AIAA 2005-0527

Multicomponent and Unsteady Velocity Measurementsof Transient Disturbances
F. Gökhan Ergin∗ and Edward B. White†
Mechanical and Aerospace Engineering
Case Western Reserve University
Cleveland, OH 44106
Two experiments are conducted to investigate transient growth and bypass transition in boundary layers with moderate- to high-amplitude roughness arrays. Unsteady and multicomponent
velocity measurements are obtained in a flat plate boundary layer downstream of a spanwise arrayof cylindrical roughness elements with roughness Reynolds numbers of Rek = 202, 264 and 334.
Measurements of the unsteady streamwise velocities in the wake indicate that growth of a 720 Hz
fluctuation leads rapidly to bypass transition at Rek = 334. The Strouhal number of this fluctuation, St = 0.17, is in good agreement with previous experiments. The locations of maximum fluctuation intensity suggest that the fluctuations result from a Kelvin–Helmholz-type instability of the
spanwise-varying wake. Additionally, steady spanwise velocities are measured in an effort to supply
the initial conditions for the downstream transient growth of the streamwise velocity disturbance.
Time-averaged streamwise and spanwise velocity components are measured using a combinationof multiple slanted and straight hotwire sensors. The multicomponent measurements reveal the
expected horseshoe vortex structure that is symmetric around the roughness centerline.

I. Introduction
Transient disturbance growth in boundary layers is a mechanism that may describe transition scenarios that lie between
classical Tollmien–Schlichting (TS) transition that occurs in low-disturbance environments and bypass transition in
high-disturbance environments. Briefly, transient growth is the algebraic growth and exponential decay of disturbances
generated by three-dimensional (3D) sources such as freestream turbulence and surface roughness. This mechanism
was first identified by Ellingsen and Palm (1975) and Landahl (1980) as an inviscid mechanism of algebraic kinetic
energy growth in shear layers. The physical picture of transient growth can be described using Landahl’s (1980) “liftup” mechanism, the process through which a weakly damped streamwise-oriented vortex transfers high-velocity fluid
from the top of the boundary layer to the lower part and simultaneously transfers low-velocity fluid from the bottomof the boundary layer to the upper part. This action creates an algebraically growing perturbation of the streamwise
velocity that consists of high- and low-speed longitudinal streaks. However, the streamwise vortex is stable and decays.
Once it disappears, the streamwise velocity disturbance cannot be sustained and this too undergoes exponential decay.
The mathematical foundations of transient growth are discussed by Schmid and Henningson (2001) and Reshotko(2001) who describe that transient growth arises because the linearized disturbance equations are not self-adjoint and,
therefore, have nonorthogonal eigenmodes. In boundary layers, disturbances are represented by the superposition of
the Orr–Sommerfeld equation’s discrete and continuous spectrum of eigenmodes. (For a detailed analysis of continuous spectrum concept see the articles by Grosch and Salwen 1978 and Ashpis and Reshotko 1990; 1998.) In classical
TS stability analysis, the most unstable discrete eigenmode represents the observed disturbance growth. However,in the subcritical region where no TS waves are unstable, the superposition of continuous spectrum is essential for
explaining transient disturbances. Because each continuous mode decays at a different rate, a disturbance that is represented by nearly canceling modes may grow as some modes decay more quickly than others and upset the initialcancellation. Different combinations of eigenmodes lead to different transient growth characteristics for the same
∗ Ph.D. Candidate, AIAA Member† Asst. Professor, AIAA Senior Member

Copyright c 2005 by the authors. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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disturbance wavenumbers and frequencies. For any combination of spanwise wavenumber and frequency one initial
disturbance leads to maximum disturbance growth and this is known as the optimal disturbance (Farrell 1988). Optimal 3D disturbances in boundary layers have been investigated by Butler and Farrell (1992), Andersson et al. (1999),
Luchini (2000) and Tumin and Reshotko (2001).
Experiments on roughness-induced transient growth have been performed by White and coworkers (White andReshotko 2002; White 2002; White and Ergin 2003; White et al. 2005) at Case Western Reserve University (Case) andby Fransson et al. (2004) at KTH. The main conclusion of these studies is that surface roughness produces stationary
disturbances that undergo transient growth but that this growth is strongly non-optimal. Moreover, the nature of the
non-optimal behavior is extremely sensitive to details of the roughness configuration (White et al. 2005). The implication is that if one is to accurately predict transient growth of physically realizable (i.e., not optimal) disturbances,initial conditions need to be determined. Fischer and Choudhari (2004) have performed a DNS of White and Ergin’s(2003) experiment and achieved remarkable agreement with the experimental data. However a DNS is costly and it
would be preferable to obtain initial conditions using other methods. Tumin (2003) proposes a method for calculating
amplitudes of discretized eigenmodes of the continuous spectrum from data using a biorthogonality principle and this
could be a means of generating the desired initial conditions. However, Tumin’s method requires knowledge of thecomplete disturbance field and this has not yet been measured in an experiment.
In addition to investigating variations in non-optimality, White et al. (2005) also examined the role of roughness
amplitude. The finding is that the energy of a stationary transient disturbance initiated by a roughness array scales
as Re2k (i.e., as k4 ); that is, the receptivity is nonlinear. The quadratic energy scaling appears to hold from Rek = 16(the smallest amplitude that produces a measurable disturbance) to Rek = 254. Beyond Rek = 254, Rice (2004)
observed unsteady bypass transition in the roughness array’s wake. This bypass is thought to be similar to the highamplitude, isolated-roughness transition scenarios investigated by Acarlar and Smith (1987) using flow visualization,
by Klebanoff et al. (1992) using hotwire anemometry and numerous earlier investigators including Dryden and Tani.
(Early references are given in the review by Sedney 1973.)
Studies of transitional flows over large-amplitude roughness elements that do not explicitly consider transient
growth provide an important background to two outstanding questions about roughness-induced transient growth.
First, beyond what roughness amplitudes is transient growth not observed because of rapid bypass transition and whatis the mechanism for this transition? Second, if bypass does not occur, how is the topology of the flow over theroughness element associated with the streamwise vortex that is the initial condition for transient growth? On the firstpoint, it is suspected that the unsteady hairpin vortices shed from the high-amplitude roughness elements are what led
to the bypass observed by Rice (2004). On the second point, the steady horseshoe vortex that is shown in photographs
and sketches by Acarlar and Smith (1987) may provide the initial streamwise vorticity that leads to transient growth.
A sketch of the horseshoe vortex is given in Fig. 1. If this vortex does exist in the roughness-array configurations
considered in recent experiments, measurements of the spanwise velocity in the array’s wake should resemble theprofiles indicated in the measurement plane on the right side of the figure.
The objective of the present paper is to continue the investigations at Case that are aimed at developing a better
understanding of roughness-induced transition and, in particular, to address the two questions raised above. This isdone in two separate experiments: one that includes measurements of unsteady streamwise velocities and one that
includes measurements of multicomponent (streamwise and spanwise) velocities. The main data analysis techniqueused previously at Case was to measure the steady streamwise disturbances, decompose these into spanwise wavelength components using a spanwise Fourier transform, and monitor each component’s downstream evolution. The
present work extends this by including measurements of the unsteady fluctuations in the wake of large-amplitude
roughness. Temporal Fourier transforms are performed to investigate the fluctuations’ frequency characteristics. The
multicomponent measurements are a preliminary experiment aimed at obtaining the data that is required as input toTumin’s (2003) biorthogonal decomposition. In the long term, multicomponent measurements will provide the initialconditions that are required for predictions of physically realizable transient growth.

II. Experiments and Data Analysis
The current experiments closely follow the previous work by White and coworkers in terms of experimental setup.
The facility used for the experiments is the Case Wind Tunnel, an open-return facility with a 0.71 m×0.71 m×2.7 m
test section and a maximum operation speed of 25 m/s. The test section features an externally-mounted, computercontrolled hotwire traverse mechanism with microstepping motors that move the sensors in three perpendicular direc2
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tions. The tunnel provides a low disturbance level for sensitive boundary-layer stability experiments. Restricted to ahigh-pass frequency of 1 Hz the turbulent u′rms level is 0.05% of the freestream speed.
Measurements are obtained in the boundary layer of a flat aluminum plate that is polished to a near-mirror finish
and has a 25-mm-long elliptical leading edge. The stagnation streamline is controlled by a 300-mm-long trailing-edge
flap. The boundary layer’s virtual leading edge location and the effective unit Reynolds number are estimated using anonlinear least-squares fit to the boundary-layer thickness as a function of streamwise position. At the experiments’nominal freestream speed of 12.2 m/s, the unit Reynolds number is Re′ = 764×103 m−1 and the virtual leading edge
location is at x = −7 mm (upstream of the physical leading edge). Great care is exercised in the alignment of the flat
plate. This is essential both to ensure a zero-pressure-gradient flow and for the multicomponent velocity measurements.
Measurements show that the smooth-plate shape factor is H = 2.56, so the flow is very slightly accelerated but within
an acceptable deviation from the zero-pressure-gradient value, H = 2.59.
Disturbances are generated by a spanwise array of cylindrical roughness elements. Roughness arrays are advantageous because they generate controlled disturbance inputs that permit the use of various data reduction techniques,
especially spatial phase-locked averaging, that greatly improve the signal-to-noise characteristics of the data. The
roughness elements consist of 6.35-mm-diameter, 101.6-µm-thick adhesive-backed labels that are stacked to provide
desired roughness heights. For both the unsteady and multicomponent experiments the roughness elements are placed
300 mm downstream of the physical leading edge and are spaced 19 mm apart in the spanwise (z) direction. This
spacing is referred to as λk .
A.

Unsteady Data Acquisition and Analysis

The previous experiments at Case consisted of time-averaged streamwise velocity measurements only. The present
work provides the unsteady velocity data to explore frequency characteristics of the transient disturbances at higher
values of Rek . For unsteady measurements, 710-, 815- and 915-µm-high roughness elements are used to yield roughness Reynolds number of Rek = 202, 264 and 334. An array of 12 of these elements is applied on the surface but the
spanwise measurement range covers only 8 elements in the middle of the array.
The hotwire velocity measurements are performed in planes perpendicular to the flow direction. Two straightwire sensors are used to measure the boundary layer and freestream velocities simultaneously. In each measurement
plane the spanwise movements of the hotwire probes are constant; the 19 mm spacing is covered in 12 steps. The
wall-normal step is variable with 400-µm steps outside the boundary layer (starting at approximately η = 8) andprogressively smaller steps within the boundary layer until a cut-off velocity is reached and the sensors are moved
away from the plate to their original position. This scanning technique is similar to what White (2002) used, however
the wall-location estimation algorithm has been adjusted to avoid errors that affected earlier results (see White and
Ergin 2004).
At each measurement location, the velocity is sampled at 10 kHz. The signal is divided into a number of segmentswith 1024 data points, which allows the use of an FFT algorithm. The number of segments increases from 8 to 12 to
18 as the boundary layer sensor approaches to the flat surface so that the measurements closest to the wall are wellresolved without requiring that all points be sampled as extensively. In order to reduce post-experiment data analysis
and computer storage requirements, the segments are Fourier transformed individually as the experiment proceeds.
After the first segment of data is obtained, it is transformed while the second segment is being gathered. The averageof the individual transforms is computed and the one-sided, magnitude-averaged, temporal power spectral density(PSD) of that particular measurement location is given by
PSD( fn ) =2Hn2.
fsampWss

In this equation, fn is the nth frequency, Hn are the discrete Fourier coefficients, fsamp is the sampling frequency(10 kHz) and Wss is the Hanning window squared and summed as defined by Press et al. (1992). This particular
expression for the PSD is selected so that a numerical integration of PSD over the frequency domain is equal to u′rms2 .
B.

Multicomponent Data Acquisition and Analysis

Tumin’s (2003) decomposition method requires knowledge of the complete disturbance field including the pressure
disturbance, all three velocity components’ disturbances and their derivatives. Obtaining the complete disturbance in3
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formation is trivial in computational studies because this information exists as part of the numerical solution. However,in an experiment, it is extremely difficult to measure the wall-normal velocity disturbances and perhaps impossible
to measure wall-normal pressure-disturbance profiles. Fortunately, partial disturbance field information, excludingpressure, can be used to decompose the flow field into a finite number of discretized eigenmodes if the disturbanceconsists of downstream vorticity eigenmodes only (Tumin 2003). For stationary transient disturbances these are theonly modes that are thought to be important. One goal of the ongoing work at Case is to obtain the complete multicomponent velocity disturbance field and to use this information as an input to Tumin’s biorthogonal decomposition
procedure. To do this, streamwise and spanwise disturbances will be measured and wall-normal disturbances will be
estimated using the continuity equation. In this preliminary work, simply measuring the streamwise and spanwise
disturbances is the objective.
Measuring the spanwise velocity requires the placement of at least two slanted hotwires with different angularorientations at the same location. For this purpose, a new hotwire sting design is used that carries a combination ofslanted and straight hotwire sensors. This sting assembly includes a shaft that rotates in a streamlined casing and allows
angular calibration of slanted hotwires. Outside the test section, the shaft is connected to an angle indicator and thesting assembly is rigidly attached to the traverse. Inside the test section, the hotwire sensors are placed perpendicular
to the shaft’s axis through a multiple-hotwire holder. The angular position of the sensors can be adjusted by rotating
the spindle of the angle controller with a precision of 0.2◦ . The multiple-hotwire holder is designed to carry four
hotwires. One hotwire is a straight sensor in the freestream; the remaining three sensors are in the boundary layer.
The boundary-layer sensors are positioned in a trident design with one straight sensor in the middle and two slanted
hotwire sensors mounted 9.5 mm one either side of the the center probe. Using a separation that is an integer multipleof the spanwise step permits all three boundary layer sensors to be placed at the same measurement location at threedifferent spanwise traverse steps. Naturally, this requires excellent alignment of the flat surface with the traverse plane
and the sensors with respect to each other and great care is exercised to achieve this alignment.
Slanted hotwire angle calibration is performed following the recommendations of Bruun (1995). First, the yaw
angle of each wire is determined by gradually rotating the sensor in the freestream and monitoring the bridge voltage.
When the bridge voltage is a maximum, the maximum cooling rate is achieved because the impinging velocity isperpendicular to the wire axis. The calibration of all four sensors is performed simultaneously. First, the sting is
positioned so that all hotwires are located in the freestream. Second, it is rotated 5◦ clockwise from its position during
an experiment and a velocity calibration is performed. Third, the sting is rotated 10◦ counterclockwise and anothervelocity calibration is performed for the same tunnel speed range. Finally, data from both velocity calibrations at
two different angles are combined and the calibration constants are computed by using the King’s Law and Hinze’s
formula (Hinze 1959):
1/2
n
.
Ve = A + B E 2 = Ṽ cos2 α + k2 sin2 α
In these equations A, B and n are calibration constants, Ṽ is the freestream speed (during the calibration procedureonly), Ve is the effective cooling velocity, E is the hotwire voltage, α is the sensor’s yaw angle and k is the sensor’s yaw
coefficient. A nonlinear least-square fit is performed to find A, B, n and k values for each sensor. The yaw coefficientis assumed to be constant following Jørgensen (1971). According to Bruun, this method predicts the effective coolingvelocity to within 1% for yaw angles between 0◦ and 70◦ and the error reaches only 15% for a yaw angle of 90◦ .
For the multicomponent velocity measurements, the roughness array with Rek = 202 elements are used. The
approach to hotwire probe movement is similar to what is used by White and Ergin (2003) and White et al. (2005).
Briefly, the sensors are positioned outside the boundary layer, they are moved in the spanwise direction with a spanwise
step of 0.5 mm (38 steps per λk ). Spanwise steps proceed until each individual sensor covers the wake of the array’s
four central elements. Once the spanwise range is completed, the sensors are moved closer to the flat surface, again
with a variable y step, and the spanwise scan is repeated. The entire process repeats until the average streamwise
velocity for a single y position falls below a cutoff velocity. Effective cooling velocities and yaw coefficients for bothslanted sensors are recorded during data acquisition for further analysis.
Multicomponent data is analyzed by decomposing the steady streamwise and spanwise velocity fields into a steady,spanwise-invariant basic state and a steady disturbance asU(x, y, z)= U(x, y) + U ′(x, y, z)W (x, y, z)= W (x, y) + W ′ (x, y, z)

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Although the basic state for a flat plate is the Blasius boundary layer profile, the present work uses an appropriate
experimental basic state which is the average of the boundary layer profiles at the midpoint of two roughness elements.
This basic state is assumed to be 2D and, therefore, W = 0.
The position of each sensor with respect to the flat surface is extremely important for multicomponent data analysis.
To account for any misalignment of sensors in the wall-normal direction, the wall location is estimated for each sensor
at between-roughness z stations. The normalized boundary layer velocity profile obtained from the middle (normal)
sensor is used to calculate the displacement thickness δ∗ which in turn gives δ and this nondimensionalizes the wall
normal position as η = y/δ. The effective cooling velocity obtained from all three sensors are interpolated onto asingle array of heights. Once this is completed, the spanwise and streamwise velocities are resolved from the effective
cooling velocity data obtained from the slanted sensors. The effective cooling velocity of a hotwire yawed in a 2D
flow is given by Hinze’s formula, Ve2 = Un2 + k2Ut2 . In this equation Un and Ut are the normal and tangential velocity
components on each slanted wire. In the present case, the flow has U and W components and Un = U cos α − W sin α
and Ut = U sin α + W cos α. The two slant wires have α = +46◦ and −38◦ and, therefore, combining measurementsof a single point from the two slant wires provides U and W .
Other experimental procedures and data analysis tools related to steady velocity data are very similar to the previous works at Case (see, e.g., White and Ergin 2003; White et al. 2005). Briefly, the experimental basic state ischosen as the average of the between-element velocity profiles and these profiles are used to calculate the displace′ )ment and momentum thickness for each streamwise location. The spanwise rms disturbance of the U velocities (Urms
is obtained from the deviation of phase-lock-averaged velocity profiles from the experimental basic state. The steadyvelocity field is spatially Fourier transformed to obtain the spatial PSD(λ) profiles. Data windowing and averaging
techniques are used to reduce the variance of the spatial power spectrum.

III. Results and Discussion
The baseline configuration for the present work is the roughness configuration with Rek = 202, which produces strong
enough disturbances that allows both unsteady and multicomponent velocity measurements. To begin, Fig. 2 gives thebasic state and steady disturbance profile at x = 310 mm. The thick solid line represents the experimental basic stateand is in exellent agreement with the Blasius profile. The agreement between the experimental and theoretical basic
state profiles is consistent throughout the streamwise domain. Also included in Fig. 2 is the streamwise rms disturbance′ , the root-mean-square deviation of the spatially phase-locked-averaged, steady velocity profiles from theprofile, Urms′
experimental basic state. Urms
reaches a maximum of approximately 8.5% of U∞ around η = 1 which is consistent
with the previous results with similar Rek values.
′ 2
The total steady disturbance energy, Erms , is obtained by integrating the steady disturbance energy, Urms
from η = 0
to the freestream. The downstream evolution of total steady disturbance energy for all three roughness amplitudes
is shown in Fig. 3. In all configurations the total disturbance energy decreases rapidly just aft of the roughness
elements followed by a slow increase farther downstream. In the Rek = 202 and 264 configurations this increase
fades and the total disturbance energy remains roughly constant throughout the remaining streamwise domain. This
general behavior is consistent with the previous results of White and Ergin (2003). However, for Rek = 334 the steady
disturbance energy undergoes a second decrease beginning at x = 425 mm. This occurs because the roughness is ofsufficient amplitude to trip the boundary layer. The resulting turbulent mixing reduces the spanwise velocity gradients
and, consequentially, the steady disturbance energy. The flow is observed to be completely turbulent downstream of
x = 475 mm.

A.

Unsteady Disturbances

The unsteady disturbance field generated by surface roughness is observed by monitoring the downstream evolutionof the fluctuation intensity. Figure 4 shows both the steady and unsteady disturbance field for the baseline Rek = 202
configuration at x = 310, 350, 400 and 500 mm. Although data is available behind eight roughness elements only halfof those are presented for better visualization. The colored contours in this figure indicate the temporal u′rms velocity
fluctuation intensity. (The contours levels follow an exponential distribution so that wide variations in u′rms can bevisualized.) The maximum value of u′rms in this figure is 0.4% of U∞ . Overlaid on the unsteady fluctuation contours
are the steady U velocity contour lines. In the near wake at x = 310 mm, the fluctuations are weak and concentrated inlobes above and downstream of the spanwise edges of the roughness elements. Moving downstream to x = 350 mm thefluctuations become stronger and cover a wider spanwise region. The lobes evolve into inverted V-shaped structures
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that are symmetric about the roughness elements’ centerlines. These structures could be interpreted as elongatedhairpin vortices such as those visualized by Acarlar and Smith (1987), tethered to the roughness elements on their
spanwise edges. At x = 400 mm the fluctuation intensity decreases as the structures split into as many as four lobes andcover an even larger spanwise region. It is possible that the structures all split into four lobes but that two or three lobes
are sometimes observed because of the measurements’ limited spanwise resolution. A critical point is well illustrated
at in the wake of the second roughness element at x = 500 mm. Here the u′rms distribution is split into four nearly
identical lobes and these are centered exactly in the regions where the spanwise gradient of the steady streamwise
velocity (∂U/∂z) is a maximum. This indicates that the fluctuations are not unsteady transient disturbances but arelikely to be a Kelvin-Helmholz instability that arises at the spanwise inflection points of the streamwise velocity. The
behavior for the Rek = 264 is very similar and is therefore omitted from this text for brevity. A second important point
is indicated by Fig. 4. Although great attention is given to produce identical roughness elements, there is great dealof variations among roughness elements’ fluctuation intensities, as much as an order of magnitude in u′rms . Therefore,
spanwise phase-locked averaging technique is not used for unsteady fluctuation intensities.
Figure 5 shows the distribution and evolution of u′rms for the transitional configuration, Rek = 334, with the same
contour levels used in Fig. 4. It is immediately apparent that both the unsteady fluctuation intensity and the steady
disturbance levels are higher than that for the baseline configuration. Immediately downstream of the roughness array,the unsteady velocity fluctuations are concentrated in three regions, one directly above the roughness centerline andtwo on both sides of the roughness element. These regions are most distinct and symmetric for third and fourth
elements from the left. The two lobes located downstream of either side of the roughness element can again beinterpreted as the two legs of the horseshoe vortex. At x = 350 mm, the unsteady fluctuations are greatly intensified
compared to the same streamwise location for the baseline configuration. Additionally, the spanwise region coveredby the fluctuations is increased such that the second and third roughness elements’ unsteady wakes are joined. At
this location, the second roughness element’s fluctuations are the most extensive in the wall-normal direction, as theyreach the edge of the boundary layer. A stronger inverted V-shaped structure is observed downstream of the rightmost
roughness element. At x = 400 mm, abrupt changes occur in both the steady and unsteady disturbances. The unsteadywakes of all roughness elements shown in this figure are joined and most of the boundary layer is contaminated byturbulent fluctuations. The mixing downstream of the middle two roughness elements is most severe and the steadyvelocity contours to from their usual laminar appearance. Further downstream, at x = 475 mm, the flow is completely
contaminated by unsteady fluctuations with high turbulence levels. The steady velocity contours are strongly affectedby turbulent mixing and are well on their way toward a fully developed turbulent boundary layer profile. The maximum
u′rms in this figure is 14% of U∞ .
A quantitative assessment of the unsteady disturbances’ evolution can be made by defining the total unsteady
disturbance energy contained in a single roughness-array wavelength, λk . This is computed by integrating the unsteady
disturbance energy across one wavelength of the disturbance array and from the wall to the freestream:erms =Z λkZ ∞
0

0

u′rms

2

dη dz.

It is obvious from Fig. 5 that the middle two roughness elements generate the strongest disturbances and trigger
turbulent flow. The second roughness element’s wake is selected for data analysis for each of the three roughness
array amplitudes. The downstream evolution of erms generated by this element is shown in Fig. 6. With increasing
downstream distance, the unsteady disturbance energy slightly increases and remains constants for Rek = 202 and 264.
For the transitional Rek = 334 configuration it experiences exponential growth prior to transition.
The important frequencies associated with the transitional disturbance input can be revealed from the frequency
spectrum of the unsteady fluctuations. Figure 7 shows the downstream evolution of the temporal power spectrum along
the centerline of the second roughness element’s wake for Rek = 334. Each plot represents a particular streamwise
location and each curve within a plot represents a particular height in the boundary layer. In the near wake, at x =310 mm, peaks are observed at 360 and a 720 Hz with the latter surrounded by a band of frequencies from 600 to
800 Hz. This band will be referred to as the 700 Hz band. It is unfortunate that these values are harmonics of 60 Hz
but contour plots of the bands’ integrated intensities are similar to those in Figs. 4 and 5 and it is concluded that thesefrequencies are associated with the disturbance input and not electronic noise.

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Between x = 330 and 360 mm, harmonics of the 700 Hz frequency band start to form as ripples in the highfrequency region of the spectra at different heights. These harmonics increase with increasing downstream distanceand gradually give the power spectrum a turbulent appearance. At x = 350 mm, the power spectrum at η = 1 appears
almost turbulent whereas the spectrum at η = 2 still has rippled structure. The spectra at η = 3 and 4 follow the same
trend, lagging the others in x-direction. This suggests that the turbulent fluctuations of roughness induced disturbances
propagate from the lower parts of the boundary layer to the upper parts with increasing downstream distance. By
x = 400 mm, power spectra at all heights indicate a turbulent flow.
Similar to the rms disturbance disturbance energy, erms , a quantitative assessment of the unsteady disturbance
growth can be given for the 700 Hz frequency band using
e700 =

Z 800 HzZ λkZ ∞600 Hz0

PSD( f ) dη dz d f .

0

The downstream evolution of e700 for the second roughness element’s wake is shown in Fig. 8. First, e700 increases
with increasing Rek . Next, the Rek = 202 and 264 configurations’ e700 grows until x = 330 mm then decay somewhat
and remain constant for the rest of the streamwise domain. On the other hand, the transitional disturbance input
experiences a exponential growth until about x = 400 mm and remains fairly constant for the next 75 mm. A plausible
interpretation of this behavior is that the near wakes of all three roughness arrays are initially unstable to the 700-Hz
disturbance. However, the stationary disturbance decreases rapidly (see Fig. 3) and, at some point, the flow becomesstable to these disturbances and their energy decreases. The exception is the transitional case where Rek = 334. Here,the steady disturbance is sufficiently large that the unsteady disturbances are able to bring about transition before they
enter a stable region.
Klebanoff et al. (1992) performed similar experiments using isolated 3.2-mm-high hemispherical and cylindricalroughness elements with diameters of 6.3 mm and 3.2 mm, respectively. Those authors report that bypass transition
is observed at Rek = 325 for hemispherical roughness elements and at Rek = 450 for cylindrical roughness elements.
In that study, the transition is defined as the first occurrence of turbulent bursts in the oscilloscope with a hotwire
positioned on the roughness centerline close to the surface. Klebanoff et al. suggest that the Strouhal number, St, is nota strong function of Rek , when calculated as St = fs δ∗ /Uk , where fs is the shedding frequency, δ∗ is the displacement
thickness and Uk is the flow velocity at roughness height. For hemispherical and cylindrical roughness elements,
Klebanoff et al. report Strouhal numbers of 0.3 and 0.25, respectively. However, Klebanoff et al.’s Fig. 23 indicatesthat for cylindrical roughness elements, St does decrease somewhat with decreasing Rek as Rek approaches values
used in the current work. The shedding frequency of fs = 720 Hz obtained here corresponds to St = 0.17 for Rek=202
which is consistent with a reasonable extrapolation of Klebanoff et al.’s data. Moreover, the height-to-diameter aspectratio of the cylindrical roughness elements is quite a bit lower in the current work than those used by Klebanoff et al.
and this could be another reason for the somewhat lower Strouhal number.
B.

Multicomponent Velocities

Multicomponent velocity measurements were performed for the baseline roughness configuration at x = 310, 320 and
330 mm. The phase-lock averaged spanwise velocity profiles obtained at these locations are presented in Fig. 9. In
each plot, the solid curves represent the spanwise velocity at the location where the curve intersects the plots’ abscissa.
Positive W velocities are to the right, negative values are to the left and 5% of the freestream speed is indicated by theheavy horizontal line in the upper right corner of each plot. Also included in these plots is the approximate locationof roughness element generating the disturbances. Immediately downstream of the roughness at x = 310 mm, thespanwise velocity field is non-zero around the downstream of the edges of the roughness. Moving downstream tox = 320 mm, the spanwise velocities in these regions increase.
Further downstream at x = 330 mm, the spanwise velocity profiles reveal several important features about theorigin of roughness-induced transient disturbances. First, it is important to note that the spanwise velocity is zero at
the roughness centerline and between the roughness elements which confirms the expected symmertries. W is also
zero along both sides of the roughness element. There are two regions on both sides of the roughness centerline thatshow significant spanwise velocities. These regions are almost perfectly symmetric and terminate exactly at the edgeof roughness. On the left-hand-side of the roughness wake, the spanwise velocity value is negative close to the surface,
below the roughness height, and positive away from the surface, above the roughness height. On the right-hand-side
the behavior is the opposite. This is the signature of a counter rotating vortex system with the vortex on the left
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hand side rotating clockwise and the vortex on the right hand side rotating counter clockwise. The rotation directions
suggest that these vortices are the legs of a horseshoe vortex system such as those visualized by Acarlar and Smith(1987) and sketched in Fig. 1. Outside of each leg of the primary horseshoe a second weaker pair of vortices can beobserved that rotate in the opposite direction of the primary pair.
It is important to note that at x = 330 mm the sense of the primary vortex system’s rotation is not consistent withwhat one might expect from the contour plot of the streamwise U velocities given in Fig. 10. This figure shows thatthe centerline velocity is decelerated relative to the flow between the roughness elements. However, the rotation of
the horseshoe vortex would tend to bring high-momentum fluid toward the wall directly behind the roughness element
and produce the opposite effect. In fact, this is exactly what is observed in the far wake both in the current work (notshown) and in previous investigations by Kendall (1981) for an isolated element and by White and Ergin (2003) andWhite et al. (2005) for roughness arrays. In the near wake, the deceleration is the result of a small separated region
on the roughness elements’ downstream side and it takes a considerable streamwise distance for the action of thehorseshoe vortex to overcome this and produce accelerated flow along the centerline. This transition from decelerationto acceleration along the centerline has been identified by Tumin and Reshotko (2004) as one cause for the discrepancy
between optimal and realizable transient growth. Transient growth of the stationary disturbance can only begin after
the primary vortex overcomes the effect of the decellerated flow just downstream of the roughness element.

IV. Summary and Conclusions
Previous transient growth experiments have concentrated mainly on the steady disturbance field and have resolved
some important characteristics such as the effects of roughness height and roughness diameter. The present study isintended to extend these efforts by measuring the unsteady and multicomponent flow fields for moderate- and highamplitude roughness conditions. Unlike the steady disturbance field, the unsteady disturbance field is observed to be
extremely sensitive to variations in roughness amplitudes and the variation of the power spectra amplitudes between
different roughness elements are large despite all the efforts to manufacture identical shapes. For this reason, phaselocked averaging technique is not used during unsteady data analysis. Instead, the roughness element with the strongest
wake is selected. The unsteady fluctuation intensity and the steady velocity contours are plotted together, revealing
signs of hairpin and horseshoe vortices and a Kelvin–Helmholz-type instability mechanism. Bypass transition occurs
because of this mechanism at Rek = 334 when the steady disturbance is sufficiently large that the unsteady disturbances
grow to transitional amplitudes before they enter as stable region as the stationary disturbance decays. The Strouhal
number calculated for the present roughness configurations are consistent with the previous studies.
Although the data for the multicomponent velocity measurements is limited at present, the results are quite interesting. The spanwise velocity measurements clearly show a horseshoe vortex system and this supports the notion thatroughness-induced transient growth is consistent with Landahl’s (1980) lift-up concept. Multicomponent velocity measurements are critical to an improved understanding of realizable transient growth because it is the streamwise-oriented
vortex (i.e., the V ′ and W ′ disturbances) that is the initiator of transient growth. Future studies will be concentrated
on estimating streamwise and spanwise velocity gradients from experimental data and calculating the wall-normal
velocity from the continuity equation. This will provide the necessary initial disturbance conditions for computational
works and will form the foundation for meaningful growth predictions for physically realizable disturbances.

Acknowledgments
This work is supported by the Air Force Office of Scientific Research under grant number F49620-02-1-0058 andis managed by Dr. Thomas Beutner. Additionally, the authors wish to thank Sam Cordaro, Wayne Schmidt, John
Weber and Ron Hudak of the Case Engineering Services Fabrication Center for their suggestions, cooperation andsupport. F.G. Ergin acknowledges the Case School of Engineering for its support through a Case Prime Fellowship.

References
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layers. Phys. Fluids 11(1), 134–150.
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Ashpis, D. and E. Reshotko (1990). The vibrating ribbon problem revisited. J. Fluid Mech. 213, 531–547.
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104

Rek = 334
Rek = 264
Rek = 202

erms

102
100
10−2
10−4
Figure 1: Schematic illustration of the horseshoe vortex
about a cylindrical roughness element. The curves on
the plane to the right of the figure indicate profiles of thesteady spanwise velocity.

10−6
300

350

400
x [mm]

450

500

Figure 6: Streamwise evolution of the unsteady disturbance energy, erms .

6
U
′
Urms

5
4
η

3
2
1
0
0

0.25

0.5
0.75
′
U, 10Urms

1
104

Figure 2: Basic state and steady disturbance profiles for
x = 310 mm, Rek = 202.
e700

0.025
Rek = 334
Rek = 264
Rek = 202

Erms

0.02

Rek = 334
Rek = 264
Rek = 202

102
100
10−2
10−4

0.015

10−6
300

0.01
0.005

350

400
x [mm]

450

500

Figure 8: Streamwise evolution of the 700 Hz band’s disturbance energy, e700 .

0
300

350

400
x [mm]

450

500

Figure 3: Streamwise evolution of the steady disturbanceenergy, Erms .
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American Institute of Aeronautics and Astronautics

5
x = 310 mm

x = 400 mm

x = 350 mm

x = 500 mm

4
3
η
2
1
0
5
4
3
η
2
1
0
0

1

2
z/λ k

3

4 0

1

2
z/λ k

3

4

3

4

Figure 4: Unsteady velocity fluctuation intensity evolution for Rek = 202.

5
x = 310 mm

x = 400 mm

x = 350 mm

x = 475 mm

4
3
η
2
1
0
5
4
3
η
2
1
0
0

1

2
z/λ k

3

4 0

1

2
z/λ k

Figure 5: Unsteady velocity fluctuation intensity evolution for Rek = 334.

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American Institute of Aeronautics and Astronautics

η=1
η=2
η=3
η=4

1E-2

PSD

1E-4
1E-6
1E-8

1E-2
1E-4
1E-6
1E-8

x = 310 mm

x = 360 mm

1E-10

1E-10

PSD

10

100

1000

10

1E-2

1E-2

1E-4

1E-4

1E-6

1E-6

1E-8

x = 370 mm

1E-10

1E-10

PSD

10

100

1000

10

1E-2

1E-2

1E-4

1E-4

1E-6

1E-6

1E-8

100

1000

1E-8
x = 380 mm

x = 330 mm
1E-10

1E-10
10

PSD

1000

1E-8
x = 320 mm

100

1000

10

1E-2

1E-2

1E-4

1E-4

1E-6

1E-6

1E-8

100

1000

1E-8
x = 390 mm

x = 340 mm
1E-10

1E-10
10

PSD

100

100

1000

10

1E-2

1E-2

1E-4

1E-4

1E-6

1E-6

100

1000

1E-8

1E-8

x = 400 mm

x = 350 mm
1E-10

1E-10
10

100

1000

10

100

f [Hz]

1000
f [Hz]

Figure 7: Temporal power spectra of the unsteady disturbances for Rek = 334.
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American Institute of Aeronautics and Astronautics

5

x = 310 mm

4
η

3
2
1
0
5

x = 320 mm

5

4
η

4

3
η

2

2

1

1

0
5

3

0
-0.5

x = 330 mm-0.167

4
η

0.167

0.5

Figure 10: Streamwise velocity contours for x = 330 mm,
Rek = 202. Contour lines indicate 10% increments of U∞ .

3
2
1
0
-0.5

0
z/λk

-0.167

0
z/λk

0.167

0.5

Figure 9: Wall-normal profiles of spanwise velocity for
Rek = 202. The rectangular boxes indicate the approximate size and location of the roughness elements and theheavy horizontal line indicates 5% U∞ .

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American Institute of Aeronautics and Astronautics