## Confined Flows

### Underlying Flow Regime 4-10

#### Abstract

Buoyant flows in simple cavities represent one further level of complexity compared with simpler unconstrained buoyant flows such as wall layers and plumes. Buoyant flows in cavities may contain both wall flows and plumes as sub-elements but it can no longer be assumed that these have no effect on their external flow. In the present case only the specific subclass of two-dimensional rectangular cavities with heated vertical walls and insulated (adiabatic) horizontal walls will be considered here as these already offer a variety of interesting phenomena. There are a wide variety of alternative buoyant cavity arrangements which have been studied both experimentally and computationally however these are beyond the scope of this review.

This type of underlying flow regime is of practical interest for example in thermal insulation, heating and ventilation of buildings, smoke spread from fires, the behaviour of solar collectors and the cooling of electronic equipment. Useful reviews of the general physics of buoyant flows in are given in standard texts and detailed reviews of buoyant cavity flows are given for example by Bejan, Gebhart et al. and Ostrach.

Figure 1 shows a schematic diagram of the general arrangement considered here, with the side walls maintained at different uniform constant temperatures, Th and Tc, and the upper and lower horizontal walls assumed to be adiabatic. The cavity has a width W and height H and the aspect ratio, H/W, is of significance in defining the various possible flow regimes. Cavities with H/W > 1 are generally termed “tall” and those with H/W < 1 “shallow”.

Figure 1. Schematic diagram of cavity arrangement

The flow is characterized by the Prandtl, Grashof and Rayleigh numbers:

 ${\displaystyle \displaystyle {Ra=GrPr={\frac {g\beta H^{3}\Delta T}{\nu ^{2}}}Pr}}$

where g is the magnitude of the gravitational acceleration, H the cavity height, ΔT the hot to cold wall temperature difference and ν the kinematic viscosity. The Grashof number represents the ratio of buoyancy to viscous forces, giving a measure of the stability of the flow regime, with low Grashof (or Rayleigh) numbers typified by stable laminar flow and high Grashof (or Rayleigh) numbers by fully turbulent flow.

A further relevant parameter is the non-dimensional temperature difference, ${\displaystyle \left.\theta \right.}$:

 ${\displaystyle \displaystyle {\theta ={\frac {\Delta T}{T_{c}+T_{k}}}}}$

which gives an indication of the size of the temperature difference relative to the mean temperature. For low values of ${\displaystyle \left.\theta \right.}$, for example ${\displaystyle \left.\theta <0.05\right.}$, the Boussinesq approximation can be considered valid for most applications (depending upon the accuracy required) whereas, as ${\displaystyle \left.\theta \right.}$ increases, the variation of density must be accounted for in all terms if an accurate solution is to be obtained.

Of particular interest is the heat transfer rate across the cavity, usually expressed as a mean Nusselt number:

 ${\displaystyle \displaystyle {{\overline {Nu}}={\frac {Q}{Q_{c}}}}}$

where Q is the total heat transfer rate and Qc the heat transfer rate due to pure conduction:

 ${\displaystyle \displaystyle {Q_{c}=k\Delta T\left({\frac {H}{W}}\right)}}$

in which k is the thermal conductivity.

Bejan defines four limiting flow regimes based on the cavity aspect ratio and Rayleigh number:

1. Conduction limit: close to linear variation of temperature across cavity with slow single-cell recirculation and mean Nusselt number close to unity, for any aspect ratio with very low Ra.
2. Tall-enclosure limit: for large aspect ratio the heat transfer is dominated by conduction and the mean Nu is again close to one.
3. High-Ra limit (boundary-layer regime): distinct thermal and hydrodynamic boundary layers are observed along the vertical walls while the core of the cavity is relatively stagnant and thermally stratified.
4. Shallow-enclosure limit: marked by heat exchange between the hot and cold horizontal streams which now have extensive contact and can act as insulation between the vertical walls.

In the following, first the case of laminar flow will be considered which provides a useful test case for evaluating numerical methods with combined fluid flow and heat transfer and for which a number of numerical benchmarks are available. In the third section turbulent flow cases are considered for which experimental data are available and one particular case is identified as a reference.

Contributors: Nicholas Waterson - Mott MacDonald Ltd