### 2002-14

#### Derivation and Analysis of Near Wall Models for Channel and Recirculating Flows

Series: 2002-14, Preprints

MSC:
76D10 Boundary-layer theory, separation and reattachment, higher-order effects

Abstract:
The problem of predicting features of turbulent flows occurs in many
applications such as geophysical flows, turbulent mixing, pollution
dispersal and even in the design of artificial hearts. One
promising approach is large eddy simulation (LES), which seeks to predict
local spacial averages $\ov{\bu}$ of the fluid's velocity $\bu$.
There are several core difficulties in LES. Closure models are very
important in applications in which the equations must be integrated over a
long time interval. In engineering applications, however, often the
equations are solved over moderate time intervals and the core difficulty
is associated with modeling near wall turbulence in complex geometries.
Thus, one important problem in LES is to find appropriate boundary conditions for the
flow averages which depend on the behavior of the unknown flow near the
wall. Inspired by early works of Navier and Maxwell, we develop such
boundary conditions of the form
$$\ov{\bu} \cdot \bn = 0 \mbox{ and } \beta(\delta,Re,|\ov{\bu} \cdot \btau|) \ov{\bu}\cdot \btau +2Re^{-1}\bn \cdot \D(\ov\bu) \cdot \btau = 0$$
on the wall. We derive effective friction coefficients $\beta$ appropriate
for both channel flows and recirculating flows and study their
asymptotic behavior as the averaging radius $\delta\to 0$ and as
the Reynolds number $Re \to \infty$. In the first limit, no--slip conditions
are recovered. In the second, free--slip conditions are recovered.

Keywords:
large eddy simulation, near wall models, turbulence, boundary layer