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Research Papers: Natural and Mixed Convection

# Patterns of Double-Diffusive Natural Convection With Opposing Buoyancy Forces: Comparative Study in Asymmetric Trapezoidal and Equivalent Rectangular Enclosures

[+] Author and Article Information
E. Papanicolaou1

“Demokritos” National Center for Scientific Research, Solar and other Energy Systems Laboratory, Aghia Paraskevi, Attiki 15310, Greeceelpapa@ipta.demokritos.gr

V. Belessiotis

“Demokritos” National Center for Scientific Research, Solar and other Energy Systems Laboratory, Aghia Paraskevi, Attiki 15310, Greecebeles@ipta.demokritos.gr

It should be noted that here $Rac$ is defined using the thermal diffusivity $α$ practice, which has been adopted by several authors (1,3), while others use the mass diffusivity for solute $D$ instead.

1

Corresponding author.

J. Heat Transfer 130(9), 092501 (Jul 14, 2008) (14 pages) doi:10.1115/1.2944241 History: Received July 16, 2007; Revised November 09, 2007; Published July 14, 2008

## Abstract

The patterns arising from instabilities of double-diffusive natural convection due to vertical temperature $T$ and solute concentration $c$ gradients in confined enclosures are investigated numerically with the finite-volume method, for mixtures with Lewis numbers Le both $Le<1$ (e.g., air-water vapor) and $Le>1$. The problem originated from the need to gain better understanding of the transport phenomena encountered in greenhouse-type solar stills. Therefore, an asymmetric, composite trapezoidal geometry is here the original geometry of interest, for which no studies of stability phenomena are available in the literature. However, this is first related to the simpler and more familiar rectangular geometry having the same aspect ratio $A$ equal to 0.3165, a value lying in a range for which available results are also limited, particularly for air-based mixtures. The case of opposing buoyancy forces is studied in particular (buoyancy ratios $N<0$), at values $N=−1$, $−0.5$ and $N=−0.1$, for which a wide spectrum of phenomena is present. The thermal Rayleigh number Ra is varied from the onset of convection up to values where transition from steady to unsteady convective flow is encountered. For $Le=0.86$ in the rectangular enclosure, a series of supercritical, pitchfork steady bifurcations (primary and secondary) is obtained, starting at $Ra≈13,250$, with flow fields with three, four, and five cells, whereas in the trapezoidal enclosure the supercritical bifurcation is always with two cells. For higher values of Ra $(Ra≥165,000)$, oscillatory phenomena make their appearance for all branches, with their onset differing between branches. The oscillations exhibit initially a simple periodic pattern, which subsequently evolves into a more complex one, with changes in the structure of the respective flow fields. For $Le=2$ and 5, subcritical branches are also encountered and the onset of convection is in most cases periodic oscillatory (overstability). This behavior manifests itself in the form of standing, traveling and modulated waves (SWs, TWs and MWs, respectively) and with an increase of Ra there is a transition from oscillatory to steady convection, either directly or, most often, through an intermediate range of Ra with aperiodic oscillations. In the trapezoidal enclosure, oscillations at onset of convection appear only for $N=−1$ in the form of traveling waves (TWs), succeeded by aperiodic and then steady convection, while for $N=−0.5$ and $−0.1$, the bifurcations are transcritical, comprising a supercritical branch with two flow cells originating at $Ra=0$ and a subcritical branch with either two or four cells.

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## Figures

Figure 5

TW pattern in the flow field for the rectangular enclosure with Le=5, N=−1 and RaH=25,000. Time history of midpoint stream function and streamlines at five different instants over a single period are shown (dimensionless wave speed=35.461).

Figure 6

Bifurcation diagrams for the rectangular enclosure at (a) Le=5, N=−1; (b) Le=2, N=−0.5. Special patterns encountered only for Le=5, N=−0.5: (c) MWs (RaH=5000) and (d) asymmetric steady solutions (streamlines and isotherms) at RaH=6500.

Figure 7

Flow patterns in the trapezoidal enclosure computed for (a) N=0, Ram=20,000 (supercritical); (b) N=0, Ram=20,000 (subcritical); (c) Le=0.86, N=−1, Ram=20,000; (d) Le=0.86, N=−1, Ram=95114; (e) corresponding bifurcation diagram in terms of Ψmax

Figure 8

Oscillatory and steady flow in the trapezoidal enclosure with Le=2, N=−1. Midpoint stream function history is shown at (a) RaH=35,000 and (b) RaH=120,000, along with streamlines and isotherms (c) for case (b).

Figure 9

Oscillatory and steady branches of the flow in the trapezoidal enclosure with N=−1 and (a) Le=2; (b) Le=5

Figure 10

Oscillatory flow- and temperature-field patterns in the trapezoidal enclosure for Le=5, N=−1, Ram=20,000 within one period T. Streamlines values are from −10.933 to 6.875, isotherms from 0 to 1 (21 values). Dimensionless wave speed=27.310.

Figure 11

Branches of solutions for the trapezoidal cavity with Le=2 and N=−0.5 (a) and N=−0.1 (b). One diagram for the maximum (a) and one for the midpoint stream function (b) are shown for each case (the others having a similar form), along with respective streamlines and isotherms for each branch at Ram=30,000(Ra0≈16).

Figure 12

Bifurcation diagrams for the trapezoidal cavity with Le=5, N=−0.5: (a) stream function at the cavity center, along with contour lines of stream function and temperature at Ram=2900 (supercritical branch) and Ram=3000 (subcritical branch); (b) mean Nusselt number, normalized with the conduction value

Figure 13

Results for the trapezoidal enclosure with Le=2 and N=−0.5: (a) Cell-strength diagram for the supercritical two-cell branch, (b) bifurcation diagrams of the normalized Nusselt number at the bottom (heated) surface, ((c) and (d)) effect of grid dimensions on Ψmax and Nusselt number (percent error) (1=31×31, 2=61×61 (base grid), 3=91×91, 4=121×121)

Figure 1

(a) Original asymmetric trapezoidal geometry (18-19) and equivalent rectangular geometry. Computational grid for (b) equivalent rectangular and (c) original trapezoidal enclosures.

Figure 2

Bifurcation diagrams of the solutions and corresponding flow-temperature field patterns for the rectangular cavity at Le=0.86, N=−1: (a) stream function at the cavity center; (b) horizontal velocity component at point (0.207, 0.102); (c) mean Nusselt number normalized by the conduction value

Figure 3

Oscillatory patterns of the stream function in the rectangular enclosure for Le=0.86, N=−1: midpoint value (Ψmid) for the five-cell branch at (a) RaH=170,000; (b) RaH=180,000; (c) instantaneous flow field evolving from the four-cell branch at RaH=220,000; (d) temporal variation of Ψmid for the three-cell branch at RaH=220,000

Figure 4

Oscillatory onset of convection in the form of standing waves in the rectangular enclosure at Le=2, N=−1: (a) time history of Ψmax for three- and two-cell modes, RaH=27,500 and RaH=40,000, respectively. Streamlines for (b) the three-cell and (c) the two-cell modes at the two limiting states of maximum amplitude within a cycle of oscillations at the saturated state.

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