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 M_τ term) and add the modeled M_τ 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 M_τ 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.

Appendix: Sensitivity of Turbulent Prandtl Number

In Section 2.2, the law of the wall for the temperature is derived as in Eq. 9 under the assumption of constant turbulent Prandtl number, which leads directly to a proposal of a method to determine the suggested value for the turbulent Prandtl number (Pr_t = 1.5 and 1.1 are suggested for p = 2 and 4 MPa cases). Nevertheless, since the sensitivity of the turbulent Prandtl number may add valuable information, Fig. 12 shows the sensitivity of the turbulent Prandtl number to the mean velocity and temperature, where the results obtained by the proposed k-ω SST+M_τ model with Pr_t = 1.5 (suggested value derived in Section 2.2) and Pr_t = 0.9 (typical value for constant-property ideal-gas flows) for the heated case (T_w = 100K) at p = 2 MPa are compared. As clearly seen, the turbulent Prandtl number affects significantly to the temperature profile, whereas the van Driest transformed mean velocity is insensitive to the choice of the turbulent Prandtl number although the density and viscosity are a function of temperature. By reducing the turbulent Prandtl number from the suggested value Pr_t = 1.5 to the typical value Pr_t = 0.9 for constant-property ideal-gas flows, the mean temperature is apart from the logarithmic law. Also, the insensitivity of the turbulent Prandtl number to the predicted mean velocity indicates that the improvement of the predicted logarithmic law for the velocity as in Fig. 9 is due to the modeling of the turbulent mass flux M_τ proposed in this study.

Fig. 12

a Van Driest transformed mean streamwise velocity and b transformed mean temperature $\overline {T}^{*}$ obtained by the proposed k-ω SST+M_τ model with different turbulent Prandtl number Pr_t = 1.5 (suggested value derived in Section 2.2) and 0.9 (typical value for constant-property ideal-gas flows) for heated case T_w = 100K at p = 2 MPa. Black lines, k-ω SST+M_τ model with Pr_t = 1.5; blue lines, k-ω SST+M_τ model with Pr_t = 0.9; circles, corresponding DNS [7]; thin gray lines, in (a) $\overline {u}_{vD}^{+}=1/0.41\log (y^{+})+5.2$, in (b) $\overline {T}^{*}=1.5/0.41\log (y^{+})+1.5$