Stochastic models of ventilation driven by opposing wind and buoyancy

Stochastic versions of a classical model for natural ventilation are proposed and investigated to demonstrate the effect of random fluctuations on stability and predictability. In a stochastic context, the well-known deterministic result that ventilation driven by the competing effects of buoyancy and wind admits multiple steady states can be misleading, due to two distinct phenomena. First, with unbounded fluctuations in the buoyancy exchanged with an external environment, such systems eventually reside in the vicinity of global minima of their potential, rather than local minima associated with meta-stable equilibria. In the particular context of one heated space with a leeward low-level and windward high-level opening, sustained buoyancy-driven flow opposing the wind direction is unlikely for wind strengths that exceed a statistically critical value, which is slightly larger than the critical value of the wind strength at which bifurcation in the deterministic system occurs. Second, fluctuations in the applied wind modify the topology of the system's potential due to the nonlinear role that wind strength has in the equation for buoyancy conservation. Sufficiently large fluctuations in the wind rule out the possibility of ventilation opposing the wind direction at large base wind strengths. Although the phenomena described above might be perceived as making prediction easier, the results also highlight that certainty in the eventual state of the system goes hand in hand with uncertainty associated with longer transient effects. The work addresses growing interest in applying stochastic analysis to problems relating to building ventilation and urban fluid mechanics by describing a mathematically accessible example of the `stochasticisation' of a canonical deterministic model, while highlighting the subtleties and challenges of developing stochastic models for ventilation in the future.