J. Fluid Mech. (2003), vol. 478, pp. 11–34.
c© 2003 Cambridge University Press
DOI: 10.1017/S0022112002003324 Printed in the United Kingdom
11
Turbulence over a compliant surface: numerical
simulation and analysis
By S. XU
1
,D.REMPFER
2
AND J. LUMLEY
1
1
Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853, USA
2
Department of Mechanical, Materials and Aerospace Engineering, Illinois Institute of Technology,
Chicago, IL 60616, USA
(Received 26 July 2002 and in revised form 8 October 2002)
In this paper we present results from a numerical investigation of turbulent channel
flow in the presence of a compliant wall. The compliant wall is modelled as a
homogeneous spring-supported plate. The simulation code is validated both by
comparison with an alternative code and by reproducing results of linear stability
theory. Our results demonstrate that with the wall compliance we used in the
simulation there is little change in the very long-time behaviour of the turbulent skin
friction drag and little modification to the near-wall turbulent coherent structures. The
values of pertinent statistical quantities of the turbulence near the compliant walls
converge to those near a rigid wall and the statistical effect of the wall compliance
on the turbulent channel flow is small.
1. Introduction
The flow in and around many technical devices creates turbulent boundary layers,
which can lead to significant losses in efficiency. Thus, reducing this turbulent drag,
which is responsible for a large part of the drag experienced by airplanes, ships and
submarines, and reducing turbulent sound production, which, among other things, is
the major source responsible for the noise inside the cabin of transport aircraft, are
particularly desirable goals. Skin friction reduction can be achieved through various
methods of boundary layer manipulation, which include active or passive transition
delay and turbulence control. A number of strategies have been proposed for this
purpose, such as polymer or particle additives, blowing and suction, LEBUs, riblets,
and compliant coatings, as well as methods for active control. Reactive boundary
layer manipulation requires sensors, actuators and control algorithms. The necessary
technology is far from mature at present and it is not clear whether the associated
development effort will ultimately pay off (Lumley & Blossey 1998). The use of
compliant walls as a simpler passive means appears an attractive alternative (Gad-el-
Hak 1998) . In particular, although the maximum amount of drag reduction that can
be achieved using compliant walls may be less than that demonstrated numerically
formethods of reactive turbulence control, compliant walls do not use any energy to
achieve their drag-reducing effect. This is in stark contrast to methods for reactive
turbulence control, which usually require significant amounts of energy, which, in all
experiments that have been done up to now, far exceed the energy that could be
saved via drag reduction.
Thus the idea of using a compliant wall with a tailored dynamic response to pressure
disturbances from a turbulent wall layer is a potentially attractive one. Having said

12 S. Xu, D. Rempfer and J. Lumley
that, we are well aware of the history of related research some twenty-five years
ago. At that time, following observations by Kramer (1961) on exceptional swimming
capabilities of dolphins, there was a long series of what now has to be described as
failed experimental attempts to verify compliant wall technology. This body of work
is described in Bushnell, Hefner & Ash (1977). After these negative results, studies
of this type were effectively discredited, at least from an experimental point of view.
With regard to this history, two more points are worth noting:
(i) The exact reasons for the failure of the experiments were not always clear,
and, in particular, to this day nothing definitive can be said about whether or not
compliant surfaces may be able to significantly reduce turbulent drag or sound
production. We believe that the obvious benefits that compliant surfaces may give
warrant taking another, more detailed look at this problem, in our case from a
more theoretical perspective and using modern approaches based on low-dimensional
modelling (Rempfer et al. 2001), and direct numerical simulations of turbulent flows.
(ii) Despite the relative wealth of experimental data, there are very few results
available on the effect of compliant walls on the structure and properties of turbulent
wall layers. Kireiko (1991) analysed the interaction of a compliant wall with near-wall
turbulence by using the monoharmonic approximation and concluded that the
interaction appears resonant in character and a considerable reduction in turbulent
skin friction may be possible for certain values of wall parameters. Semenov (1991)
proposed a set of conditions for modelling and choosing viscoelastic coatings for
drag reduction according to a hydraulic smoothness requirement and interference
theory, in which the linear harmonic solution to the interaction is obtained for a
given pressure fluctuation spectrum using a simplified linear near-wall turbulence
model. The first of his conditions is a requirement of quick attenuation or absence
of free vibrations of the coating. The second one requires a limitation of the coating
compliance such that the allowable amplitude of the wall deformation remains below
r
+
=6(wherer
+
is the wall roughness height in wall units). The third condition states
that the natural frequency of the wall should be chosen in order to ensure a large
phase–frequency region of favourable interaction. Kang & Choi (2000) have studied
active walls deformed according to successful control strategies for drag reduction
in turbulent channel flows. They found that overall 13%−17% drag reduction can
be obtained with active wall motions. Turbulence intensity and near-wall streamwise
vortices can be significantly weakened and instantaneous wall shapes are elongated
in the streamwise direction. From these results it is natural to ask whether it might
be possible to design a compliant surface with elastic properties such that the wall,
driven by pressure disturbances from the flow, would move ‘by itself ’ in the same way
as the one used in Kang & Choi’s study. It is this question that we will study in more
detail below.
We also note that a number of investigations have been aimed at delaying boundary
layer transition (which is a relevant flow regime in the case of the dolphin) using
compliant coatings. Results of numerical studies indicating benefits from compliant
walls in the area of transition delay were reported by Carpenter & Morris (1990)
and by Davies & Carpenter (1997). The case of transition manipulation is interesting,
because in that case there is quite an extensive body of theoretical material on
this problem, which is described by the linear stability theory and the asymptotic
theory of transitional flows over compliant walls or in compliant channels (see Gajjar
&Sibanda 1996; Larose & Grotberg 1996; Lucey & Carpenter 1995; Reutov &
Rybushkina 1998; Riley, Gad-el-Hak & Metcalfe 1988; Yeo, Khoo & Zhao 1999).
In fact, linear stability theory can provide satisfactory explanations for the failure of