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

THE RELATION BETWEEN WALL PRESSURE AND VELOCITY FLUCTUATIONS IN A
TURBULENT SEPARATION BUBBLE
Sen Wang
Department of Mechanical Engineering
University of Alberta
Edmonton, AB
sen6@ualberta.ca

Bradley Gibeau
Department of Mechanical Engineering
University of Alberta
Edmonton, AB
bgibeau@ualberta.ca

Sina Ghaemi
Department of Mechanical Engineering
University of Alberta
Edmonton, AB
ghaemi@ualberta.ca
ABSTRACT
We investigated the relationship between velocity andwall-pressure fluctuations in a turbulent separation bubble
formed just upstream of an airfoil trailing edge. The
experiments were conducted at a chord-based Reynolds
number of 720,000 and an angle of attack of 9.7º.
Synchronized measurements of wall-pressure fluctuations andtime-resolved planar particle image velocimetry were
performed in a streamwise-wall-normal plane along themidspan of the airfoil. A turbulent separation bubble that wastriangular in shape formed between two shear layers thatemerged from the suction and pressure sides of the wing. The
flow unsteadiness was characterized using the Strouhal
number, Stl, defined based on the characteristic length of theseparation bubble and freestream velocity. The results showthat wall-pressure fluctuation upstream of the detachment
correlates with the unsteadiness in the shear layer above theseparation bubble at Stl of approximately 0.04. The positive
and negative pressure fluctuations are related to the expansion
and contraction of the TSB (i.e., the breathing motion) throughvariations of the adverse pressure gradient.

INTRODUCTION
Trailing-edge (TE) separation typically refers to thedetachment of a turbulent boundary layer from the aft sectionof an airfoil due to an adverse pressure gradient. The separated
flow forms a turbulent separation bubble that features anintermittent separation front and shear layers that roll up into
spanwise vortices (Thompson and Whitelaw, 1985). The
incoming turbulent boundary layer (TBL) contains highfrequency unsteadiness, while the separation bubble exhibits
lower frequencies due to the vortex shedding and breathing
motions (Mohammed-Taifour and Weiss, 2016). The lattermotion results in the large-scale spatial variation of the
turbulent separation bubble (TSB) and has been subjected torecent investigations (Weiss et al., 2015; Mohammed-Taifour
and Weiss 2016; Le Floc'h et al., 2020).
Several explanations were proposed to address the lowfrequency breathing motion and the hypotheses were mainly
categorized into two groups based on the location of
unsteadiness. One group attributed the breathing of theseparation bubble to the unsteadiness downstream of thedetachment point. Numerous studies pointed out that the
instantaneous imbalance of the entrainment and reinjection of1

fluid due to the propagation of vortices is responsible for lowfrequency breathing (Eaton & Johnston, 1982; Na & Moin,
1998). Another group ascribed the breathing motion to theperturbations in the incoming flow. In the DNS of Wu et al.
(2020), the breathing motion was attributed to Gӧrtler-typeinstabilities upstream of the detachment by examining the
low-frequency flow structures using dynamic mode
decomposition. A recent experimental work by MohammedTaifour & Weiss (2021) provided further evidence by showing
the suppression of TSB expansion through controlled
perturbation in the upstream flow. Both groups of mechanisms
were broadly investigated in geometry-induced and pressureinduced separation. However, it is still unclear whether thesemechanisms are applicable in TE separation since the flow
condition is more complex.
The low-frequency unsteadiness in a separated flow has aStrouhal number, Stl, that is several orders of magnitude lower
than the fluctuations in the TBL, and the corresponding Stl isusually on the order of 0.01. This result is supported bynumerous investigations of TSBs induced by pressure
gradients on flat plates (Weiss et al., 2015; MohammedTaifour & Weiss 2016; Le Floc'h et al., 2020) and by forwardfacing steps (Camussi et al., 2008; Graziani et al., 2018).
There are also some documented results suggesting that lowfrequency breathing features a Stl on the order of 0.1 (Eaton
and Johnston 1982; Wu et al. 2020). The discrepancy in Stl is
not well-understood and may be a result of varying flow
conditions.
In the context of airfoil separation, several studies havehinted at a possible connection between the low-frequency
breathing of the TSB and the force fluctuations on the airfoil
by comparing their dominant Stl (Broeren & Bragg, 1998;
Weiss et al., 2015, Liu & Xiao, 2020). It, therefore, seems
pertinent to further investigate the unsteadinesses of TSBs if
we wish to fully understand the vibration and noise associatedwith TE separation. In particular, investigating therelationship between the velocity field and wall-pressure
fluctuations will improve our understanding of breathing
motion and how it impacts airfoil performance.
The objective of the present investigation is to determine
the relationship between the wall-pressure fluctuations and theflow motions in the vicinity of the separation bubble. The
study was carried out using synchronized wall-pressure andtime-resolved planar particle image velocimetry (PIV)
measurements. The investigation provides insight into the12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022
origin of the pressure fluctuations within different frequency
bands.

EXPERIMENTAL SETUP
The present experiments were conducted in a two-story,
closed-loop wind tunnel with a cross-section of 2.4 × 1.2 m2.
A NACA 4418 airfoil was installed vertically in the windtunnel at an angle of attack of 9.7º. The airfoil had a chord
length of c = 975 mm and a span of 1190 mm. A tripwire wasinstalled 0.2c downstream of the leading edge. The
experiments were conducted at a freestream velocity (U∞) of
11.2 m/s, which corresponds to a chord-based Reynolds
number of 720,000. The origin O of the coordinate system islocated at the center of the trailing edge as shown in Figure1(a). The streamwise, wall-normal, and spanwise directions
are denoted by x, y, and z, respectively.
Time-resolved planar PIV measurements were conductedat the trailing-edge region of the airfoil. The measurements
along the midspan in the x-y plane employed two high-speed
cameras (Phantom v611). The stitched field of view, as shownby the blue dash-line boxes in Figure 1(a), was 500 × 125 mm2(x × y) with a digital resolution of 0.2 mm/pixel. An Nd:YLF
laser (Photonics Industries DM20-527-DH) was used toilluminate the measurement plane. The data acquisition was
performed at 2 kHz. In total, ten time-resolved datasets werecollected; each set was 3.45 s. The signal-to-noise ratio of the
PIV images was improved by removing the ensemble
minimum and normalizing with the ensemble average. The
vector fields were obtained by performing a sliding sum-ofcorrelation using three successive image pairs and a final
interrogation window size of 32 × 32 pixels with 75% overlap.
Wall-pressure
measurements
were
conducted
simultaneously with the time-resolved PIV measurements
using infrasound microphones (1/2-inch Brüel & Kjær 4964)
paired with pre-amplifiers (Brüel & Kjær 2669). The
microphones were calibrated using a constant-frequency
calibrator (Brüel & Kjær 4231) at 1 kHz. The sampling
frequency was set to 20 kHz, and the pressure signals wererecorded using Simulink Real-Time through a Speedgoat
Target Machine with a 16-bit input/output module (model
IO135). The trigger signals of the PIV system were also
recorded for the purpose of data synchronization in postprocessing. The microphones were placed behind pinholes toavoid spatial filtering of the pressure signals. The wallpressure measurements were carried out at seven streamwise
locations along the midspan. A detailed view of the pinholeson the wing surface is shown in the zoomed-in view within
Figure 1(a). The most upstream pinhole was at x/c = −0.30,
and the most downstream pinhole was at x/c = −0.08. The
spacing between the pinholes was approximately 0.035c. The
freestream velocity was carefully adjusted to have the mean
separation point coincide with the pinhole at x/c = −0.19 suchthat there were three pinholes upstream and downstream of themean separation point.
The cross-section views of the microphone installation in
the wall-normal-spanwise and streamwise-wall-normal planes
are provided in Figure 1(b) and (c), respectively; a local
coordinate system is illustrated with its local origin O′ at thecenter of the pinhole, on the wing surface. The local
coordinate axes (x′, y′, and z′) are in the same directions as
those associated with origin O. As shown in Figure 1(b), themicrophone is threaded into the dark grey region, and its
diaphragm is highlighted by the blue line. A cylindrical cavity

2with a radius of 5 mm and a depth of 3 mm is formed as shownin the light grey region in Figure 1(b) and (c). The pinhole hasa length of 3.5 mm and a diameter of 0.5 mm that connects thecavity to the wing surface. The length of the pinhole is morethan twice the size of its diameter, as suggested by Shaw
(1960). The pinhole-cavity device acts as a Helmholtz
resonator (Tsuji et al., 2007), resulting in amplitude and phase
distortion. These distortions were corrected using the transfer
function of the resonator which was determined using a
dynamic calibration technique. A Wiener filter was employed
to remove the wind tunnel background noise using an
additional microphone placed in the freestream.

RESULTS AND DISCUSSION
In this work, the instantaneous velocities in the x and ydirections are denoted by U and V, respectively. The
corresponding fluctuating velocity components are u and v,
and the fluctuating wall pressure is p. An ensemble average intime is represented by 〈…〉 and the subscript 'rms' is used toindicate the root-mean-square value of a variable.
The power spectral density (PSD) of p/prms at various
pinhole locations are presented in Figure 2. The Strouhal
number Stl is defined as fl/U∞, where l is the characteristic
length of the mean separation bubble. In this work, l is definedas the distance between the most upstream and downstream
points of the mean TSB and is equal to 0.25c. The PSDs
upstream of the mean detachment point (x/c = -0.19) for thepressure sensors located at x/c = -0.30, -0.26, and -0.23 are
relatively flat until Stl ≈ 1, followed by a rapid drop in power
at higher Stl. For the pressure sensors located at x/c = -0.16, 0.12, and -0.08, a broad peak in the PSDs is present at Stl ≈0.3. For these locations, the rapid decay of PSD starts at alower Stl around 0.5. With increasing x/c, the mid-frequency pappears to be stronger, and the high-frequency p is lesssignificant. The results of Figure 3 generally agree with theobservations of Weiss et al. (2015). In their work, they
concluded that the high-frequency pressure fluctuations are
related to small-scale turbulence fluctuations, the midfrequency (Stl = 0.2 to 0.4) pressure fluctuations are inducedby the flapping of the shear layer, and the low-frequencypressure fluctuations are associated with the breathing motion.
In the present work, the difference in power between lowfrequency and mid-frequency pressure fluctuations is not well
pronounced as seen in the work of Weiss et al. (2015). The
measured fluctuations in the TSB appear to be more energetic
at Stl ≈ 0.3, which suggests the flapping of shear layer may
play a more important role in varying wall-pressure
fluctuation.
A contour of normalized instantaneous velocity, U/U∞, isillustrated in Figure 3(a). The boundary of the TSB isindicated by the contour of U = 0 using black lines. The
irregular upper border of the boundary in the x-y view isassociated with the roll-up of the shear layer. The velocity
field reveals the existence of two shear layers, developed
along the suction and pressure surfaces of the wing. The
contours of forward-flow probability, γ, are plotted in Figure3(b). The mean boundary of the TSB generally overlaps withthe contour of γ = 0.5. The upper and lower shear layers are
identified using the local minimum and maximum of the mean
spanwise vorticity 〈Ωz⟩, respectively. They are illustrated by
the blue dashed lines in the figure. The two shear layers forge
the triangular shape of the TSB.

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022
Since the dynamics of the TSB are highly affected by theshear layers, we performed a spectral analysis of u along theupper shear layer to investigate the associated unsteadiness.
The PSD of u/U∞ as a function of x/c along the upper shear
layer is shown in Figure 4(a). Energetic unsteadiness is visible
downstream of the mean detachment point (i.e., x/c > -0.2) at
Stl < 0.04. A secondary peak is also observed downstream of
the TE (i.e., x/c > 0) at Stl = 0.14. The spectral analysis of flow
unsteadiness in the upper shear layer reveals that the dominantpeak is at Stl = 0.04, which is potentially related to the lowfrequency breathing of the TSB.
To identify the relationship between wall-pressure
fluctuation and the unsteadiness in the upper shear layer, thecoherence function, Cpu, between p at x/c = -0.30 and u along
the upper shear layer is investigated in Figure 4(b). The resultshows that the highest local Cpu is observed at Stl = 0.04. This
matches the dominant frequency of unsteadiness observed in
Figure 4(a) and is similar to the frequency of the breathing
motion reported in investigations of TSBs on flat plates(Weiss et al., 2015; Mohammed-Taifour and Weiss 2016; Le
Floc'h et al., 2020). It also appears that Cpu downstream of x/c= -0.30 is much higher than upstream of x/c = -0.30. The
dominant peak of Cpu is seen at -0.03 < x/c < 0.09. The factthat a strong coherence is detected downstream of thereference point suggests that the low-frequency pressure
fluctuations generated by the shear layer unsteadinesspropagate upstream of the mean TSB.
To identify the motions responsible for the low-frequency
content in wall-pressure fluctuations, two-point correlations
between low-pass-filtered p at x/c = -0.30 and fluctuating
velocities (u and v) in the measured domain were performed.
Based on the results of Cpu in Figure 4(b), a low-pass cut-off
frequency at Stl = 0.1 (6.5 Hz) is used to extract the lowfrequency content of p. The correlations are further separatedusing positive (superscript '+') and negative pressure(superscript '‒') to isolate the motions associated with eachsign of fluctuation. The correlations with p < 0 are multipliedby ‒1 to retain the true velocity directions in the visualizations.
The correlations are described by+ (Δ𝑥, Δ𝑦) =𝑅𝑝𝑢〈𝑝(𝑡)𝑢(Δ𝑥,Δ𝑦,𝑡)〉− (Δ𝑥, Δ𝑦) =𝑅𝑝𝑢

𝑝rms 𝑢rms

|

, and

(1)

,

(2)

𝑝>0
〈𝑝(𝑡)𝑢(Δ𝑥,Δ𝑦,𝑡)〉𝑝rms 𝑢rms|

𝑝<0

where Rpu is the streamwise component of the correlation
coefficient. A similar correlation was carried out using p andv to obtain Rpv as the wall-normal component. The spatial
correlations of Rpu associated with p > 0 and p < 0 are
presented in Figure 5(a) and (b), respectively. The contour of〈U〉 = 0 is illustrated using a green line to show the position of
the mean TSB. The results of Figure 5 demonstrate that lowfrequency positive and negative p are correlated with largescale streamwise flow motions downstream of the reference
point (x/c = -0.30). In particular, the low-frequency positive p
is correlated with a large-scale backward flow motion, i.e.,
low-frequency expansion of the TSB. The negative pcorrelates with the low-frequency forward flow motion, whichis the contraction of the TSB. The breathing of the TSB cantherefore increase/reduce the adverse pressure gradient
upstream of the TSB, and produce low-frequency positive andnegative pressure fluctuations, respectively.
It is of interest to note that the low-frequency positive andnegative p do not show a strong correlation with any local

3

wall-normal flow motions as seen in the work of Gibeau &Ghaemi (2021), where the positive and negative p at thelowest frequencies are induced by wall-normal velocity
associated with very-large-scale motions. The different
observations reveal that there is a fundamental difference inthe mechanisms that induce low-frequency positive andnegative p in an attached TBL versus a TSB with breathing
motions. The present result is more likely due to the effect ofsuction/compression above the pinhole created by thecontraction/expansion of the TSB.

CONCLUSION
The present study demonstrated that the PSD of wallpressure fluctuations in the TSB features a broad peak at Stl =0.3. The high-frequency pressure fluctuation within the same
region rapidly decayed with increasing Stl. The most energetic
unsteadiness in the upper shear layer above the separation
bubble was observed at Stl = 0.04. Through the investigationof the coherence between the unsteadiness in the upper shear
layer and low-frequency wall-pressure fluctuation at x/c=0.30, it was found that the low-frequency wall-pressure
fluctuations upstream of the detachment point correlated withthe unsteadiness in the upper shear layer at Stl of 0.04. The
associated flow motion was identified as the breathing of the
turbulent separation bubble. The positive and negative wallpressure fluctuations were related to the expansion andcontraction of the separation bubble through the variation of
the adverse pressure gradient.

REFERENCES
Broeren, A., & Bragg, M., 1998, "Low-frequency
Flowfield Unsteadiness During Airfoil Stall and the Influenceof Stall Type", 16th AIAA Applied Aerodynamics Conference,
Albuquerque, N.M., U.S.A., pp. 2517.
Camussi, R., Felli, M., Pereira, F., Aloisio, G. & Dimarco,
A., 2008, "Statistical Properties of Wall Pressure Fluctuations
over a Forward-facing Step", Phys. Fluids, Vol. 20 (7), pp.
075113.
Eaton, J. K., & Johnston, J. P., 1982, "Low Frequency
Unsteadiness of a Reattaching Turbulent Shear Layer",
Turbulent shear flows 3, Springer, Berlin, Heidelberg, pp.
162-170.
Gibeau, B., & Ghaemi, S., 2021, "Low- and Midfrequency Wall-pressure Sources in a Turbulent Boundary
Layer", J. Fluid Mech., Vol. 918.
Graziani, A., Kerhervé, F., Martinuzzi, R.J. & Keirsbulck,L., 2018, "Dynamics of the Recirculating Areas of a Forwardfacing Step", Exp. Fluids, Vol. 59 (10), pp. 154.
Le Floc'h, A., Weiss, J., Mohammed-Taifour, A., &Dufresne, L., 2020, "Measurements of Pressure and Velocity
Fluctuations in a Family of Turbulent Separation Bubbles", J.
Fluid Mech., Vol. 902.
Liu J. & Xiao Z., 2020, "Low-frequency Oscillation over
NACA0015 Airfoil Near Stall at High Reynolds Number",
AIAA Journal, Vol. 58, pp. 53-60.
Mohammed-Taifour, A., & Weiss, J., 2016, "Unsteadinessin a Large Turbulent Separation Bubble", J. Fluid Mech., Vol.
799, pp. 383-412.
Mohammed-Taifour, A., & Weiss, J., 2021, "Periodic
forcing of a large turbulent separation bubble", J. Fluid Mech.,Vol. 915.

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022
Na, Y., & Moin, P., 1998, "Direct Numerical Simulationof a Separated Turbulent Boundary Layer", J. Fluid Mech.,Vol. 374, pp. 379-405.
Shaw, R., 1960, "The Influence of Hole Dimensions on
Static Pressure Measurements", J. Fluid Mech., Vol. 7, pp.
550-564.
Thompson, B. E., & Whitelaw J. H., 1985,"Characteristics of a trailing-edge flow with turbulent
boundary-layer separation", J. Fluid Mech., Vol. 157, pp. 305326.

(a)Tsuji, Y., Imayama, S., Schlatter, P., Alfredsson, P. H.,
Johansson, A. V., Marusic, I., Hutchins, N. & Monty, J. 2012,"Pressure Fluctuations in high-Reynolds-number Turbulent
Boundary Layer: Results from Experiments and DNS", J.
Turb., Vol. 13 (50), pp. 1-19.
Weiss, J., Mohammed-Taifour, A., & Schwaab, Q., 2015,"Unsteady Behavior of a Pressure-Induced Turbulent
Separation Bubble", AIAA Journal, Vol. 53(9), pp. 2634-2645.

(b)

(c)

y′

Wing surface

Wing surface
O′

Flow
FOV

y′

z′

O′

x′

O
x
y
z

Diaphragm

Figure 1. (a) Schematics of the time-resolved planar PIV where the combined FOV is represented by the blue dash-line boxes.
The cross-section views of the microphone installation in (b) the streamwise-spanwise, and (c) streamwise-wall-normal plane.

Figure 2. PSD of p/prms at seven streamwise locations along the centreline of the 2D wing.

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

Figure 3. (a) Sample contours of U/U∞ with the TSB boundary, represented by the contour of U = 0, shown in black lines. (b)The contours of the forward-flow probability γ are presented using black lines. The wall-normal locations of 〈Ωz⟩min/max as a functionof x/c are shown using blue dashed lines to represent the upper and lower shear layers.

Figure 4. (a) The contours of PSD of u/U∞ in the upper shear layer. (b) The coherence Cpu between p at x/c = -0.30 and u fromthe upper shear layer.

512th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19-22, 2022

Figure 5. Visualizations of the pressure-velocity correlations associated with (a) p > 0 and (b) p < 0. A low-pass filter of Stl <0.1 was applied to extract the low-frequency p before computing the correlations. The reference point of p is at x/c = −0.30 on thewall and is represented by Δx = 0 here. The vector field is constructed using Rpu and Rpv. An overlay of the contour of 〈U〉 = 0 (greenline) is added to show the position of the mean TSB.

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