# Transient flow of air through rectangular vents in a horizontal partition.

INTRODUCTION

The problem of natural convection heat and mass transfer has been studied extensively by many authors (Juluria 1980). The problem of natural convection in enclosures with the top plate colder than the bottom plate has been treated extensively in the literature as well (Sleiti 2008). However, the problem of heavier fluid on top of lighter fluid separated by a narrow vent has not been dealt with analytically or numerically. Such a flow configuration occurs in buildings where elevator shafts, stairwells, service shafts, etc. can act as vents connecting two floors. The flow in these vents is buoyancy driven, due to a fire in the bottom enclosure, and transient in nature. This transient flow may have several modes, and the heat transfer mechanism is of interest, especially when the flow becomes oscillatory.

Flow through apertures connecting two enclosures has been a subject of study for more than four decades. Early studies were limited to openings in vertical partitions. The fundamental difference between flow through a vent in a vertical partition and flow through a horizontal partition is the stable stratification of fluid in the former case and the instability of fluid in the latter case. In spite of this fundamental difference, these studies offer unique insight into flow mechanisms through apertures and different methodologies available to model them.

Prahl and Emmons (1975) conducted an experimental and theoretical study of fire-induced flow through an opening in a vertical partition. The experiments involved steady flows through a single opening with a reduced-scale kerosene/water analog. Inflow and outflow orifice coefficients were determined. These were found to be significantly different at low values of Reynolds number, based on flow height, but reached an asymptotic value of approximately 0.68 at large values of Reynolds number.

In the same year, Leach and Thompson (1975) carried out an experimental investigation of flows in horizontal circular tubes. For the whole range of 0.5 [less than or equal to] L/D [less than or equal to] 9.4 and 3 x [10.sup.4] [less than or equal to] Re [less than or equal to] 1.5 x [10.sup.5] investigated, they found that [C.sub.D] = 0.09, a constant independent of L/D and [Re.sub.[DELTA]]. A large-scale experiment with carbon dioxide and water as working fluids was used to verify the above results for gases. Leach and Thompson also investigated the forced flow rate requirements to prevent the counter current flow in the tube.

Brown (1962) was first to study both analytically and experimentally the flow through square openings in a horizontal partition with heavier fluid above the partition. In his analysis, he assumed that all heat transfer is due to advection only, and no mixing occurs within the vent. Assuming friction to be negligible, he invoked Bernoulli's equation to predict that the Nusselt number (Nu) is proportional to the product of the square root of the Grashof number (Gr) and the Prandtl number (Pr), where the reference length is the height of the vent. Brown and Salvason (1962) conducted a similar analysis for a vent in a vertical partition to derive a theoretical relationship between Nu and Gr.

Epstein (1988) reported an experimental study of buoyancy-driven exchange flow through small openings in horizontal partitions for the same geometric configuration as that in this study. The experimental apparatus consisted of two enclosures, one on top of the other, with brine and fresh water as working fluids. The study concentrated on finding the effect of height-to-diameter aspect ratio of the vent (L/D) on the dimensionless exchange flow rate as defined by Mercer and Thompson (1975) in their study of inclined ducts. Four different flow regimes were identified as L/D and varied from 0.01 to 10. The first of these was named oscillatory exchange flow regime, where plumes of fluids periodically broke through the opening. Epstein (1988) used Taylor's wave theory to predict the motion of the interface and showed that the Froude number (Fr) is a constant for this regime. The second flow regime was called the Bernoulli flow regime due to the fact that data showed the same trends as those predicted by Brown (1962). For this regime, Epstein (1988) found that Froude number Q = 0.23[(L/D).sup.1/2] (length/width of the vent). For large L/D, the flow rate was observed to be smaller than that of the other two regimes due to violent mixing within the vent. He used an analysis similar to that reported by Gardner (1977) to provide correlations for his regime. An intermediate regime was also identified as having combined characteristics of turbulent diffusion and Bernoulli flow.

Brine-water analog was also studied experimentally by Conover and Kumar (1993), and Conover et al. (1995) studied experimentally the buoyant countercurrent exchange flow through a vented horizontal partition using a two-component laser doppler velocimeter. Detailed measurements of velocity were made at a small distance above a circular tube with an aspect ratio of one. Even for this aspect ratio and a Reynolds number ranging between 2400 and 7700, the mixing in the vent was found to be turbulent and unsteady. Conover and Kumar (1993) also found that the flow coefficient was nearly constant for a Reynolds number as low as 2400. This finding was in contrast with the work of Stecker et al. (1984, 1986) who showed that exchange flows reached self similarity only for Reynolds numbers greater than 10,000. Tan and Jaluria (1992) and Jaluria et al. (1993) studied the cases where the vent flow was governed by both pressure and density differences across the vent.

Myrum (1990) conducted heat transfer experiments using water in a top-vented enclosure heated by a disk on the enclosure floor. He observed four modes of flow, which were unstable and oscillated randomly from one to the next. More on this flow configuration is given in the "Results and Discussion" section.

A numerical study of unsteady buoyant flow through a horizontal vent placed slightly asymmetrically between two enclosed vents was performed by Singhal and Kumar (1995). They observed and described several flow regimes.

In all studies discussed above, only high Rayleigh number (Ra) ranges were investigated so that the effect of viscosity could be neglected, and, in general, flow coefficient or Froude number was considered a function of L/D only, independent of the driving potential. Here, Ra number is defined as the ratio of buoyancy forces and (the product of) thermal and momentum diffusivities and is equal to [g[beta][DELTA]TL.sup.3]/[nu][alpha]. Also, the scaled brine/fresh water models used may not give comparable results to full-scale air models due to large changes in the Pr number and the Schmidt number values at low Ra. Therefore, these models may not be applicable to the range of Ra investigated in this study.

The objective of this study is to identify the different flow regimes encountered for small-to-high Ra numbers in a vented enclosure and to discuss the physical mechanisms using the time trace of temperature and fluid flow results for a range of vent aspect ratios. Understanding the different flow regimes in such vented enclosures is a key to understanding airflow design in different designs, including for tall buildings.

Governing Equations and Formulation

The time-dependent governing equations for the laminar, two-dimensional flow were derived from fundamental laws of conservation of mass, momentum, and energy as follows:

[[[partial derivative][rho]]/[[partial derivative]t]] + [[[partial derivative]([rho]u)]/[[partial derivative]x]] + [[[partial derivative]([rho]v)]/[[partial derivative]y]] = 0 (1)

[rho]([[partial derivative]u]/[[partial derivative]t] + u[[[partial derivative]u]/[[partial derivative]x]] + v[[partial derivative]u]/[[partial derivative]y]) = - [[[partial derivative]p]/[[partial derivative]x]] + [mu]([partial derivative]/[[partial derivative]x](2[[[partial derivative]u]/[[partial derivative]x]] - [2/3]([nabla]*[[vector].v])) + [[partial derivative]/[[partial derivative]y]]([[partial derivative]u]/[[partial derivative]y] + [[partial derivative]v]/[[partial derivative]x])) (2)

[rho]([[partial derivative]v]/[[partial derivative]t] + u[[[partial derivative]v]/[[partial derivative]x]] + v[[partial derivative]v]/[[partial derivative]y]) = - [[[partial derivative]p]/[[partial derivative]y]] + [mu]([partial derivative]/[[partial derivative]y](2[[[partial derivative]v]/[[partial derivative]y]] - [2/3]([nabla]*[[vector].v])) + [[partial derivative]/[[partial derivative]y]]([[partial derivative]u]/[[partial derivative]y] + [[partial derivative]v]/[[partial derivative]x])) + [rho]g (3)

[[[partial derivative]T]/[[partial derivative]t]] + u[[[partial derivative]T]/[[partial derivative]x]] + v[[[partial derivative]T]/[[partial derivative]y]] = [alpha]([[[partial derivative].sup.2]T]/[[partial derivative][x.sup.2]] + [[[partial derivative].sup.2]T]/[[partial derivative][y.sup.2]]) + [[[beta]T]/[[rho][C.sub.p]]]([[partial derivative]p]/[[partial derivative]t] + u[[[partial derivative]p]/[[partial derivative]x]] + v[[partial derivative]p]/[[partial derivative]y]) (4)

L and [alpha]/L are used to normalize length and velocity, respectively; [alpha]/[L.sup.2] is used to nondimensionalize time; and [[DELTA]T.sub.i] is used to nondimensionalize temperature.

Except for the buoyancy term in the momentum equation, the transient calculation approach considered in this study doesn't use the Boussinesq model that treats density as a constant value in all solved equations. Instead, in the approach used here, the initial density is computed from the initial pressure and temperature, so the initial mass is known. As the solution progresses over time, this mass is properly conserved. When this approach is used, the operating density [[rho].sub.o] appears in the body-force term in the momentum equation as ([rho] - [[rho].sub.o])g. The definition of the operating density [[rho].sub.o] is thus important for this buoyancy-driven flow.

No-slip conditions were used for velocity boundary conditions at all walls, including the vent walls. Initially, fluid is at rest everywhere in the domain; therefore, velocity components were set equal to zero at nondimentionalized time [tau] = 0. All the walls were treated adiabatically and, hence, the normal gradients of temperature were set equal to zero at all fluid-wall interfaces. At [tau] = 0, the lower chamber contains hot fluid with [theta] = 1, while the upper chamber and the vent contain cold fluid with [theta] = 0. The flow parameters of interest are Ra and Pr. Pr is maintained constant at 0.7. Referring to Figure 1, the geometric parameters are vent aspect ratio, L/D, H1/L, and the enclosure aspect ratio H1/H2. In this study, the investigated parameters are fixed at L/D of 1, 0.5, and 2; H1/L of 10, 20, and 5; and H1/H2 of 1, 2, and 0.5, as shown in Table 1.

[FIGURE 1 OMITTED]

Although the real problem of natural convection in a horizontal cavity with bottom heating is three dimensional, it can be treated as two-dimensional problem for very long vents, which is the case considered in this paper.

It is important to state that the building air conditioning and airflow organization usually deal with high a Ra problem (turbulent natural convection), which is very difficult to be simulated numerically.

Numerical Procedure

Equations 1-4 are discretized over the computational domain using a control volume approach, as documented by Patankar (1980). The resulting algebraic equations for velocity components and temperature were solved using the SIMPLE algorithm, which involves the use of pressure correction equations for enforcing mass conservation. FLUENT (ANSYS 2006) is used for this simulation. A segregated solver with first-order implicit unsteady formulation is employed for this study. The air properties (i.e., density, thermal conductivity, specific heat, and dynamic viscosity) are varied as piecewise functions of temperature. By default, the solver will compute the operating density [[rho].sub.o], which appears in the body-force term in the momentum equations as ([rho] - [[rho].sub.o])g, by averaging over all cells. The discretization schemes used are second order for pressure, SIMPLE for pressure-velocity coupling and second order upwind for momentum and energy. The under-relaxation factor is 0.3 for pressure; 1 for density, body forces, and energy; and 0.7 for momentum. The code was tested against certain benchmark solutions to validate the results. The results obtained by Patterson and Imberger (1980) for the unsteady case, and the results obtained by de Vahl Davis (1983) for the steady case, were selected for comparison with the results predicted. Both of these papers deal with enclosure geometries with heated and cooled vertical walls as well as adiabatic bottom and top walls. The predicted results were within 3% of these benchmark solutions.

Figure 2a shows the numerical grid generated for this study using a Gambit (ANSYS 2006) grid generator. Since FLUENT is an unstructured solver, it uses internal data structures to assign an order to the cells, faces, and grid points in a mesh and to maintain contact between adjacent cells. It does not, therefore, require i,j,k indexing to locate neighboring cells. This gives the flexibility of using the grid topology that is best for the problem, since the solver does not force an overall structure or topology on the grid. For the 2D problem considered here, quadrilateral cells are used to form multiblock structured mesh. More details on grid generation are provided in FLUENT documentation (ANSYS 2006). A grid-independent study is performed using three nonuniform grid distributions: coarse grid with 91 x 63 grid points used for every enclosure and 21 x 13 grid points used for the vent, medium grid with 133 x 91 grid points used for every enclosure and 31 x 19 grid points used for the vent, and fine grid with 201 x 135 grid points used for every enclosure and 47 x 29 grid points used for the vent. The results for temperature and the two components of velocity at the center of the vent and at different locations inside the two enclosures (see Figure 2a) for different Ra values were compared using all three grids. The maximum difference in temperature and velocities between coarse and medium grids was found to be less than 1.5% when results from both grids were compared over the full time domain. The maximum difference between medium and fine grids was found to be less than 0.2%. Finally, the nonuniform grid of 133 x 91 grid points was used for every enclosure, and a grid of 31 x 19 grid points was selected for the vent with grid points placed near every sidewall inside the boundary layer. One complete unsteady solution for one geometry and average Ra number involved computation of up to 70,000 time steps, with approximately 30 to 40 iterations for each time step; although, for more confidence in results, a convergence usually is achieved after 10 to 15 iterations.

[FIGURE 2 OMITTED]

Convergence criteria were set to ensure converged results for continuity, x-velocity, y-velocity, and energy. A solution was assumed converged if the maximum scaled residuals of the continuity, x-velocity, y-velocity, and energy equal [10.sup.-4], [10.sup.-5], [10.sup.-6], and [10.sup.-7], respectively. The residual is scaled using a scaling factor representative of the flow rate of [phis] through the domain. This scaled residual is defined as

[R.sup.[phis]] = [[[[SIGMA].sub.cellsP]|[[SIGMA].sub.nb][a.sub.nb][[phis].sub.nb] + b - [a.sub.P][[phis].sub.P]|]/[[[SIGMA].sub.cellsP]|[a.sub.P][[phis].sub.P]|]]. (5)

For the momentum equations, the denominator term [a.sub.P][[phis].sub.P] is replaced by [a.sub.P][v.sub.P], where [v.sub.P] is the magnitude of the velocity at cell P.

The scaled residual is a more appropriate indicator of convergence. For the continuity equation, the solver's scaled residual is defined as

[R.sub.[iteration].sup.c]N/[R.sub.[iteration].sup.c]5, (6)

where [R.sub.iteration.sup.c]N is the residual of the n-th iteration, and [R.sub.iteration.sup.c]5 is the largest absolute value of the continuity residual in the first five iterations.

For the enclosure under consideration, a fixed time-stepping method is used for a time-marching scheme. The time step used was based on estimating the time constant as

[t.sub.c] = [L/U][approximately equal to][[L.sup.2]/[alpha]][(PrRa).sup.[-1/2]] = [L/[square root of [g[beta][DELTA][T.sub.i]L]]],

where L and U are the length and the velocity scales, respectively (Bejan 1984).

Then the time step is determined such that [DELTA]t = [t.sub.c]/20. Note that this time-step value used in this study is five times less than the recommended time step in (ANSYS 2006) to ensure time convergence. After oscillations with a typical frequency have decayed, the solution reaches steady state.

The time-step independence was tested prior to obtaining solution for the desired problem. The time-independent study is established by tracking the changes in temperature and the two components of velocity at a point located at the center of the vent by increasing and decreasing the time step for all cases of Ra studied. Decreasing the time step from [DELTA]t = [t.sub.c]/20 to [DELTA]t = [t.sub.c]/25 resulted in less than 0.05% difference in temperature and velocities when results from both time steps are compared over the full time domain. Increasing the time step from [DELTA]t = [t.sub.c]/20 to [DELTA]t = [t.sub.c]/15 resulted in less than 0.1% difference in temperature and velocities when results from both time steps are compared over the full time domain. Based on this, the time step used for all simulations is [DELTA]t = [t.sub.c]/20. Figure 2b shows sample convergence plots of the scaled residual for Ra of 2500 at different time steps.

Quantities of Interest

The Ra number was defined as Ra = (g[beta][[DELTA]T.sub.i][L.sup.3]/[nu][alpha]), where [[DELTA]T.sub.i] is the initial temperature difference between the two enclosures (i.e., at [tau] = 0). As the interaction between the fluids in the two enclosures proceeds with time, the effective driving potential is the temperature difference across the vent and not [[DELTA]T.sub.i]. The magnitude of the temperature difference across the vent decreases with time. Therefore, new quantities need to be defined that reflect this change in driving potential with time. To this end, we define the following quantities:

1. Instantaneous averaged nondimensionalized temperature ([[theta].sub.av]) on left- and right-side walls of the vent are defined as [[theta].sub.av] = ([T(t).sub.av] - [T.sub.c])/[[DELTA]T.sub.i], where [T(t).sub.av] is the average temperature as a function of time, [T.sub.c] is the initial temperature in the upper enclosure, and [[DELTA]T.sub.i] is the initial temperature difference between the upper and lower enclosures. Thus ([[theta].sub.av]) varies from 0.0 to 1.0, where close to 0.0 values indicate no difference in averaged temperature between the two enclosures.

2. Instantaneous bulk (mass weighted average) nondimensionalized temperature ([[theta].sub.b]) along a cross section on the center of the vent at L/2 is defined as: [[theta].sub.b] = ([T(t).sub.b] - [T.sub.c])/[[DELTA]T.sub.i]. Thus, ([[theta].sub.b]) varies from 0.0 to 1.0, where close to 0.0 values indicate no difference in bulk temperature between the two enclosures.

3. Instantaneous vertical velocity ([[PSI].sub.c]) at a point located at the center of the gap (i.e., at D/2 and L/2.) is normalized by the maximum vertical velocity at the same location ([[PSI].sub.c] =[V.sub.y]/[V.sub.max]), where the maximum velocity is the highest velocity found on the domain at any time. Thus, ([[PSI].sub.c]) varies from -1.0 to 1.0. Values of ([[PSI].sub.c]) close to 0.0 mean the flow field is stabilizing.

4. Instantaneous mass weighted average velocity magnitude ([[PSI].sub.b]) along a cross section located at the center of the gap (i.e., at L/2) is normalized by the maximum velocity magnitude along the same location ([[PSI].sub.b] =[V.sub.bmag]/[V.sub.max]), where the maximum velocity magnitude is the highest velocity magnitude found on the domain at any time. Thus ([[PSI].sub.b]) varies from 0.0 to 1.0. Values close to zero mean the flow field is stabilizing.

All quantities described above are then calculated at every time step throughout the simulation.

Results and Discussion

The interaction between the two enclosures takes place through the vent, and the flow patterns within the vent determine the mode of heat transfer and the rate of heat and mass transfer across it. These flow patterns in the vent depend on the vent geometry, the magnitude of the buoyancy force that drives the flow across it, and, to a great extent, the flow patterns in enclosures themselves. The last category mentioned above makes the flow configuration difficult to analyze. However, from the results presented, an attempt has been made to explain the localized phenomena. Based on the mode of heat transfer and the associated flow characteristics, three categories have been identified: the conduction regime, the countercurrent regime, and the oscillatory regime. The Ra number largely governs these regimes, as explained below.

The Conduction Regime (Ra [less than or equal to] 1500). For small Ra numbers, the buoyancy forces are not strong enough to overcome the viscous forces in the vent where the vertical walls are closer to each other as compared to the enclosure's vertical walls. Consequently, there is no bulk fluid motion within the vent, as shown in Figure 3, where the instantaneous vertical velocity ([[PSI].sub.c]) is almost constant, and all the heat transfer is solely by conduction for Ra [less than or equal to] 1500.

[FIGURE 3 OMITTED]

Flow patterns for Ra = 1000 at different time steps are given in Figure 4. In this regime, the vent acts like a heater plate for the upper chamber, which is filled with colder fluid, and as a cold plate for the lower chamber, which is filled with warmer fluid. Thus, in the enclosures, a bulk fluid motion ensues immediately after the initial establishment of the temperature gradient in the vent. A plume rises steadily into the upper chamber, forcing the colder fluid to move along the side walls to replace the rising fluid and forming two cells of equal strength on either side of it. For a considerable length of time, the cells stay symmetric in each chamber. After that, the cells in the upper chamber become asymmetric before merging into a single cell. There is a time delay before the lower chamber behaves exactly like the upper one, exhibiting a single cell. At large [tau], the fluid in the chambers is accelerated enough to drive the fluid in the vent by entrainment/shear.

[FIGURE 4 OMITTED]

This singular phenomenon of no flow at low Ra numbers is analogous to the Benard convection problem, with free boundaries at the top and bottom. At [tau] = 0, fluid is at rest everywhere, and the vent marks the region of separation of large bodies of hot and cold fluid. With respect to the vent, the denser fluid at the top has a natural tendency to exchange places with the lighter fluid at the bottom. However, this exchange is inhibited by its own viscosity, and, for the flow to ensue, the adverse temperature gradient must exceed a certain value. The onset of instability beyond a critical value of the Ra number was thoroughly analyzed using techniques such as linear stability theory and the power integral method. Chandrasekhar (1962) presented a comprehensive treatment of the linear stability theory where he showed mathematically the thermodynamic significance of [Ra.sub.critical]. In his words, "Instability occurs at the minimum temperature gradient at which a balance can be steadily maintained between the kinetic energy dissipated by viscosity and the internal energy released by the buoyancy force." [Ra.sub.critical] corresponds to the above-mentioned minimum temperature gradient in conjunction with the dynamic conditions at the two bounding planes. For the case of rigid constant temperature surfaces, [Ra.sub.critical] was calculated to be 1708; for one rigid and one free surface it is 1100; and for both free surfaces, [Ra.sub.critical] is found to be 658. The same line of reasoning can be applied to the flow in the vent. This is justified if one takes a closer look at Figure 5, which shows a magnified view of velocity vectors in the vent. The magnitude of velocity in the vent is extremely small compared to those in the enclosures. In the middle of the vent, velocities are negligible, which suggests that the fluid movement at the top and bottom edges of the vent is due to bulk fluid motion in the enclosures. Two planes of symmetry divide the vent into four similar parts. Few complications arise in this problem, which makes stability analysis difficult. One is the effect of side walls, and the other is the influence of bulk fluid motion in upper and lower chambers, where the later is more complex than the former. To date, several researchers have addressed the issue of the effect of side walls on [Ra.sub.critical]. The first complete analysis was given by Yih (1959) for the stability of a viscous fluid between insulated vertical plates. Davis (1983) was the first to consider the fully confined fluid; he used the Galerkin method for his analysis. Later, Catton (1970) and Catton and Edwards (1967) improved on Davis' results and produced a chart for [Ra.sub.critical] as a function of H1 and H2 as parameters, where H1 and H2 are spanwise dimension and depth, respectively. Both studies considered the side walls to be perfectly conducting walls. Latter, Catton and Edwards (1967) performed an experimental study on the effects of side walls on natural convection. They were able to obtain [Ra.sub.critical] as a function of height-to-width ratio for insulating as well as conducting lateral walls. For an aspect ratio of 1.0 and insulating lateral walls, they reported that [Ra.sub.critical] was in the range of 1 x [10.sup.4] and 2 x [10.sup.4]. They also developed a heat transfer correlation.

[FIGURE 5 OMITTED]

No prior work exists about the influence of bulk fluid motion in enclosures on the vent. Figures 6 and 7 give the change in [[theta].sub.av] on right and left walls of the vent, respectively, as a function of time. Instantaneous bulk (mass weighted average) nondimensionalized temperature ([[theta].sub.b]) along a cross section on the center of the vent at L/2 is given by Figure 8, and the instantaneous mass weighted average velocity magnitude ([[PSI].sub.b]) is shown in Figure 9. Curves for Ra of 1200 and 1300 shown in the above figures demonstrate a limiting value for Ra for different regimes.

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

For very small values of [tau], a sharp gradient appears in ([[PSI].sub.b]) because of the initial conditions imposed in the enclosures. However, the flow adjusts quickly to decrease this gradient as ([[theta].sub.b]) steadily decreases with time. Since all the heat transfer is by conduction only, the average wall temperature [[theta].sub.av] has a constant value of about 0.5 for all values of [tau].

In the conduction regime, there is no flow across the vent, as the viscous forces are as large as the buoyancy forces. This is analogous to the Benard convection problem, where the Ra number must exceed a critical value before the flow can start, as explained analytically by Chandrasekhar (1961) using linear stability theory. However, no closed-form solution is available for the configuration under investigation to determine the exact value of critical Ra.

The Countercurrent Regime (1600 [less than or equal to] Ra < 3600). This regime is characterized by a down flow along approximately one half of the vent and upflow along the rest to satisfy continuity, as shown in Figure 10.

[FIGURE 10 OMITTED]

One essential difference in the flow patterns between Ra = 1000 and Ra = 2500 (Figures 4 and 10) is that the cells in the top chamber for Ra = 2500 become asymmetric at early dimensionless time. This asymmetry arises at [tau] = 5.0 during the onset of the countercurrent flow at the vent. A closer look at the vent (Figure 11) clearly shows the upflow on the right side and downflow on the left side of the vent at [tau] = 2.2. With time, the flow gradually reverts itself to the conduction regime, with one large cell in each chamber, as seen for Ra = 1000.

[FIGURE 11 OMITTED]

As Ra number increases in this flow regime, the flow patterns present the most interesting phenomenon. In Figure 10 at [tau] = 2.2, the flow patterns are strictly countercurrent in the vent; however, at [tau] = 8, the flow everywhere in the enclosure slows down. This is evident from the magnified view of the vent in Figure 11. Following this deceleration, a flow reversal occurs at the vent at approximate [tau] value of 11 and lasts as seen at [tau] = 19, and the flow accelerates again as evident by the magnitude of velocity vectors. This is the only flow reversal that occurs in this flow regime.

Such a phenomenon was also seen by Myrum (1990). He conducted an experimental study to find the effect of vent size and Ra on natural convection heat transfer from a heated disk. This disk was located at the bottom of a top-vented enclosure. During the experiments, he also studied the flow patterns in the vent and reported four basic modes. In mode I, flow exited the vent along its axis and entered around its circular perimeter. In mode II, regions of outflow and inflow formed concentric rings within the vent. In mode III, inflow occurred through one half of the vent along the perimeter and the corresponding outflow occurred through the rest. A nonperiodic shift in sides of inflow and outflow was also observed.

Although a direct comparison cannot be made between the current study and Myrum's (1990), the time history of heat transfer along with numerical flow visualization reveal mode III. At the time of this change in direction near [tau] = 8, as discussed above, the exchange flow rate decreases drastically with corresponding decrease in heat transfer across the vent, as seen in Figures 8 and 9. This change, therefore, appears in [[theta].sub.b] and [[PSI].sub.b] time history as a sudden decrease. A sharp decrease and increase in the rate of change of instantaneous [[theta].sub.av] on the right and left walls of the vent, respectively, occur due to the flow reversal, as shown in Figures 6 and 7. As soon as the change in sides is complete, the average and bulk temperatures and velocities resume at almost the same values as before the change. At large values of [tau], flow in the vent gradually dies out due to lack of driving potential, namely the bulk-mean temperature difference between the two enclosures. As this happens, the plumes are no longer strong enough to sustain two cells on either side, and gradually the two cells merge to form a single cell. Since at this point in time there is no flow across the vent, the heat transfer is by conduction only. Hence, the flow configuration is exactly as seen in Figure 4 for large [tau]. Figures 3, 6, and 7 confirm this observation, where [[PSI].sub.c] and [[theta].sub.av] are the same as for the conduction regime.

The Oscillatory Flow Regime (Ra [greater than or equal to] 3600). This regime is characterized by sudden bursts of both upflow and downflow, which show up as a periodic-like response in [[PSI].sub.c], [[PSI].sub.b], and [[theta].sub.av], [[theta].sub.b] time history for large Ra numbers only. This type of flow originates at nearly Ra = 3600.

The oscillatory flow with a single cell in the vent is dominant at high Ra numbers of 5000, 10,000, and 50,000. The flow patterns are qualitatively similar and, hence, the flow and temperature time traces for Ra = 10,000 and the flow patterns for Ra = 50,000 in the enclosures and the magnified vent were chosen for discussion and are given in Figures 12 and 13. Unlike the other two regimes, the flow in the bottom chamber reaches a unicellular pattern earlier than the top enclosure. A close look at the flow patterns within the vent for Ra = 50,000, as given in Figure 13, reveals that the cell that develops in the vent persists for a longer period before the flow patterns undergo a flow reversal as seen before. Temperature and velocity plots for Ra = 10,000 at different times (Figures 3, 6, 7, 8, and 9) give a unique perspective of occurrences in the vent. Temperature and velocity inversions occur at different times, corroborating the fact that the flow oscillates with a well-defined frequency in the entire domain.

[FIGURE 12 OMITTED]

[FIGURE 13 OMITTED]

It is suspected that these bursts are periodic in nature and the frequency of bursts is a function of driving potential. Time traces of [[PSI].sub.c], [[PSI].sub.b], and [[theta].sub.av], [[theta].sub.b] for Ra = 10,000 (Figures 3, 6, 7, 8, and 9) suggest that there are varying amplitude oscillations in a narrow frequency range that eventually die out and essentially reproduce the flow behavior quantified in the countercurrent and conduction flow regimes at large nondimensional time. The Nu number trace was observed by Mitchell and Quinn (1966) in a confined layer heated from below. They found that, as the plate temperature was increased, the fluid oscillated in a narrow frequency band. They also noted that the oscillations were stable over a large Ra number range. Once again, a direct comparison of our results with those of Mitchell and Quinn (1966) cannot be made, since their mean flow was steady.

Based on the oscillatory flow regime results above, it is apparent that, for the building air conditioning and airflow organization, we usually deal with a high Ra problem (turbulent natural convection), which is very difficult to simulate numerically.

Case of L/D = 0.5. Figures 14 and 15 show overall velocity field and magnified view of the velocity vectors inside the vent for L/D = 0.5 at Ra = 10,000. At low [tau] = 0.8, the heavier and slower flow from the top enclosure tends to penetrate through the vent into the lower enclosure. The resistance from the lighter hot fluid in the lower enclosure is increased and finally inversed at [tau] = 4.3. At this time level, the highest velocities are found in the upper enclosure. At [tau] = 10.73, the fluid of the upper enclosure is trying to penetrate to the lower enclosure, where, at this time level, the highest velocities are found in lower enclosure. It is found that the flow at this L/D = 0.5 does not show signs of the oscillatory flow regime observed for L/D = 1, where a large vortex formed in the vent. Instead, the flow oscillates with higher frequency between the two enclosures until it reaches the conducting flow regime at high time levels of [tau] > 50. The full symmetry of the flow in the upper and lower enclosures is reached at time levels of around [tau] = 63.

[FIGURE 14 OMITTED]

[FIGURE 15 OMITTED]

Case of L/D = 2. Figures 16 and 17 show overall velocity field and magnified view of the velocity vectors inside the vent for L/D = 2 at Ra = 10,000. The flow for this case is solely conducting at all time levels. At low [tau] = 0.135, the heavier cold fluid from the top enclosure tends to penetrate through the vent towards the lower enclosure, but the high resistance from the lighter hot fluid in the lower enclosure prevents its penetration. The first mode of symmetry of the flow in the two enclosures is reached at low [tau] = 0.5. This symmetry is characterized by a large vortex that occupies the whole enclosure. At [tau] = 0.2 (not shown), a second vortex starts to form on the upper half of the upper enclosures as a result of the motion of the flow, which comes back from the vent region. This flow, too weak to travel all the way to the upper wall of the upper enclosure, turns back at the location above the center of the enclosure, causing two counter current vortices of different size and velocity. At this time level, no second vortex is found in the lower enclosure. This second vortex in the lower enclosure starts to appear only at [tau] = 4 (not shown) and is completely shaped at [tau] = 10 (not shown). The flow in both enclosures at all higher time levels is characterized by those four vortices, while the flow exchange in the vent is solely by conduction.

[FIGURE 16 OMITTED]

[FIGURE 17 OMITTED]

CONCLUSION

A numerical study of unsteady, laminar, buoyant flow through a horizontal rectangular vent between two large enclosures was performed. Based on the flow patterns in the vent and the corresponding modes of heat transfer through it, three flow regimes were identified: the conduction regime (Ra [less than or equal to] 1500), the countercurrent flow regime (1600 [less than or equal to] Ra < 3600), and the oscillatory flow regime (Ra [greater than or equal to] 3600). In the conduction regime, there is no flow across the vent, as the viscous forces are as large as the buoyancy forces.

In the countercurrent flow regime, the fluid moves through the vent in two distinct streams, with one going up and the other going down. A flow reversal takes place in the vent soon after flow initiation, which is associated with a flow deceleration and a significant decrease and increase in the rate of change of instantaneous [[theta].sub.av] on the right and left walls of the vent, respectively. As soon as the change in sides is complete, the average and bulk temperatures and velocities resume at almost the same values as before the change.

The third regime is characterized by sudden bursts of upflow and corresponding downflow with a well-defined frequency range. Temperature and velocity results associated with such bursts display interesting characteristics of increasing amplitudes but nearly the same frequency.

Decreasing partition thickness from L/D = 1 to 0.5 increases the flow exchange between the upper and lower enclosures with increasing frequency of the oscillatory flow pattern. Increasing partition thickness from L/D = 1 to 2 decreases the flow exchange between the two enclosures, and flow regime becomes conduction with low frequency.

NOMENCLATURE

D = width of the vent

Gr = Grashof number (A dimensionless number approximates the ratio of the buoyancy to viscous force acting on a fluid.)

g = acceleration due to gravity

H1 = width of the enclosure

H2 = height of the enclosure

L = height of the vent

L/D = vent aspect ratio

Nu = Nusselt number (This parameter provides a measure of the convection heat transfer occurring at the surface.)

p = total pressure less the hydrostatic pressure

P = dimensionless pressure = p/([rho][[alpha].sup.2]/[L.sup.2])

Pr = Prandtl number (Dimensionless number approximating the ratio of momentum diffusivity [kinematic viscosity] and thermal diffusivity.)

Ra = Rayleigh number = g[beta][DELTA][TL.sup.3]/[nu][alpha] (The ratio of buoyancy forces and the product of thermal and momentum diffusivities.)

[Ra.sub.critical] = critical value of Ra for onset of flow through the vent

Re = Reynolds number

[Re.sub.[DELTA]] = densimetric Reynolds number = [V.sub.[DELTA]]H/[nu]

t = time

[t.sub.c] = time constant

T = temperature

[T.sub.cold], [T.sub.c] = initial temperature of fluid in upper chamber

[T.sub.hot] = initial temperature of fluid in lower chamber

[T.sub.ref] = reference temperature

[[DELTA]T.sub.i] = initial temperature difference between upper and lower chamber

u,v = components of velocity in x and y directions, respectively

U,V = dimensionless components of velocity in x and y directions, respectively

x,y = horizontal and vertical coordinates

X,Y = dimensionless horizontal and vertical coordinates

Greek Symbols

[alpha] = thermal diffusivity of the fluid

[beta] = coefficient of thermal expansion of the fluid

[nu] = kinematic viscosity of the fluid

[theta] = dimensionless temperature = (T - [T.sub.ref])/[[DELTA]T.sub.i]

[[theta].sub.av] = (T[(t).sub.av] - [T.sub.c])/[[DELTA]T.sub.i] = instantaneous averaged nondimensionalized temperature on left- and right-side walls

[[theta].sub.b] = (T[(t).sub.b] - [T.sub.c])/[[DELTA]T.sub.i] = instantaneous bulk (mass weighted average) nondimensionalized temperature along a cross section on the center of the vent at L/2

[[rho].sub.o] = reference density of the fluid.

[tau] = t[alpha]/[L.sup.2] = dimensionless time

[[PSI].sub.c] = [V.sub.y]/[V.sub.max] = instantaneous vertical velocity at a point located at the center of the vent i.e. at D/2 and L/2 (This velocity is normalized by the maximum vertical velocity at the same location, where the maximum velocity is the highest velocity found on the domain at any time.)

[[PSI].sub.b] = [V.sub.bmag]/[V.sub.max] = instantaneous mass weighted average velocity magnitude along a cross section located at the center of the gap--i.e., at L/2. (This velocity is normalized by the maximum velocity magnitude along the same location, where the maximum velocity magnitude is the highest velocity magnitude found on the domain at any time.)

REFERENCES

ANSYS. 2006. Fluent 6.3 Reference Manual. Canonsburg, PA: ANSYS Inc.

Bejan, A. 1984. Convection Heat Transfer. New York: John Wiley and Sons.

Brown, W. 1962. Natural convection through rectangular openings in partitions--Part 2: Horizontal partitions. International Journal of Heat and Mass Transfer 5:869-81.

Brown, W., and K. Salvason. 1962. Natural convection through rectangular openings in partitions--Part 1: Vertical partitions. International Journal of Heat and Mass Transfer 5:859-68.

Catton, I. 1970. Convection in closed rectangular region: The onset of motion. Journal of Heat Transfer 92:186-88.

Catton, I., and D.K. Edwards. 1967. Effect of side walls on natural convection between horizontal plates heated from below. Journal of Heat Transfer 90:295-99.

Chandrasekhar, S. 1961. Hydrodynamic and Hydromagnetic Stability. London: Oxford Clarendon Press.

Conover, T.A., and R. Kumar. 1993. LDV study of buoyant exchange flow through a vertical tube. Proceedings of 5th International Conference on Laser Anemometry, Advances and Applications, Koningshof, Veldohven, The Netherlands.

Conover, T. A., R. Kumar, and J.S. Kapat. 1995. Buoyant pulsating exchange flow through a vent. Journal of Heat Transfer 117(33):641-48.

de Vahl Davis, G. 1983. Natural convection of air in a square cavity: A bench mark numerical solution. International Journal of Numerical Methods in Fluids 3:249-64.

Epstein, M., 1988. Buoyancy driven exchange flow through small openings in a horizontal partitions. Journal of Heat Transfer 110:885-93.

Gardner, G. 1977. Motion of miscible and immiscible fluids in closed horizontal and vertical ducts. International Journal of Multiphase Flow 3:305-18.

Jaluria, Y., Natural Convection Heat and Mass Transfer. Oxford: Pergamon Press.

Jaluria, Y., S.H.-K. Lee, G.P. Mercier, and Q. Tan. 1993. Visualization of transport across a horizontal vent due to density and pressure difference. Proceedings of the 1993 National Heat Transfer Conference, New York, NY, pp. 1-17.

Leach, S., and H. Thompson. 1975. An investigation of some aspects of flow into gas cooled nuclear reactors following and accidental depressurization. Journal of the British Nuclear Energy Society 14:243-50.

Mercer, A., and H. Thompson. 1975. An experimental investigation of some further aspects of the buoyancy-driven exchange flow between carbon dioxide and air following a depressurization accident in a Magnox reactor, Part 1: The Exchange Flow in Inclined Ducts. Journal of the British Nuclear Energy Society 14:327-34.

Mitchell, W.T., and J.A. Quinn. 1966. Thermal convection in a completely confined fluid layer. AIChE Journal 12:1116-124.

Myrum, T. 1990. Natural convection from a heat source in a top-vented enclosure. Journal of Heat Transfer 112:632-39.

Patankar, S.V. 1980. Numerical Heat Transfer and Fluid Flow. Washington: Hemisphere Publishing Corp.

Patterson, J., and J. Imberger. 1980. Unsteady natural convection in a rectangular cavity. Journal of Fluid Mechanics 100:65-86.

Prahl, J., and H.W. Emmons. 1975. Fire induced flow through an opening. Combustion and Flame 25:369-85.

Singhal, M., and R. Kumar. 1995. Unsteady buoyant exchange flow through a horizontal partition. Journal of Heat Transfer 117:515-20.

Sleiti, A.K. 2008. Effect of vent aspect ratio on unsteady laminar buoyant flow through rectangular vents in large enclosures. International Journal of Heat and Mass Transfer 51:4850-861.

Steckler, K., H. Baum, and J. Quintiere. 1984. Fire induced flows through room openings-flow coefficients. Proceedings of the Twentieth Symposium on Combustion, Ann Arbor, Michigan, pp. 1591-1600.

Steckler, K., H. Baum, and J. Quintiere. 1986. Twenty first Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, PA.

Tan, Q. and Y. Jaluria. 1992. Flow through a horizontal vent in an enclosure fire. Fire and Combustion Systems, ASME HTD 199:115-22.

Yih, C.S. 1959. Thermal instability of viscous fluids. Quarterly of Applied Mathematics 17(1):25-42.

Ahmad K. Sleiti, PhD

Member ASHRAE

Received December 5, 2008; accepted August 19, 2009

Ahmad K. Sleiti is an assistant professor of mechanical engineering technology, Department of Engineering Technology, The William States Lee College of Engineering, University of North Carolina Charlotte, Charlotte, NC.

The problem of natural convection heat and mass transfer has been studied extensively by many authors (Juluria 1980). The problem of natural convection in enclosures with the top plate colder than the bottom plate has been treated extensively in the literature as well (Sleiti 2008). However, the problem of heavier fluid on top of lighter fluid separated by a narrow vent has not been dealt with analytically or numerically. Such a flow configuration occurs in buildings where elevator shafts, stairwells, service shafts, etc. can act as vents connecting two floors. The flow in these vents is buoyancy driven, due to a fire in the bottom enclosure, and transient in nature. This transient flow may have several modes, and the heat transfer mechanism is of interest, especially when the flow becomes oscillatory.

Flow through apertures connecting two enclosures has been a subject of study for more than four decades. Early studies were limited to openings in vertical partitions. The fundamental difference between flow through a vent in a vertical partition and flow through a horizontal partition is the stable stratification of fluid in the former case and the instability of fluid in the latter case. In spite of this fundamental difference, these studies offer unique insight into flow mechanisms through apertures and different methodologies available to model them.

Prahl and Emmons (1975) conducted an experimental and theoretical study of fire-induced flow through an opening in a vertical partition. The experiments involved steady flows through a single opening with a reduced-scale kerosene/water analog. Inflow and outflow orifice coefficients were determined. These were found to be significantly different at low values of Reynolds number, based on flow height, but reached an asymptotic value of approximately 0.68 at large values of Reynolds number.

In the same year, Leach and Thompson (1975) carried out an experimental investigation of flows in horizontal circular tubes. For the whole range of 0.5 [less than or equal to] L/D [less than or equal to] 9.4 and 3 x [10.sup.4] [less than or equal to] Re [less than or equal to] 1.5 x [10.sup.5] investigated, they found that [C.sub.D] = 0.09, a constant independent of L/D and [Re.sub.[DELTA]]. A large-scale experiment with carbon dioxide and water as working fluids was used to verify the above results for gases. Leach and Thompson also investigated the forced flow rate requirements to prevent the counter current flow in the tube.

Brown (1962) was first to study both analytically and experimentally the flow through square openings in a horizontal partition with heavier fluid above the partition. In his analysis, he assumed that all heat transfer is due to advection only, and no mixing occurs within the vent. Assuming friction to be negligible, he invoked Bernoulli's equation to predict that the Nusselt number (Nu) is proportional to the product of the square root of the Grashof number (Gr) and the Prandtl number (Pr), where the reference length is the height of the vent. Brown and Salvason (1962) conducted a similar analysis for a vent in a vertical partition to derive a theoretical relationship between Nu and Gr.

Epstein (1988) reported an experimental study of buoyancy-driven exchange flow through small openings in horizontal partitions for the same geometric configuration as that in this study. The experimental apparatus consisted of two enclosures, one on top of the other, with brine and fresh water as working fluids. The study concentrated on finding the effect of height-to-diameter aspect ratio of the vent (L/D) on the dimensionless exchange flow rate as defined by Mercer and Thompson (1975) in their study of inclined ducts. Four different flow regimes were identified as L/D and varied from 0.01 to 10. The first of these was named oscillatory exchange flow regime, where plumes of fluids periodically broke through the opening. Epstein (1988) used Taylor's wave theory to predict the motion of the interface and showed that the Froude number (Fr) is a constant for this regime. The second flow regime was called the Bernoulli flow regime due to the fact that data showed the same trends as those predicted by Brown (1962). For this regime, Epstein (1988) found that Froude number Q = 0.23[(L/D).sup.1/2] (length/width of the vent). For large L/D, the flow rate was observed to be smaller than that of the other two regimes due to violent mixing within the vent. He used an analysis similar to that reported by Gardner (1977) to provide correlations for his regime. An intermediate regime was also identified as having combined characteristics of turbulent diffusion and Bernoulli flow.

Brine-water analog was also studied experimentally by Conover and Kumar (1993), and Conover et al. (1995) studied experimentally the buoyant countercurrent exchange flow through a vented horizontal partition using a two-component laser doppler velocimeter. Detailed measurements of velocity were made at a small distance above a circular tube with an aspect ratio of one. Even for this aspect ratio and a Reynolds number ranging between 2400 and 7700, the mixing in the vent was found to be turbulent and unsteady. Conover and Kumar (1993) also found that the flow coefficient was nearly constant for a Reynolds number as low as 2400. This finding was in contrast with the work of Stecker et al. (1984, 1986) who showed that exchange flows reached self similarity only for Reynolds numbers greater than 10,000. Tan and Jaluria (1992) and Jaluria et al. (1993) studied the cases where the vent flow was governed by both pressure and density differences across the vent.

Myrum (1990) conducted heat transfer experiments using water in a top-vented enclosure heated by a disk on the enclosure floor. He observed four modes of flow, which were unstable and oscillated randomly from one to the next. More on this flow configuration is given in the "Results and Discussion" section.

A numerical study of unsteady buoyant flow through a horizontal vent placed slightly asymmetrically between two enclosed vents was performed by Singhal and Kumar (1995). They observed and described several flow regimes.

In all studies discussed above, only high Rayleigh number (Ra) ranges were investigated so that the effect of viscosity could be neglected, and, in general, flow coefficient or Froude number was considered a function of L/D only, independent of the driving potential. Here, Ra number is defined as the ratio of buoyancy forces and (the product of) thermal and momentum diffusivities and is equal to [g[beta][DELTA]TL.sup.3]/[nu][alpha]. Also, the scaled brine/fresh water models used may not give comparable results to full-scale air models due to large changes in the Pr number and the Schmidt number values at low Ra. Therefore, these models may not be applicable to the range of Ra investigated in this study.

The objective of this study is to identify the different flow regimes encountered for small-to-high Ra numbers in a vented enclosure and to discuss the physical mechanisms using the time trace of temperature and fluid flow results for a range of vent aspect ratios. Understanding the different flow regimes in such vented enclosures is a key to understanding airflow design in different designs, including for tall buildings.

Governing Equations and Formulation

The time-dependent governing equations for the laminar, two-dimensional flow were derived from fundamental laws of conservation of mass, momentum, and energy as follows:

[[[partial derivative][rho]]/[[partial derivative]t]] + [[[partial derivative]([rho]u)]/[[partial derivative]x]] + [[[partial derivative]([rho]v)]/[[partial derivative]y]] = 0 (1)

[rho]([[partial derivative]u]/[[partial derivative]t] + u[[[partial derivative]u]/[[partial derivative]x]] + v[[partial derivative]u]/[[partial derivative]y]) = - [[[partial derivative]p]/[[partial derivative]x]] + [mu]([partial derivative]/[[partial derivative]x](2[[[partial derivative]u]/[[partial derivative]x]] - [2/3]([nabla]*[[vector].v])) + [[partial derivative]/[[partial derivative]y]]([[partial derivative]u]/[[partial derivative]y] + [[partial derivative]v]/[[partial derivative]x])) (2)

[rho]([[partial derivative]v]/[[partial derivative]t] + u[[[partial derivative]v]/[[partial derivative]x]] + v[[partial derivative]v]/[[partial derivative]y]) = - [[[partial derivative]p]/[[partial derivative]y]] + [mu]([partial derivative]/[[partial derivative]y](2[[[partial derivative]v]/[[partial derivative]y]] - [2/3]([nabla]*[[vector].v])) + [[partial derivative]/[[partial derivative]y]]([[partial derivative]u]/[[partial derivative]y] + [[partial derivative]v]/[[partial derivative]x])) + [rho]g (3)

[[[partial derivative]T]/[[partial derivative]t]] + u[[[partial derivative]T]/[[partial derivative]x]] + v[[[partial derivative]T]/[[partial derivative]y]] = [alpha]([[[partial derivative].sup.2]T]/[[partial derivative][x.sup.2]] + [[[partial derivative].sup.2]T]/[[partial derivative][y.sup.2]]) + [[[beta]T]/[[rho][C.sub.p]]]([[partial derivative]p]/[[partial derivative]t] + u[[[partial derivative]p]/[[partial derivative]x]] + v[[partial derivative]p]/[[partial derivative]y]) (4)

L and [alpha]/L are used to normalize length and velocity, respectively; [alpha]/[L.sup.2] is used to nondimensionalize time; and [[DELTA]T.sub.i] is used to nondimensionalize temperature.

Except for the buoyancy term in the momentum equation, the transient calculation approach considered in this study doesn't use the Boussinesq model that treats density as a constant value in all solved equations. Instead, in the approach used here, the initial density is computed from the initial pressure and temperature, so the initial mass is known. As the solution progresses over time, this mass is properly conserved. When this approach is used, the operating density [[rho].sub.o] appears in the body-force term in the momentum equation as ([rho] - [[rho].sub.o])g. The definition of the operating density [[rho].sub.o] is thus important for this buoyancy-driven flow.

No-slip conditions were used for velocity boundary conditions at all walls, including the vent walls. Initially, fluid is at rest everywhere in the domain; therefore, velocity components were set equal to zero at nondimentionalized time [tau] = 0. All the walls were treated adiabatically and, hence, the normal gradients of temperature were set equal to zero at all fluid-wall interfaces. At [tau] = 0, the lower chamber contains hot fluid with [theta] = 1, while the upper chamber and the vent contain cold fluid with [theta] = 0. The flow parameters of interest are Ra and Pr. Pr is maintained constant at 0.7. Referring to Figure 1, the geometric parameters are vent aspect ratio, L/D, H1/L, and the enclosure aspect ratio H1/H2. In this study, the investigated parameters are fixed at L/D of 1, 0.5, and 2; H1/L of 10, 20, and 5; and H1/H2 of 1, 2, and 0.5, as shown in Table 1.

[FIGURE 1 OMITTED]

Table 1. Investigated Parameters L/D H1/L H1/H2 Case 1 1 10 1 Case 2 0.5 20 2 Case 3 2 5 0.5

Although the real problem of natural convection in a horizontal cavity with bottom heating is three dimensional, it can be treated as two-dimensional problem for very long vents, which is the case considered in this paper.

It is important to state that the building air conditioning and airflow organization usually deal with high a Ra problem (turbulent natural convection), which is very difficult to be simulated numerically.

Numerical Procedure

Equations 1-4 are discretized over the computational domain using a control volume approach, as documented by Patankar (1980). The resulting algebraic equations for velocity components and temperature were solved using the SIMPLE algorithm, which involves the use of pressure correction equations for enforcing mass conservation. FLUENT (ANSYS 2006) is used for this simulation. A segregated solver with first-order implicit unsteady formulation is employed for this study. The air properties (i.e., density, thermal conductivity, specific heat, and dynamic viscosity) are varied as piecewise functions of temperature. By default, the solver will compute the operating density [[rho].sub.o], which appears in the body-force term in the momentum equations as ([rho] - [[rho].sub.o])g, by averaging over all cells. The discretization schemes used are second order for pressure, SIMPLE for pressure-velocity coupling and second order upwind for momentum and energy. The under-relaxation factor is 0.3 for pressure; 1 for density, body forces, and energy; and 0.7 for momentum. The code was tested against certain benchmark solutions to validate the results. The results obtained by Patterson and Imberger (1980) for the unsteady case, and the results obtained by de Vahl Davis (1983) for the steady case, were selected for comparison with the results predicted. Both of these papers deal with enclosure geometries with heated and cooled vertical walls as well as adiabatic bottom and top walls. The predicted results were within 3% of these benchmark solutions.

Figure 2a shows the numerical grid generated for this study using a Gambit (ANSYS 2006) grid generator. Since FLUENT is an unstructured solver, it uses internal data structures to assign an order to the cells, faces, and grid points in a mesh and to maintain contact between adjacent cells. It does not, therefore, require i,j,k indexing to locate neighboring cells. This gives the flexibility of using the grid topology that is best for the problem, since the solver does not force an overall structure or topology on the grid. For the 2D problem considered here, quadrilateral cells are used to form multiblock structured mesh. More details on grid generation are provided in FLUENT documentation (ANSYS 2006). A grid-independent study is performed using three nonuniform grid distributions: coarse grid with 91 x 63 grid points used for every enclosure and 21 x 13 grid points used for the vent, medium grid with 133 x 91 grid points used for every enclosure and 31 x 19 grid points used for the vent, and fine grid with 201 x 135 grid points used for every enclosure and 47 x 29 grid points used for the vent. The results for temperature and the two components of velocity at the center of the vent and at different locations inside the two enclosures (see Figure 2a) for different Ra values were compared using all three grids. The maximum difference in temperature and velocities between coarse and medium grids was found to be less than 1.5% when results from both grids were compared over the full time domain. The maximum difference between medium and fine grids was found to be less than 0.2%. Finally, the nonuniform grid of 133 x 91 grid points was used for every enclosure, and a grid of 31 x 19 grid points was selected for the vent with grid points placed near every sidewall inside the boundary layer. One complete unsteady solution for one geometry and average Ra number involved computation of up to 70,000 time steps, with approximately 30 to 40 iterations for each time step; although, for more confidence in results, a convergence usually is achieved after 10 to 15 iterations.

[FIGURE 2 OMITTED]

Convergence criteria were set to ensure converged results for continuity, x-velocity, y-velocity, and energy. A solution was assumed converged if the maximum scaled residuals of the continuity, x-velocity, y-velocity, and energy equal [10.sup.-4], [10.sup.-5], [10.sup.-6], and [10.sup.-7], respectively. The residual is scaled using a scaling factor representative of the flow rate of [phis] through the domain. This scaled residual is defined as

[R.sup.[phis]] = [[[[SIGMA].sub.cellsP]|[[SIGMA].sub.nb][a.sub.nb][[phis].sub.nb] + b - [a.sub.P][[phis].sub.P]|]/[[[SIGMA].sub.cellsP]|[a.sub.P][[phis].sub.P]|]]. (5)

For the momentum equations, the denominator term [a.sub.P][[phis].sub.P] is replaced by [a.sub.P][v.sub.P], where [v.sub.P] is the magnitude of the velocity at cell P.

The scaled residual is a more appropriate indicator of convergence. For the continuity equation, the solver's scaled residual is defined as

[R.sub.[iteration].sup.c]N/[R.sub.[iteration].sup.c]5, (6)

where [R.sub.iteration.sup.c]N is the residual of the n-th iteration, and [R.sub.iteration.sup.c]5 is the largest absolute value of the continuity residual in the first five iterations.

For the enclosure under consideration, a fixed time-stepping method is used for a time-marching scheme. The time step used was based on estimating the time constant as

[t.sub.c] = [L/U][approximately equal to][[L.sup.2]/[alpha]][(PrRa).sup.[-1/2]] = [L/[square root of [g[beta][DELTA][T.sub.i]L]]],

where L and U are the length and the velocity scales, respectively (Bejan 1984).

Then the time step is determined such that [DELTA]t = [t.sub.c]/20. Note that this time-step value used in this study is five times less than the recommended time step in (ANSYS 2006) to ensure time convergence. After oscillations with a typical frequency have decayed, the solution reaches steady state.

The time-step independence was tested prior to obtaining solution for the desired problem. The time-independent study is established by tracking the changes in temperature and the two components of velocity at a point located at the center of the vent by increasing and decreasing the time step for all cases of Ra studied. Decreasing the time step from [DELTA]t = [t.sub.c]/20 to [DELTA]t = [t.sub.c]/25 resulted in less than 0.05% difference in temperature and velocities when results from both time steps are compared over the full time domain. Increasing the time step from [DELTA]t = [t.sub.c]/20 to [DELTA]t = [t.sub.c]/15 resulted in less than 0.1% difference in temperature and velocities when results from both time steps are compared over the full time domain. Based on this, the time step used for all simulations is [DELTA]t = [t.sub.c]/20. Figure 2b shows sample convergence plots of the scaled residual for Ra of 2500 at different time steps.

Quantities of Interest

The Ra number was defined as Ra = (g[beta][[DELTA]T.sub.i][L.sup.3]/[nu][alpha]), where [[DELTA]T.sub.i] is the initial temperature difference between the two enclosures (i.e., at [tau] = 0). As the interaction between the fluids in the two enclosures proceeds with time, the effective driving potential is the temperature difference across the vent and not [[DELTA]T.sub.i]. The magnitude of the temperature difference across the vent decreases with time. Therefore, new quantities need to be defined that reflect this change in driving potential with time. To this end, we define the following quantities:

1. Instantaneous averaged nondimensionalized temperature ([[theta].sub.av]) on left- and right-side walls of the vent are defined as [[theta].sub.av] = ([T(t).sub.av] - [T.sub.c])/[[DELTA]T.sub.i], where [T(t).sub.av] is the average temperature as a function of time, [T.sub.c] is the initial temperature in the upper enclosure, and [[DELTA]T.sub.i] is the initial temperature difference between the upper and lower enclosures. Thus ([[theta].sub.av]) varies from 0.0 to 1.0, where close to 0.0 values indicate no difference in averaged temperature between the two enclosures.

2. Instantaneous bulk (mass weighted average) nondimensionalized temperature ([[theta].sub.b]) along a cross section on the center of the vent at L/2 is defined as: [[theta].sub.b] = ([T(t).sub.b] - [T.sub.c])/[[DELTA]T.sub.i]. Thus, ([[theta].sub.b]) varies from 0.0 to 1.0, where close to 0.0 values indicate no difference in bulk temperature between the two enclosures.

3. Instantaneous vertical velocity ([[PSI].sub.c]) at a point located at the center of the gap (i.e., at D/2 and L/2.) is normalized by the maximum vertical velocity at the same location ([[PSI].sub.c] =[V.sub.y]/[V.sub.max]), where the maximum velocity is the highest velocity found on the domain at any time. Thus, ([[PSI].sub.c]) varies from -1.0 to 1.0. Values of ([[PSI].sub.c]) close to 0.0 mean the flow field is stabilizing.

4. Instantaneous mass weighted average velocity magnitude ([[PSI].sub.b]) along a cross section located at the center of the gap (i.e., at L/2) is normalized by the maximum velocity magnitude along the same location ([[PSI].sub.b] =[V.sub.bmag]/[V.sub.max]), where the maximum velocity magnitude is the highest velocity magnitude found on the domain at any time. Thus ([[PSI].sub.b]) varies from 0.0 to 1.0. Values close to zero mean the flow field is stabilizing.

All quantities described above are then calculated at every time step throughout the simulation.

Results and Discussion

The interaction between the two enclosures takes place through the vent, and the flow patterns within the vent determine the mode of heat transfer and the rate of heat and mass transfer across it. These flow patterns in the vent depend on the vent geometry, the magnitude of the buoyancy force that drives the flow across it, and, to a great extent, the flow patterns in enclosures themselves. The last category mentioned above makes the flow configuration difficult to analyze. However, from the results presented, an attempt has been made to explain the localized phenomena. Based on the mode of heat transfer and the associated flow characteristics, three categories have been identified: the conduction regime, the countercurrent regime, and the oscillatory regime. The Ra number largely governs these regimes, as explained below.

The Conduction Regime (Ra [less than or equal to] 1500). For small Ra numbers, the buoyancy forces are not strong enough to overcome the viscous forces in the vent where the vertical walls are closer to each other as compared to the enclosure's vertical walls. Consequently, there is no bulk fluid motion within the vent, as shown in Figure 3, where the instantaneous vertical velocity ([[PSI].sub.c]) is almost constant, and all the heat transfer is solely by conduction for Ra [less than or equal to] 1500.

[FIGURE 3 OMITTED]

Flow patterns for Ra = 1000 at different time steps are given in Figure 4. In this regime, the vent acts like a heater plate for the upper chamber, which is filled with colder fluid, and as a cold plate for the lower chamber, which is filled with warmer fluid. Thus, in the enclosures, a bulk fluid motion ensues immediately after the initial establishment of the temperature gradient in the vent. A plume rises steadily into the upper chamber, forcing the colder fluid to move along the side walls to replace the rising fluid and forming two cells of equal strength on either side of it. For a considerable length of time, the cells stay symmetric in each chamber. After that, the cells in the upper chamber become asymmetric before merging into a single cell. There is a time delay before the lower chamber behaves exactly like the upper one, exhibiting a single cell. At large [tau], the fluid in the chambers is accelerated enough to drive the fluid in the vent by entrainment/shear.

[FIGURE 4 OMITTED]

This singular phenomenon of no flow at low Ra numbers is analogous to the Benard convection problem, with free boundaries at the top and bottom. At [tau] = 0, fluid is at rest everywhere, and the vent marks the region of separation of large bodies of hot and cold fluid. With respect to the vent, the denser fluid at the top has a natural tendency to exchange places with the lighter fluid at the bottom. However, this exchange is inhibited by its own viscosity, and, for the flow to ensue, the adverse temperature gradient must exceed a certain value. The onset of instability beyond a critical value of the Ra number was thoroughly analyzed using techniques such as linear stability theory and the power integral method. Chandrasekhar (1962) presented a comprehensive treatment of the linear stability theory where he showed mathematically the thermodynamic significance of [Ra.sub.critical]. In his words, "Instability occurs at the minimum temperature gradient at which a balance can be steadily maintained between the kinetic energy dissipated by viscosity and the internal energy released by the buoyancy force." [Ra.sub.critical] corresponds to the above-mentioned minimum temperature gradient in conjunction with the dynamic conditions at the two bounding planes. For the case of rigid constant temperature surfaces, [Ra.sub.critical] was calculated to be 1708; for one rigid and one free surface it is 1100; and for both free surfaces, [Ra.sub.critical] is found to be 658. The same line of reasoning can be applied to the flow in the vent. This is justified if one takes a closer look at Figure 5, which shows a magnified view of velocity vectors in the vent. The magnitude of velocity in the vent is extremely small compared to those in the enclosures. In the middle of the vent, velocities are negligible, which suggests that the fluid movement at the top and bottom edges of the vent is due to bulk fluid motion in the enclosures. Two planes of symmetry divide the vent into four similar parts. Few complications arise in this problem, which makes stability analysis difficult. One is the effect of side walls, and the other is the influence of bulk fluid motion in upper and lower chambers, where the later is more complex than the former. To date, several researchers have addressed the issue of the effect of side walls on [Ra.sub.critical]. The first complete analysis was given by Yih (1959) for the stability of a viscous fluid between insulated vertical plates. Davis (1983) was the first to consider the fully confined fluid; he used the Galerkin method for his analysis. Later, Catton (1970) and Catton and Edwards (1967) improved on Davis' results and produced a chart for [Ra.sub.critical] as a function of H1 and H2 as parameters, where H1 and H2 are spanwise dimension and depth, respectively. Both studies considered the side walls to be perfectly conducting walls. Latter, Catton and Edwards (1967) performed an experimental study on the effects of side walls on natural convection. They were able to obtain [Ra.sub.critical] as a function of height-to-width ratio for insulating as well as conducting lateral walls. For an aspect ratio of 1.0 and insulating lateral walls, they reported that [Ra.sub.critical] was in the range of 1 x [10.sup.4] and 2 x [10.sup.4]. They also developed a heat transfer correlation.

[FIGURE 5 OMITTED]

No prior work exists about the influence of bulk fluid motion in enclosures on the vent. Figures 6 and 7 give the change in [[theta].sub.av] on right and left walls of the vent, respectively, as a function of time. Instantaneous bulk (mass weighted average) nondimensionalized temperature ([[theta].sub.b]) along a cross section on the center of the vent at L/2 is given by Figure 8, and the instantaneous mass weighted average velocity magnitude ([[PSI].sub.b]) is shown in Figure 9. Curves for Ra of 1200 and 1300 shown in the above figures demonstrate a limiting value for Ra for different regimes.

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

For very small values of [tau], a sharp gradient appears in ([[PSI].sub.b]) because of the initial conditions imposed in the enclosures. However, the flow adjusts quickly to decrease this gradient as ([[theta].sub.b]) steadily decreases with time. Since all the heat transfer is by conduction only, the average wall temperature [[theta].sub.av] has a constant value of about 0.5 for all values of [tau].

In the conduction regime, there is no flow across the vent, as the viscous forces are as large as the buoyancy forces. This is analogous to the Benard convection problem, where the Ra number must exceed a critical value before the flow can start, as explained analytically by Chandrasekhar (1961) using linear stability theory. However, no closed-form solution is available for the configuration under investigation to determine the exact value of critical Ra.

The Countercurrent Regime (1600 [less than or equal to] Ra < 3600). This regime is characterized by a down flow along approximately one half of the vent and upflow along the rest to satisfy continuity, as shown in Figure 10.

[FIGURE 10 OMITTED]

One essential difference in the flow patterns between Ra = 1000 and Ra = 2500 (Figures 4 and 10) is that the cells in the top chamber for Ra = 2500 become asymmetric at early dimensionless time. This asymmetry arises at [tau] = 5.0 during the onset of the countercurrent flow at the vent. A closer look at the vent (Figure 11) clearly shows the upflow on the right side and downflow on the left side of the vent at [tau] = 2.2. With time, the flow gradually reverts itself to the conduction regime, with one large cell in each chamber, as seen for Ra = 1000.

[FIGURE 11 OMITTED]

As Ra number increases in this flow regime, the flow patterns present the most interesting phenomenon. In Figure 10 at [tau] = 2.2, the flow patterns are strictly countercurrent in the vent; however, at [tau] = 8, the flow everywhere in the enclosure slows down. This is evident from the magnified view of the vent in Figure 11. Following this deceleration, a flow reversal occurs at the vent at approximate [tau] value of 11 and lasts as seen at [tau] = 19, and the flow accelerates again as evident by the magnitude of velocity vectors. This is the only flow reversal that occurs in this flow regime.

Such a phenomenon was also seen by Myrum (1990). He conducted an experimental study to find the effect of vent size and Ra on natural convection heat transfer from a heated disk. This disk was located at the bottom of a top-vented enclosure. During the experiments, he also studied the flow patterns in the vent and reported four basic modes. In mode I, flow exited the vent along its axis and entered around its circular perimeter. In mode II, regions of outflow and inflow formed concentric rings within the vent. In mode III, inflow occurred through one half of the vent along the perimeter and the corresponding outflow occurred through the rest. A nonperiodic shift in sides of inflow and outflow was also observed.

Although a direct comparison cannot be made between the current study and Myrum's (1990), the time history of heat transfer along with numerical flow visualization reveal mode III. At the time of this change in direction near [tau] = 8, as discussed above, the exchange flow rate decreases drastically with corresponding decrease in heat transfer across the vent, as seen in Figures 8 and 9. This change, therefore, appears in [[theta].sub.b] and [[PSI].sub.b] time history as a sudden decrease. A sharp decrease and increase in the rate of change of instantaneous [[theta].sub.av] on the right and left walls of the vent, respectively, occur due to the flow reversal, as shown in Figures 6 and 7. As soon as the change in sides is complete, the average and bulk temperatures and velocities resume at almost the same values as before the change. At large values of [tau], flow in the vent gradually dies out due to lack of driving potential, namely the bulk-mean temperature difference between the two enclosures. As this happens, the plumes are no longer strong enough to sustain two cells on either side, and gradually the two cells merge to form a single cell. Since at this point in time there is no flow across the vent, the heat transfer is by conduction only. Hence, the flow configuration is exactly as seen in Figure 4 for large [tau]. Figures 3, 6, and 7 confirm this observation, where [[PSI].sub.c] and [[theta].sub.av] are the same as for the conduction regime.

The Oscillatory Flow Regime (Ra [greater than or equal to] 3600). This regime is characterized by sudden bursts of both upflow and downflow, which show up as a periodic-like response in [[PSI].sub.c], [[PSI].sub.b], and [[theta].sub.av], [[theta].sub.b] time history for large Ra numbers only. This type of flow originates at nearly Ra = 3600.

The oscillatory flow with a single cell in the vent is dominant at high Ra numbers of 5000, 10,000, and 50,000. The flow patterns are qualitatively similar and, hence, the flow and temperature time traces for Ra = 10,000 and the flow patterns for Ra = 50,000 in the enclosures and the magnified vent were chosen for discussion and are given in Figures 12 and 13. Unlike the other two regimes, the flow in the bottom chamber reaches a unicellular pattern earlier than the top enclosure. A close look at the flow patterns within the vent for Ra = 50,000, as given in Figure 13, reveals that the cell that develops in the vent persists for a longer period before the flow patterns undergo a flow reversal as seen before. Temperature and velocity plots for Ra = 10,000 at different times (Figures 3, 6, 7, 8, and 9) give a unique perspective of occurrences in the vent. Temperature and velocity inversions occur at different times, corroborating the fact that the flow oscillates with a well-defined frequency in the entire domain.

[FIGURE 12 OMITTED]

[FIGURE 13 OMITTED]

It is suspected that these bursts are periodic in nature and the frequency of bursts is a function of driving potential. Time traces of [[PSI].sub.c], [[PSI].sub.b], and [[theta].sub.av], [[theta].sub.b] for Ra = 10,000 (Figures 3, 6, 7, 8, and 9) suggest that there are varying amplitude oscillations in a narrow frequency range that eventually die out and essentially reproduce the flow behavior quantified in the countercurrent and conduction flow regimes at large nondimensional time. The Nu number trace was observed by Mitchell and Quinn (1966) in a confined layer heated from below. They found that, as the plate temperature was increased, the fluid oscillated in a narrow frequency band. They also noted that the oscillations were stable over a large Ra number range. Once again, a direct comparison of our results with those of Mitchell and Quinn (1966) cannot be made, since their mean flow was steady.

Based on the oscillatory flow regime results above, it is apparent that, for the building air conditioning and airflow organization, we usually deal with a high Ra problem (turbulent natural convection), which is very difficult to simulate numerically.

Case of L/D = 0.5. Figures 14 and 15 show overall velocity field and magnified view of the velocity vectors inside the vent for L/D = 0.5 at Ra = 10,000. At low [tau] = 0.8, the heavier and slower flow from the top enclosure tends to penetrate through the vent into the lower enclosure. The resistance from the lighter hot fluid in the lower enclosure is increased and finally inversed at [tau] = 4.3. At this time level, the highest velocities are found in the upper enclosure. At [tau] = 10.73, the fluid of the upper enclosure is trying to penetrate to the lower enclosure, where, at this time level, the highest velocities are found in lower enclosure. It is found that the flow at this L/D = 0.5 does not show signs of the oscillatory flow regime observed for L/D = 1, where a large vortex formed in the vent. Instead, the flow oscillates with higher frequency between the two enclosures until it reaches the conducting flow regime at high time levels of [tau] > 50. The full symmetry of the flow in the upper and lower enclosures is reached at time levels of around [tau] = 63.

[FIGURE 14 OMITTED]

[FIGURE 15 OMITTED]

Case of L/D = 2. Figures 16 and 17 show overall velocity field and magnified view of the velocity vectors inside the vent for L/D = 2 at Ra = 10,000. The flow for this case is solely conducting at all time levels. At low [tau] = 0.135, the heavier cold fluid from the top enclosure tends to penetrate through the vent towards the lower enclosure, but the high resistance from the lighter hot fluid in the lower enclosure prevents its penetration. The first mode of symmetry of the flow in the two enclosures is reached at low [tau] = 0.5. This symmetry is characterized by a large vortex that occupies the whole enclosure. At [tau] = 0.2 (not shown), a second vortex starts to form on the upper half of the upper enclosures as a result of the motion of the flow, which comes back from the vent region. This flow, too weak to travel all the way to the upper wall of the upper enclosure, turns back at the location above the center of the enclosure, causing two counter current vortices of different size and velocity. At this time level, no second vortex is found in the lower enclosure. This second vortex in the lower enclosure starts to appear only at [tau] = 4 (not shown) and is completely shaped at [tau] = 10 (not shown). The flow in both enclosures at all higher time levels is characterized by those four vortices, while the flow exchange in the vent is solely by conduction.

[FIGURE 16 OMITTED]

[FIGURE 17 OMITTED]

CONCLUSION

A numerical study of unsteady, laminar, buoyant flow through a horizontal rectangular vent between two large enclosures was performed. Based on the flow patterns in the vent and the corresponding modes of heat transfer through it, three flow regimes were identified: the conduction regime (Ra [less than or equal to] 1500), the countercurrent flow regime (1600 [less than or equal to] Ra < 3600), and the oscillatory flow regime (Ra [greater than or equal to] 3600). In the conduction regime, there is no flow across the vent, as the viscous forces are as large as the buoyancy forces.

In the countercurrent flow regime, the fluid moves through the vent in two distinct streams, with one going up and the other going down. A flow reversal takes place in the vent soon after flow initiation, which is associated with a flow deceleration and a significant decrease and increase in the rate of change of instantaneous [[theta].sub.av] on the right and left walls of the vent, respectively. As soon as the change in sides is complete, the average and bulk temperatures and velocities resume at almost the same values as before the change.

The third regime is characterized by sudden bursts of upflow and corresponding downflow with a well-defined frequency range. Temperature and velocity results associated with such bursts display interesting characteristics of increasing amplitudes but nearly the same frequency.

Decreasing partition thickness from L/D = 1 to 0.5 increases the flow exchange between the upper and lower enclosures with increasing frequency of the oscillatory flow pattern. Increasing partition thickness from L/D = 1 to 2 decreases the flow exchange between the two enclosures, and flow regime becomes conduction with low frequency.

NOMENCLATURE

D = width of the vent

Gr = Grashof number (A dimensionless number approximates the ratio of the buoyancy to viscous force acting on a fluid.)

g = acceleration due to gravity

H1 = width of the enclosure

H2 = height of the enclosure

L = height of the vent

L/D = vent aspect ratio

Nu = Nusselt number (This parameter provides a measure of the convection heat transfer occurring at the surface.)

p = total pressure less the hydrostatic pressure

P = dimensionless pressure = p/([rho][[alpha].sup.2]/[L.sup.2])

Pr = Prandtl number (Dimensionless number approximating the ratio of momentum diffusivity [kinematic viscosity] and thermal diffusivity.)

Ra = Rayleigh number = g[beta][DELTA][TL.sup.3]/[nu][alpha] (The ratio of buoyancy forces and the product of thermal and momentum diffusivities.)

[Ra.sub.critical] = critical value of Ra for onset of flow through the vent

Re = Reynolds number

[Re.sub.[DELTA]] = densimetric Reynolds number = [V.sub.[DELTA]]H/[nu]

t = time

[t.sub.c] = time constant

T = temperature

[T.sub.cold], [T.sub.c] = initial temperature of fluid in upper chamber

[T.sub.hot] = initial temperature of fluid in lower chamber

[T.sub.ref] = reference temperature

[[DELTA]T.sub.i] = initial temperature difference between upper and lower chamber

u,v = components of velocity in x and y directions, respectively

U,V = dimensionless components of velocity in x and y directions, respectively

x,y = horizontal and vertical coordinates

X,Y = dimensionless horizontal and vertical coordinates

Greek Symbols

[alpha] = thermal diffusivity of the fluid

[beta] = coefficient of thermal expansion of the fluid

[nu] = kinematic viscosity of the fluid

[theta] = dimensionless temperature = (T - [T.sub.ref])/[[DELTA]T.sub.i]

[[theta].sub.av] = (T[(t).sub.av] - [T.sub.c])/[[DELTA]T.sub.i] = instantaneous averaged nondimensionalized temperature on left- and right-side walls

[[theta].sub.b] = (T[(t).sub.b] - [T.sub.c])/[[DELTA]T.sub.i] = instantaneous bulk (mass weighted average) nondimensionalized temperature along a cross section on the center of the vent at L/2

[[rho].sub.o] = reference density of the fluid.

[tau] = t[alpha]/[L.sup.2] = dimensionless time

[[PSI].sub.c] = [V.sub.y]/[V.sub.max] = instantaneous vertical velocity at a point located at the center of the vent i.e. at D/2 and L/2 (This velocity is normalized by the maximum vertical velocity at the same location, where the maximum velocity is the highest velocity found on the domain at any time.)

[[PSI].sub.b] = [V.sub.bmag]/[V.sub.max] = instantaneous mass weighted average velocity magnitude along a cross section located at the center of the gap--i.e., at L/2. (This velocity is normalized by the maximum velocity magnitude along the same location, where the maximum velocity magnitude is the highest velocity magnitude found on the domain at any time.)

REFERENCES

ANSYS. 2006. Fluent 6.3 Reference Manual. Canonsburg, PA: ANSYS Inc.

Bejan, A. 1984. Convection Heat Transfer. New York: John Wiley and Sons.

Brown, W. 1962. Natural convection through rectangular openings in partitions--Part 2: Horizontal partitions. International Journal of Heat and Mass Transfer 5:869-81.

Brown, W., and K. Salvason. 1962. Natural convection through rectangular openings in partitions--Part 1: Vertical partitions. International Journal of Heat and Mass Transfer 5:859-68.

Catton, I. 1970. Convection in closed rectangular region: The onset of motion. Journal of Heat Transfer 92:186-88.

Catton, I., and D.K. Edwards. 1967. Effect of side walls on natural convection between horizontal plates heated from below. Journal of Heat Transfer 90:295-99.

Chandrasekhar, S. 1961. Hydrodynamic and Hydromagnetic Stability. London: Oxford Clarendon Press.

Conover, T.A., and R. Kumar. 1993. LDV study of buoyant exchange flow through a vertical tube. Proceedings of 5th International Conference on Laser Anemometry, Advances and Applications, Koningshof, Veldohven, The Netherlands.

Conover, T. A., R. Kumar, and J.S. Kapat. 1995. Buoyant pulsating exchange flow through a vent. Journal of Heat Transfer 117(33):641-48.

de Vahl Davis, G. 1983. Natural convection of air in a square cavity: A bench mark numerical solution. International Journal of Numerical Methods in Fluids 3:249-64.

Epstein, M., 1988. Buoyancy driven exchange flow through small openings in a horizontal partitions. Journal of Heat Transfer 110:885-93.

Gardner, G. 1977. Motion of miscible and immiscible fluids in closed horizontal and vertical ducts. International Journal of Multiphase Flow 3:305-18.

Jaluria, Y., Natural Convection Heat and Mass Transfer. Oxford: Pergamon Press.

Jaluria, Y., S.H.-K. Lee, G.P. Mercier, and Q. Tan. 1993. Visualization of transport across a horizontal vent due to density and pressure difference. Proceedings of the 1993 National Heat Transfer Conference, New York, NY, pp. 1-17.

Leach, S., and H. Thompson. 1975. An investigation of some aspects of flow into gas cooled nuclear reactors following and accidental depressurization. Journal of the British Nuclear Energy Society 14:243-50.

Mercer, A., and H. Thompson. 1975. An experimental investigation of some further aspects of the buoyancy-driven exchange flow between carbon dioxide and air following a depressurization accident in a Magnox reactor, Part 1: The Exchange Flow in Inclined Ducts. Journal of the British Nuclear Energy Society 14:327-34.

Mitchell, W.T., and J.A. Quinn. 1966. Thermal convection in a completely confined fluid layer. AIChE Journal 12:1116-124.

Myrum, T. 1990. Natural convection from a heat source in a top-vented enclosure. Journal of Heat Transfer 112:632-39.

Patankar, S.V. 1980. Numerical Heat Transfer and Fluid Flow. Washington: Hemisphere Publishing Corp.

Patterson, J., and J. Imberger. 1980. Unsteady natural convection in a rectangular cavity. Journal of Fluid Mechanics 100:65-86.

Prahl, J., and H.W. Emmons. 1975. Fire induced flow through an opening. Combustion and Flame 25:369-85.

Singhal, M., and R. Kumar. 1995. Unsteady buoyant exchange flow through a horizontal partition. Journal of Heat Transfer 117:515-20.

Sleiti, A.K. 2008. Effect of vent aspect ratio on unsteady laminar buoyant flow through rectangular vents in large enclosures. International Journal of Heat and Mass Transfer 51:4850-861.

Steckler, K., H. Baum, and J. Quintiere. 1984. Fire induced flows through room openings-flow coefficients. Proceedings of the Twentieth Symposium on Combustion, Ann Arbor, Michigan, pp. 1591-1600.

Steckler, K., H. Baum, and J. Quintiere. 1986. Twenty first Symposium (International) on Combustion, The Combustion Institute, Pittsburgh, PA.

Tan, Q. and Y. Jaluria. 1992. Flow through a horizontal vent in an enclosure fire. Fire and Combustion Systems, ASME HTD 199:115-22.

Yih, C.S. 1959. Thermal instability of viscous fluids. Quarterly of Applied Mathematics 17(1):25-42.

Ahmad K. Sleiti, PhD

Member ASHRAE

Received December 5, 2008; accepted August 19, 2009

Ahmad K. Sleiti is an assistant professor of mechanical engineering technology, Department of Engineering Technology, The William States Lee College of Engineering, University of North Carolina Charlotte, Charlotte, NC.

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Author: | Sleiti, Ahmad K. |
---|---|

Publication: | HVAC & R Research |

Article Type: | Report |

Geographic Code: | 1USA |

Date: | Nov 1, 2009 |

Words: | 7583 |

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