12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

STABILITY AND TRANSITION OF A LAMINAR SEPARATION BUBBLE ON
A FINITE WING

Connor Toppings
Department of Mechanical and Mechatronics Engineering
University of Waterloo
200 University Avenue W, Waterloo, Ontario, Canada N2L 3G1
ctoppings@uwaterloo.ca

Serhiy Yarusevych
Department of Mechanical and Mechatronics Engineering
University of Waterloo
200 University Avenue W, Waterloo, Ontario, Canada N2L 3G1
syarus@uwaterloo.ca

ABSTRACT

sensitive to changes in the pressure distribution on a wing or
turbine blade resulting from changes in angle of attack.
In most practical applications, three-dimensional end effects are present, and the effective angle of attack changes inthe spanwise direction. Consequently, changes in LSB transition dynamics may be expected to occur along the span ofa wing or turbine blade. In common design analysis methods, such as lifting line theory and blade element momentum
theory, spanwise variations are modelled under the assumption that the flow at a given spanwise location is analogous
to the flow over a two-dimensional airfoil at an equivalenteffective angle of attack. This assumption enables the useof empirical lift and drag data from two-dimensional airfoil
tests to predict the performance of practical three-dimensional
lifting surfaces. Although numerous studies have consideredtransition induced by an LSB on two dimensional airfoil orflat plate geometries (e.g., Brendel & Mueller 1988; Watmuff
1999; Marxen et al. 2012, 2013), there are few investigations
into the transition process on a finite lifting surface. In fact,
recent work by the authors suggests that LSB topology on finite lifting surfaces can differ notably from that expected due
to effective angle of attack variation along the span (Toppings
et al., 2021; Toppings & Yarusevych, 2021). Therefore, thegoal of this work is to study stability and transition within an
LSB on a finite wing so as to enable more accurate transition
prediction methods for situations in which three-dimensional
wing or blade tip/root effects are present.

Disturbance amplification in a laminar separation bubble
forming on the suction surface of a cantilevered NACA0018
wing at an angle of attack of 6◦ and Reynolds number of
1.25 × 105 is studied using a combination of particle image
velocimetry measurements and local and non-local linear stability analyses. The experimental measurements reveal the formation of largely two-dimensional roll-up vortices in the separated shear layer at a nearly constant wavenumber and frequency, despite the substantial reduction in the effective angleof attack near the wing tip. The most amplified wavenumber
and its growth rate are accurately predicted by linear stability
theory outside of the regions of the separation bubble dominated by three-dimensional end effects, with negligible differences between local and non-local stability analyses.

INTRODUCTION
On lifting surfaces at low chord Reynolds numbers (Rec <5 × 105 ), boundary layer transition often occurs downstreamof the location of minimum pressure (Carmichael, 1981). If
the laminar boundary layer separates, transitions to turbulence,and reattaches, a laminar separation bubble is formed (Tani,
1964). The reduction in stability of the separated laminar shear
layer leads to transition through the rapid convective amplification of disturbances (Marxen et al., 2012). Once shear layer
perturbations reach substantial magnitudes and non-linear interactions begin, the dominant disturbances evolve into shear
layer vortices, which are shed near the location of maximum bubble thickness (Brendel & Mueller, 1988). Threedimensional deformations of the shear layer vortices eventually cause breakdown to turbulence and the development ofa turbulent boundary layer downstream of the LSB (Marxen
et al., 2013). Relative to the fully attached flow condition
seen when transition occurs prior to separation at higher chord
Reynolds numbers, LSB formation reduces lifting surface efficiency (Mueller & DeLaurier, 2003). Because the strength of
the adverse pressure gradient strongly influences the locationof laminar separation, the stability characteristics of the separated shear layer, and the possibility of reattachment, LSBs are

EXPERIMENTAL SETUP
Experiments were conducted on a rectangular cantilevered NACA 0018 wing model with an aspect ratio of
AR = 2.5 and a chord length of c = 0.2 m at a chord Reynolds
number of Rec = 1.25 × 105 and an angle of attack of 6◦ inthe recirculating wind tunnel at the University of Waterloo.
The test section has a square 0.6 m × 0.6 m cross section and afreestream turbulence intensity of less than 0.08%.
To measure the mean three-dimensional velocity fieldof the LSB on the wing, stereo particle image velocimetry(PIV) was performed in multiple z-normal planes along the1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
In local stability analysis, the base flow is assumed to beparallel, thus ∂ U/∂ x, ∂ P/∂ x = 0, and for the case of twodimensional disturbances, following ansatz is applied to the
disturbance modes:
q0 = q̂(y)ei(αx−ωt) + c.c.

where q̂(y) is a complex function of only y that describes
the mode shape, α = αr + iαi is the complex wavenumber,ω = ωr + iωi is the complex frequency, and c.c. is the complex conjugate. When analysing convective instabilities suchas those that dominate the transition process in LSBs, ωi isassumed to be zero and the disturbance mode is convectively
unstable if αi < 0. Local stability analysis does not accountfor streamwise changes in the base flow or disturbance modes.
In non-local stability analysis, the base flow and the disturbance mode shape functions are allowed to vary slowly
in the streamwise direction. By neglecting only the termν∂ 2 u0 /∂ x2 in eq. (1), the parabolised stability equations (PSE)
are obtained. In non-local stability analysis using the PSE,
two-dimensional disturbance modes take the following form:

Figure 1: PIV light sheet orientations and surface at-tached coordinate system definition.
wingspan, termed the side-view configuration, as illustrated in
fig. 1 using two LaVision Imager sCMOS 5.5Mpx cameras at asampling frequency of 52.35 Hz. In addition, two-component
time-resolved measurements were performed in planes locatedat z/c = 0.95, 1.30, 1.66 and 2.01 using two Photron FastCam
SA4 1Mpx cameras at a sampling frequency of 3.88 kHz. The
flow was seeded with water-glycol fog particles, which were
illuminated with a Photonics DM20-527 Nd:YLF pulsed laser
and imaged in dual frame mode. Near the wing root and tip,where end effects substantially influence LSB development,
the spanwise distance between side-view stereo-PIV planes
was 0.05c.
Another PIV configuration was used to observe the spanwise structure of shear layer vortex shedding in the aft portionof the LSB near the wing tip, where the most significant threedimensional effects are expected to occur. Two-component
PIV using two side-by-side LaVision Imager sCMOS cameras
was performed in a single plane tangent to the suction surfaceof the wing, referred to as the top-view configuration.
Measurements using each PIV configuration were also
performed on a nominally two-dimensional NACA 0018 airfoil model as a baseline case. For all PIV measurements, image acquisition and processing were performed using the LaVision DaVis 10 software. Velocity fields were calculated using
an iterative multi-pass cross-correlation procedure with a final
interrogation window size of 16px × 16px and 24px × 24px,
resulting in a vector pitch of 5 × 10−4 c and 1.9 × 10−3 c, forthe side-view stereo-PIV and top-view configurations, respectively, with 75% overlap between windows.

i( xx α(ξ )dξ −ωt)q0 = q̂(x, y)eZ ∞0∇ · u0 = 01+ u0 · ∇ U + (U · ∇) u0 = − ∇p0 + ν∇2 u0
ρ0+ c.c.

(3)

q̂∗ ·

∂ q̂
dx = 0∂x(4)

where ∗ denotes complex conjugation. This condition ensures that changes in the energy of the disturbance are contained within α(x), whereas q̂(x, y) only contains changes tothe shape of the mode.
Local and non-local stability calculations were performed
using the mean velocity fields at each side-view stereo-PIV
measurement plane to explore how the stability characteristicsof the separated laminar shear layer vary along the span. To reduce the sensitivity of wall-normal velocity derivatives to experimental noise, the mean velocity fields were preprocessedby applying a smoothing spline fit to the measured velocity
profiles at each x location. The linearised Navier-Stokes equation was discretised in the wall-normal direction with a spectral collocation technique using Chebyshev polynomials. For
non-local analysis using the PSE, streamwise derivatives of the
disturbance modes were evaluated using first order backward
differencing, and the solution for q0 was obtained by marching in the streamwise direction. More details of the numericalmethods employed in the stability calculations can be found inWesterbeek (2020). Due to the limited extent of the field ofview of PIV measurements, calculation of amplification factors (van Ingen, 2008) by integrating αi starting from the critical point is not possible. Since the most rapid disturbance
amplification is expected to occur in the separated shear layer
contained entirely within the streamwise extent of the PIV fieldof view, amplification factors were estimated starting the integration from the upstream end of the PIV field of view. The
uncertainty of the linear stability calculations was estimatedusing a Monte-Carlo method whereby the mean velocity fields
were perturbed by normally distributed noise of the same amplitude as the uncertainty in the velocity field measurements.

Previous studies of instability mechanisms in LSBs haveshown that the initial stages of disturbance amplification are
well-modelled by linear stability theory (e.g., LeBlanc et al.
1989; Dovgal et al. 1994; Diwan & Ramesh 2009). Linear stability theory assumes that disturbance amplitudes are infinitesimally small so that non-linear interactions between disturbance modes can be neglected (e.g., Herbert & Spalart 1992).
The linearised incompressible Navier-Stokes equations govern
the evolution of linear disturbances (q0 = [u0 , p0 ]) in a steadybase flow (U, P) (Herbert, 1994):∂t

RAn additional condition is required to partition the streamwise
growth between the shape function q̂ and the complex exponential (Herbert, 1994). This is usually done using the condition:

STABILITY ANALYSIS METHODS

∂ u0

(2)

(1)

In this study, both local and non-local linear stability analyses
are conducted to study the influence of non-parallel effects on
the stability characteristics of the separated shear layer of the
LSB on the wing model.

212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
enced by the distance of the separated shear layer from thewing surface Dovgal et al. (1994) which varies substantiallyin the spanwise direction.
The spanwise and streamwise variations in the frequencyof the most unstable disturbance mode in the separated laminar shear layer predicted by local stability analysis are illustrated in fig. 4a, which shows relatively small spanwise variations for z/c < 2.00. For z/c > 2.25, the most unstable frequency is substantially reduced because the boundary layer
remains attached near the wing tip. The expected frequencyof the most amplified disturbances on the wing based on theeffective angle of attack variation and empirical correlationfrom time-resolved PIV measurements on the same NACA
0018 two-dimensional airfoil is shown by the dashed line in
fig. 4b, with the shaded area representing the experimental uncertainty bounds. The markers indicate the central frequencyof the most amplified disturbances on the wing estimated fromspectral analysis of time-resolved PIV measurements of wallnormal velocity fluctuations. Whereas a substantial reductionin the most amplified frequency near the wing tip is expectedfrom the two-dimensional airfoil measurements as a result of
the reduction in effective angle of attack near the wing tip, themeasurements on the wing do not show a corresponding reduction near the wing tip, indicating that the transition process
in the LSB does not follow the trend expected from sectionalvariation of the effective angle of attack along the span. The
solid line in fig. 4b indicates the frequency of the disturbancemode with the largest amplification factor at the approximate
location of shear layer roll-up (x/c ≈ 0.40, fig. 3) from local linear stability analysis. On the average, the results follow
the trend of the experimentally measured frequencies on thewing. This suggests that spanwise changes in the mean LSB
structure on the wing are at least partially responsible for thehigher frequency of the most amplified disturbances measured
on the wing relative to the two-dimensional airfoil at the same
effective angle of attack. Specifically, the local thickening of
the LSB near the wing tip (fig. 2) is expected to contribute tothe observed increase the most amplified frequency in the outboard portion of the bubble.

Figure 2: Skin friction lines and streamsurface of separa-tion coloured by wall-normal height. Red line: locationof data presented in fig. 5
RESULTS AND DISCUSSION
The three-dimensional shape of the LSB formed on thesuction surface of the wing model is illustrated in fig. 2, which
presents skin friction lines calculated from the near-wall velocity gradient and the streamsurface of separation that emanatesfrom the separation line. The topology is that of an open bubble, whereby spanwise flow enters the LSB at the wing root
and tip, resulting in a spanwise flow within the LSB near its
ends. The colouring of the streamsurface of separation indicates the wall-normal extent of the LSB, revealing that local thickening of the bubble occurs near the wing root and tip
where fluid flows into the bubble.
As detailed in Toppings et al. (2021) and Toppings &Yarusevych (2021), the LSB forming on the finite wing differs from the LSB forming on the same two-dimensional airfoil section at the same effective angles of attack. In contrast to
the downstream shift in LSB location expected due to the substantial reduction in effective angle of attack near the wing tip,
the LSB location on the wing remains largely uniform along
the span, except in relatively close proximity to the tip. Alongthe majority of the span, boundary layer separation occurs at
x/c ≈ 0.25, and reattachment at x/c ≈ 0.45.
The formation and development of coherent structures on
the outboard portion of the wing is depicted in fig. 3, which
presents instantaneous streamwise velocity contours from the
top-view PIV configuration. Because the light sheet for the
top-view PIV configuration was positioned to intersect the top
halves of the shear layer vortices, the roll-up vortices appear
as spanwise bands of increased streamwise velocity. Shearlayer roll-up occurs near x/c ≈ 0.40, and the spanwise uniformity of the vortices for z/c < 2.15 suggests that the transition process in a three-dimensional LSB on a wing does not
respond to local changes in the effective angle of attack, butthat the vortex shedding locks onto a single instability modeacross the entire LSB. For z/c > 2.20, vortex shedding ceases
abruptly, and transition does not occur within the field of view
because downwash from the wing tip vortex suppresses boundary layer separation. The high degree of spanwise uniformity
in the roll-up vortices is unexpected, since the growth rates andwavenumbers of unstable disturbances are known to be influ-

Wavelet analysis using the morse wavelet (Olhede &Walden, 2002) was performed to obtain the wavenumbers andgrowth rates of the most unstable disturbances in the separated shear layer using the wall-normal velocity fluctuation
measurements from each side-view stereo-PIV measurement
plane. The wavenumber of the most amplified disturbancemode in the separated laminar shear layer is plotted in fig. 5afor the spanwise region within which vortex shedding occurs(0.10 ≤ z/c ≤ 2.20). The data are taken at the streamwise location depicted by the red line in fig. 2, located three-quartersof the streamwise distance from the separation point to thepoint of maximum LSB height and at y = δx∗ . This location
was chosen so that disturbance amplitudes remain measurable
but relatively small (< 0.05U∞ ) to ensure that the assumptionsof linear stability theory are valid for neglecting non-linear
interactions. Within 0.15 < z/c < 2.00, the most amplified
wavenumber obtained using wavelet analysis from the experiment is largely invariant, in agreement with the top-view PIV
results (fig. 3). However, a large reduction in the most amplified wavenumber occurs near the wing root at z/c = 0.10,
and a progressive increase in the most amplified wavenumber
is seen in the experimental data near the wing tip (z/c ≥ 1.75).
These changes are attributed to three-dimensional effects near
the ends of the LSB. Away from the wing root and tip, the mostamplified wavenumbers predicted by local and non-local linear
stability maintain a nearly constant value across the LSB. The

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

Figure 3: Snapshot of shear layer vortex development in the aft portion of the LSB near the wing tip from top view PIV.

Spanwise vortices appear as bands of increased streamwise velocity.

0

5

10

15

20

25

30

0.45

20

0.40
15

0.35
10

0.30
5

0.25

0.20
0.00

0.50

1.00

1.50

2.00

2.50

0
0.00

0.50

1.00

1.50

2.00

Figure 4: Frequency of most unstable normal disturbance mode calculated from local stability analysis (a), and comparisons of the spanwise variation of the most amplified frequency expected from two-dimensional airfoil measurements,
finite aspect ratio wing measurements, and local stability analysis (b).

differences between the most amplified wavenumbers measured in the experiment and those predicted by local and nonlocal analyses are of similar magnitude as the experimental
uncertainty for 0.15 ≤ z/c ≤ 2.0, but near the ends of the LSB,
the changes in the experimentally measured wavenumber are
not well predicted by either local or non-local linear stability analysis. These results suggest that non-parallelism andboundary layer growth do not play a major role in disturbance
amplification within the LSB on the finite wing.

reduction in growth rate is also observed near the wing root,
where the interaction with junction flow reduces the size of
the LSB. The maximum growth rates from local and non-local
stability analyses do not show any substantial differences, providing further evidence that non-local effects do not play a major role in the LSB on the finite wing. There is a good agreement between the maximum growth rates of the most amplified wavenumber from the experiment and from linear stability
analyses for z/c ≤ 2.00. Closer to the wing tip, the linear stability analysis tends to over-predict the maximum growth rateof the most amplified wavenumber relative to the experiment,
although the differences fall within the expected uncertainty ofstability predictions (Boutilier & Yarusevych, 2013).

The changes in frequency (fig. 4b) and wavenumber(fig. 5a) of the most amplified disturbance mode near the wing
tip imply an increase in phase speed. Since fig. 3 shows that theroll-up vortices become progressively weaker for z/c > 2.10,
the changes in the most amplified frequency and wavenumber near the wing tip are likely the result of slower growth ofthe primary instability mode in the region influenced by downwash from the wing tip vortex.

CONCLUSION
The LSB forming on the suction surface of a rectangular,cantilevered NACA0018 wing with an aspect ratio of 2.5 wasinvestigated using particle image velocimetry and linear stability analyses at an angle of attack of 6◦ and a chord Reynolds
number of Rec = 1.25 × 105 . Except for the regions in the immediate vicinity of the wing root and tip, the locations of sep-

In fig. 5b, the growth rates of the most amplified
wavenumber at each side-view stereo-PIV measurement location are plotted. Near the wing tip, a substantial reduction in
the growth rate occurs where the LSB thickness decreases asa result of downwash from the tip vortex. A relatively smaller

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2.50

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022

200

150

100

100

50

0
0

0.5

1

1.5

2

2.5

Figure 5: Spanwise variation of most amplified wavenumber (a), and maximum growth rate of most amplified wavenumber(b).

aration and reattachment and the frequencies and wavenumbers of the most amplified disturbances are largely spanwise
uniform, despite the significant reduction in effective angle ofattack near the wing tip and the spanwise variations in LSB
thickness. This leads to the formation of essentially twodimensional roll-up vortices in the aft portion of the LSB at anearly spanwise invariant wavenumber in the midspan regionof the wing. Near the wing root and tip, the most amplified
wavenumber decreases and increases, respectively, but thesechanges are not accurately predicted by linear stability analysis. However, a reduced growth rate of the most amplified
wavenumber in the regions influenced by three-dimensional
effects near the wing root and tip is seen in both the experimental measurements and linear stability analysis. The growthrate of the most amplified wavenumber is accurately predictedby local and non-local stability analyses over a majority of thewing span, except for the region within 0.35c of the wing tip
where linear stability analysis over-predicts the growth rate of
the most amplified disturbance mode. The close agreement between local and non-local linear stability analyses leads to theconclusion that non-parallel effects do not significantly influence disturbance amplification in the typical short LSB forming on the finite wing at aspect ratios for which the span sufficiently separates the flow regions affected by the end conditions.

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Osaka, Japan (Online), July 19-22, 2022
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