By Oberbeck-Boussinesq flows, we mean fluid flows governed by approximation equations of motion for heat-conducting fluid flows. In this talk, we will introduce a stochastic functional integral representation for solutions of Oberbeck-Boussinesq  equations in the form of McKean-Vlasov-type mean field equation, which can be used to design numerical schemes for calculating solutions and implementing Monte-Carlo simulations of Oberbeck-Boussinesq flows. Our approach is based on the duality of conditional  laws for a class of diffusion processes associated with solenoidal vector fields, which allows us to obtain a novel integral representation theorem for the solution of some linear parabolic equation in terms of the Green function and the pinned measure of  the associated diffusion. I will also present some numerical results to show the efficiency of the numerical schemes, which are capable of revealing numerically the details of Oberbeck-Boussinesq flows within their thin boundary layer.

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