APTEFF, 43, (2012) UDK: :66.011:004.4 DOI: /APT M BIBLID: (2012) 43, Original scientific paper

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1 Original scientiic 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 o Novi Sad, Faculty o Technology, Bulevar Cara Lazara 1, Novi Sad, Serbia This paper is concerned with numerical study o the two-sided lid-driven luid low in a staggered cavity. The ANSYS FLUENT commercial sotware was used or the simulation, In one o 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 or various Re numbers are presented in the orm o low patterns and velocity proiles along the central lines o the cavity. The results are compared with the existing data rom the literature. In general, a good agreement is ound, especially in the antiparallel motion, while in the parallel motion the same low pattern is ound, but the velocity proiles are slightly dierent. KEY WORDS: cavity benchmark; luid low; two-sided lid driven cavity; parallel motion; antiparallel motion INTRODUCTION In the past decades, low in a lid-driven cavity has been studied extensively as one o the most popular luid problems in the computational luid dynamics (CFD). This classical problem has attracted considerable attention because the low coniguration is relevant to a number o industrial applications. ANSYS FLUENT uses conventional algorithms or calculation o macroscopic variables. Computational advantages o this commercial sotware are simplicity o the problem setup, parallel computing and higher precision. Two-sided lid-driven staggered cavity appears to be a synthesis o two benchmark problems: a lid-driven cavity and backward acing step. Furthermore, it has all the main eatures o 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 or the low in a staggered cavity obtained by using wavelet-based discrete singular convolution. Nithiarasu and Liu solved the same problem using the artiicial compressibility-based * Corresponding author: Jelena Marković, University o Novi Sad, Faculty o Technology, Bulevar Cara Lazara 1, Novi Sad, Serbia, jmarkovic@t.uns.ac.rs 169

2 Original scientiic paper characteristic-based split scheme. Tekic et al. solved this problem by using the lattice- Boltzmann method. The aim o this work was to study two-sided lid-driven staggered cavity utilizing the commercial sotware package FLUENT. Solutions are presented in the parallel and antiparallel motion o the lid and the low pattern which develops under these conditions. Figure 1. Schematic diagram o two-sided lid-driven staggered cavity: (a) antiparallel; (b) parallel motion. 170 MATHEMATICAL FORMULATION General Scalar Transport Equation: Discretization and Solution - ANSYS FLU- ENT 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 o the integration o the transport equation about each control volume, yielding a discrete equation that expresses the conservation law on a control-volume basis. Discretization o the governing equations can be illustrated most easily by considering the unsteady conservation equation or transport o a scalar quantity Φ. This is demonstrated by the ollowing equation written in integral orm or an arbitrary control volume V as ollows: dv v d A d A S dv [1] t V where ρ is the density, v - velocity vector; A - surace area vector; - diusion coeicient or Φ, S source o 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 o such a control volume. Discretization o Equation [1] on a given cell yield N aces N aces V v A A SV [2] t V

3 Original scientiic paper where N aces represents the number o aces enclosing the cell, Φ is the value o convected through the ace, A is the area o the ace and V is the cell volume. The equations solved by ANSYS FLUENT take the same general orm as the one given above and apply readily to multi-dimensional, unstructured meshes composed o arbitrary polyhedra. Figure 2. Control volume used to illustrate discretization o a scalar transport equation. For relatively uncomplicated problems (laminar lows with no additional models activated) in which convergence is limited by the pressure-velocity coupling, a converged solution can oten be obtained more quickly using SIMPLEC. With SIMPLEC, the pressure-correction under-relaxation actor 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 o the momentum and continuity equations and their solution by means o the pressure-based solver is most easily described by considering the steady-state continuity and momentum equations in the integral orm: v d A 0 [3] v v d A pi d A d A FdV [4] where I is the identity matrix, is the stress tensor, and F is the orce vector. Discretization o the Momentum Equation - previously described a discretization scheme or a scalar transport equation is also used to discretize the momentum equations. For example, the x-momentum equation can be obtained by setting u : ^ a u a u p A i S P nb nb [5] nb I the pressure ield and ace mass luxes are known, Equation [5] can be solved in the previously outlined manner, and a velocity ield can be obtained. However, the pressure ield and ace mass luxes are not known a priori and have to be obtained as a part o the solution. There are important issues with respect to the storage o pressure and the discretization o 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 o the pressure at the ace between cells c 0 and c 1, shown in Figure 2. V 171

4 Original scientiic paper Thereore, an interpolation scheme is required to compute the ace values o pressure rom the cell values. Discretization o continuity equation- Equation [1] may be integrated over the control volume to yield the ollowing discrete equation N aces J 0 [6] where J is the mass lux through the ace v. In order to proceed urther, it is necessary n to relate the ace values o the velocity, v n, to the stored values o velocity at the cell centers. Linear interpolation o cell-centered velocities to the ace results in an unphysical checker-boarding o pressure. ANSYS FLUENT uses a procedure similar to that outlined by Rhie and Chow (4) to prevent checkerboarding. The ace value o velocity is not averaged linearly; instead, momentum-weighted averaging, using weighting actors based on the a P coeicient rom the equation [5], is perormed. Using this procedure, the ace lux, J, may be written as: a v a v ^ p, c0 n, c0 p, c1 n, c1 J d (( p ( p) r ) ( p ( p) r )) J d ( p p ) c0 c0 0 c1 c1 1 c0 c1 [7] a a p, c0 p, c1 where p c, p 0 c and v 1 n,c, v 0 n,c1 A within the two cells on either side o the ace, and in these cells (Figure 2). The term, are the pressures and normal velocities, respectively, J d is a unction o ^ contains the inluence o velocities a P, the average o the momentum equation o the a coeicients or the cells on either side o the ace. P Spatial Discretization - By deault, FLUENT stores discrete values o the scalar at the cell centers (c 0 and c 1 in Figure 2). However, the ace values are required or the convection terms in Equation [2] and they have to be interpolated rom the cell center values. This is accomplished using an upwind scheme. Upwinding means that the ace value is derived rom quantities in the cell upstream, or upwind, relative to the direction o the normal velocity v n in Equation [2]. The diusion terms are centraldierenced and are always second-order accurate. When second-order accuracy is desired, the quantities at cell aces are computed using a multidimensional linear reconstruction approach (5,6). In this approach, higherorder accuracy is achieved at cell aces through a Taylor series expansion o the cellcentered solution about the cell centroid. Thus, when second-order upwinding is selected, the ace value is computed using the ollowing expression:, r [8] SOU where and are the cell-centered value and its gradient in the upstream cell, and r is the displacement vector rom the upstream cell centroid to the ace centroid. This ormulation requires the determination o the gradient in each cell. Finally, the gradient is limited so that no new maxima or minima are introduced. 172

5 APTEFF, 43, (2012) UDK: :66.011:004.4 DOI: /APT M BIBLID: (2012) 43, Original scientiic paper Simulation setup - Mesh was created with 140x140 number o elements with grid reand viscosity to inement adjacent to the walls. Density o the luid was set to 1 kg/m 3, Pas. Reynolds number was calculated as Re = ul/. where ρ represents the luid density; μ is dynamic visocity o the luid; L is the characteristical length o cavity, and u lid velocity in the x direction. The velocity o the moving lid was calculated based on deright wall, and sired Re number. Boundary conditions were set as no-slip or the let and or the upper and bottom moving lid as moving walls with deined velocity and direction o moving depending on the case (parallel or antiparallel). Starting conditions or the irst-order upwind scheme were taken as 0.5 velocity o the moving lids, and results were used as starting conditions or the second-order upwind scheme. RESULTS AND DISCUSSION Validation or results o one-sided lid driven square cavity In order to validate the simulation mehod, a popular benchmark problem o one-sided lid driven square cavity is simulated or dierent Re numbers and compared with the resulst in the available literature. Figure 3 shows the u- and v-velocity proile, through the geometric center o the cavity. The obtained results are in good agreement with the results o Chen et al. (6) and Ghia et al (7). Figure 3. Velocity proiles u and v- along the vertical and horizontal centerlines o the square cavity. Antiparallel motion o the lids The results or antiparallel motion o lids are listed in Table 1. Streamunction conobtained or the tours at various Re numbers are presented in Figure 4, while the results velocity u and v-proiles through the mid-section o the staggered cavity are given in Figure 5. For comparison sake, the results obtained by Tekic et al. (3) are also presented. 173

6 APTEFF, 43, (2012) UDK: :66.011:004.4 DOI: /APT M BIBLID: (2012) 43, Original scientiic paper It is evident that with the increase in the Re number, extreme values o the velocity components also increase in magnitude. Furthermore, the inertial orces are dominant compared to the viscous ones. As a result, the gradients close to the moving lids are stronger or higher Re. As previously mentioned, three studies on staggered cavity, (1)-(3) showed unsteady behavior or Re numbers above In the present study, symmetric and asymmetric patterns are achieved even at Re numbers lower than Multiple vortices are ormed, more precisely there are three primary vortices, although in Table 1 the third vortex is re- are all verti- erred to as secondary or easier comparison o the results. Primary vortices cally aligned along the mid-section o the cavity. Opposed to this, secondary vortices are located in the let and right bottom corner o the cavity. Figure 4. Streamunction contours at various Re numbers antiparallel motion. With the increase o Re, the primary vortex located in the let bottom corner grows at the expense o the primary vortex located in the upper right corner. With the urther in- are two pri- crease o the Re number, the bottom let corner vortex disappears, and there mary vortices along the long diagonal o the cavity, secondary vortices appear in the corners next to the moving lid. Figure 5. Velocity proiles u and v- along the vertical and horizontal centerlines o the staggered cavity antiparallel motion (Re a Tekic et al. results (3)) Velocity proiles along the vertical centerline o the cavity dier or some Re values. The most notable dierence is or Re =100. While the results o Tekic et al. (3) show 174

7 APTEFF, 43, (2012) UDK: :66.011:004.4 DOI: /APT M BIBLID: (2012) 43, Original scientiic paper more lattened proiles, ANSYS FLUENT results show the existence o a sine-like curve, more similar to the proiles which Tekic et al. (3) showed or higher Re numbers (Re= 1000). The velocity proiles along the horizontal centerline are almost identical. Considering that there is a very good agreement between the present study and the work o Ghia et al. (6) and Chen et al. (7), and also between results o Tekic et al. (3) and previously mentioned authors, the reasons or disagreements with present study could be ound in dierent Re number deinitionn and dierent boundary conditions implementa- are listed in tions caused by the dierent numerical approach. To summarize the results, the locations o the centers o the vortices Table 1 and compared with (1) and (3). It can be noticed that the results or primary vortices are in good agreement with the results in the available literature. Table 1. Locations and secondary vortices antiparallel motion, a Zhou et al. (1), b Tekic et al.(3), c present study Re 50 a 50 b 50 c 100 a 100 b 100 c Primary vortex (xc1,yc1) (0.9781, ) (0.4219, ) (0.9637, ) (0.4494, ) ( , ) ( , ) (1.0172, ) (0.3828, ) (1.0031, ) (0.4082, ) ( , ) ( , ) First secondary vortex (xc2,yc2) (1.3556, ) (0.0444, ) (1.3484, ) (0.0460, ) ( , ) ( , ) (1.3556, ) (0.0444, ) (1.3502, ) (0.0460, ) ( , ) ( , ) Primary vortex Re (xc1,yc1) 400 (0.7000, ) 400 (0.6822, ) 400 ( , ) 1000 (0.7000, ) 1000 (0.7000, ) 1000 (0.6934, ) First secondary vortex (xc2,yc2) (1.3500, ) (0.0500, ) (1.3522, ) (0.0554, ) ( , ) ( , ) (1.3250, ) (0.0750, ) (1.3250, ) (0.0750, ) (1.3371, ) (0.0722, ) Second secondary vortex (xc3,yc1) (0.4703, 1.625) (0.9219, ) (0.4382, ) (0.9824, ) ( , ) (0.7256,0.2000) (0.5339, ) (0.8811,0.2167) (0.5301, ) ( , ) ( , ) Parallel motion As expected, parallel motion o the opposite lids develops a dierent low pattern compared to the antiparallel motion. Figure 6 shows the streamunction contours or dierent Re numbers. Figure 6. Streamunction contours at various Re numbers parallel motion. It can be noticed that two primary counter-rotating vortices are present and that a ree shear layer orms between them. Compared to the previous studies o low inside rectan- (6), (8), gular cavities, where the ree shear layer is ormed along a horizontal centerline 175

8 APTEFF, 43, (2012) UDK: :66.011:004.4 DOI: /APT M BIBLID: (2012) 43, Original scientiic paper in the staggered cavity the ree shear layer is ormed along the shorter diagonal. The low is no longer symmetrical due to the upper lid moving rom the oset, while the lower lid moves towards the oset. At low Re numbers, a secondary vortex is present close to the corner o the right wall and oset. As the Re number increases, this vortex gains in strength at the cost o the upper primary vortex. At higеr Re numbers, secondary vortex causes splitting o the primary vortex and ormation o a second secondary vortex as shown in Figure 6. Both primary vortices have become more prominent and larger in sito the walls o ze, so that viscous eects are conined to the thin boundary layers close the cavity (9). As mentioned by Sahin and Owens (10), luid begins to rotate like a rigid body with a constant angular velocity at high Re numbers. Figure 7 shows the u- and v- velocity proiles along mid sections o the cavity. As discussed in the previous section, with the increase in the Re number, extreme velocity values also increasee in magnitude. Further, the ree shear layer ormed between the two primary vortices shrinks with the increase in the Re number due to turbulence. The proiles conirm asymmetrical low about the horizontal centerline o the cavity, as previously mentioned. Compared with the results o Tekic et al., it can be seen thatt the obtained proiles are quite similar. Figure 7. Velocity proiles u and v- along the vertical an horizontal centerlines o the staggered cavity parallel motion (Re a Tekic et al. results (3)). As in the case o antiparallel motion, the results o Tekic et al. (3) give more lattened proiles, while the present study shows the existence o a minimum velocity pitch. These dierences occur at lower values o Re number (50 and 100). Velocity proiles along the horizontal centerline show relatively good agreement or the Re values 50, 400 and 1000, while or the Re=100 there is a more signiicant dierence. In general, the velocity proihave more pro- les obtained by simulation in the present study are more symmetrical and nounced minimum and maximum velocity pitch. These dierences, as previously mentio- o ned, could be a result o the dierent Re calculation procedure, and implementation the dierent boundary conditions. 176

9 Original scientiic paper CONCLUSION Results o the ANSYS FLUENT commercial sotware simulation o two-sided liddriven low inside a staggered cavity are presented in this article. Both antiparallel and parallel motions o two acing lids are investigated. The benchmark results obtained with ANSYS FLUENT are in good agreement with the results available in the literature. For antiparallel motion o lids in a staggered cavity results show symmetrical and asymmetrical low patterns. Velocity proiles along the horizontal centerline are in a good agreement with existing data rom the literature, while the proiles along the vertical centerline are slightly dierent rom those used or comparison, especially or Re=50 and Re=100. These dierences could be explained by the dierent Re calculation procedures and dierent boundary conditions implementation methods, considering the dierent numerical approach. The situation is quite similar in case o parallel motion o lids. Unlike or antiparallel motion, steady-state asymmetric patterns are obtained or all investigated Re numbers. It can be noticed that a ree shear layer is ormed along the short diagonal o the staggered cavity. All the main eatures o the low are shown, streamline contours, horizontal and vertical velocity components along the mid sections o the cavity are visually presented, while the location o vortices is presented in Table 1. Acknowledgement This research was inancially supported by the Ministry o Science and Technological Development o the Republic o Serbia (Project No ) REFERENCES 1. Zhou, Y.C., Patnaik, B.S.V., Wan, D.C., and Wei, G.W.: Dsc Solution or Fow in a Staggered Double Lid Driven Cavity. Int. J. Num. Meth. Eng. 57 (2003) Nithiarasu, P. and Liu, C.-B.: Steady and Unsteady Incompressible Flow in a Double Driven Cavity Using the Artiicial Compressibility (Ac)-Based Characteristic-Based Split (Cbs) Scheme. Int. J. Num. Meth. Eng. 63 (2005) Tekić, P., Rađenović, J., Lukić, N., and Popovic, S.: Lattice Boltzmann Simulation o Two-Sided Lid-Driven Flow in a Staggered Cavity. Int. J. Comp. Fluid Dyn. 24 (2010) Rhie, C.M. and Chow, W.L.: Numerical Study o the Turbulent Flow Past an Airoil with Trailing Edge Separation, AIAA 21 (1983) Barth, J. and Jespersen, D.: The Design and Application o Upwind Schemes on Unstructured Meshes, AIAA , AIAA 27th Aerospace Sciences Meeting, Reno, Nevada, (1989) Chen, S., Tolke, J., and Kraczyk, M.: A New Method or the Numerical Solution o Vorticity-Streamunction Formulations. Comp. Meth. Applied Mech. Eng., 198 (2008)

10 Original scientiic paper 7. Ghia, U., Ghia, K.N., and Shin, C.T.: High-Re Solutions or Incompressible Flow Using the Navier Stokes Equations and a Multigrid Method. J. Comput. Phys. 48 (1982) Perumal, D.A. and Dass, A.K.: Simulation o Flow in Two-Sided Lid-Driven Square Cavities by the Lattice Boltzmann Method, Advances in luid mechanics VII. Boston, MA: WIT Press (2008) Patil, D.V., Lakshmisha, K.N., and Rogg, B.: Lattice Boltzmann Simulation o Lid- Driven Flow in Deep Cavities, Comput. Fluids 35 (2006) 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) 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, 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