Is there convection between two volumes of air at different temperatures?

For volumes of air of different temperatures separated by a vertical opening such as a doorway, the physics is different to that of convective flow over a heated surface.

If we have two volumes of air at different temperatures separated by a vertical opening, the interface in the plane of the opening is not stable and a flow will result. The flow is driven by the difference in hydrostatic pressure between the two sides of the opening and is structured, with the warmer, lighter air occupying the top portion of the opening and the cooler, denser air occupying the bottom portion. For a rectangular opening of width $W$ and height $H$ separating rooms with air of density $rho_1$ and $rho_2>rho_1$, the flow rate $Q_{ex}$ can be shown (Epstein 1989) to be

$$ Q_{ex}=frac{C_D}{3}WH^{3/2}g'^{1/2}, $$

where

$$ g'=frac{rho_2-rho_1}{bar{rho}} $$

is the reduced gravity, $bar{rho}$ is the average density and $C_D=O(1)$ is a loss coefficient for the opening.

The analysis of Epstein (1989) actually considered the case when there is a forced flow through the door, the result above being a limiting value of no forcing. As the forcing is increased, initially there is still flow in both directions, though now the exchange volume flow against the forcing is less than that with the forcing. When the forcing reaches a critical value, sometimes called the flooding rate, there is no longer an exchange and the flow through the opening becomes unidirectional. Epstein (1989) calculated the flooding rate $Q_F$ to be

$$ Q_F=2^{3/2}Q_{ex}. $$

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