Theor. Comput. Fluid Dyn. (2012) 26:565–581 DOI 10.1007/s00162-011-0246-6
O R I G I NA L A RT I C L E
A. K. Nayak · S. Bhattacharyya
Double-diffusive convection in a cubical lid-driven cavity with opposing temperature and concentration gradients
Received: 18 May 2009 / Accepted: 20 November 2011 / Published online: 7 December 2011 © Springer-Verlag 2011
Abstract A numerical study of three-dimensional incompressible viscous flow inside a cubical lid-driven cavity is presented. The flow is governed by two mechanisms: (1) the sliding of the upper surface of the cavity at a constant velocity and (2) the creation of an external gradient for temperature and solutal fields. Extensive numerical results of the three-dimensional flow field governed by the Navier-Stokes equations are obtained over a wide range of physical parameters, namely Reynolds number, Grashof number and the ratio of buoyancy forces. The preceding numerical results obtained have a good agreement with the available numerical results and the experimental observations. The deviation of the flow characteristics from its two-dimensional form is emphasized. The changes in main characteristics of the flow due to variation of Reynolds number are elaborated. The effective difference between the two-dimensional and three-dimensional results for average Nusselt number and Sherwood number at high Reynolds numbers along the heated wall is analyzed. It has been observed that the substantial transverse velocity that occurs at a higher range of Reynolds number disturbs the two-dimensional nature of the flow. Keywords Binary mixture · Buoyancy force · Grashof number · Transverse flow List of symbols B C CH CL D g Gr Le Nu Nu p p Pr Re Ri
Buoyancy ratio Dimensionless species concentration Concentration at the hot wall (K) Concentration at the cold wall (K) Mass diffusivity Acceleration due to gravity (m/s2 ) Grashof number Lewis number Nusselt number Average Nusselt number Dimensional pressure (=kg/ms2 ) Dimensionless pressure Prandtl number Reynolds number Richardson number (=Gr/Re2 )
Communicated by Zang. A. K. Nayak (B) · S. Bhattacharyya Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, India E-mail:
[email protected]
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Sc Sh Sh T TH TL u v w
Schmidt number Sherwood number Average Sherwood number Dimensionless temperature Temperature at the hot wall (K) Temperature at the cold wall (K) Dimensionless x-component of velocity Dimensionless y-component of velocity Dimensionless z-component of velocity
Greek β β∗ ν μ θ κ βT βC ρ ρ0
Volumetric coefficient of thermal expansion Volumetric coefficient of solutal expansion Kinematic viscosity Dynamic viscosity Dimensionless temperature Thermal diffusivity Volumetric coefficient of thermal expansion Volumetric coefficient of solutal expansion Fluid density Fluid density at zero temperature (Reference density)
1 Introduction Double-diffusive convection, which refers to the convection driven by a combination of concentration and temperature gradients, has been the subject of much study because of its numerous practical importance. Effects of thermal and solutal buoyancy forces lead to complex flow structures. Double-diffusive convection occurs in a wide variety of fields such as oceanography, astrophysics, vulcanology, drying chambers, material processing and metallurgy. It also occurs in many engineering applications such as solar ponds, natural gas storage tanks, crystal manufacturing and metal solidification processes. The studies on double-diffusive convection with gradients of thermal and solutal fields parallel to the buoyancy force are motivated due to its application in oceanography. The situation in which the gradients are perpendicular to the buoyancy force appears in solidification convection. Among the few geometries that have been studied in detail is the study of cavity flow driven by its lid. Despite its simple geometry, flow in the cavity exhibits features of more complex flows. Numerical investigation on the flow inside a cavity was initiated by Burggraf [1]. Subsequently, the lid-driven cavity flow has been extensively investigated by several authors. These two-dimensional studies revealed that the global flow structures could be characterized by a primary eddy and secondary eddies that formed near the lower solid walls. While some fundamental flow phenomena have been revealed through two-dimensional solutions, many of the subtleties of the three-dimensional characteristics of the flow are missing. Shankar and Deshpande [2] have provided an excellent review on this topic. Sheu and Tasi [3] have also provided a detailed numerical study on the three-dimensional flow in a cubical cavity. The first extensive experimental work on the lid-driven cavity problem was undertaken by the Stanford group (Koseff and Street [4,5]), who used flow visualizations and velocity measurements as their main tools for studying the square-section cavity flow for various values of spanwise aspect ratios and Reynolds numbers. Their studies were very detailed, and it is presently well established that, in cross-sections, there exist a large primary eddy (PE) and two secondary eddies: the downstream secondary eddy (DSE) and the upstream secondary eddy (USE). It is also noteworthy that a third secondary eddy develops when the Reynolds number reaches 1,500. Moreover, it has been shown that the lid-driven cavity flow exhibits three main 3D structures that are induced by the no-slip conditions imposed at the end-walls or by the flow instability, namely the corner vortices located near the end-walls, the outward and inward spanwise currents, and the counter-rotating vortices. With regard to the counter-rotating vortex pairs, Koseff and Street [4] contended that those vortices develop on account of a Taylor–Gortler-like (TGL) centrifugal instability induced by the concave separation pathline between the PE and the DSE. In addition, these researchers also demonstrated that the TGL vortices are issued from initial transient toroidal vortices that develop during the start-up phase on account of a Taylor-like (TL) instability of the PE. It was also shown that this instability is linked to the concave pathline
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curvature of the primary vortex and not to the concave curvature of the separation line between the PE and secondary eddies. The time development of counter-rotating vortices, which are formed in the early stages of impulsively started laminar lid-driven cavity flows, has been studied via experimental visualizations using the particle streak technique by Migeon [6]. Thermohaline double-diffusive convection that arises in solar ponds occurs as one of the diffusion agents (salt) maintains a stable density gradient, while the other agent (heat) produces an unstable gradient. The first study of a double-diffusive, thermohaline system was made by Turner [7]. He considered a mixed layer growth in a linearly stratified salt solution resulting from vigorous bottom heating. The similarity solution of natural convection flows arising from the combined buoyancies due to thermal and chemical species diffusion has been obtained by Gebhart and Pera [8] and later Pera and Gebhart [9]. Numerical study of double-diffusive convection with competing buoyancy forces was made by Mahajan and Angirasa [10]. They considered the flow adjacent to a vertical surface and found that the heat and mass transfer rates undergo complex changes with the variation of buoyancy forces ratio. The phenomena of natural convection caused by combined temperature and concentration buoyancy effects in a vertical enclosure was considered by Trevisan and Bejan [11]. Lee et al. [12], Hyun and Lee [13–15] and Lee and Hyun [16–18] presented a detailed study on double-diffusive natural convection in a rectangular cavity. One of the striking features of this flow field in an enclosed cavity with fixed side walls is the emergence of the layered structure under certain physical conditions. The layer is separated by very thin interfaces, and the density field varies rapidly across theme. Recently, Mergui and Gobin [19] studied the natural convection with competing thermal and isothermal buoyancy forces in a cavity. Younis et al. [20] discussed the double-diffusive natural convection in an enclosure filled with a liquid and subjected to differential heating and differential species concentration. They used an open lid operation to check the hydraulic effect of the upper lid on the heat, mass transfer and flow configurations. Almost all of the numerical work mentioned previously was performed in two dimensions. The threedimensional analysis inside an enclosure filled with porous medium or homogeneous liquid has been analyzed by Mohamad et al. [21–24]. The three-dimensional aspects of the thermosolute natural convection in a cubic enclosure subject to horizontal gradients of heat and solute concentration have been investigated by Sezai and Mohamad [23]. The experimental studies in which the container is extended in the direction transverse to the imposed gradients reveal a similar picture to the two-dimensional numerical simulations. Sezai and Mohamad [23] studied the three-dimensional effect in a cubic enclosure subjected to opposing and horizontal gradients. They found that secondary flow structures evolve on planes perpendicular to the main flow rotation which cannot be predicted by two-dimensional models. Subsequently, Sezai and Mohamad [24] made a three-dimensional, numerical simulation of the Rayleigh–Benard convection in a cavity. The present paper deals with the heat and mass transfer in a cubical lid-driven cavity which is filled with an ideal binary mixture of non-condensable gas and solvent vapor. We considered the flow to be governed by two mechanisms: (1) shear force due to sliding of the top lid and (2) buoyancy forces due to horizontal thermal and solutal gradients. The thermal and solutal gradients are created by imposing a constant temperature and constant concentration along the two opposite side walls. All other walls are assumed to be impermeable and insulated. The hot side wall is maintained with high concentration. Our specific aim in this paper is to investigate the three-dimensional characteristics of the flow fields, heat and mass transfer in a binary mixture due to the variation of shear and buoyancy forces. We consider that the thermal and solutal buoyancy forces are competing, that is, the thermal and solutal buoyancy ratio (B) is negative. The pathlines, isotherms, iso-solutal lines and the heat and mass transfer rates are presented for various values of flow parameters. 2 Governing equations We consider the flow within a closed cubical cavity of length l filled with an ideal binary mixture of a solvent vapor and a non-condensable gas. The flow is assumed to be laminar, and the binary fluid is assumed to be Newtonian and incompressible. Initially, the binary mixture is considered to be at rest with a uniform temperature TL∗ and concentration C L∗ . The side walls are insulated and impermeable to solute. The prescribed ∗ ) and (T ∗ , C ∗ ), respectively, temperature and concentration at the left and right vertical walls are (TH∗ , C H L L ∗ ∗ ∗ ∗ where TH > TL and C H > C L . The flow is induced by the shear force due to the motion of the top lid with a constant velocity U as shown in Fig. 1. The binary fluid is assumed to be Newtonian with constant density everywhere, except in the gravitational force terms in the Navier–Stokes equation, where it varies linearly with the local temperature and solute mass fraction [8], that is,
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v
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Fig. 1 Geometry of the cubical cavity. u, v, w are the fluid velocities in the x, y, z directions, respectively
ρ(T, C) = ρ0 [1 − βT (T ∗ − TL∗ ) − βC (C ∗ − C L∗ )]
(1)
where ρ0 is the density in the undisturbed fluid. The volumetric coefficient of thermal expansion βT = 1 ∂ρ − ρ1 ∂∂ρ T ∗ > 0. The solutal expansion coefficient βC = − ρ ∂C ∗ < 0, that is, the molecular weight of the solute is higher than that of the gas. The governing Navier–Stokes equations with the heat and mass transport equations in dimensional form with the Boussinesq-fluid assumption are given by
∂V ∗ ∂t ∗
∇ · V∗ = 0 + (V ∗ · ∇)V ∗ = −∇ p ∗ + ν∇ 2 V ∗ + (gβT (T ∗ − TL∗ ) + gβC (C ∗ − C ∗ L ))j ∂T ∗ + (V ∗ · ∇)T ∗ = α∇ 2 T ∗ ∂t ∗ ∂C ∗ + (V ∗ · ∇)C ∗ = D∇ 2 C ∗ . ∂t ∗
(2) (3) (4) (5)
where j is the unit vector along y-direction and V ∗ = (u, v, w) is the velocity vector. The characteristic length scale and velocity scale are considered as l and U , respectively. The dimensionless variables p, t, θ and C for pressure, time, temperature and concentration, respectively, are
t =
C ∗ − C L∗ T ∗ − TL∗ t ∗ .U p∗ , p = , C = , θ = ∗ ∗ − C∗ . l TH − TL∗ ρ(U )2 CH L
(6)
Here the dimensional variables are denoted with superscript ∗. The flow, heat and mass transfer are characterized by four dimensionless parameters, namely (i) Reynolds number Re, (ii) Prandtl number Pr , (iii) Schmidt number Sc and (iv) Grashof number Gr , which are given by
Re =
gβT (TH∗ − TL∗ )l 3 Ul ν ν . , Pr = , Sc = , Gr = ν κ D ν2
(7)
The buoyancy force ratio is defined as
B =
∗ − C∗ ) βC (C H L . βT (TH∗ − TL∗ )
(8)
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The boundary conditions for time t > 0 are given by u = v = w = 0, θ = C = 1 at x = 0, 0 ≤ y, z ≤ 1 u = v = w = 0, θ = C = 0 at x = 1, 0 ≤ y, z ≤ 1 ∂C ∂θ = 0, =0 at y = 0, 0 ≤ x, z ≤ 1 u = v = w = 0, ∂y ∂y ∂C ∂θ = 0, =0 at y = 1, 0 ≤ x, z ≤ 1 u = 1, v = w = 0, ∂y ∂y ∂C ∂θ =0 =0 at z = 0, 0 ≤ x, y ≤ 1 u = v = w = 0, ∂z ∂z ∂θ ∂C u = v = w = 0, =0 =0 at z = 1, 0 ≤ x, y ≤ 1. ∂z ∂z
(9) (10) (11) (12) (13) (14)
The average rate of heat and mass transfer is obtained in terms of the non-dimensional coefficients, the Nusselt and Sherwood numbers, defined, respectively, as
1 1 Nu =
− 0
0
∂θ ∂x
dydz
(15)
dydz.
(16)
x=0
and 1 1 Sh = 0
0
∂C − ∂x
x=0
3 Numerical method The governing equations along with the specified boundary conditions are solved numerically using a control volume approach. This method involves integrating the continuity, momentum and energy equations over a control volume on a staggered grid. Different control volumes are used to integrate different equations. In the staggered grid arrangement, the velocity components are stored at the midpoints of the cell faces to which they are normal. The scalar quantities such as the pressure, temperature and concentration are stored at the center of the cell. The discretized form of the governing equations are obtained by integrating over the control volumes using the finite volume method. A third-order accurate QUICK (quadratic upstream interpolation for convective kinematics, Leonard [25]) scheme is employed to discretize the convective terms in the Navier–Stokes equations. A third-level implicit scheme is used for discretization of time derivatives. We use a pressure correction-based iterative algorithm SIMPLE [26] for solving the discretized equations. This algorithm is based on a cyclic series of guess-and-correct operations to solve the governing equations. The velocity components are first calculated from the momentum equations using a guessed pressure field. The pressure and velocities are then corrected so as to satisfy continuity. This process continues until the solution converges. The pressure link between the continuity and momentum equations is accomplished by transforming the continuity equation into a Poisson equation for pressure. The Poisson equation implements a pressure correction for a divergent velocity field. Iteration at each time step continues until the divergence-free velocity field is obtained. At each iteration, the resulting block tri-diagonal system of algebraic equations is solved through a block elimination method. A time-dependent numerical solution is achieved by advancing the flow field variables through a sequence of shorter time steps of duration 0.001. For the range of parameter values considered here, the flow field achieves a steady state after a transient state, and this steady state is independent of the initial conditions prescribed.
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Grid consideration and algorithm testing The grid-independent tests were performed by varying the grids between 61 × 61 × 61 and 121 × 121 × 121. Fig. 2 presents the effect of grid size on the velocity profile at Re = 400. We found that the changes in solution due to halving the grid size occur in the third decimal place. The grids 81 × 81 × 81 were found to be optimal. First, we tried to compare our two-dimensional simulated results with some validated previously published two-dimensional numerical results, and then we compared our three-dimensional simulated results with some of the three-dimensional published results considering the same geometry. To validate the accuracy of our two-dimensional results, we have compared our results for the average Nusselt number and Sherwood number with Maiti et al. [28]. They studied the double-diffusive convection in a square cavity with a sliding top lid. We have shown our comparison in Fig. 3, when buoyancy ratio B = 5.0 and Grashof number is −106 for different Re values. It is seen that the agreement between the published data by Maiti et al. [28] and our numerical result is very good. We have compared our results for three-dimensional flow with the two-dimensional results due to Moallemi and Jang [29]. They considered the mixed convection in a lid-driven cavity combined with the buoyancy force due to heating the lower lid of the cavity with no species diffusion (C = 0, B = 0). Figure 4 shows the comparison of local Nusselt number on the plane z = 0.5 along the top (y = 1) and bottom lids (y = 0) for the case of no species diffusion when Re = 500, Gr = 105 . The maximum percentage difference of Nusselt number on upper lid from the result due to Moallemi and Jang [29] is 5.5%. We have compared our results for natural convection with the results due to Sezai and Mohamad [23] for Ra = 105 , B = −0.5 (Fig. 5). The average Nu and Sh on the heated plane for different values of Le (−2.3 ≤ Le ≤ 4.6) are compared in Fig. 5. Our results are in excellent agreement with those of Sezai and Mohamad [23]. 4 Results and discussion The flow, heat and mass transfer characteristics are governed by the parameters Re, Gr, Sc, Pr and B. The Richardson number, Ri = Gr/Re2 , measures the importance of buoyancy-driven natural convection relative to lid-driven forced convection. The quantity B measures the relative importance of solute and thermal buoyancy forces in causing the density differences. It may be noted that the buoyancy force ratio, B, is zero for no species diffusion, positive for both effects combining to drive the flow and negative for the effects opposed. We considered the values of B between 0 and −2.0 (opposing solutal buoyancy). In our study, we choose Prandtl
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18 16 14
Nu Sh
Nu, Sh
12 10 8 6 4 2 0
1000
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Re Fig. 3 Comparison of present 2-D average Nusselt number and Sherood number with Maiti et al. [28], when B = 5.0 and Grashof number is −106 for different Re values. Solid lines represent the present result, and filled circles represent the result due to Maiti et al. [28] 40 35 Present result Moallemi and Jang
30
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x Fig. 4 Comparison of present Nusselt number at upper and lower lids with Moallemi and Jang [29] when Pr = 1.0, B = 0.0 and C = 0. Grashof number is 105 and Re = 500
number, Pr = 0.72, and Schmidt number, Sc = 5.0. In a gaseous mixture, the Schmidt number remains below 10.0 [8]. We obtained the solutions for Reynolds number varying between 500 and 5,000 and Gr from 104 to 106 . Thus, in our study, the Ri varies from 0.4 to 4. Figures 6, 7, 8 present the pathlines, temperature and concentration lines in the xy-plane at different z (=0.25, 0.5, 0.75) at Re = 500, Gr = 105 (Ri < 1) with B = −2.0. The projection of pathlines on a particular plane is obtained from the in-plane components of the velocity field in that plane. The direction of the flow due to the thermal buoyancy force is clockwise since the left vertical wall is hotter than the right vertical
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Le Fig. 5 Comparison of present average Nusselt number and Sherood number along the heated wall with the result due to Sezai and Mohamad [23] for Ra = 105 , B = −0.5
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wall TH > TL . The concentrations on the vertical boundaries are assigned such that C H > C L . Since we considered the molecular weight of the solute to be higher than that of the gas (B < 0), the solutal buoyancy will induce a flow which will oppose the flow due to thermal buoyancy. At low buoyancy ratio, B = −2.0, the flow is mainly driven by the shear force due to the sliding of the top lid and the thermal buoyancy force. Thus, the main flow direction is clockwise in the xy-plane. The flow in the xy-plane is characterized by a primary eddy and a secondary eddy near the lower downstream corner. At Re = 500, we find that the primary flow is invariant due to the variation of z. As Reynolds number is increased (Fig. 9), the secondary eddies at both the upstream and downstream corners appear. The isotherms and lines of constant concentrations in the xy-plane are presented in Fig. 9 for different Re (=500, 2,500, 5,000) at Gr = 105 and B = −2.0. The thermal boundary layers form along the vertical walls, while the horizontal walls are insulated. The variation of temperature and concentration within the core of the cavity is slow at a lower range of Reynolds number. In Fig. 10, we have presented the pathlines at various z = constant planes for Re = 5,000, Gr = 105 and B = −2.0. The dependence of the flow, thermal and solutal fields on z is evident from these results. The core fluid is well mixed at these high values of Reynolds number. To show the three-dimensionality of the flow, we present the distribution of the transverse velocity (w) at different constant z-planes for Re = 500, Gr = 105 and B = −2.0 in Fig. 11. At this Reynolds number, the w-velocity is nonzero but weak. In Fig. 12, we have presented the w-velocity profiles at different z cross-sections (z = 0.25, z = 0.5, z = 0.75) at high Reynolds number, that is, Re = 5,000 with Gr = 105 and B = −2.0. Figure 12 shows that the variation of w with the variation of z-coordinate is prominent. The w-velocity is low in the symmetry plane z = 0.5. Figures 11 and 12 suggest that three-dimensionality is observed at higher values of Re due to the end wall effects. In Fig. 13, we have presented the w-velocity profile in the yz-plane at different x-stations (x = 0.25, x = 0.5, x = 0.75) for Re = 500, Gr = 105 and B = −2.0. In the midplane x = 0.5, the w-velocity shows a wavy pattern that is due to the formation of vortices near the corner zone. Figure 14 presents the w-velocity profile in the y = 0.5 plane at different x-stations (x = 0.25, x = 0.5, x = 0.75) at Re = 500, Gr = 105 and B = −2.0. It is obvious that the magnitude of the transverse flow, w, is not small and therefore has three-dimensionality even at low Re. Due to the buoyancy effect, the value of w along the center line is significant. At higher Re values, substantial transverse flow close to the end wall is observed. For the comparison of two-dimensional results with the three-dimensional results for a certain range of parameters, Fig. 15 presents the average Nusselt and Sherwood numbers at different values of buoyancy ratio. The Nu and Sh are obtained over the heated vertical wall (x = 0.0). The maximum difference between the two-dimensional and three-dimensional results occurs for B = −0.5. In Fig. 16, the results for the average Nusselt and Sherwood numbers of the vertical plane x = 0.0 (heated wall) for the case of two-dimensional and three-dimensional flows have been presented. We have also shown the comparison of N u and Sh for fixed value of the buoyancy ratio (B = −2.0) at different values of Re. Here the Grashof number corresponds to Gr = 105 . Results show that there exists a significant difference between the two-dimensional and three-dimensional results for average Nusselt number and Sherwood number at higher Reynolds number along the heated wall.
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0.125
0.01 2
0.75
96
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2.13659E-05
0
0.3
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023
75
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0.2 -0.0
4 08
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7 0.38
y
0.6
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2
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0 0
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0 0
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7
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3 0.
32 0.
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28 0.4 0. 35
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13 00
0.009
01
y 0.2
09 0.0
0.0
0.4
0.1
-0.0
0.2 0 0
0.4
-0.018
0.4
0.32
0.6
0.35
y
-0.023
y
-0.031
0.6
0.8
-0.038
0.6
8
41
7
0.8 0.3
-0.0
0.35
0.8
0.8
1
x
Fig. 9 Pathlines, temperature lines and concentration lines on the x y- plane at z = 0.5, for Re = 500 (top), Re = 2,500 (middle), Re = 5,000 (bottom), Gr = 105 and B = −2.0
The effect of Reynolds number on the average Nusselt number and Sherwood number is shown in Fig. 17 at different values of Grashof numbers, Gr = 104 , 105 and 106 , and buoyancy ratio B = −2.0. A significant increase in both heat transfer and mass transfer is observed when Gr is increased from 104 to 106 . However, there is not much difference in values of Nu and Sh between Gr = 105 and Gr = 104 even at high values of Reynolds number. With the increase in Reynolds number for a fixed value of Gr, the rates of heat and mass transfer increase monotonically. The pathlines, isotherms and concentration lines at various z cross-sections (z = 0.25, 0.5, 0.75) are illustrated in Figs. 18, 19, 20 when Gr = 106 , Re = 500 (Ri = 4.0) and B = −2.0. The usual flow structures like formation of primary eddy, upstream and down stream secondary eddies have been observed for this range of parameters. At z = 0.5, we note that the downstream secondary eddy size increases, and the upstream secondary eddy vanishes. For these values of parameters, the flow is buoyancy dominated, as Ri > 1. At z = 0.75, the flow field is similar to the flow at the z = 0.25 plane. The temperature and concentration lines at different planes show that they remain invariant with the variation of z for Ri > 1 when Re = 500.
Double-diffusive convection in a cubical lid-driven cavity
575
1
0.8
0.8
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87
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36
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0.23 6
5 0.34
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87
99
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01
67
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8
7
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1
1
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0.5
x
0.2
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5
0.28 4
0.2
2
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0 0
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0.2
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0 0
3
y
y 0.0
11
5
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-0.0
3 35 -0.0
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y
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3
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6
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2
3
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48
0.8
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0.8
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0.75
0 0
1
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x
0.75
1
x
Fig. 10 Pathlines, temperature lines and concentration lines on the x y- plane at z = 0.25, z = 0.5 and z = 0.75 for Re = 5,000, Gr = 105 and B = −2.0
0.005 0.04 0.02 w 0 -0.02 0
0.25
0.5 x 0.75
1 1
0.5 0.75
(a)
0.25
y
0
0 0
0.25
0.5 x 0.75
1 1
0.5 0.75
(b)
0.25
y
0
w 0
0.25
0.5 0.75
x
1 1
0.5 0.75
0.25
0.04 0.02 0 w -0.02 -0.04 0
y
(c)
Fig. 11 w-velocity plots on x y- plane at different z cross-sections, a z = 0.25, b z = 0.5, c z = 0.75, for Re = 500, Gr = 105 and B = −2.0
576
A. K. Nayak, S. Bhattacharyya
0.06 0
0.05
0.025
w
0
w
0
-0.025 -0.06 0
0.25
0.5 0.75
x
1 1
0.5 0.75
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0
0
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0.5 x 0.75
y
1 1
(a)
0.5 0.75
0.25
w
-0.05
0
0
0.25
0.5 x 0.75
y
(b)
1 1
0.25 0.5 0.75 y
0
(c)
Fig. 12 w-velocity plots on xy- plane at different z cross-sections, a z = 0.25, b z = 0.5, c = 0.75, for Re = 5,000, Gr = 105 and B = −2.0
0.05
w
0
-0.05 0
0 0.5
0.5
z
y 1
1
(a)
0.05
0
0
0 0.5
w
-0.05
0.5
y
z 1
1
(b)
0.04 0
w
-0.04 0
0
y
0.5
0.5 1
z
1
(c) Fig. 13 w-velocity plots on yz- plane at x = 0.25, x = 0.5 and x = 0.75 for Re = 500, Gr = 105 and B = −2.0
Double-diffusive convection in a cubical lid-driven cavity
577
0.1
0.1 x = 0.25 x = 0.5 x = 0.75
0.05
0.05
0.025
0.025
0
0
-0.025
-0.025
-0.05
-0.05
-0.075
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x = 0.25 x = 0.5 x = 0.75
0.075
w
w
0.075
-0.1
0.75
0.25
0.5
z
z
(a)
(b)
0.75
Fig. 14 Results for w-velocity profile at y = 0.5 at different x-stations a Re = 500, Gr = 105 and B = −2.0 b Re = 5,000, Gr = 105 and B = −2.0
Nu , Sh
15
Nu (2-D) Sh (2-D) Nu (3-D) Sh (3-D)
10
5 -2
-1.5
-1
-0.5
0
B Fig. 15 The average Nusselt and Sherwood number versus buoyancy ratio. Where the Grashof number corresponds to 105 and Re = 500
In Fig. 21, we have presented the w-velocity profiles at different xy-plane (z = 0.25, 0.5, 0.75) for Re = 500, Gr = 106 (Ri = 4.0) and B = −2.0. It is observed from those distributions for w that the transverse velocity depends on z. Thus, the three-dimensionality of the flow occurs for these parameter values. 5 Conclusions The three-dimensional aspect of mixed convection is studied in a cubical lid-driven cavity with the horizontal gradients of temperature and concentration. The effects of Grashof number (Gr ) as well as Reynolds number (Re) and buoyancy ratio (B) on the flow structures are investigated. Solutions are obtained when the flow is dominated by buoyancy (Ri > 1) as well as by shear force (Ri < 1). The fluid in the core of the cavity
578
A. K. Nayak, S. Bhattacharyya
Nu (3-D) Sh (3-D) Nu (2-D) Sh (2-D)
30
Nu , Sh
25
20
15
10
5
1000
2000
3000
4000
5000
Re Fig. 16 The average Nusselt and Sherwood number versus Reynolds number. Where the Grashof number corresponds to Gr = 105 and B = −2.0 35
4
Gr = 10 5 Gr = 10 6 Gr = 10
30
Nu, Sh
25
20
15
10
5 1000
2000
3000
4000
5000
Re Fig. 17 The average Nusselt and Sherwood number versus Reynolds number for various Grashof numbers. Buoyancy ratio B = −2.0
rotates clockwise. For Ri > 1, the heat transfer is mostly due to conduction when Re ≤ 500. Increase in −B produces reduction in heat and mass transfer from the lids for Ri < 1, but variation in B does not produce any significant change on heat and mass transfer when Ri > 1. The buoyancy effects on the flow are prominent when Re ≤ 500. In this range of Re, buoyancy produces an increment in the downstream eddy length, whereas the size of the upstream eddy decreases in the presence of buoyancy force. We have presented the steady flow in various transverse planes, and the main rotational cell in the x y-plane has been found to be non-symmetric while previous two-dimensional studies have assumed a centro-symmetric flow structure. The temperature distribution in the flow is influenced in the vicinity of the sliding wall. The heat and mass transfer increases monotonically with the increase in Reynolds number with fixed value of buoyancy ratio and Grashof number. As the Grashof number is increased at a fixed buoyancy ratio, a large difference is found between the results
Double-diffusive convection in a cubical lid-driven cavity
1 0.8
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and B = −2.0. a Pathline, b temperature and c concentration lines on the
0.4
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6
Fig. 19 Contour plots for Re = 500, Gr = xy- plane at z = 0.5
0.00
1
y
0 0
1
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1
and B = −2.0. a Pathline, b temperature and c concentration lines on the
0.8
-0.068
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(c)
1
0.5
0.75
(b)
0.9
0.6
0.5
x
1
-0.089
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x
0.9 0.7
0.46
y 25 0.1
0.5
0.68
Fig. 18 Contour plots for Re = 500, Gr = xy- plane at z = 0.25
106
0.3
0.3
0.44
0.94
0.25
0.3
0 0
0.55
1
(a)
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136
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y
4
y
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1
0.7
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-0.003
0.6
-0.0
64
0.8
0.1 1
0 0
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0.5
x
x
x
(a)
(b)
(c)
0.75
1
Fig. 20 Contour plots for Re = 500, Gr = 106 and B = −2.0. a Pathline, b temperature and c concentration lines on the xy- plane at z = 0.75
580
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0.2 0
w
0
0
x
0.5
0.5
y
1
(a)
0.01 0
w
-0.01 0
0
x
0.5
0.5 1
y
1
(b)
0.1 0
w
0
0 0.5
x 0.5 1
y
1
(c) Fig. 21 w-velocity plots on xy- plane at different z cross-sections, a z = 0.25, b z = 0.5, c z = 0.75, for Re = 500, Gr = 106 and B = −2.0
for two-dimensional and three-dimensional flow cases. Our result shows that the complex three-dimensional flow structure arises at high values of Re and Gr .
References 1. 2. 3. 4. 5. 6. 7. 8. 9.
Burggraf, O.R.: Analytical and numerical studies of steady separated flows. J. Fluid Mech. 24, 113–151 (1966) Shankar, P.N., Despande, M.D.: Fluid mechanics in the driven cavity. Ann. Rev. Fluid Mech. 32, 93–136 (2000) Sheu, T.W.H., Tsai, S.F.: Flow topology in a steady three-dimensional lid-driven cavity. Comput. Fluids 31, 911–934 (2002) Koseff, J.R., Street, R.L.: The lid driven cavity flow: a synthesis of qualitative and quantitative observations. J. Fluids Eng. 106, 390–398 (1984) Koseff, J.R., Prasd, A.K., Street, R.L.: Complex cavities: are two dimensions sufficient for computation? Phys. Fluid A 2, 619–622 (1990) Migeon, C.: Details on the start-up development of the Taylor-Gortler-like vortices inside a square-section lid-driven cavity for 1,000 = Re = 3,200. Exp. Fluids 33, 594–602 (2002) Turner, J.S.: The behavior of a stable salinity gradient heated from below. J. Fluid Mech. 33, 183–200 (1968) Gebhart, B., Pera, L.: The nature of vertical natural convection flows resulting from the combined buoyancy effect of thermal and mass diffusion. Int. J. Heat Mass Transf. 14, 2025–2050 (1971) Pera, L., Gebhart, B.: Natural convection flows adjacent to horizontal surfaces resulting from the combined buoyancy effects of thermal and mass diffusion. Int. J. Heat Mass Transf. 15, 269–278 (1972)
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