An order-of-magnitude approximation for the hybrid terms in the compressible hybrid RANS/LES governing equations

Abstract

The present work follows up our previous development in the compressible hybrid Reynolds-averaged Navier–Stokes (RANS)/large eddy simulation (LES) methodology (M. Sánchez-Rocha and S. Menon, The compressible hybrid RANS/LES formulation using an additive operator, J. Comput. Phys. 228 (2009), pp. 2037–2062). In that original work, the exact compressible hybrid RANS/LES (HRL) equations were formally derived revealing the existence of additional hybrid terms (HT) that depend explicitly on the LES and RANS variables. The importance of the exact equations was established in flat-plate turbulent boundary layers, demonstrating that the law-of-the-wall could not be accurately reproduced if the HT were not included. Unfortunately, the new HT could not be directly computed since it is required to reconstruct the LES variables from the hybrid field—a task that is equivalent to an inverse filtering operation that in general is ill-conditioned. Therefore, in our original work, parallel LES simulations were conducted to provide the LES field to close the HRL equations. In the present work, a simple model is proposed to reconstruct the LES field from the hybrid variables to close the exact hybrid equations. The model is obtained by manipulating the HRL operator and by order-of-magnitude estimations. Here it is stressed that the approximated LES field is used only to compute the HT and not to compute flow statistics. Numerical calculations for a flat-plate turbulent boundary layer at Re θ=1400 are conducted to evaluate the performance of the approach proposed against previous hybrid studies. Simulations using the approximated HT are compared against hybrid calculations without the HT, simulations using the exact HT, and LES simulations. Overall, the results obtained with the approximated HT favorably compare with LES and experimental data. The sensitivity of the new approach to the blending function implemented is also addressed, and it is found that with the modeled HT, the mean velocity profile is correctly predicted independently of the blending function used. Finally, the effect of the grid resolution on the model proposed is evaluated by increasing the Reynolds number (Re θ=3330); results for this case are found in reasonable agreement with experimental data and LES calculations.