Modeling near-wall effects in second-moment closures by elliptic relaxation

The elliptic relaxation method for modeling near-wall turbulence via second-moment closures (SMC) is compared to direct numerical simulation (DNS) data for channel flow at Re-tau = 395. The agreement for second-order statistics, and the terms in their balance equation is quite satisfactory, confirming that essential kinematic effects of the solid boundary on near-wall turbulence are accurately modeled by an elliptic operator. Additional viscous effects, immediately next to the surface, can be added via Kolmogoroff scales. In combination, elliptic relaxation and Kolmogorov scaling provide a general formulation to extend high Reynolds number SMC to wall-bounded flows. This formulation was easily applied to the nonlinear Craft-Launder and Speziale-Sarkar-Gatski (SSG) pressure-strain models. It is observed that the boundary conditions of the relaxation operator dominate the homogeneous pressure-strain model in the near-wall region. While looking at high-Reynolds number channel flows, it was found necessary to modify the effect of the relaxation operator throughout the log-layer by accounting for gradients of the turbulent lengthscale; this brings the velocity gradient into perfect agreement with the von Karman constant. The final form of the model based on the SSG homogeneous closure was then successfully applied to rotating channel flows, including relaminarization. The paper merges and updates two contributions by the five authors to the 10th Turbulent Shear Flow Conference.