A sub-grid-scale model for polydisperse bubbly flows with heat and mass transfer

Ensemble-averaged models of polydisperse bubbly flows require statistics of the evolving bubble population. Prior quadrature-based moment formulations close bubble pressure with a polytropic relation that omits heat and mass transfer at the bubble wall. We formulate constant-transfer equations for bubble pressure and vapor mass within a conditional hyperbolic quadrature method. Second-order conditional inversion produces four joint radius--radial-velocity nodes per equilibrium-radius bin. Bubble pressure and vapor mass are advanced at each node. The node values close the ensemble-averaged flow equations without adding mixed pressure or vapor-mass moments to the transported moment set. The model is implemented in MFC. Monte Carlo calculations verify the evolution of quadrature nodes and the mean bubble variables for a harmonically forced population. Bubble-screen calculations quantify closure error as the equilibrium radius is discretized and the initial distributions vary. The constant-transfer calculation does not exhibit the high-frequency pressure oscillations observed with the polytropic closure under the conditions considered. 3D bubble-screen calculations give a 1.5% relative root-mean-square error between the Euler--Euler center pressure and the mean of 40 volume-averaged Euler--Lagrange realizations.