Spatially averaged turbulent stress and its partitioning
Dubravka Pokrajac Æ Ian McEwan Æ
Received: 5 April 2007 / Revised: 13 December 2007 / Accepted: 17 January 2008 / Published online: 3 March 2008
Ó Springer-Verlag 2008
Abstract Double averaging of the fundamental ﬂow
equations is an attractive methodology for investigating
spatially heterogeneous ﬂows. The resulting double-aver-
aged equations can be used as a framework for development
of turbulence models. In order to fully explore the potential
of such models the stress terms that appear in the momen-
tum equation as a result of each averaging step needs to be
parameterised. We investigate the stress terms resulting
from the double-decomposition of instantaneous velocity.
The experimental values of these stress terms were deter-
mined from the laboratory experiments involving PIV
measurements of shallow open channel ﬂows over rough
beds. Special attention is paid to the two terms that together
form the spatially averaged turbulent stress, which have so
far not been reported in the literature.
In spatially inhomogeneous ﬂows, point turbulence statis-
tics depend on the location of the observation point.
However some ﬂows such as ﬂow near a rough boundary
can be rendered globally homogeneous when they are
expressed in terms of their spatially averaged properties.
For such ﬂows using both temporal and spatial-averaging
of the fundamental equations and ﬂow quantities, called
double-averaging, is an attractive methodology which has a
potential to provide new insights into the ﬂow structure and
initiate a new generation of turbulence models.
The theory of spatial averaging was ﬁrst developed for
general conservation equations in multiphase systems in
porous media ﬂows (Whitaker 1967, 1973, Gray 1975;
Hassanizadeh and Gray 1979, etc.). The idea of applying
spatial averaging to time-averaged ﬂow equations was
gradually developed through the contributions of Smith
and McLean (1977), Wilson and Shaw (1977), Raupach
and Shaw (1982), Finnigan (1985), Raupach et al. (1991),
Wang and Tackle (1994), Gimenez-Curto and Corniero
Lera (1996), etc. Application of the double-averaging
methodology to open channel ﬂows over rough surfaces
began only recently (Nikora et al. 2001, 2007).
Applying two averaging steps (one in time one in space)
to the fundamental Navier–Stokes equation results in a
double-averaged Navier–Stokes equation, DANS, which
can be used as a framework for modelling. However, at each
of those two steps, averaging non-linear momentum term
creates an additional term, which can be regarded as a stress
(sometimes called apparent stress) term. In order to fully
explore the potential of turbulence models based on the
double-averaged equations these stress terms need to be
parameterised. This paper investigates the terms using
experimental data obtained by Partcile Image Velocitmetry.
The resulting discussion clariﬁes the meaning of the terms,
originally introduced by Pedras and de Lemos (2001).
2 Stress terms in the double-averaged momentum
We consider ﬂows where the hydrodynamic variables
represent a stationary and ergodic random ﬁeld so the term
‘time-averaging’ is used throughout the paper and regarded
as equivalent to ‘ensemble-averaging’. Furthermore only
uniform (in spatially averaged properties) ﬂows are
D. Pokrajac (&) Á I. McEwan Á V. Nikora
Department of Engineering, University of Aberdeen,
Aberdeen AB24 3UE, UK
Exp Fluids (2008) 45:73–83