NUMERICAL ANALYSIS OF A MODIFIED SMAGORINSKY MODEL ∗ FARJANA SIDDIQUA † AND XIHUI XIE ‡ Abstract. The classical Smagorinsky model’s solution is an approximation to a (resolved) mean velocity. Since it is an eddy viscosity model, it cannot represent a flow of energy from unresolved fluctuations to the (resolved) mean velocity. This model has recently been modified to incorporate this flow and still be well-posed. Herein we first develop some basic properties of the modified model. Next, we perform a complete numerical analysis of two algorithms for its approximation. They are tested and proven to be effective. Key words. Eddy Viscosity, Modified Smagorinsky, Complex turbulence, Backscatter AMS subject classifications. 65M06, 65M12, 65M22, 65M60, 76M10 1. Introduction. Consider the Smagorinksy model [33] 1 , with prescribed body force f , kinematic viscosity ν in the regular and bounded flow domain Ω ⊂ R d (d = 2, 3), which was later advanced independently by Ladyzhenskaya [19, 20]: ∇ · w =0 and (1.1) w t + w · ∇w − ν Δw + ∇q − ∇· ( (C s δ) 2 |∇w|∇w ) = f (x). Here (w,q) approximate an ensemble average of Navier-Stokes solutions, ( u, p). This is an eddy viscosity model with turbulent viscosity, ν T =(C s δ) 2 |∇w|, where C s ≈ 0.1, Lilly [24], δ is a length scale (or grid scale). Like all eddy viscosity models, the Smagorinsky model represents a flow of energy from means to unresolved fluctuations (u ′ = u − u, for a precise formula see Definition 2.14) and has errors by not represent- ing any intermittent energy flow from fluctuations back to means. Corrections have recently been made representing this flow in Jiang and Layton [15] and Rong, Layton and Zhao [32]. Following their ideas, we develop a modified model in section 3. We also analyze and test numerical algorithms for effective approximation of the resulting modified model: ∇· w = 0 and (1.2) w t − C 4 s δ 2 µ −2 Δw t + w · ∇w − ν Δw + ∇q − ∇·  (C s δ) 2 |∇w|∇w  = f (x). Here µ is a constant from Kolmogorov-Prandtl relation [18, 31]. The main result of this paper is the complete numerical analysis and computa- tional testing of effective algorithms for this model. This paper gives detailed numeri- cal analyses in section 4 and section 5. This model is able to capture the phenomenon of transferring energy from fluctuation to means, which is tested numerically in sub- section 6.2. There were few attempts made for extending model that represents flow at statistical equilibrium to non-equilibrium. For instance, in a previous work by Jiang and Layton [15], there was a fitting parameter β in the second term of (1.2) which is needlessly complicated. But in our paper, we use a different idea to model it, which results in a simpler model with no fitting parameter. * The research was partially supported by NSF grant DMS-2110379. † Department of Mathematics, University of Pittsburgh, Pittsburgh, PA-15260 (fas41@pitt.edu ). ‡ Department of Mathematics, University of Pittsburgh, Pittsburgh, PA-15260 (xix55@pitt.edu). 1 The mechanically correct formulation is with the ∇ s w instead of ∇w in the term −∇· ( (Csδ) 2 |∇w|∇w ) where ∇ s is the symmetric part of the gradient tensor. But since the estimates are same and analyses are simpler with ∇w due to Korn’s inequality ‖v‖ 2 H 1 (Ω) ≤ C[‖v‖ 2 L 2 (Ω) + ‖∇ s v‖ 2 L 2 (Ω) ], we use ∇w throughout the paper. 1 This manuscript is for review purposes only. arXiv:2201.08914v1 [math.NA] 21 Jan 2022