A note on fitting a generalized Moody diagram for wall modeled Large Eddy Simulations

Motivated by the needs of wall modeled Large Eddy Simulation (LES), we introduce fits to numerical solutions of the Reynolds Averaged Navier-Stokes equations in their simplest near-wall, boundary layer approximation including a mixing-length model. We formulate the problem such that independent dimensionless variables are those directly available in LES. We provide practical fits for the dependent variable, fits that encompass a smooth transition between the viscous sublayer and inertial logarithmic layer, and then progress first considering moderate pressure gradients as well as roughness effects under the assumption that the mixing-length is not affected by the pressure gradient. An alternative fit based on the empirical wall model of Nickels (J. Fluid Mech. vol.512, pp. 217-239, 2004) is also provided, taking into account possible effects of pressure gradient on turbulence near-wall structure. We then consider the case of general pressure gradients, both favorable and adverse, up to conditions of separation, for both smooth and rough surfaces. The proposed fitting functions constitute a generalized Moody chart, comply with analytical solutions valid in various asymptotic regimes, and obviate the need for numerical iterative solution methods or near-wall numerical integration of ordinary differential equations during LES.