Unifying formalism and closures for coarse-grained approaches to turbulence
A. Cimarelli, N. Marras, B. Niceno, Y. Tessier Urrecha
TL;DR
This work introduces a temporally filtered Navier–Stokes framework that unifies statistical (RANS) and scale-resolving turbulence closures by treating turbulence-induced smoothing as an explicit time filter $\langle \cdot \rangle_\tau$. It derives a dynamic RANS closure from a two-filter identity, yielding an adaptive eddy-viscosity coefficient $c_\mu$ via $c_\mu\,\mathcal{M}_{ij}=\mathcal{L}_{ij}$ with $c_\mu = \dfrac{\mathcal{M}_{ij}\mathcal{L}_{ij}}{\mathcal{M}_{ij}\mathcal{M}_{ij}}$, applicable to any two-equation closure. The approach demonstrates improved flow physics for transitional and separated flows (e.g., laminar separation, transition, reattachment) on both fine-scale and coarsened grids, and even enhances statistical steady-state predictions in 2D. The results indicate robust, regime-spanning performance and reveal how the dynamic coefficient adapts to near-wall and bulk turbulence, enabling better capture of laminar–turbulent transitions and sensitivity to free-stream turbulence. Overall, the temporally filtered paradigm provides a solid foundation for advancing coarse-grained turbulence modeling and suggests directions for extending the methodology to other closures and flow regimes.
Abstract
We propose the use of an unifying paradigm for the assessment and development of closed forms of the coarse-grained Navier-Stokes equations in approaches ranging from the statistical to the scale-resolving ones. It consists in the exact formalism provided by the temporally filtered Navier-Stokes equations. The fundamental idea is that the smoothing action of turbulent stresses can be described as a temporal filtering operator implicitly applied to the solution. Contrary to the average and spatial filtering operators, the temporal filter is an unifying operator smoothly varying within the statistical and scale-resolving realms. The potential of the temporal filtering paradigm is here highlighted by unveiling relevant algebraic properties and by deriving a new class of turbulence closures. A dynamic procedure is derived and shown to provide an unifying closure for both scale-resolving and statistical approaches. Results show that an improved physics is captured. Challenging phenomena such as the laminar to turbulence transition and the dependence of separation and reattachment on free-stream turbulence applied through boundary conditions are captured.