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

STATISTICS OF UNSTEADY FAVORABLE PRESSURE GRADIENTTURBULENT BOUNDARY LAYERS

Aadhy Parthasarathy
Aerospace Engineering
University of Illinois at Urbana-Champaign
104 S Wright St, Urbana, Illinois 61801, USA.
aadhysp2@illinois.edu

Theresa Saxton-Fox
Aerospace Engineering
University of Illinois at Urbana-Champaign
104 S Wright St, Urbana, Illinois 61801, USA.
tsaxtonf@illinois.edu

ABSTRACT

have received relatively less attention, despite their practicalrelevance. Flows over rotating machinery, maneuvering aircraft, and vehicles operating in gusty environments, are someexamples. Difficulties in studying these flows range from thecomplexity in generating unsteady TBLs in a repeatable manner, covering a wide parametric space relevant to engineering situations, to obstacles in meaningfully comparing resultsfrom different facilities and simulations (Carr, 1981; Brereton
et al., 1990). The current industry standard for full-scale predictions is to use Unsteady Reynolds-Averaged Navier–Stokes(URANS) simulations along with turbulence models for steady
PG TBLs, implicitly making the assumption that the unsteadyflow is pseudo-steady. Various researchers have shown thatsuch assumptions result in poor predictions due to the effectsof complex interactions between the imposed unsteadiness andturbulent fluctuations being absent in these models that are
tuned for steady PGs (Park et al., 2021; Rezaeiha et al., 2019).
The focus of this paper is on understanding the effect of
unsteady FPGs on the statistics of TBLs in comparison to corresponding statistics of steady FPG TBLs, for different timerates of change of PGs. The goal is to provide insights into
key features of unsteady FPG TBLs that can be used to evaluate if lower-fidelity simulations of these flows predict physically consistent features. Additionally, since FPG followedby APG is a common geometric configuration in engineeringflows, and considering that history effects play an importantrole in the dynamics of PG TBLs (Vinuesa et al., 2017), propagating the features of unsteady FPG flows into a succeeding
APG region is also of significance in accurately predicting unsteady APG flows.

The statistics of temporally-strengthening favorable pressure gradient (FPG) turbulent boundary layers (TBLs) arestudied, in comparison to corresponding stationary FPG TBLs.
The experiments were carried out in a wind tunnel facility
capable of imposing unsteady pressure gradients of different
timescales on a flat plate TBL through the use of a rapidly deformable ceiling mechanism. The responses of the transient
TBL were captured using phase-locked time-resolved particle
image velocimetry. A series of steady-state experiments werealso performed for 22 discrete states of the ceiling within the
deformation range, to help separate the effects of unsteadinessand spatial pressure gradients. The ensemble-averaged mean
and Reynolds stresses at a location at the exit of spatial acceleration are presented in this paper, for 3 different timescalesof unsteadiness corresponding to reduced frequency, kx (defined as the ratio of convective timescale to imposed unsteadytimescale) = 4.5, 1.5, 0.6. The development of salient features of the steady-state pressure gradient flow and the effectof unsteadiness on these features are discussed. Significant deviations from steady-state behavior are seen for kx = 4.5 and
1.5, indicating that pseudo-steady assumptions in their numerical simulation using lower-fidelity tools will lead to inaccurate
predictions.

INTRODUCTION
Pressure gradient (PG) TBLs are of immense interest to
researchers and engineers due to their ubiquitous nature. Increasingly sophisticated experiments (Schröder et al., 2018;
Shehzad et al., 2021) and high-fidelity numerical simulations(Piomelli & Yuan, 2013; Gungor et al., 2021) have furnishedreliable statistics of favorable and adverse PG TBLs overa large parametric space, allowing lower-fidelity turbulencemodels to be continually evaluated and improved based onthese statistics. Current limitations of these models are wellidentified, for example, near flow separation or in sequences ofpressure gradients of opposing signs, and these are active areasof research in the turbulence community (Elyasi & Ghaemi,
2019; Slotnick, 2019). Unsteady PG TBLs, on the other hand,

Accelerated TBLs
A spatially accelerated TBL under FPGs exhibits several interesting features that differ from a ZPG TBL, some ofwhich are reported in Sreenivasan (1982), Bourassa & Thomas(2009), and Dixit & Ramesh (2010). The mean velocity develops an enlarged viscous sublayer and significant deviations
from log and wake law behaviors are seen for moderately
strong PGs (K > 2.8 × 10−6 ). The standard-law deviations

1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
ated flat plate compressible TBL was studied by Saavedra et al.
(2020), who saw a significant acceleration effect on the shear
stress and heat flux through a sudden overshoot dominated byinertia and a subsequent relaxation to quasi-steady state.
The studies to-date on non-periodic unsteady TBLs havebeen valuable in the identification of interesting physics andin questioning the adequacy of lower-fidelity simulations. Advancing our understanding of complex mechanisms in theseflows and their prediction calls for significantly more researchthrough systematic experiments (physical or numerical) characterizing the unsteady effects over a wide range of parameters relevant to engineering flows of interest. The successof extensive research on rapidly accelerated or decelerated
pipe/channel flows serves as additional motivation.

mark the onset of relaminarization, a process of reverse transition to laminar flow. The process is triggered by a sufficiently
strong suppression of turbulence intensities and of turbulence
production, especially in the outer region. A shift in the energy spectrum towards lower wavenumbers is apparent, andthe 2-point correlation structures appear elongated and more
inclined to the wall. Prolonged application of the FPG cancause the boundary layer to eventually laminarize, but the flow
quickly bypass-retransitions to turbulence when the stabilizing
effect of the FPG is removed. Turbulence models often fail topredict some features of relaminarizing TBLs, especially pertaining to the behaviors of Reynolds shear stress and turbulent
kinetic energy (Saltar & Araya, 2020). More complex dynamics result when the FPG region succeeds an APG, such as in
the bump / hill flows of Uzun & Malik (2021), Balin & Jansen(2021), Baskaran et al. (1987) that impose a mild APG at thefoot of the bump / hill, followed by a strong FPG until the apex.
The pressure gradient sign change triggers an internal boundary layer that dominates the near-wall behavior including skinfriction, and the Reynolds stress profiles develop footprints ofthe formation and growth of the internal layer, exhibiting features like knee points and outer peaks that are otherwise absent
in FPG flows that succeed a ZPG region.

EXPERIMENTThe experiments reported in this paper were performed in
an open-return boundary layer wind tunnel at the University
of Illinois. Quiescent air is sucked through a 10.19 cm thickhoneycomb straightener, followed by four 24-mesh turbulence
reducing screens, and a contraction section with an area ratio of 27:1, before reaching a 0.381 m × 0.381 m × 3.687 m(15” × 15” × 12′ ) test section. The resulting freestream turbulence intensity is approximately 0.5%. A fully-developed
TBL is obtained over a 3 m long flat plate with a leading-edge
trip, mounted at the test section half-height. The test area was
located at 2.3 m from the leading-edge. The zero pressure gradient 99% boundary layer thickness, δ , at the test area was 42mm and the friction Reynolds number, Reτ , was 990.
A recently built and characterized installation for the ceiling of the wind tunnel was used to generate the unsteady pressure gradients. A schematic of this installation is shown in
Fig.1a. It houses an electro-pneumatic actuation mechanismthat rapidly deforms a flexible ceiling panel to the shape of an
inverted convex bump. The dimensions of the ceiling panel
are 0.45 m × 0.375 m × 0.001 m. As the ceiling deforms, aspatially- and temporally-varying favorable and adverse pressure gradient sequence is imposed on the flat plate TBL beneath it. The spatial profile of the PG imposed and the temporal rate at which it is imposed can be varied as desiredby providing suitable control inputs to the actuation mechanism. A detailed description of this facility can be found inParthasarathy & Saxton-Fox (2022).
This paper is focused on studying the temporal responseof the TBL at a 1D plane located at x0 (marked in Fig.1a),where the flow has undergone spatial acceleration whosestrength increases with time as the ceiling deforms. The estimated spatial PG profile imposed and its variation in time
are shown in Fig.1b over the region (0 < x/L < 1), in termsof the acceleration parameter, K. L is the length of the ceiling panel, and the plane x0 corresponds to x/L = 0.5. K iscdefined as Uν2 dU, where ν is the kinematic viscosity, and Uc
c dx
is the local bulk velocity. Uc was computed as the bulk velocity variation required to satisfy continuity at all geometricstates of the ceiling, and was validated using discrete pressure measurements. The temporal strength or unsteadiness of
the PG imposed is quantified in terms of reduced frequency,
kx = tc /t f , where tc is the convective timescale and t f is theimposed unsteady timescale, equal to the ceiling deformation
time. kx = 4.38, 1.53, and 0.61 for the cases discussed in thispaper. The convective length and velocity scales used in thedefinition of kx are chosen to be the boundary layer development length and freestream velocity, consistent with literatureon unsteady pitching airfoil experiments and simulations. A

Literature on temporally-accelerated TBLs is sparse, withmost unsteady TBL studies focusing on periodic unsteadiness
imposed by means of an oscillatory freestream flow, periodically accelerating and decelerating the boundary layer. Karlsson (1959) imposed sinusoidal oscillations of large amplitudes(upto 34% of the mean) and of a wide range of frequencies (0.33 - 40 Hz) about a constant mean, and reported thatthe time-averaged mean showed negligible variations compared to the quasi-steady mean, and that the variations instreamwise turbulence intensity were not systematic with frequency and amplitude. Similar experiments by Parikh et al.
(1981), Menendez & Ramaprian (1989), and Brereton et al.
(1990) supported the observation of insignificant effects of unsteadiness on time-averaged quantities, but additionally identified strong non-quasi-steady effects when probing the oscillatory (phase-averaged) quantities. Phase leads and lags in
the turbulent stresses were noted, and were shown to be afunction of the frequency of oscillation. Low frequency oscillations elicited responses throughout the boundary layer,
whereas high-frequency oscillations modulated the near-wall
most, with the outer region turbulence and the boundary layer
thickness remaining effectively frozen at their mean values
throughout the cycles. These behaviors were similar in character to that of turbulent pipe and channel flows under periodic
unsteadiness.
A non-periodic, rapidly accelerated TBL over a flat platewith non-zero pressure gradients (an FPG-APG sequence imposed by a converging-diverging slip wall) was simulated by
Kharghani & PasandidehFard (2022). The effect of temporal acceleration bore similarities to spatially-accelerated flows,
resulting in stronger suppression of fluctuations and delays inturbulence propagation due to their simultaneous application.
The similarities of temporal acceleration effects to spatial acceleration have previously been noted in rapidly accelerated
pipe and channel flows (He & Jackson, 2000; Talha & Chung,2015). Saavedra & Paniagua (2021) studied the effects of asudden flow acceleration over a wall-mounted hump (FPGAPG sequence), in the compressible subsonic flow regime.
They noted a boost of near-wall momentum due to the sudden
acceleration, enough for the TBL to overcome APG-induced
flow detachment, suggesting that mean flow transients can beleveraged to modulate separation events. A rapidly acceler-212th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022(a)(b)
× 10-!

ELECTROPNEUMATIC
ACTUATION MECHANISM

x=x#

FLAT PLATE
SURFACE

Acceleration parameter, K

x

t*
0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1
1.35

4

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1.15
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-4
-6

Ue/U∞

RAPIDLY DEFORMINGCEILING PANEL
y

6

0

1.05
1
0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

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1

x/L

Figure 1. Experimental details. (a) Schematic of unsteady the pressure gradients installation. The geometric states of the ceiling at
the 6 instances chosen for analysis in Fig. 2 are shown as gray curves over yellow curves. Flow is from left to right. (b) Left axis :Variation of acceleration parameter in space and in time, where the darker curves indicate later in time. Right axis: Variation of edge
velocity in time at location x0 , scaled by freestream velocity (U∞ ) and unsteady time (t f ). t f = 0.07, 0.25, 0.5 s respectively for k1, k2,
k3. (——) k1, (——) k2, (——) k3, ( ) Steady.

more suitable comparison of the timescales involved in thisboundary layer flow is captured by kτ = τt /t f , where τt is thelarge turbulent timescale. For the cases discussed, kτ = 1.79,
0.63, and 0.25, and will be referred to as cases k1, k2, and k3
respectively henceforth.
The time-varying 1D data at x0 are sourced from a recent experimental campaign to capture the spatio-temporal response of the TBL to unsteady PGs using phase-locked, timeresolved particle image velocimetry at a rate of 3.755 kHz.
A mineral-oil based seeding was introduced at the tunnel inlet, and a Continuum®Terra PIV 527-80-M high-speed laser
was used along with a Phantom®VEO 710L camera to collectthe particle image pairs. For the unsteady cases, 400 independent cycles of the boundary layer response were recorded ina phase-locked manner. Each independent cycle consisted of
280 image pairs for k1, 480 for k2, and 1020 for k3. The number of image pairs per cycle are naturally higher for higher t f .
The resulting vector fields calculated using LaVision’s DaVis
10 software had a spatial resolution of 42δν , where δν = ν/uτ0
and uτ0 is the initial zero pressure gradient friction velocity
computed using the Clauser method. The temporal resolution
was 2τν , where τν = ν/uτ 2 . With 400 phase-locked ensem0
bles, the mean field converged to within 0.1%, the normal
Reynolds stresses to within 1%, and the Reynolds shear stressto within 3%. The variation of non-dimensional edge velocity, Ue /U∞ , at x0 with respect to non-dimensional time, t/t f , isshown in Fig.1b for each rate of imposition of pressure gradient. The edge velocities and local boundary layer thicknesses
used in the outer scaled plots of Fig. 2 and Fig. 3 were calculated using the diagnostic plot technique of Vinuesa et al.
(2016). For the steady-states, 10, 000 velocity fields each wereacquired at 0.2 kHz for 22 static configurations of the ceiling.

the ZPG, steady FPG, and unsteady FPG profiles collapse, as
expected. The unsteady Reynolds stress profiles are less converged compared to the unsteady mean profiles, but are able tocapture the trends in the statistics (validated at t ∗ =0), permitting their use in interpreting the trends in unsteady behavior(t ∗ >0).
The steady FPG mean velocity profiles, shown in blackin Fig. 2a, show the effect of increasingly stronger spatial acceleration on the TBL. The spatial gradient at the final profile
shown corresponds to Ue /U∞ = 1.36 (and t ∗ = 1 in the unsteady case), as shown in Fig. 1b. The steady FPG profiles are
significantly fuller compared to the ZPG profiles as a result of
the acceleration. In the unsteady case, as the ceiling deforms,
the mean initially overshoots (t ∗ = 0.36), then undershoots (t ∗= 0.68) the corresponding steady-state profiles, before starting
to approach the steady-state mean towards the end of the transient (t ∗ = 1). With the rapid application of the FPG, mass conservation dictates the external flow to immediately accelerate,
while the inner boundary layer takes a finite time to negotiate
the no slip at the wall. During this initial stage, a large velocity
gradient develops near the wall. As time progresses, this inertial effect relaxes through viscous diffusion. This behavior has
previously been observed in temporally-accelerating pipe andchannel flows (Seddighi et al. (2011)), where an initial stage
dominated by inertial forces resulted in large velocity gradients near the wall and hence a rapid increase in viscous shear
stress, which later relaxed as the frozen turbulence responds
to the acceleration, causing the flow to approach steady-statebehavior.
The station x0 is located at the exit of a strong FPG region,
immediately followed by a strong APG region. The steady
FPG Reynolds stress profiles at x0 , plotted in black in Fig. 2bd, show many interesting features characteristic of such pressure gradient sign changes. The first is the formation of a knee
point in the streamwise Reynolds stress (Fig. 2b) in the third
profile (corresponding to the unsteady case t ∗ = 0.53) at y/δ ∼0.23. The knee point is attributed to the formation of an internal layer triggered by the pressure gradient sign change at
x0 , as previously observed in bump/hill flows (Baskaran et al.,1987) or when other boundary condition changes occurs. Below this point (sometimes taken as the edge of the internallayer), the streamwise stress peaks, and above, the stress weakens under the effect of the FPG. The presence of the internallayer also causes the formation of outer peaks in the wallnormal (Fig. 2c) and shear stress profiles (Fig. 2d), whose

RESULTS AND DISCUSSION
The ensemble-averaged mean velocity and Reynolds
stresses at station x0 for k1 are shown in Fig. 2 in red for6 instances in time (t ∗ = 0, 0.36, 0.53, 0.68, 0.78, 1). The
profiles at subsequent t ∗ are horizontally shifted such that timeflows left to right. The corresponding steady-state statistics areplotted in black, showing results from a boundary layer witha static spatial pressure gradient that instantaneously matches
that of the time-varying case at time t ∗ . ZPG profiles at the initial freestream velocity are plotted in grey to highlight differences in the BL response to zero and favorable PGs. At t ∗ = 0,3

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
1

y/δ

0.8
0.6
0.4
0.2
0

0.5

1 0.5

1 0.5

1 0.5

1 0.5

1

U/Ue

10-3

10
< u’2 > / Ue2

1 0.5

8
6
4
2
0

0

< v’2 > / Ue2

2

10-3

10

10

10

10

10

1

10

10

10

10

10

1

10

10

10
y/δ

10

10

1

1.5
1
0.5
0

0

10-3

- < uv > / Ue2

1.5
1
0.5

0

Figure 2. Outer-scaled turbulent statistics at t ∗ = 0, 0.36, 0.53, 0.68, 0.78, 1. (a) Mean streamwise velocity profiles. (b) Streamwise
Reynolds stress (c) Wall-normal Reynolds stress (d) Reynolds shear stress. Profiles at subsequent t ∗ values are shifted by 0.5 units for(a) and 1.1 units for (b), (c), and (d) along the x-axis for visual clarity. ( ) ZPG, ( ) steady FPG , ( ) unsteady FPG.
profile, so that at t ∗ = 1, the unsteady peak is located at y/δ =0.26, whereas the steady peak is at y/δ = 0.16.
The statistics at discrete time instances in Fig. 2 suggest asignificant impact of the unsteadiness on features of the pressure gradient flow, including the formation and wall-normal
growth of the internal layer, and the suppression of outer region turbulence. The continuous time behavior of the features
discussed is quantified next and the effect of rate of pressure
gradient imposition is studied. Fig. 3 shows the temporal evolution of one chosen feature each in the mean and the Reynolds
stresses for k1, k2, and k3, along with the variations in the corresponding steady-states. The inset in each figure illustrates
the quantity chosen to be tracked in time.
The outer-scaled mean velocity at a point where the mean
velocity changes slope is plotted with non-dimensional timein Fig. 3a, to study the over- and under-responses to acceleration seen in Fig. 2a for k1. A similar evolution to k1 isseen in case k2, with the mean velocity overshooting the corresponding steady-state mean for 60% of the unsteady time. The
mean velocity undershoots the steady-state for the reminder of
unsteady time, and at t ∗ = 1, shows a possibility of reachingmagnitudes increase with increasing pressure gradient, eventually overtaking the corresponding inner peak magnitudes (evidently visible starting in the fourth profile shown, corresponding to the unsteady cases with t ∗ > 0.68). In the unsteady case,the streamwise stress profiles show an initial rapid suppression
in magnitude throughout the boundary layer, causing an overshooting of the corresponding steady profiles, followed by aperiod of under-response (t ∗ > 0.68). The knee point forms
earlier at t ∗ = 0.36 (at a weaker spatial pressure gradient or
lower Ue /U∞ than the steady case), suggesting that the sudden temporal change has aided the mechanism of formationof the internal layer. The depth of the knee point, taken as
an indication of how defined/strong the internal layer is, becomes weaker in the unsteady case for t ∗ > 0.68. The unsteadywall-normal and shear stress profiles deviate from the bi-modal
structures of the corresponding steady-state responses. The
unsteady wall-normal stress exhibits a stronger inner peak andan absence of the outer peak, which can be associated with aweaker internal layer in the unsteady case as the flow evolves.
The single shear stress peak shifts further away from the wallcompared to the first peak of the corresponding steady-state4

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

(b)

U/Ue

0.75
0.7

y/δ

0
×10

0.8 0.91y/δ

0<v’2>/Ue2

1.6
1.4

0.1 0.2 0.3 0.4 0.5 0.6 0.7

0.8 0.91

0.8 0.9

1

(d)
0.26

1.8
<v’2>/Ue2

0.1 0.2 0.3 0.4 0.5 0.6 0.7
t*
-3

4
3.5
3
2.5
2
1.5
1
0.5

-<u’v’>/Ue2

(c)

0.24
0.22

y/δ

U/Ue

0.8

<u’2>/Ue2

k1
k2
k3
Steady

0.85

×10-3

<u’2>/Ue2

(a)

y/δ

1.2

y/δ

0.2
0.18

1

0.16

0.8

0.14
0

0.1 0.2 0.3 0.4 0.5 0.6 0.7
t*

0.8 0.910

0.1 0.2 0.3 0.4 0.5 0.6 0.7
t*Figure 3. Temporal evolution of select statistical features. (a) Acceleration in the mean. (b) Depth of knee point in the streamwise
Reynolds stress. (c) Magnitude of wall-normal Reynolds stress peak. (d) Wall-normal location of Reynolds shear stress peak. The inset
in each figure illustrates the definition of the quantity chosen to be tracked in time.

CONCLUSION

steady-state shortly after. The closest to steady-state behavior is exhibited by case k3, where the temporal evolution of
the chosen mean quantity does not exhibit discernible over- or
under-responses to the sudden acceleration.

The mean and Reynolds stresses of a flat plate turbulent boundary layer (TBL) encountering unsteady favorablepressure gradients (FPG) of 3 different timescales (reduced
frequency, kx = 4.38, 1.53, 0.61 or kτ = 1.79, 0.63, 0.25)
are studied in this paper, in comparison to the statistics of
22 steady FPG TBLs. The experiments were conducted in aboundary layer wind tunnel facility that employed a rapidly
deformable ceiling panel to impose an FPG-APG sequence
on a TBL, whose strength increased in time as the ceilingdeformed. The time rate of imposition of pressure gradient
was varied by suitably prescribing the rate of ceiling deformation. The spatio-temporal response of the TBL was captured
using phase-locked, time-resolved particle image velocimetry. The focus of this study was a station x0 at the exit of
the FPG region, where the TBL had encountered temporallystrengthening spatial acceleration.
Outer-scaled statistics in the wall-normal direction for thehighest reduced frequency case (k1) at discrete times were presented along with the statistics of corresponding steady-states.
The unsteady mean flow comparatively exhibited a significant period of over-response followed by a period of underresponse to the imposed rapid acceleration. The developmentof key features characteristic of steady TBLs at the apex of
an FPG-APG sequence, like the formation and growth of aninternal layer, and development of outer peaks in the wallnormal and shear stress profiles were discussed. The differences in these characteristic features for the unsteady case k1
were highlighted. The continuous temporal evolution of selectfeatures were shown for k1, k2, k3, along with the steady-state
responses, to understand the effect of rate of pressure gradientimposition. k3 showed the least deviation from steady-statebehavior. k1 (strongest unsteady case with kτ = 1.79) elicitedthe most response from the TBL, with significant variationsin the Reynolds stresses including a delay in the suppression

Fig. 3b shows the depth of the knee point in the streamwise Reynolds stress, identified as the first local minimum inthe stress profile. The wall-normal location of the knee point(not tracked here) is considered as the edge of the internallayer (Baskaran et al. (1987); Efros & Krogstad (2011)). The
depth of the knee point is interpreted as indicating how defined/strong the internal layer is, and is tracked here to study
the formation of the internal layer in time. Cases k1 and k2 develop an internal layer at t ∗ = 0.075 upon rapid application of
the pressure gradient, which persists in a more defined mannerthan the corresponding steady pressure gradient internal layer
upto t ∗ = 0.56 and 0.67 respectively for k1 and k2. After this,the depth of the knee point rises higher than the corresponding
steady-state, indicating a weaker internal layer as the boundary
layer has negotiated the initial condition change.
The first peak of the wall-normal Reynolds stress profile istracked in Fig. 3c. Cases k1 and k2 exhibit less suppression ofouter-scaled wall-normal stress throughout the unsteady time
relative to the steady-states, and more interestingly, show novariation for a significant 20% of unsteady time (t ∗ <0.2) despite being scaled by the square of an increasing quantity (i.e,Ue2 ). Overall, the effects of unsteadiness seem stronger in k1
compared to k2. Similar behavior is observed in Fig. 3d,
where the shifting of the wall-normal location of shear stresspeak seen in Fig. 2c for k1 is plotted as the y/δ location of
the first maxima in time for all cases. k2 exhibits a significant
shifting away from the wall, less than k1 and more than k3
and the steady-states. In Fig. 3b-d, case k3 shows the least
variation from steady-state behavior.

5

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan, July 19–22, 2022
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Vinuesa, Ricardo, Örlü, Ramis, Sanmiguel Vila, Carlos,
Ianiro, Andrea, Discetti, Stefano & Schlatter, Philipp 2017
Revisiting history effects in adverse-pressure-gradient turbulent boundary layers. Flow, turbulence and combustion
99 (3), 565–587.

of wall-normal stress due to the acceleration and a shift awayfrom the wall of the shear stress peak. The deviations in TBL
behavior for k2 were similar to k1, but weaker in comparison.
The similarities of some of the observations to rapidly accelerated pipe and channel flows, which have been studied indetail by the turbulence community, were noted. But as highlighted by PasandidehFard et al. (2022), findings on unsteadyinternal flows cannot be directly applied to external unsteady
TBLs. This study contributes to the limited experimental observations on unsteady FPG TBLs over slow, medium, andfast unsteady timescales (relevant to the turbulent timescale,τt ), highlighting key features of the flow that can potentiallybe used to test physical consistency of predictions by lower
fidelity simulations.

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