavvity, as previouslyy mentioned. Com mpared with the resultts of Tekic et al., it i can be seen thatt the obtained proffiles are quite sim milar.

Figu ure 7. Velocity prrofiles u – and v- aalong the vertical an horizontal cen nterlines of the staggered cavity – parallel m motion (Rea – Tekkic et al. results (3 3)). As A in the case of anntiparallel motionn, the results of Teekic et al. (3) give more flattened profilles, while the pressent study shows tthe existence of a minimum velociity pitch. These differrences occur at lower values of Re number (50 and 100). Velocity pro ofiles along the horizzontal centerline shhow relatively good agreement for the Re values 50,, 400 and 1000, whilee for the Re=100 there t is a more siggnificant differencce. In general, thee velocity profiles ob btained by simulaation in the presennt study are more symmetrical s and have h more pronounced minimum andd maximum velocity pitch. These diifferences, as prev viously mentioned, could be a result of the different R Re calculation prrocedure, and imp plementation of the diifferent boundary conditions. 176

APTEFF, 43, 1-342 (2012) DOI: 10.2298/APT1243169M

UDK: 532.54:66.011:004.4 BIBLID: 1450-7188 (2012) 43, 169-178 Original scientific paper

CONCLUSION Results of the ANSYS FLUENT commercial software simulation of two-sided liddriven flow inside a staggered cavity are presented in this article. Both antiparallel and parallel motions of two facing lids are investigated. The benchmark results obtained with ANSYS FLUENT are in good agreement with the results available in the literature. For antiparallel motion of lids in a staggered cavity results show symmetrical and asymmetrical flow patterns. Velocity profiles along the horizontal centerline are in a good agreement with existing data from the literature, while the profiles along the vertical centerline are slightly different from those used for comparison, especially for Re=50 and Re=100. These differences could be explained by the different Re calculation procedures and different boundary conditions implementation methods, considering the different numerical approach. The situation is quite similar in case of parallel motion of lids. Unlike for antiparallel motion, steady-state asymmetric patterns are obtained for all investigated Re numbers. It can be noticed that a free shear layer is formed along the short diagonal of the staggered cavity. All the main features of the flow are shown, streamline contours, horizontal and vertical velocity components along the mid sections of the cavity are visually presented, while the location of vortices is presented in Table 1. Acknowledgement This research was financially supported by the Ministry of Science and Technological Development of the Republic of Serbia (Project No. 46010) REFERENCES 1. Zhou, Y.C., Patnaik, B.S.V., Wan, D.C., and Wei, G.W.: Dsc Solution for Fow in a Staggered Double Lid Driven Cavity. Int. J. Num. Meth. Eng. 57 (2003) 211-234. 2. Nithiarasu, P. and Liu, C.-B.: Steady and Unsteady Incompressible Flow in a Double Driven Cavity Using the Artificial Compressibility (Ac)-Based Characteristic-Based Split (Cbs) Scheme. Int. J. Num. Meth. Eng. 63 (2005) 380-397. 3. Tekić, P., Rađenović, J., Lukić, N., and Popovic, S.: Lattice Boltzmann Simulation of Two-Sided Lid-Driven Flow in a Staggered Cavity. Int. J. Comp. Fluid Dyn. 24 (2010) 383-390. 4. Rhie, C.M. and Chow, W.L.: Numerical Study of the Turbulent Flow Past an Airfoil with Trailing Edge Separation, AIAA 21 (1983) 1525-1532. 5. Barth, J. and Jespersen, D.: The Design and Application of Upwind Schemes on Unstructured Meshes, AIAA-89-0366, AIAA 27th Aerospace Sciences Meeting, Reno, Nevada, (1989) 10-13 6. Chen, S., Tolke, J., and Krafczyk, M.: A New Method for the Numerical Solution of Vorticity-Streamfunction Formulations. Comp. Meth. Applied Mech. Eng., 198 (2008) 367-376.

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APTEFF, 43, 1-342 (2012) DOI: 10.2298/APT1243169M

UDK: 532.54:66.011:004.4 BIBLID: 1450-7188 (2012) 43, 169-178 Original scientific paper

7. Ghia, U., Ghia, K.N., and Shin, C.T.: High-Re Solutions for Incompressible Flow Using the Navier–Stokes Equations and a Multigrid Method. J. Comput. Phys. 48 (1982) 387-411. 8. Perumal, D.A. and Dass, A.K.: Simulation of Flow in Two-Sided Lid-Driven Square Cavities by the Lattice Boltzmann Method, Advances in fluid mechanics VII. Boston, MA: WIT Press (2008) 45-54. 9. Patil, D.V., Lakshmisha, K.N., and Rogg, B.: Lattice Boltzmann Simulation of LidDriven Flow in Deep Cavities, Comput. Fluids 35 (2006) 1116-1125. 10. Sahin, M. and Owens, R.G.: A Novel Fully Implicit Finite Volume Method Applied to the Lid-Driven Cavity Problem – Part I: High Reynolds Number Flow Calculations., Int. J. Numer. Methods Fluids 42 (2003) 57-77. СИМУЛАЦИЈА ТОКА У ДВОСТРАНО ВОЂЕНОМ ПОКРЕТНОМ КАНАЛУ ПОМОЋУ ANSYS FLUENT ПРОГРАМСКОГ ПАКЕТА

Јелена Ђ. Марковић, Наташа Љ. Лукић, Јелена Д. Илић, Бранислава Г. Николовски, Милан Н.Совиљ и Ивана М. Шијачки Универзитет у Новом Саду, Технолошки факултет, Булевар цара Лазара 1, 21000 Нови Сад, Србија

Рад се бави проблематиком нумеричке анализе струјања флуида у каналима у којима струјање флуида настаје услед кретања горње и доње странице канала. Комерцијални софтвер ANSYS FLUENT је коришћен за симулацију двострано вођеног струјања флуида. Симулација је урађена за два случаја, први – када се горња и доња страна крећу у супротним смеровима (антипаралелно струјање) и други – када се горња и доња страна крећу у истом смеру. Резултати прорачуна за низ вредности Рејнолдсовог броја приказани су у виду путања струјања флуида и профила брзина дуж хоризонталне и вертикалне централне линије канала. Добијени резултати су упоређени са потојећим подацима у литератури. Генерално уочено је добро слагање са резултатим претходних истраживања, нарочито када се ради о антипаралелном струјању. У случају паралелног струјања, визуелно ток флуида је исти, али потоји мала разлика у профилима брзина. Кључне речи: симулација, Ansys Fluent, струјање флуида, двострано вођени покретни канали Received: 6 July 2012 Accepted: 14 September 2012

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