Differential and Integral Equations Volume 15, Number 11, November 2002, Pages 1395–1407 ON “VERIFIABILITY” OF MODELS OF THE MOTION OF LARGE EDDIES IN TURBULENT FLOWS M. Kaya 1 Gazi University, Faculty of Science and Arts Department of Mathematics, Ankara, Turkey W.J. Layton 2 Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260 (Submitted by: Reza Aftabizadeh) Abstract. If the Navier-Stokes equations are averaged by a local, spa- cial filter (denoted by an over-bar) the following system results: ∇· u = 0 and ut + u ·∇ u − Re −1 Δ u + ∇· R(u)+ ∇ p = f, where R(u) := uu − u u denotes the Reynold’s stresses. To close this system, various models for the Reynold’s stresses of the form R(u) ∼ = ˜ R( u) are used. The desire is that the solution to the resulting system (call it w) be close to u. However, models are often validated instead by calculating u and explicitly checking ||R(u) − ˜ R( u)||. This report studies conditions under which the latter being small implies w is close to u. (we call this verifiability of a model.) Since u → u as δ → 0, we also study when it can be proven that w → u as δ → 0. 1. Introduction The pointwise fluid velocity u(x, t) and pressure p(x, t) describing the turbulent flow of an incompressible, viscous fluid satisfy the Navier-Stokes equations in the flow domain Ω ⊂ R 3 , given by u t + ∇· (uu) − Re −1 Δu + ∇p = f, and ∇· u =0. (1.1) For the Reynolds number, Re, large enough, solutions to (1.1) display the complex spacial patterns and extreme sensitivity typical of turbulent flow. Thus, it is common to seek not to predict the pointwise fluid velocity but rather suitable averages of it. Accepted for publication: April 2002. AMS Subject Classifications: 76F65, 76D03. 1 The research of M. Kaya was partially conducted during a visit to the University of Pittsburgh. 2 Partially supported by NSF Grants INT-98 14115, INT-9805563 and DMS 9972622. 1395