AIAA JOURNAL
Vol. 44, No. 11, November 2006

Unsteady and Transitional Flows Behind Roughness Elements
F. Gökhan Ergin∗ and Edward B. White†
Case Western Reserve University, Cleveland, Ohio 44106
The role of surface roughness in boundary layers continues to be a topic of significant interest, especially withregard to how controlled roughness might be used to delay laminar-to-turbulent transition. Although it may be
useful for control, large-amplitude roughness may itself lead to transition. In an effort to understand the breakdown
mechanics associated with large-amplitude surface roughness, experiments are conducted to investigate the steadyand unsteady disturbances generated by three-dimensional roughness elements whose amplitudes are close to thecritical roughness-based Reynolds number Rek for roughness-induced transition. Measurements are obtained ina flat-plate boundary layer downstream of a spanwise array of cylindrical roughness elements at both subcriticaland supercritical values of Rek . The steady disturbance field has strong shear in the wall-normal and spanwise
directions, and the unsteady streamwise velocities in the roughness elements’ wake show evidence of hairpinvortices. The locations of maximum fluctuation intensity correspond to the locations of inflection points in thesteady flow streamwise velocity, and this suggests that the fluctuations may result from a Kelvin–Helmholtz-typeinstability. Temporal power spectra indicate an unstable band of frequencies from 300 to 800 Hz. The Strouhal
number associated with the 650-Hz fluctuations that are often observed to be the strongest give Sr = 0.15, a value thatis in good agreement with previous findings. At supercritical Rek , rapid transition takes place when the unsteady
disturbances reach nonlinear amplitudes. The disturbance growth rates indicate that in this situation transition
can be understood as a competition between the unsteady disturbance growth and the rapid relaxation of thesteady flow that tends to stabilize these disturbances.

NomenclaturessampvleDEef= roughness diameter= steady disturbance energy= unsteady disturbance energy in a frequency band
centered at ff= frequency
H= shape factor, δ ∗ /θk= roughness heightN= number of samples
Rek= roughness-based Reynolds number, Ū (k)k/ν
Re= unit Reynolds number, U∞ /ν
Sr= Strouhal number of unsteady vortex
shedding, f δ ∗ /U∞Ū= spanwise-invariant streamwise basic state velocityU= steady streamwise disturbance velocityU∞= freestream velocityu= unsteady streamwise disturbance velocity
x, y, z = streamwise, wall-normal, and spanwise coordinatesxk= streamwise location of the roughness arrayα= spatial growth rateδ= boundary-layer length scale, [(x − xvle )/Re ]1/2
∗δ= displacement thickness
λk= roughness spacingη= Blasius coordinate, y/δθ= momentum thickness
ν= kinematic viscosity

I.

TIntroduction

HE study of laminar-to-turbulent transition on surfaces withsignificant roughness levels has had a long history motivatedby the need to understand transition on realistic surfaces. Laboratory
studies on low-speed flows have tended to treat three main classesof roughness: two-dimensional roughness strips; three-dimensional,
isolated roughness elements or arrays of three-dimensional elements; and random distributed three-dimensional roughness fields.
The distinctions among these arise from how each is accommodated as a disturbance source in stability theory and from practicalconsiderations of what actually exists on aerodynamic surfaces.
In two-dimensional boundary layers, the role of two-dimensionalroughness is well understood as a generator of two-dimensional
Tollmien–Schlichting (TS) waves. Goldstein1 uses a triple-deck
analysis to show that the streamwise variations of the mean flowthat are produced by two-dimensional roughness scatter acoustic
waves into TS wavelengths and, thus, provide a receptivity mechanism for unstable TS waves. Additional details on the receptivityof TS waves are in the review by Saric et al.2 Because the TSwave amplitudes depend on the roughness amplitude, increasing
the roughness amplitude should be expected to move gradually thetransition location toward the two-dimensional roughness, and thisis exactly what was observed in numerous transition experiments of
the 1940s and 1950s.3,4
The role of isolated and distributed three-dimensional roughness
in two-dimensional boundary layers has not been as well understoodon a theoretical basis because the stationary, spanwise varying disturbances created by three-dimensional roughness are not unstablewithin the context of a normal-mode stability analysis.5 Typically,transition in the wake of isolated three-dimensional roughness elements is predicted using correlations based on a critical roughnessbased Reynolds number Rek,crit , established in the 1950s.4,6−10
Within the context of the correlation approach, if a roughness element exceeds Reynolds number Rek,crit , then transition is expected
to occur at or just aft of the roughness element, but if the criticalvalue is not exceeded, then no effect is expected.
The correlation approach based on a critical Reynolds number
Rek is well established, but provides no guidance on the more subtle

Subscriptsccritrms= shedding= sampling= virtual leading edge

= centerline= critical= root-mean-square

Received 30 April 2005; revision received 11 October 2005; accepted
c 2006 by the American Infor publication 12 October 2005. Copyright stitute of Aeronautics and Astronautics, Inc. All rights reserved. Copies ofthis paper may be made for personal or internal use, on condition that thecopier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc.,
222 Rosewood Drive, Danvers, MA 01923; include the code 0001-1452/06$10.00 in correspondence with the CCC.
∗ Postdoctoral Research Associate, Department of Mechanical andAerospace Engineering. Member AIAA.
† Assistant Professor, Department of Mechanical and Aerospace Engineering. Senior Member AIAA.
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