International Journal of Engineering & Technology Sciences (IJETS) 1 (1): 8-23, 2013 ISSN xxxx-xxxx Academic Research Online Publisher Research Article Simulation of Thermo-Hydraulic Behavior of a Lid-Driven Cavity Considering Gas Radiation Effect and a Heat Generation Zone Mohammad Mehdi Keshtkar* Department of Mechanical Engineering, Islamic Azad University, Kerman Branch, Iran. * Corresponding author. Tel.: +983413201139; fax: +983413201010 E-mail address: mkeshtkar54@yahoo.com ARTICLE INFO Article history Received:1March2013 Accepted:12march2013 Keyords: Lid-driven cavity Thermal radiation Heat generation zone Transient state Discrete ordinates method A b s t r a c t This paper presents a numerical study on the influence of thermal radiation in a lid-driven square cavity filled ith participating gases by considering a nonuniform heat generation zone inside the media. The fluid is treated as a gray, absorbing, emitting and scattering medium. The to-dimensional Cartesian coordinate system is used to solve the governing equations hich are the conservations of mass, momentum and energy. Since the gas is considered as a radiating medium, besides the convective and conductive terms in the energy equation, the radiative term is also presented. The finite difference method has been adopted to solve the governing equations in transient form and the discrete ordinates method (DOM) is used for computation of the radiative heat flux distribution inside the participating media. The crucial influence of gas thermal radiation effect on the cavity thermo-hydrodynamic (THD) characteristics is thoroughly explored. Comparison beteen the present numerical results ith theoretical data in literatures shos a good consistency. Academic Research Online Publisher. All rights reserved. 1. Introduction The problem of lid-driven flos in square cavities has been received considerable interest due to its ide applications in the area of engineering applications. Examples of such systems can be traced to oil extraction, thermal management of electronic cooling and improvement of performance of heat exchangers. The simple rectangular geometry is often considered for analyzing the momentum and energy transport processes inside the cavity. The flo is induced by sliding the top horizontal all at a constant speed, hile the heat transfer is triggered by sustaining a temperature gradient beteen the top and bottom alls. Cavity flo 8 Page
simulation as introduced in early 1980 by Ghia et al. [1]. The vorticity-stream function formulation of the to-dimensional incompressible Navier-Stokes equations as used to study the effectiveness of the coupled strongly implicit multigrid (CSI-MG) method in the determination of high-re fine-mesh flo solutions. Koseff and Street [2] conducted a comprehensive study on this subject. The flo as three-dimensional and as eaker at the symmetry plane than that predicted by accurate to-dimensional numerical simulations. They found that the local three-dimensional features, such as corner vortices in the end-all regions and longitudinal Taylor-Görtler-like vortices, have significant influences on the flo. Similar experimental studies ere done by other investigators to obtain the THD behavior of the lid-driven cavities [3-5]. Cortes and Miller [6] investigated the comparison of fluid flo inside to configurations cavities. The centerline velocity profiles obtained from the solution of to- and three-dimensional representations of the lid driven cavity flo problem ere compared for different Reynolds numbers. Fusegi et al. [7] studied three-dimensional steadystate natural convection of air in a lid-driven cavity. Their graphs revealed the threedimensional behavior of laminar flo in this geometry. Also, Barakos et al. [8] studied the laminar and turbulence flos in a square cavity and compared their orks ith experimental data and previous numerical results. A benchmark quality solution presented for flo in a staggered double lid driven cavity obtained using the avelet-based discrete singular convolution (DSC) by Zhou [9]. The DSC algorithm tested on the single lid driven cavity flo and the Taylor problem ith a closed form solution. The stability of the todimensional, steady, incompressible flo in a rectangular square cavity as investigated experimentally for the parallel motion of to facing alls by Siegmann-Hegerfeld [10]. The critical Reynolds numbers for the onset of three-dimensional steady flo, its structure, and the bifurcation diagram of the velocity field, ere measured by LDV, hich ere in agree ith numerical predictions. Albensoeder [11] introduced a Chebyshev-collocation method in space, hich allos an accurate calculation of three-dimensional lid-driven cavity flos. The time integration as carried out by an Adams-Bashforth backard-euler scheme. The accuracy of the method relies on the representation of the solution as a superposition of stationary local asymptotic solutions and a residual flo field. A numerical study carried out by Chen [12] to investigate the effects of cavity shape on flo and heat transfer characteristics of the lid-driven cavity flos. Dependence of flo and thermal behavior on the aspect ratio of the cavities as also evaluated. Three types of the cross-sectional shape, namely, circular, triangular, and rectangular, and four aspect ratios, ere taken into account to construct telve possible combinations, such that, attention as focused on the small- 9 Page
aspect-ratio situations. Direct numerical simulation of the flo in a lid-driven cubical cavity at high Reynolds numbers (based on the maximum velocity on the lid), has been carried out by Leriche [13]. An efficient Chebyshev spectral method has been implemented for the solution of the incompressible Navier-Stokes equations in a cubical domain. Rousse [14] and Rousse et al. [15] used the Control Volume Finite Element Method (CVFEM) for the solution of combined mode of heat transfer in to-dimensional cavities. Collapse dimension method (CDM) has been used to analyze combined conduction-radiation problem by Talukdar et al. [16]. An incompressible lattice Bhatnagar Gross Krook (LBGK) model studied by Zhang [17] as used to simulate lid-driven flo in a to-dimensional isosceles trapezoidal cavity. In their numerical simulations, the effects of Reynolds number (Re) and the top angle on the strength, center position and number of vortices in the isosceles trapezoidal cavities ere studied. Numerical results shoed that, as Re increases, the phenomena in the cavity becomes more and more complex, and the number of vortexes increases. Amiri et al. [18] studied the problem of combined conduction and radiation heat transfer in 2D irregular geometries by using discrete ordinates method ith temperature and heat flux boundary conditions. Simulating of Mixed Convection in a Lid-Driven cavity ith an open side by the LBM as carried out by jafari et al. [19]. Result shoed that by increasing of aspect ratio, the average Nusselt number on lid- driven all decreases and ith same Reynolds number by increasing of aspect ratio, Richardson number plays more important role in heat transfer rate. Guermond [20] validated a ne highly parallelizable direction splitting algorithm in solving cavity problems. The parallelization capabilities of this algorithm ere illustrated by providing a highly accurate solution for the start-up flo in a three-dimensional impulsively started lid-driven cavity at different Reynolds numbers. Numerical simulation of radiative heat transfer effect by considering absorbing-emitting media, on natural convection heat transfer in a square cavity under normal room conditions as studied by Lari et al. [21]. The results shoed that even under normal room conditions ith a lo temperature difference, the radiation plays a significant role on temperature distribution and flo pattern in the cavity. Also, several interesting effects of radiation are observed such as a seep behavior on the isotherms, streamlines and velocity distributions of the cavity along the optical thickness and a reverse behavior on maximum stream function and convective Nusselt number at different Rayleigh numbers. The revie of literature shos that, the THD characteristics of lid-driven cavities ith considering the gas radiation effect as not studied by any investigator. Thereby the present ork deals to solve this problem by numerical solution of the governing equations by CFD techniques, hile the radiative transfer equation 10 Page
is solve using DOM to find the distribution of radiative heat flux inside the participating media. 2. Analysis 2.1. Problem description The configuration described in present investigation is shon in Figure 1. The geometry is essentially a to-dimensional square cavity ith a side length L x. In addition the lid-driven cavity is filled ith a participating media that is homogeneous and anisotropic. Also, a nonuniform heat generation zone is considered in the region x1 x x ith a parabolic 2 variation in y-direction, such that maximum heat released takes place at the centerline and the minimum value near the solid alls. The thermo-physical properties of the orking fluid, is taken to be constant. The sliding top all, hich move from left to right side at a constant speed,u, is maintained at a higher temperature than the three stationery alls at bottoms and sides. Fig. 1. The configuration of the problem under consideration 2.2. Governing equations The governing conservation equations for a to-dimensional, unsteady, laminar and constant property flo may be ritten as follos: u v Continuity : 0, x y (1) 11 Page
x momentum : u u u p 2 u 2 u 1 u v ( ), t x y x x y (2) y momentum : v 1 u v v v p ( v v ), t x y y x y (3) Energy : 1 u v ( ) Q( y) ( x). q. T T T T T t x y x y cp r (4) In the above equations, u and v are the velocity components in x- and y-directions, respectively, the density, p the pressure, T the temperature, the kinematics viscosity, the thermal diffusivity, and q is the radiative flux vector. The term ( x ) is the delta r function defined as unity for x 1 x x 2 and zero elsehere. The heat source term. qr radiation in energy equation, is calculated as follos: due to. q k [4 I ( r) I( r, ) d] r a b 4 (5) Where k a is the absorption coefficient, I( r, ) the directional intensity, the direction of radiation, and I b the black body intensity. To find I ( r, ), the radiative transfer equation (RTE) is solved for a gray medium [22]: s (. ) I( r, ) KaIb( r) I I( r, ) ( ) d 4 (6) 4 In hich s is the scattering coefficient, k a the extinction coefficient, and ( ss, ) is s the scattering phase function for the radiation from incoming direction s and confined ithin the solid angle d to scattered direction s confined ithin the solid angled. In this study, isotropic scattering medium is considered, in hich the phase function is equal to unity. The boundary condition for diffusely emitting and reflecting gray alls is: (1 ) I( r, s) I ( r ) I( r, s) n. sd b n. s0 n. s 0 (7) 12 Page
Where is the all emissivity, I ( r ) is the black body radiation intensity at the b temperature of the boundary surface, and n is the outard unit vector normal to the surface. Since the RTE depends on the temperature field through the emission term, I ( r ), thus it must be solved simultaneously ith overall energy equation. Here the DOM is used to solve the RTE. In the DOM, Eq. 6 is solved for a set of n different directions, s i, i 1, 2,3,..., n and integrals over solid angle are replaced by the numerical quadrature, that is: b 4 n f ( s) d ( ) i 1 if si (8) here i are the quadrature eights associated ith the directions s i. By this method, Eq. 6 is approximated by a set of n different equations, as follos: s n ( si. ) I( r, si) KaIb( r) I I( r, s ) (, ) j 1 j s j si i 4 (9) i 1,2,3,..., n subjected to the boundary conditions: (1 ) I( r, si ) Ib( r ) I( r, ). n. si 0 s j n s j j (10) n. s 0 i and the divergence of the radiative heat flux is expressed as:. q k [4 I ( r ) I ( r, s ) ] (11) r a b i 1 i i n For the to-dimensional Cartesian geometry, the RTE can be expressed for each individual ordinate direction m, as: m m m I m I m s m m k aib I I (12) 4 x y m m m Where m and m denote outgoing and incoming directions and, are the direction cosines of a discrete direction. For parametric study, the non-dimensional parameters are defined as follos: 2 U ul /, V vl /, X x / L, Y y / L, t t / L 13 Page
( T Tc)/ T, 2 4 P pl /, I I T C, Q q / QL 2 / pr, pl kt / QL, c 2 QL p1 k T, T / Tc Using these dimensionless groups, the non dimensional forms of the governing equations can be ritten as follos: U V Continuity : 0, X Y (13) x momentum : U U U P U U U V Pr( ), t X Y X X Y (14) y momentum : V V V p U V Pr( V V ), t X Y Y X Y (15) Energy : 1 U V ( ) p 1( x). Q r. t X Y X Y pl (16) 4 Divergence of the radiative flux :. Q r (1 )[4( 1) Id ] (17) RTE : 1 (1 ) 4 (. ) I ( 1) I I d 4 (18) The equations (14) - (18) should be solved by the appropriate boundary conditions. In accordance ith the problem description, the initial and boundary conditions can be ritten as follos: U V 0, 0 for t 0 4 4 U 1, V 0, 1 at y L, 0 x L for t 0 y x U V 0, 0 at y 0, 0 x L and at x 0, L 0 y L for t 0 x x y 3. Results and discussion The governing equations are discretized in a uniform structural mesh by a finite difference method. A false transient technique is used for solving of the Navier-Stokes and energy equations (Eqs. (14)-(16)). Discretized forms of the governing equations ere numerically 14 Page
solved by a numerical program in MATLAB. Grid independency is checked for different mesh sizes and finally a uniform mesh of 4040 is selected. The finite difference technique is used to discretize the spatial part of the RTE and an iterative procedure is employed to determine the radiating intensity field. The folloing operating splitting technique is applied to solve the problem: 1- A first approximation for velocity, temperature and intensity distributions is assumed. 2- For one time step, the coupled pressure-velocity fields are obtained from the momentum equations. 3-The RTE equation is solved by the iteration method to calculate the radiative flux distribution for each nodal point using S 6 approximation. 4-Using the values of radiative flux distribution hich ere obtained in step 2, the energy equation is solved to determine the temperature field inside the cavity. 5- Steps 2 to 4 are repeated until convergence is obtained. 6- For subsequent time step, until to achieving steady state condition, steps 2 to 6 are repeated Repeated computations ere performed until all the calculated variables satisfy certain 4 imposed convergence criteria and a very small under- relaxation factor ( 10 ) as employed at each grid point. At first the validation of the present computational code has been verified for the fluid flo in a lid-driven cavity. Figure 2 shos the distributions of velocity components along the mid-plane of the cavity ith comparison to theoretical data given [1]. The consistency beteen these results shos the accuracy of the present numerical procedure. At the present ork, the effect of gas radiation on thermo-hydraulic behavior of a time-dependent square lid-driven cavity is studied, hile a non-uniform heat generation zone inside the media is presented. 15 Page
(a) y=ly/2 (b) x=lx/2 Fig 2. Distributions of x- and y- velocity components inside the cavity In the present computations, all thermophysical properties of the gas are considered to be constant and temperature independent. It is assumed that, the cavity is filled ith a gas of pr 0.717. The hot and cold alls are maintained at temperature 773 K and 373 K, respectively. All alls are black and the gas is absorbing, emitting and scattering medium against thermal radiation. It should be noted that in all test cases, the values of L x and L y is considered to be 1 cm from the bottom to top all. The heat generation zone is situated in the middle of the cavity ith a thickness of 0.1 L x and height of 1 cm. The THD behavior of radiating gas flo in a lid-driven cavity has been presented in Figure 3 by plotting the streamlines, vector field and isotherm lines inside the enclosure. In this test case, the top all constantly slides to the right at constant velocity. Streamlines, isotherms and velocity vectors field at steady state condition at Re 100 are shon in this figure, hen the gas radiation effect is considered in the computations. It is found from Figure 3(a), that the recirculation due to shearing action of upper fluid is clearly visible hich identical ith conventional all shear cavity flo [1]. The possibility of the reverse flo inside the cavity is demonstrated in velocity contour for Re = 100 in Figure 3(b). Isotherms are displayed in Figure 3(c) in the presence of internal heat generation at mid-plan section, hich is clarified that, near the heat generation zone, the isotherms are collapsed together. It is due to reverse flo at the top of cavity. In order to sho the flo pattern at transient time period, the streamlines are plotted in Figure 4 for flo in lid-driven cavity at Re=100 and for three different time steps in a board time range from 0 to 34 (sec). 16 Page
(a) (b) (c) Fig. 3. (a) Streamlines pattern; (b) velocity vectors field; and (c) temperature contour in lid-driven cavity at steady state. ( Re 100, 1, 0.5 ) The upper lid started its moving at t 0 and the computations shos the time of steady state condition equal to 34 (sec). It is seen from Figure 4, by increasing in time from starting time to steady state condition, the center of the recirculated zone near to top all moves toard the center of the cavity, ith a little approach to the right hand side. 17 Page
(a) (b) (c) Fig. 4. Streamlines pattern at three different time steps (a) t 0, (b) t 17 s, (c) t 34s. ( Re 100, 1, 0.5 ) Figure 5 shos the distributions of isotherm lines in three different times at Re=100. It is assumed that, the top all is hot and other alls are cold. The cold and hot alls are maintained at temperature 773 K and 373 K, respectively. Also, a non-uniform heat generation zone is considered in the mid-plan of cavity. This figure shos a very high temperature region near the heat generation zone, such that by spending in time and removing thermal energy from the source zone toard all other parts in the cavity, the value of maximum temperature get a decreasing trend. As another result hich is depicted in Figure 5, one can notice to the center of high temperature zone hich is inside the heat generation zone but moves toard the left hand side by increasing in time. This behavior is due to the follo pattern in the cavity hich as shon before in Figure 4. 18 Page
(a) (b) Fig.5. Temperature contour at three different time steps (a) t 0, (b) t 17 s, (c) t 34s. ( Re 100, 1, 0.5 ) For the purpose of shoing more clear figures about the thermal behavior of the lid driven cavity system, the variations of gas temperature along the y-direction at different cross sections ( x L x 0.2, 0.5 and 0.8) are shon in Figure 6 at three different time levels. The temperature increases in the region near to the heat generation zone is clearly seen from this figure. Besides, it is depicted in Figure 6 that the amount of maximum gas temperature inside the cavity decreases by increasing in time from starting to steady state condition. It is due to this fact that more thermal energy is removed from the heat generation zone toard the surrounding alls by both convection and radiation mechanisms ith spending in time. Also, more uniform temperature field obtained in the cavity as the thermal system approaches to its steady state condition. (c) 19 Page
(a) (b) Fig 6. Temperature distribution at three different places aspect to x, (a) t 0, (b) t 17, 0.5 ) (c) t 34s. (Re 100, 1 Finally, to obtain the effect of gas radiation effect on thermal behavior of the lid-driven cavity, the gas temperature distribution in x- direction at to time levels ith and ithout considering the radiative term in energy equation are shon in Fig 7. This figure shos the temperature distribution at to different Reynolds number and at to different times. It is seen that the maximum temperature inside the cavity has a decreasing trend by time increasing. Also Figure 7 depicted that T max in the cavity has a very greater value in the case of considering gas radiation effect compared to the case ith neglecting radiant heat transfer. It is due to this fact that the radiating gas inside the heat generation zone absorb more radiant energy that lead to having high value for T max. s, (c) 20 Page
(a) (b) Fig 7. Temperature distribution at to different time steps ith and ithout radiating effect: (a)re=100 (b)re=1000 ( y 0.8L y, 1, 0.5 ) 4. Conclusions A numerical investigation of forced convection flo of radiating gases in a 2-D rectangular time-depended lid-driven cavity as carried out. The continuity, momentum and energy equations ere solved by a finite difference method and the radiative transfer equation by discrete ordinate method. A heat generation zone in the middle of cavity from the bottom to top all is considered inside the lid-driven cavity. The medium as considered emitting, absorbing and scattering are surrounded ith black alls. The THD characteristics of the liddriven cavity ere studied at a board range of time. References [1] Ghia, U, Ghia KN Shin CT. High-Re solutions for incompressible flo using the Navier- Stokes equations and a multigrid method. Computational Physics 1982; 48: 387-411. [2] Koseff JR, Street R L. The Lid-Driven Cavity Flo: A Synthesis of Qualitative and Quantitative Observations. Fluids Engineering 1984; 106: 390-98. [3] Migeon C, Pineau G, Texier A. Three-dimensionality development inside standard parallel pipe lid-driven cavities at Re=1000. Fluids and Structure 2003; 17: 717-38. [4] Ilegbusi OJ, Mat MD, A comparison of predictions and measurements of kinematics mixing of to fluids in a 2D enclosure. App. Mathematical Modelling 2000; 24: 199-213. [5] Vogel MJ, Hirsa, AH, Lopez J M. Spatio-temporal dynamics of a periodically driven cavity flo. Fluids Mechanics 2003; 478: 197-226. 21 Page
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