Commutation error mitigation in variable-resolution PANS closure: Proof of concept in decaying isotropic turbulence

Abstract

Two quintessential attributes desired in variable-resolution (VR) bridging turbulence calculations are (i) ability to perform flow simulations at any prescribed level of resolution with the highest possible fidelity; and (ii) capability to vary the resolution in space and time while retaining high degree of physical accuracy through out the computational domain. Many VR models strive to satisfy the first requirement and have been successful to varying degrees. Spatio-temporal variation in resolution, the very attribute that make VR methods attractive for practical application, renders closure modeling rather complicated due to the so-called commutation error. In this paper, in the context of partially-averaged Navier-Stokes (PANS) VR approach, we develop a physics-based closure model for the commutation residual arising from spatio-temporal variation in flow resolution. The PANS model is then tested in a decaying isotropic turbulence flow field. It is demonstrated that (i) in constant resolution simulations, PANS achieves the prescribed resolution level and adequately captures the corresponding energy spectrum; and (ii) when the resolution is varied suddenly during the course of a simulation, the commutation model comes into effect yielding the required change in the calculations. The advent of variable-resolution (VR) turbulence computational approach has brought to fore a new set of closure modeling challenges not encountered in the more traditional Reynolds averaged NavierStokes (RANS) and large-eddy simulation (LES) methods. The VR methods can be broadly classified into zonal and bridging approaches and the closure challenges are specific to each one. Zonal methods combine RANS computations in some zones of the flow domain with LES in other parts in an attempt to best utilize the computational resources for a given problem. Bridging methods aim to achieve the same outcome with a different strategy: a single closure model is employed throughout the flow domain and the variation in resolution is accomplished with smooth variations in the model parameters. Two of the key modeling challenges facing bridging closures are: (i) scale-dependent VR constitutive relationship with demonstrated ability to yield prescribed level of flow resolution; and (ii) capability to retain high-degree of physical accuracy in regions of scale resolution variation. The issue of computational fidelity in regions of changing scale resolution is of critical importance in all VR methods as well as LES. In diverse disciplines – eg., particle physics, quantum electrodynamics, fluid turbulence and nanotechnology – there exists a compelling need to describe multiple-scale phenomena at different levels of resolution. Many of these multi-scale systems exhibit scale-invariance property implying that the mathematical description is self similar at any scale of description. In the context of fluid turbulence, the scale invariance property of Navier-Stokes equations has been established for incompressible flows in