Turbulence Modeling for Turbulent Boundary Layers at Supercritical Pressure: A Model for Turbulent Mass Flux

Abstract

Based on the analysis of the direct numerical simulation (DNS) database of the heated and unheated turbulent boundary layers at supercritical pressures (Kawai J. Fluid Mech. 865, 563 2019), this paper proposes a Reynolds-averaged Navier-Stokes (RANS) turbulence modeling for predicting the turbulent boundary layers at supercritical pressure where large density fluctuations are induced by the pseudo-boiling phenomena. The proposed approach is to model the mass flux contribution term $M_{\tau }=\overline {u_{i}^{\prime \prime }} \partial \overline {\tau _{ij}}/\partial x_{j}$ in the turbulent kinetic energy equation (more specifically the turbulent mass flux $\overline {u_{i}^{\prime \prime }}= -\overline {\rho ^{\prime } u_{i}^{\prime }}/\overline {\rho }$ in Mτ term) and add the modeled Mτ to the k-transport equation in the RANS model in order to incorporate the effects of the large density fluctuations on turbulence observed in the DNS. The key idea of modeling the turbulent mass flux in Mτ is to employ the gradient diffusion hypothesis and we propose to model $\overline {u_{i}^{\prime \prime }}$ as a function that is proportional to the density gradient (i.e. $\overline {u_{i}^{\prime \prime }} \propto \overline {\mu }_{t} \partial \overline {\rho }/\partial x_{j}$). The proposed RANS model shows significant improvements over existing models for predicting the logarithmic law for the mean velocity and temperature in the turbulent boundary layers at supercritical pressure, something that existing RANS models fail to do robustly.