Analytical insights into the partially integrated transport modeling method for hybrid Reynolds averaged Navier-Stokes equations-large eddy simulations of turbulent flows

Abstract

The basis of the partially integrated transport modeling (PITM) method was introduced by Schiestel and Dejoan [“Towards a new partially integrated transport model for coarse grid and unsteady turbulent flow simulations,” Theor. Comput. Fluid Dyn. 18, 443 (2005)]10.1007/s00162-004-0155-z and Chaouat and Schiestel [“A new partially integrated transport model for subgrid-scale stresses and dissipation rate for turbulent developing flows,” Phys. Fluids 17, 065106 (2005)]10.1063/1.1928607. This method provides a continuous approach for hybrid RANS-LES (Reynolds averaged Navier-Stokes equations-large eddy simulations) simulations with seamless coupling between RANS and LES regions. The main ingredient of the method is the new dissipation-rate equation that can be applied as a subfilter scale turbulence model. Then, it becomes easy to convert almost any usual RANS transport model into a subfilter scale model. In particular, the method can be applied to two equation models and to stress transport models as well. In the derivation of the method, the partial integration technique allows to keep a link between the spectral space and the physical space of the resulting model. The physical turbulent processes involving the production, dissipation, and flux transfer of the turbulent energy are introduced in the equations. The present work, after recalling the main building steps of the PITM method, brings further insight into the physical interpretation of the method, its underlying hypotheses and its internal acting mechanisms. In particular, the finiteness of the coefficients used in the dissipation-rate equation is discussed in detail from a theoretical point of view. Then, we consider the analytical example of self-similar turbulent flow for analyzing the dissipation-rate equation. From an analytical solution obtained by Taylor series expansions taking into account the Kovasznay hypothesis for evaluating the transfer term, we compute the functional coefficients \documentclass[12pt]{minimal}\begin{document}$c_{{\epsilon }_2}$\end{document}cε2 and \documentclass[12pt]{minimal}\begin{document}$c_{sfs{\epsilon }_2}$\end{document}csfsε2 used in RANS and LES methodologies, respectively, and we demonstrate that both coefficients take on finite values when the Reynolds number goes to infinity. Finally, after briefly mentioning some flow illustrations to get a real appraisal of the PITM method in its capabilities to simulate unsteady flows on relatively coarse grids with a sufficient accuracy for engineering computations, we study the coefficient \documentclass[12pt]{minimal}\begin{document}$c_{sfs{\epsilon }_2}$\end{document}csfsε2 through one chosen example.