Time-dependent ventilation flows driven by opposing wind and buoyancy

Abstract

We consider transient flow in a box containing an isolated buoyancy source, ventilated by a windward high-level opening and a leeward low-level opening, so that prevailing wind acts to oppose buoyancy-driven flow. Hunt & Linden (J. Fluid Mech., vol. 527, 2005, p. 27) demonstrated that two stable steady states can exist above a critical wind strength: buoyancy-driven displacement ventilation with a two-layer stratification and wind-driven mixing ventilation with the whole interior contaminated by buoyant fluid. We present two time-dependent models for this system: a nonlinear ordinary differential equation (ODE) model following Kaye & Hunt (J. Fluid Mech., vol. 520, 2004, p. 135), assuming ‘perfect’ vertical mixing of fluid within each layer, and a partial differential equation model assuming zero vertical mixing, following Germeles (J. Fluid Mech., vol. 71, 1975, p. 601).We apply these models to an initial-value problem – the filling box with constant opposing wind. The interface between the upper hot plume fluid and the lower cool ambient air can dramatically overshoot its final level before relaxing to equilibrium; in some cases, a fully contaminated transient can occur before the buoyancy-driven two-layer steady state is reached. However, we find that for an initially completely uncontaminated box, the system converges to a stable wind-driven steady state whenever it exists. By analysing phase diagrams of the ODE model for the flow, we establish a general method of determining which final state is attained and also explain the hysteresis observed by Hunt & Linden (2005). We confirm these transient behaviours by conducting salt bath experiments in a recirculating flume tank and establish quantitative agreement between theory and experiment. Our ‘zero mixing’ model is more accurate than our ‘perfect mixing’ model for our experiments, as the upper layer remains stratified for a substantial time.