A Note on Taylor-Eddy and Kovasznay Solutions of NS- -Deconvolution and Leray- -Deconvolution Models

We show that both the Taylor-eddy and Kovasznay exact solutions of the Navier-Stokes equations are also exact solutions of both the NS-α-deconvolution and Leray-α-deconvolution models, but with modified pressures that converge to the Navier-Stokes pressure solution as or the order of deconvolution tends to infinity. The existence of these exact model solutions will provide for better benchmark testing and validation of numerical codes and also shows that the models preserve these special structures. 1. Introduction The Leray- and NS- models and variations thereof have become of significant interest in both the mathematical and engineering communities interested in reduced-order fluid flow modeling. It is the purpose of this paper to derive exact solutions for these models, specifically those of Taylor-eddy and Kovasznay type, both for the purpose of providing better benchmark solutions for computational testing and to show that these models preserve some of the special structures of Navier-Stokes solutions. Solutions of Taylor-eddy type have been shown to exist for the Stolz-Adams approximate deconvolution model by Layton in [1] and for the Rational model by Berselli in [2], thus showing that existence of such solutions for models is important for model comparisons. To our knowledge, no other model has been shown to admit exact Kovasznay solutions. Denoting by overbar the -filter , the models are defined by the following: Leray- NS- In this work, we will consider these models in , or 3. The solutions we develop will satisfy the models pointwise, and so the models could also be equipped with boundary conditions, provided they are consistent with the solutions. The Leray model was developed by Leray in 1934 (but using a Gaussian filter instead of the -filter) as a theoretical tool to better understand the Navier-Stokes equations [3]. The model was then revisited by Cheskidov et al. in [4] with the -filter, and they proved fundamental properties of the model including well-posedness and agreement of the energy spectrum with that of true fluid flow on the large scales and an increased rate on the small scales (thus showing that the model is more computable). Work in [5] proved a microscale for the model, which better quantified its advantage in computability versus the Navier-Stokes equations. All of these properties are also valid for NS- [6, 7], but NS- also has several advantages over Leray- from the theoretical point of view such as helicity conservation [6], frame invariance [8], and adherence to Kelvin’s circulation theorem [6]. Numerous numerical