Differential and Integral Equations

On "verifiability" of models of the motion of large eddies in turbulent flows

M. Kaya and W. J. Layton

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Abstract

If the Navier-Stokes equations are averaged by a local, spacial filter (denoted by an over-bar) the following system results: $$\nabla \cdot {\overline{u}} = 0 \ \text{and}\ {\overline{u}}_t + {\overline{u}} \cdot \nabla {\overline{u}} - Re^{-1} \ \Delta {\overline{u}} + \nabla \cdot \mathbb R(u) + \nabla {\overline{p}} = {\overline{f}},$$ where $\mathbb R(u): = {\overline{u \ u}} - {\overline{u}} \ {\overline{u}}$ denotes the Reynold's stresses. To close this system, various models for the Reynold's stresses of the form $\mathbb R(u) \cong \tilde {\mathbb R}({\overline{u}})$ are used. The desire is that the solution to the resulting system (call it $w$) be close to ${\overline{u}}$. However, models are often validated instead by calculating $u$ and explicitly checking $||\mathbb R(u) - \tilde {\mathbb R}({\overline{u}})||$. This report studies conditions under which the latter being small implies $w$ is close to ${\overline{u}}$. (we call this {\it verifiability} of a model.) Since ${\overline{u}} \rightarrow u$ as $\delta \rightarrow 0$, we also study when it can be proven that $w\rightarrow u$ as $\delta \rightarrow 0$.

Article information

Source
Differential Integral Equations, Volume 15, Number 11 (2002), 1395-1407.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060729

Mathematical Reviews number (MathSciNet)
MR1920694

Zentralblatt MATH identifier
1161.76498

Subjects
Primary: 76F02: Fundamentals
Secondary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10] 76D17: Viscous vortex flows

Citation

Kaya, M.; Layton, W. J. On "verifiability" of models of the motion of large eddies in turbulent flows. Differential Integral Equations 15 (2002), no. 11, 1395--1407. https://projecteuclid.org/euclid.die/1356060729


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