Filtered gas-solid momentum transfer models

To account for meso-scale phenomena in coarse grid simulations, correlation terms appearing in the filtered gas-solid flow equations have to be modeled. Filtered models for the gas-solid momentum transfer are focused on. The latter consists of the drag force on the one hand and the distribution of the gas phase pressure gradient over the phases on the other hand. Effective interphase momentum transfer coefficient closure models have been proposed for the filtered drag force [1]. For the correlation between the solids volume fraction and the gas phase pressure gradient, Zhang and VanderHeyden proposed a generalized added mass closure model [2], introducing a generalized added mass coefficient. The closure models for the filtered gas-solid momentum transfer proposed so far are analyzed with a mixture speed of sound test. Whereas the closure terms in filtered models may alter the calculated mixture speed of sound at frequencies higher than the filter frequency, they should not alter the calculated mixture speed of sound at frequencies lower than the filter frequency. The mixture speed of sound test shows an acceptable behavior for the generalized added mass approach. Theoretically, it is shown that a generalized added mass term appears directly from the correlation between the solids volume fraction and the gas phase pressure gradient and that it corresponds to a redistribution of the filtered gas phase pressure gradient over the phases. This direct contribution scales according to the mean square of the solid volume fraction fluctuations. Two-dimensional dynamic meso-scale simulations (DMS) over a broad solid volume fraction range and for two domain sizes and two grid resolutions are carried out to calculate the magnitude of the generalized added mass effect. A second, indirect contribution to the generalized added mass term from the correlation between the solids volume fraction and the gas phase pressure gradient is shown to be statistically significant, but one order of magnitude smaller than the direct contribution. A further quantification of the maximum generalized added mass effect as a function of the filter frequency is obtained from the mixture speed of sound test. The results support the generalized added mass approach [2], but show that a large generalized added mass coefficient, as previously reported by Zhang and VanderHeyden [2], is justified only in case the filter frequency is low (< 20 Hz), i.e. if the grid is, either spatially or temporally, sufficiently coarse