Explicit algebraic relation for calculating Reynolds normal stresses in flows dominated by bubble-induced turbulence

Two new algebraic turbulence models for flows dominated by bubble-induced turbulence (BIT) are presented. They combine different elements of existing models that are considered superior to their alternatives. Both models focus on the core region of a channel flow, where the flow can be assumed to be in local equilibrium and the void fraction is approximately homogeneous. The first model, referred to as the algebraic Reynolds normal stress model, is derived from the differential Reynolds stress model of Ma et al. [J. Fluid Mech. 883, A9 (2020)]. The second model utilizes the original two-equation turbulence model for bubbly flows [Ma et al., Phys. Rev. Fluid 2, 034301 (2017)] to achieve algebraic expressions for k and epsilon in BIT-dominated cases. If both models are combined, it results in a purely algebraic (i.e. not involving any differential equations), explicit relation for the Reynolds normal stresses, which depends only on the mean flow parameters, namely, the mean gas void fraction and mean liquid and gas velocities. We find that the model can well predict the Reynolds normal stresses, compared with direct numerical simulation and experimental data.