ISSN 0015-4628, Fluid Dynamics, 2008, Vol. 43, No. 3, pp. 333–340. © Pleiades Publishing, Ltd., 2008. Original Russian Text © I.A. Ermolaev, A.I. Zhbanov, S.V. Otpushchennikov, 2008, published in Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, 2008, Vol. 43, No. 3, pp. 3–11.

Investigation of Low-Intensity Convection Regimes in a Rectangular Cavity with a Heat Flux on the Boundary I. A. Ermolaev, A. I. Zhbanov, and S. V. Otpushchennikov Received July 23, 2007

Abstract—The characteristics of heat transfer during natural thermogravitational fluid convection of low intensity in a rectangular cavity heated from below (cooled from above) are investigated. Local convection effects are studied. The dependence of local superheating (supercooling) on the Grashof number and the cavity side ratio is found for single-, two- and three-vortex steady motions. The limits of the convection regimes are estimated. DOI: 10.1134/S0015462808030010 Keywords: natural thermal convection, rectangular cavity, local heat transfer, finite element method.

Natural thermal convection in closed and open spaces has been studied theoretically and experimentally for a long time [1–3]. In most cases, developed convection regimes for Rayleigh numbers higher than 103 have generally been considered. Nevertheless, in modern microelectronic systems, as a result of miniaturization, heat transfer is often accompanied by low-intensity convection characterized by Rayleigh numbers Ra < 104 . Moreover, there are other important applications of low-intensity convective flows [4]. Convection at moderate Ra values is also of theoretical interest due to the fundamental features of local heat transfer, such as the superheating (supercooling) effect [5]. For example, in problems with isothermal boundaries on part of the wall the heat flux is smaller than in the absence of convection. In problems with a uniform heat flux, this corresponds to an increase in the local temperature as compared with the heat conduction regime. Low-intensity convection effects have been the subject of relatively few studies. For example, for low values of the Ra number an increase, as compared with the absence of convection, in the maximum fluid temperature in a cylindrical vessel with heat supplied to the lateral surface was detected in [6]. Low-intensity compressible-gas convection in a square cavity with isothermal and adiabatic boundaries heated from one side or from below and the convection in a porous layer heated from one side at various relative elongations were investigated in [5]. The effect of local superheating (supercooling) for a horizontal compressible gas layer heated from below at given boundary temperatures was mentioned in [7]. In [8], the unsteady compressible-gas convection in a rectangular region with heat flux supplied to a lateral wall was investigated. An increase in the local temperature as compared with the heat conduction regime was observed at small Ra and during the development of unsteady convection at higher Rayleigh number values. The local superheating and supercooling effects were also confirmed by the results of three-dimensional simulation [9–11]. In the present study the low-intensity fluid convection in narrow and wide rectangular cavities heated from below (cooled from above) with a given heat inflow or outflow on one horizontal wall at a fixed temperature of the opposite wall is investigated.

333

334

ERMOLAEV et al.

1. FORMULATION OF THE PROBLEM. BASIC EQUATIONS. METHOD OF SOLUTION The calculation domain considered is a two-dimensional rectangular cavity of width L and height H with rigid impermeable walls. We will assume the fluid to be a viscous, thermally compressible medium with constant thermophysical characteristics, for which the Boussinesq approximation holds. The problem is solved in the Cartesian coordinate system with the origin coinciding with the left lower corner of the cavity and the x and y axes directed horizontally and vertically. In the numerical calculations, on the lower boundary a constant heat flux is given, the lateral boundaries are adiabatic, and on the upper wall the temperature is constant. At the initial moment the temperature field is uniform at the upper boundary temperature, the medium is in hydrostatic equilibrium in the field of the gravity force which acts vertically downward, and the heated wall is supplied with a constant heat flux. In solving the problem described the nonstationary two-dimensional convection equations in the Boussinesq approximation [1] are used. As the distance, time, velocity and temperature scales H, H 2 /ν , ν /H, and q0 H/λ are taken. The dimensionless variables are defined as X = x/H, Y = y/H, τ = ν t/H 2 , U = uH/ν , V = vH/ν , and θ = λ ϑ /q0 H, where x and y are the coordinates, t is time, ν is the kinematic viscosity, u and v are the velocity projected on the x and y axes, respectively, ϑ = T − T0 , T0 = 0, λ is the thermal conductivity, and q0 is the heat flux scale. This makes it possible to write down the dimensionless Boussinesq equations in vorticity–stream function–temperature variables as follows:

∂ω ∂ψ ∂ω ∂ψ ∂ω ∂θ − = Δω − Gry , + ∂τ ∂Y ∂ X ∂ X ∂Y ∂X

(1.1)

Δψ = ω ,

(1.2)

1 ∂θ ∂ψ ∂θ ∂ψ ∂θ − = Δθ . + ∂τ ∂Y ∂ X ∂ X ∂Y Pr

(1.3)

Here, ω and ψ are the vorticity and the stream function, Gry = gy β q0 H 4 /λ ν 2 is the Grashof number, Pr = ν /χ is the Prandtl number, gy is the y-component of the gravity acceleration (gx = 0), β is the volume thermal expansion coefficient, and χ is the thermal diffusivity. The dimensionless boundary conditions for system of equations (1.1)–(1.3) have the form:

∂ ψ (0, Y, τ ) ∂ θ (0, Y, τ ) = = 0, ∂X ∂X ∂ ψ (L, Y, τ ) ∂ θ (L, Y, τ ) = = 0, X = L : ψ (L, Y, τ ) = ∂X ∂X ∂ ψ (X , 0, τ ) ∂ θ (X , 0, τ ) = 0, = 1, Y = 0 : ψ (X , 0, τ ) = ∂Y ∂Y ∂ ψ (X , H, τ ) = θ (X , H, τ ) = 0. Y = H : ψ (X , H, τ ) = ∂Y On the walls the vorticity was determined according to the Woods formula [3]. At the initial moment, ω (X , Y, 0) = ψ (X , Y, 0) = θ (X , Y, 0) = 0. The flow was formed as a result of the growth of the small perturbations that ensued due to approximation errors. Initial perturbations of a special shape to form single, two- and three-vortex flows were not assigned. The Prandtl number was fixed: Pr = 1. The problem was solved using the Galerkin finite element method [2]. The temperature, the vorticity, and the stream function were approximated by linear combinations of time-independent basis (shape) functions on linear triangular finite elements, which implied a piecewise-linear approximation of ω , ψ , and θ inside the computation domain and on its boundaries. For the temporal approximation an implicit two-layer method was used. Equations (1.1)–(1.3) were solved successively: each time step started with the calculation of the temperature field (1.3), then the boundary conditions and the vorticity field (1.1) were determined, after which X = 0 : ψ (0, Y, τ ) =

FLUID DYNAMICS

Vol. 43

No. 3

2008

INVESTIGATION OF LOW-INTENSITY CONVECTION REGIMES

335

the vorticity value was corrected and the stream function field (1.2) was determined. The calculations were performed using finite-element software which realized this algorithm. A more detailed description of the algorithm and software is presented in [12, 13]. Stationary solutions were obtained using the stabilization method, by solving the nonstationary problem (1.1)–(1.3). The stabilization criterion was the inequality       k+1 θm − θmk  + ωmk + 1 − ωmk  + ψmk + 1 − ψmk  < ε , where θm , ωm , and ψm are the extreme values of the temperature, vorticity and stream function. The superscript k is the time step number and ε was varied on the interval 10−6 –10−5 . The time step was equal to 10−3 . The calculations were performed on uniform grids measuring from 27 × 27 to 27 × 108, depending on the cavity side ratio L/H. Testing on a grid measuring 40 × 40 showed that the relative variation of the maximum temperature was not greater than 1%. 2. DISCUSSION The numerical calculations were mainly performed on the parameter range 0 ≤ Gr ≤ 104 , 0.5 ≤ L/H ≤ 4. In the cavity, depending on the L/H and Gr values, single-, two- or three vortex steady flow is established. At Grashof numbers lower than 103 the convective flows have almost no effect on heat transfer in the cavity for all L/H and the temperature fields to a large extent correspond to the heat conduction regime. The isotherms are parallel to the horizontal walls; a small distortion can be observed only near the heated lower boundary, where a nonzero horizontal component of the conductive heat flux appears. With increase in the intensity of the buoyancy forces the horizontal temperature stratification intensifies and the horizontal heat flux component increases, which leads to an increase in temperature in a certain local region. The local superheating can be characterized by the excess of the maximum dimensionless temperature over that in the heat conduction regime. For all the flows investigated the temperature maximum was located in one of the lower corners of the cavity. A typical development of the local heat transfer with increase in the Grashof number for single-vortex flow is shown in Fig. 1. At Gr = 1.25 × 103 , along the heated boundary the temperature distribution is linear and the length of the local superheating zone is equal to about one third of the cavity width. Far from the heated wall the temperature field differs only slightly from that in the heat conduction regime. With increase in the convection intensity the horizontal temperature stratification intensifies, the superheating zone shortens, and the maximum temperature increases, reaching a maximum at Gr ≈ 2 × 103 (curves 1–4). With further increase in Gr, the superheating zone length and the maximum temperature increase (curves 4–6). At Gr ≈ 3 × 103 the superheating and the length of the corresponding zone vanish, which can be treated as the end of the transition regime and the beginning of developed convection. The dependence of the local superheating on the Grashof number is shown in Fig. 2 for cavities with a side length ratio 0.5 ≤ L/H ≤ 1.75. Here (and in all the figures), the line 1 indicates the maximum temperature corresponding to the heat conduction regime. The excess of the maximum temperature over that in the heat conduction regime is insignificant up to a certain Gr value, after which θm rapidly increases to its maximum level, falls to the initial value, and continues to decrease with increase in Gr. The superheating increases with increase in the cavity side length ratio, its maximum being displaced in the direction of lower Grashof numbers. We note that in narrow cavities (L/H ≤ 1) the transition from the heat conduction to the developed convection regime occurs at higher Gr values. For such cavities the superheating is also considerably greater. This must be because of the stabilizing effect of the lateral boundaries. For wide cavities (L/H > 1) the superheating is insignificant up to Gr ≈ 103 . Then the temperature begins to grow rapidly, reaches a maximum at Gr ≈ 2 × 103 , after which the superheating decreases and is already equal to zero at Gr ≈ 3 × 103 . In Fig. 2, special interest attaches to the curve for L/H = 1.75. Here the single-vortex flow loses stability and breaks down at Gr ≈ 2.5 × 103 , forming a two-vortex flow. Figure 3 demonstrates the changes in the convective flow intensity with increase in Gr. The form of the functions ψm (Gr) corresponds to the changes FLUID DYNAMICS

Vol. 43

No. 3

2008

336

ERMOLAEV et al.

Fig. 1. Temperature distributions over the heated wall for a cavity with L/H = 1.5: heat conduction regime (1); Gr = 1.25 × 103 , 1.5 × 103 , 2.0 × 103 , 2.5 × 103 , 3.0 × 103 (2–6).

Fig. 2. Changes in the maximum temperature θm with increase in Gr for a single-vortex flow: L/H = 0.5, 0.75, 1.0, 1.25, 1.5, 1.75 (2–7).

in temperature in Fig. 2. In narrow cavities the stream function grows at higher Gr values. For wide cavities the functions ψm (Gr) almost coincide. The characteristic features of low-intensity convection development in two- and three-vortex flows are illustrated by Figs. 4 and 5. Here, as for the single-vortex flow, the temperature excess over the heat conduction regime is insignificant up to Gr ≈ 103 , then the temperature rapidly increases, and the maximum superheating is reached at Gr ≈ 2 × 103 . Then the temperature decreases, returning to the heat conduction regime level at Gr ≈ 3 × 103 . The maximum superheating increases with the cavity side length ratio and, in contrast to the flow in Fig. 2. is displaced in the direction of higher Grashof numbers. The displacement is more clearly expressed for the two-vortex flows. In Fig. 4, note curve 2, where a single-vortex flow still exists up to Gr ≈ 1.6 × 103 . The transition from the two- to the three-vortex regime occurs at 3.25 < L/H < 3.5. Figure 6 shows the changes in the superheating with increase in the ratio L/H for two Grashof numbers. Dependence 2 illustrates the regime of a small convection effect (heat conduction regime). The maximum temperature increases insignificantly and almost monotonically with increase in L/H. The crises are almost indiscernible at 1.75 < L/H < 2 and 3.25 < L/H < 3.5, where the one- and two-dimensional flows, respectively, lose stability. Dependence 3 is close to the maximum superheating and illustrates the transition FLUID DYNAMICS

Vol. 43

No. 3

2008

INVESTIGATION OF LOW-INTENSITY CONVECTION REGIMES

337

Fig. 3. Absolute value of the stream function extremum vs. Gr for a single-vortex flow. Notation same as in Fig. 2.

Fig. 4. Maximum temperature θm vs. Gr for a two-vortex flow: L/H = 2.0, 2.25, 2.5, 2.75, 3.0, 3.25 (2–7).

regime. Note the significant increase in temperature within the limits of the same flow type with increase in L/H. At the crisis points the temperature falls sharply. In general, the maximum superheating decreases with increase in the number of vortices. Note the decrease in the angle of inclination of dependence 3 with increase in the number of vortices, which suggests a further decrease in the maximum superheating for multivortex flows. The conditions of the problem imply the existence of an equilibrium solution that loses stability relative to small perturbations at a certain sequence of critical Rayleigh number values [1]. In Fig. 7, for comparison and as a reference equilibrium limit, the minimum critical Rayleigh number Ram = 1296 (for the wave number 2.56) [14] for a layer with an isothermal upper wall and constant heat flux density on the lower wall is indicated (line I). The vertical lines approximately indicate the boundaries of the different flow types. Curve III is the lower boundary of the developed convection regime and has minima at L/H = 2.25 (B), 3.5 (C), and 1.25 (A). Here, developed convection ensues at lower Gr. FLUID DYNAMICS

Vol. 43

No. 3

2008

338

ERMOLAEV et al.

Fig. 5. Maximum temperature θm vs. Gr for a three-vortex flow: L/H = 3.5, 3.75, 4.0 (2–4).

Fig. 6. Changes in the maximum temperature θm with increase in the cavity side length ratio: Gr = 1 × 103 , 2 × 103 (2–3).

The local superheating effect in a cavity heated from below corresponds to the local supercooling effect in a cavity with heat removed from above. Figure 8 depicts the temperature distributions in the two-vortex convective flow in a cavity with L/H = 2.5 for q(H, Y, τ ) = −1 and an isothermal lower wall at a constant temperature θ (0, Y, τ ) = 0. The lateral boundaries are adiabatic. Here, the fluid sinks along the lateral walls and rises at the cavity center. With increase in Gr, as with the results presented in Fig. 1, we observe an increase in the supercooling and a decrease in the lengths of the supercooling zones (curves 1–3). Then both the supercooling and the supercooling zone lengths decrease (curves 3–5). Curve 6 corresponds to the developed convection regime. Summary. In rectangular cavities heated from below (cooled from above) with a heat inflow (outflow) given on the boundary, heat conduction and transitional natural convection regimes can be distinguished, depending on the value and distribution of the conductive and convective heat fluxes. FLUID DYNAMICS

Vol. 43

No. 3

2008

INVESTIGATION OF LOW-INTENSITY CONVECTION REGIMES

339

Fig. 7. Boundaries of the superheating region for single- (A), two- (B) and three-vortex (C) flows: (I) [14], (II) maximum superheating, (III) boundary of the superheating region.

Fig. 8. Temperature distributions over the upper boundary for cooling from above: Gr = 1.5 × 103 , 1.75 × 103 , 2.0 × 103 , 2.25 × 103 , 2.5 × 103 (2–6).

The regime of transition to developed convection is characterized by the appearance of zones of local superheating (supercooling), whose lengths decrease with increase in Gr. With increase in the Grashof number the superheating increases up to a certain maximum and then decreases to the initial level. With increase in the cavity side length ratio the maximum superheating increases within the limits of the same flow type, decreasing sharply at its breakdown points. In narrow cavities the developed convection regime ensues at higher Gr number values and in this case the superheating is low. In wide cavities the developed convection regime boundary corresponds to lower Grashof numbers and the superheating is higher. The maximum superheating is reached at Gr values similar for different L/H ratios.

FLUID DYNAMICS

Vol. 43

No. 3

2008

340

ERMOLAEV et al.

REFERENCES 1. G.Z. Gershuni and E.M. Zhukhovitskii, Convective Stability of an Incompressible Fluid [in Russian] (Nauka, Moscow, 1972). 2. V.I. Polezhaev, A.V. Bune, N.A. Verezub, et. al., Mathematical Modeling of Convective Heat and Mass Transfer on the Basis of the Navier–Stokes Equations [in Russian] (Nauka, Moscow, 1987). 3. E.L. Tarunin, Numerical Experiment in Free Convection Problems [in Russian] (Irkutsk University Press, Irkutsk, 1990). 4. V.I. Polezhaev, “Convection and Heat/Mass Transfer Processes under Space Flight Conditions,” Fluid Dynamics 41 (5), 736–754 (2006). 5. V.S. Avduevskii and V.I. Polezhaev, “Several Characteristics of Natural Gas and Liquid Convection,” in Selected Problems of Applied Mechanics [in Russian] (VINITI, Moscow, 1974), 11–20. 6. V.I. Polezhaev, “Convective Interaction in a Cylindrical Vessel Partially Filled with Liquid and with Heat Supplied to the Side Surface, Free Surface, and Bottom,” Fluid Dynamics 7 (4), 606–615 (1972). 7. V.I. Polezhaev and M.P. Vlasyuk, “Cellular Convection in an Infinitely Long Horizontal Gas Layer Heated from Below,” Dokl. Akad. Nauk SSSR 195 (5), 1058–1061 (1970). 8. V.I. Polezhaev, “Nonstationary Laminar Heat Convection in a Closed Domain for a Given Heat Flux,” Fluid Dynamics 5 (4), 621–627 (1970). 9. O.A. Bessonov, V.A. Brailovskaya, S.A. Nikitin, and V.I. Polezhaev, “Test for Numerical Solutions of the ThreeDimensional Problem of Natural Convection in a Cubic Cavity,” Mat. Modelirovanie 11 (12), 51–58 (1999). 10. T. Fusegi, J.M. Hyun, and K. Kuwahara, “Transient Three-Dimensional Natural Convection in a Differentially Heated Cubical Enclosure,” Intern. J. Heat and Mass Transfer 34 (6), 1559–1564 (1991). 11. T. Fusegi, J.M. Hyun, K. Kuwahara, and B. Farouk, “A Numerical Study of Three-Dimensional Natural Convection in a Differentially Heated Cubical Enclosure,” Intern. J. Heat and Mass Transfer 34 (6), 1543–1557 (1991). 12. I.A. Ermolaev, A.I. Zhbanov, and V.S. Koshelev, “Solution of the Two-Dimensional Nonstationary Problem of Heat and Mass Transfer by the Finite Element Method,” in Problems of Applied Physics [in Russian] (Saratov University Press, Saratov, 2002), Is. 8, p. 60. 13. I.A. Ermolaev and A.I. Zhbanov, “Formation of Structures during Multivortex Natural Convection in a Horizontal Layer Heated from Below,” Teplofizika Vysokikh Temperatur 41 (4), 561–567 (2003). 14. E.M. Sparrow, R.J. Goldstein, and V.K. Jonsson, “Thermal Instability in a Horizontal Fluid Layer: Effect of Boundary Conditions and Nonlinear Temperature Profile,” J. Fluid Mech. 18 (4), 513–528 (1964).

FLUID DYNAMICS

Vol. 43

No. 3

2008