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

TOLLMIEN-SCHLICHTING WAVES OVER FORWARD-FACING STEPS:AN EXPERIMENTAL AND NUMERICAL STUDY

M. Barahona1∗ , A. F. Rius-Vidales1∗ , F. Tocci2 , P. Ziegler3 , S. Hein2 and M. Kotsonis1
1: Section of Aerodynamics, Delft University of Technology, Delft, The Netherlands.
2: Institute of Aerodynamics and Flow Technology, German Aerospace Center (DLR), Göttingen, Germany.
3: University of Stuttgart, Stuttgart, Germany.
*Correspondent authors: m.barahonalopez@student.tudelft.nl and a.f.riusvidales@tudelft.nl

ABSTRACT

the specific test conditions makes Reh,crit alone not sufficient
to determine when FFSs become critical.
The experimental results of Wang & Gaster (2005) suggested that transition with FFS could be largely described by
the exponential growth of disturbances downstream. Their results were in agreement with observations from other authors,whose focus was on developing a semi-empirical model inspired on the eN -method to predict transition location (Perraud
et al., 2004; Crouch et al., 2006). These models attempt to isolate the effect that the step has on transition in one single parameter, ∆N. Large discrepancies are observed when the data
fits and models proposed by different authors are compared,
as noticed in (Costantini, 2016). The outcomes of the aforementioned studies raise the question of whether the effect ofan FFS on transition can be uniquely described by ∆N.
Recent DNS and experiments in 3D boundary layers dominated by Crossflow Instabilities (CFI) in the presence of FFS
point to different growth mechanisms at the step. These could
explain the observed delay on transition for some step heights(Rius-Vidales & Kotsonis, 2021), the unsteadiness of the shear
layer downstream (Eppink, 2020) or the onset of near-wall vortices immediately after the step (Casacuberta et al., 2021). Despite the different incoming instability in these works, compared to TS waves, their findings emphasize the importance of
close-examining the disturbances evolution in the vicinity of a
FFS to understand how this could influence transition downstream.
From the aforementioned works the mechanisms governing the growth of disturbances at the step remain not fully understood. In turn, the lack of comparison between numericaland experimental data further hinders the problem understanding. On the one hand, existing experimental works on TS-FFS
dominated transition were mainly concerned with transition
prediction, providing limited measurements in the vicinity ofthe step. On the other hand, previous numerical studies were
not designed to match other experimental works. In light of
this, the aim of this study is to close-examine the TS waves
at the step to identify which are the relevant mechanisms thatmodify their growth and move transition upstream.

This work presents an experimental and numerical investigation jointly conducted by TU Delft and DLR on TollmienSchlichting (TS) waves interaction with a Forward-Facing
Step (FFS). Experiments are conducted at the TU Delft lowturbulence anechoic wind tunnel on an unswept flat platemodel. Single-frequency disturbances are introduced using
controlled acoustic excitation. The temporal response of theflow in the vicinity of the step is measured using Hot-Wire
Anemometry (HWA). In addition, the global effect of thestep on laminar-turbulent transition is captured using Infrared
Thermography (IR). Two-dimensional (2-D) Direct Numerical
Simulations (DNS) performed at DLR provide detailed information at the step. Experimental and DNS results in clean andstep case conditions present very good agreement. Both methods predict large distortion of the TS waves downstream of thestep, where DNS results present different growth trends between |û| and |v̂| components of the TS waves. Furthermore,negative and positive regions of the production term are observed to correlate with streamwise positions where the disturbances appear tilted in and against the mean shear, respectively. These findings point towards the presence of different
growth mechanisms triggered by the step which could modify
the level of amplification of disturbances far downstream.

INTRODUCTION
The presence of panel discontinuities on aircraft wings
can lead to laminar-turbulent transition advancement and anincrease in skin-friction drag. The recent interest in low-speed
aircraft for regional air mobility has drawn attention to the effect of surface irregularities on unswept wings, where transition is dominated by Tollmien-Schlichting (TS) waves. Panel
discontinuities are commonly modeled in the form of twodimensional steps. Prior subsonic studies in TS dominatedtransition have shown that a Backward-Facing Step (BFS)
configuration is more detrimental than a Forward-Facing Step
(FFS) (Wang & Gaster, 2005; Crouch et al., 2006). Therefore,
in the design of laminar flow components, FFSs are preferredover BFSs.
First research efforts concerning FFS-induced transition
aimed at finding critical Reynolds heights (e.g. Reh,crit ) to
provide manufacturers with a rapid estimate of the transition
location and the maximum permissible height between panel
discontinuities. However, it was soon proven from the experiments of the Northrop Grumman Corporation (Drake et al.,2010) that the large dependency of the transition location on

EXPERIMENTAL SET-UP
The experiments were performed in the A-tunnel, an
open-jet, closed-circuit, subsonic, vertical wind tunnel. The
anechoic test chamber features acoustic absorbing foam which
reduces sound reflections. This creates a suitable environment
to force and study TS waves since these are highly responsive

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
IR camera
Traverse in (x, y, z)-dir.

Kevlar fabricMicrophone
Top wallPressure taps

FFS

Collector

Hot wire
Acoustic system

Pitot static tube

(benchmark subwoofer)
Bottom wall

Silicon heater thermostat

Figure 1: Experimental set-up. Schematic showing the FFS location, the acoustic unit and the different measurement techniques used.

to acoustic emissions. A nozzle is used to adapt the circular
inlet test section of diameter 600 mm to a rectangular one withdimensions 500× 500 mm2 containing an unswept flat platemodel with a chord of c = 950 mm at zero angle-of-attack. The
flat plate features a Modified Super Ellipse (MSE) at the leading edge (LE) with aspect ratio AR = 6. This design moves the
suction peak towards the LE and reduces the adverse pressure
gradient in the LE region (Lin et al., 1992), ensuring a rapid
development towards a Blasius boundary layer. The stagnation
point and the LE pressure distribution can be adjusted with a
trailing edge (TE) flap. The flow on the bottom side is forced
to turbulence using a zig-zag tape to avoid uncontrolled shedding. The external velocity over the measurement side is set
to Uref = 22.5 m/s which results in a Reynolds number basedon the chord of Rec = 1.32 × 106 and a freestream turbulence
intensity below 0.1% (bandwidth 5 − 2 × 105 Hz). Finally,
adjustable FFSs at x/c = 0.4 are created using a built-in mechanism (see Figure 1). The step heights are characterized bytraversing a Micro-Epsilon 3010-25BL laser scanner (resolution of 1.5 µm) along the span. The step is located at xh = 380mm.
Single-frequency TS waves are excited using acoustic
forcing. A 120W benchmark subwoofer located 2.15 m downstream of the LE is used to generate a continuous acoustic signal at the forcing frequency of interest. The model’s MSE LE
removes the curvature discontinuity between the flat plate andLE juncture, ensuring receptivity only at the nose of the LE(Lin et al., 1992). The acoustic wave amplitude and frequency
are monitored using a GRAS 46BE analog free-field microphone flushed-mounted close to the LE. In addition, Kevlar
type 120 fabric covers the side of the microphone facing theflow to reduce the permeability to hydrodynamic fluctuations
with respect to sound waves. In these conditions, the signalto-noise ratio (SNR) was always higher than 10.
Fifteen boundary layer profiles were measured using HotWire Anemometry (HWA) along the model’s midspan (themeasurement region covers 40 mm upstream and 40 mm
downstream of the step) with a single-wire probe (Dantec
Dynamics 55P15). The HWA sampling frequency was set
to 51.2 kHz with a total measuring time per point of 3 seconds to ensure statistical convergence. The distance from thelast HWA point to the wall was measured using a TaylorHobson microalignment telescope. In addition, during thepost-processing the wall distance was refined using velocity
profiles from the DNS. Finally, the amplitude of the forced
TS wave is acquired from the Fast-Fourier Transformed (FFT)HWA signal at the excitation frequency.
A silicon heater, electronically controlled by a thermostat, actively heats1 a portion of the flat plate surface downstream of the step, with dimensions of 500×300 mm. An

Optris PI640 camera (640×480 pixels of optical resolution
and a thermal sensitivity of 75 mK) equipped with a wideangle lens ( f = 10.5 mm) is used to capture the temperaturechanges across the black acrylic plate (surface emissivity of0.95) constituting the step. For each measurement 100 frames
are recorded at a frequency rate of 3.5 Hz at two different
Reynolds numbers to perform Differential Infrared Thermography (DIT). This technique has already proved to be succesful
to increase the SNR of the transition front identification (RiusVidales & Kotsonis, 2021).

DIRECT NUMERICAL SIMULATIONS
Direct Numerical Simulations (DNS) are performed at
DLR using the Nek5000 code by Fischer et al. (2008) which isbased on the Spectral Element Method (SEM). The SEM decomposes the physical domain into (spectral) elements where
the flow field solution is given by a sum of Lagrange interpolants defined by an orthogonal basis of Legendre polynomials up to degree P within each element. The results presented
here were obtained with a polynomial order P = 9 within the
spectral elements.
A boundary layer code developed at the Aerodynamics
section in TU Delft is initially used to compute a reference solution for the clean case based on the measured pressure distribution. This solution is used to obtain the boundary conditions
imposed in the DNS, allowing to reproduce the experimental
conditions, see Table 1.
The DNS investigations focus on the beginning of thetransition process and therefore are limited to 2-D domains to
reduce the computational costs. The calculations consisted ofa DNS of the steady laminar 2-D base flow and succeeding
unsteady DNS for the disturbances propagation. For the latter, a harmonic blowing and suction strip centered at x = 200mm is applied at the wall to introduce periodic TS waves. The
blowing and suction strip introduces a wall-normal velocity
component at a selected frequency with zero net-mass flow.
Its amplitude is prescribed such that the developing TS wave
matches the amplitude measured with HWA at x = 340 mm.

RESULTS
The step heights (h) and forcing conditions used for DNS
and experiments are indicated in Table 1. A comprehensive
parametric study is performed by means of IR measurements
for seven different steps at two forcing frequencies and no forcing. HWA and DNS results will focus on one step case. The
forcing frequency of F = 90 constitutes a TS mode with highest amplitude at the step region while F = 110 undergoes decay at the step.

1 The surface temperature difference was kept below half a degree
to avoid interfering with the transition process.

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12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
Table 1: Experimental test cases. Step located at x/c = 0.4 andstep height spanwise non-uniformity, σh = 7 µm. All cases
were also run without acoustic
forcing. F denotes reduced fre
2 × 10−6 , and δ ∗ is the displacement
quency, F = 2π f ν/Uref
thickness in no-step case at x/c = 0.4.

2.5



−0.008† , 0.775† , 1.075, 1.261, 1.462‡ , 1.654, 1.950‡h, µm[−8, 732, 1014, 1190, 1380, 1561, 1840]

F

(a)

T

2

Rex

(h/δ ∗ )

10 6

1.5
1

[90, 110]0.5
6† : cases measured with HWA and computed with 2D-DNS.

(b)‡ : cases measured with HWA.

N

4

Transition Front Movement with Step Height2

The transition location is defined in this work as the intersection between the model’s centerline and a linear fit to thefront identified from DIT images (Rius-Vidales & Kotsonis,2021). Figure 2 (a) shows the Reynolds number based on thetransition location (RexT = xT Uref /ν∞ ) obtained in the present
experiments. In addition, the experimental data from Wang
& Gaster (2005) is included given the similarities of their setup and freestream conditions with the present experiments. Inturn, in Wang & Gaster (2005) experiments transition is driven
by the model’s receptivity to background disturbances while inthis work also forced conditions are addressed.
Large differences are observed between the RexT valuesof Wang & Gaster (2005) and this work. There are two plausible reasons for which the transition location in clean case conditions takes place farther upstream in these experiments compared to the ones of Wang & Gaster (2005). First, the turbulent
wedges of the test section side-walls could be interferring with
the natural transition front, promoting early transition. Secondly, receptivity at the LE could be affected by other acoustic
sources coming from resonance effects in the open-jet facility,e.g. standing waves (sec. 16.1.4 in Tropea et al. (2007)), orfrom a global feedback loop generated by vortex shedding at
the TE.
In FFS-TS transition related literature, ∆N is used to isolate the effect of the step on transition from the freestream conditions, enabling proper comparison between different experimental works. This parameter represents the decrease of the
N-factor at transition with respect to the clean case N-factor
due to the presence of a roughness element, i.e. ∆N = NT0 − N.
The clean transition front takes place outside of the heatingplate in these experiments (see Figure 1). Hence, NT0 is instead
computed using the N-factor (from linear stability theory) obtained at a higher Reynolds number, for which transition location could be measured.
Figure 2 (b) shows the ∆N results from this work compared to the different models and fits for ∆N from Wang &Gaster (2005), Crouch et al. (2006) and Costantini (2016).
The present results show a nearly linear dependence of ∆Nwith the relative step height (h/δ ∗ ), similar to the model proposed by Crouch et al. (2006) but with a lower slope, i.e. the
FFS appears more detrimental for laminar-turbulent transitionin Crouch et al. (2006) experiments.
From Figure 2 (b) it can be observed that neither of themodels presented nor the present experiment results present
good correlation. For instance, while the model of Crouch
et al. (2006) predicts a gradual movement of the transition
front with h/δ ∗ , the data fit of Wang & Gaster (2005) depicts
an exponential-like increase of ∆N, indicating that higher steps
quickly advance transition towards the step location comparedto smaller steps. Other authors (Edelmann, 2014; Costan-

0
0

0.5

1

1.5

2

2.5

3

Figure 2: (a) Reynolds transition (RexT ) as a function of therelative step height (h/δ ∗ ). Data digitized from Fig. 4 inWang & Gaster (2005) () and these experiments (◦). (b) Nfactor decrease at transition, ∆N, as a function of the relative
step height. Data fit digitized from Fig. 7 in Wang & Gaster(2005) (), Fig. 6.4.2.1 (d) in Costantini (2016) (),∆N = 1.6h/δ ∗ model from Crouch et al. (2006) () andthe present experiments (◦). Unforced (black) and forced conditions at F = 90 (blue) and F = 110 (green). Shaded regions
indicate confidence bands.

tini, 2016) have already noticed the large differences on ∆Nfrom different experimental works. Costantini (2016) argued
that possible explanations for this could comprise the different
measurement techniques used to identify transition location or
the effect of the step location. The latter was shown to havealmost no effect on ∆N in Crouch & Kosorygin (2020). These
results indicate that there appears to be no universal criteria
to represent FFS-transition based on a single step or transition parameter. Instead, the effect of the step seems to largely
depend on the specific test conditions or transition detection
techniques used in each experiment.

Meanflow Modulation at the Step
Figure 3 (a) shows the pressure coefficient at the wallfrom DNS results. The pressure imposed in the DNS at the
top boundary condition is set to match the pressure developingat the wall during experiments in no-step conditions. However,since the pressure exhibits a weak variation in wall-normal direction (∂ p/∂ y ≈ 0), small differences are observed in Figure 3 (a) when comparing pressure tap measurements and wallpressure from DNS for h/δ ∗ = 0. Wall pressure in step conditions could not be measured during the experiments since
the available pressure taps were located outside the step region(see Figure 1).
The boundary layer is subject to a progressively growing
adverse pressure gradient as it approaches the step (Figure 3(a)). Regions of low momentum close to the wall are subject to large curvature (∂ 2U/∂ y2 ) changes due to the pressure
gradient. This creates an inflection point within the boundary
layer which moves away from the wall as the adverse pressure
gradient increases (note contour lines representing the inflection points in Figure 5). The loss of streamwise momentum3

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

(a)

4
-0.2

2
0

H

-0.3

(b)

3

2

2

0
-50

0

50

100

2

2

3

4

5

0.05
0.02
-5

0

5

0.00

tions are shown in Figure 7. Given the existence of additional
lobes in the TS shape function (observed in both |û| and |v̂|components) and the difficulty to distinguish the original TS
lobe from the emerging ones, only the TS maxima is tracked
to define the N-factor. This explains the abrupt changes in the
N-factor curvature observed for F = 110 at x ≈ 38 mm in Figure 6.
Very good agreement is observed between DNS and experimental results in terms of TS amplification and shape functions from both Figure 6 and Figure 7, respectively. Althoughexperiments do not reach to measure the near-wall peak downstream of the step, they accurately capture the highly distorted
TS shape at x − xh = 40 mm, which, interestingly, takes place
in a region where the meanflow has already recovered to clean
case conditions (see Figure 3 (b)).
Upstream of the step the fundamental mode grows exponentially (Figure 6). It is observed from Figure 7 that the TS
wave becomes thicker in regions where the shear stress is maximum upstream of the step. In front of the step, |û| decays
while |v̂| rapidly increases due to the distortion prompted by
the step geometry (Figure 6). After the step edge, a small peak
emerges near the wall in |û| and |v̂| which coincides with thenear-wall peak observed in ∂U/∂ y (see Figure 7). Phase contours (not presented here) show in this region a feature with
opposite phase to the incoming TS wave. Zoomed streamlines and vorticity contours in Figure 8 further evince the presence of a vortical structure counterrotating the TS wave on top.
Note that this near-wall vortex emerges in regions where no
recirculating flow is predicted from DNS. When the near-wallpeak amplitude is tracked (not shown here), it is observed todecay in streamwise direction, indicating that the existing TS
wave prevails downstream (Figure 8). Within this region the
TS wave undergoes a pronounced stabilization (Figure 6). Further downstream the fundamental mode keeps on stabilizing
more gradually up to x − xh = 40 mm, where it starts to destabilize again. Interestingly, all higher harmonics (not shownhere) grow significantly just upstream of x − xh = 40 mm, indicating that non-linear effects might become important in thevicinity of this region. However, given their low amplitudes
compared to the fundamental mode (20 times larger than thesecond harmonic), the growth of disturbances remains mainly
linear upstream and downstream of the step.
From Figure 6 it can be noticed that the growth of the fundamental mode becomes highly complex downstream of thestep. The appearance of several inflection points in the Nfactor curve together with the emergence of additional lobes
in the shape function constitute some indicators of this. Furthermore, slightly upstream and far downstream of the step |û|

4

1

0.07

Figure 5: Contours of U (a) and V (b) meanflow components from DNS. Contour lines of ∂ 2U/∂ y2 = 0 in (a) and∂ 2V /∂ y2 = 0 in (b) are indicated with dashed lines ().

Figure 3: (a) Pressure coefficient at the wall. (b) Shape factor computed from DNS and HWA boundary layer profiles inclean case (black) and step (orange) conditions. Symbols denote experimental data and full lines DNS. Confidence bands
for experimental data are included for step (orange) and nostep (black) cases.

0
0

(b)

4

1.0
0.8
0.5
0.3
0.0

6

Figure4: Normalized boundary layer profiles (Q =√U 2 +V 2 ) in clean (black) and step (orange) conditions from
HWA measurements (◦) and DNS results () at x − xh =[−40, −7, 1, 20, 40] mm. Profiles of ∂ Q/∂ y are shown indashed lines. Experimental uncertainty is within symbol size.

close to the wall, forces the boundary layer to expand in y,increasing its shape factor, as observed in Figure 3 (b).
At the step, a strong favorable pressure gradient takes
place that is immediately followed by a gradually decreasing
adverse pressure gradient downstream. Similar to the experiments of Eppink (2020) with CFI, a kink in U is observed athe step edge concurrent with the explosive growth in V (Figure 5). Figure 3 (b) shows that the shape factor of the boundary
layer suddenly drops as a consequence of the strong favorablepressure gradient but smoothly recovers to clean case values
downstream. Velocity profiles in Figure 4 provide further insight into the modulation undergone as a consequence of thesudden jump in pressure. After the step the boundary layer
presents a dual structure characterized by two different maxima in ∂U/∂ y (also observed from inflection contour lines in
Figure 5 (a)). This modular composition of the boundary layer
is the result of the curvature change undergone due to the presence of two quasi-simultaneous opposite pressure gradients. It
would appear that the effect of such a strong and delimited favorable pressure gradient persists after the step imprinted in
the boundary layer curvature. Further downstream this is supressed and the adverse pressure gradient prevails, note themerging of the two ∂U/∂ y peaks from Figure 4 and Figure 5(a).

Growth of Tollmien-Schlicthing Waves
Figure 6 shows the growth of two different fundamental
modes, F = 90 and F = 110, in clean and step case conditions. TS wave shape functions at different streamwise loca-

412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
40
24

1

1

8
-8

0.5
0

-24

0
-2

-1

-40

-20

0

20

40

60

-1

0

1

2

3

-40

Figure 8: Contours of the normalized spanwise vorticity anddisturbance streamlines around the step corner. Forcing frequency F = 90 and phase φ = π rad. Wall-normal axis isscaled by 5/3 for visualization purposes.

80

Figure 6: N-factor evolution of the TS wave |û| maximum inclean case () and step conditions () for forcing frequencies F = 90 (blue) and F = 110 (green). Lines denote
DNS data and symbols HWA measurements. N-factor basedon the maximum of |v̂| is represented by a light-colored line.

term andRthe total value integrated in wall-normal direction,i.e. P̃ = 0hDNS P̂dy with hDNS being the DNS domain height.
From Figure 9 (b) it is observed that the dominant term is P̂12 ,which can be expressed in terms of the disturbances amplitude
and phase:(a)4P̂12 = −2 |û| |v̂| cos (ϕu − ϕv )

2
0

(b)

2

1

2

3

4

5

Figure 7: TS shape function at x − xh = [−40, −7, 1, 20, 40]mm for forcing frequencies F = 90 (a) and F = 110 (b) instep case conditions. Lines denote DNS data and symbols
HWA measurements. Dashed lines correspond to profiles ofthe mean shear, ∂U/∂ y.
and |v̂| maxima exhibit different growth rates, indicating thatthe growth of TS waves cannot be uniquely described in exponential form in these regions2 .
Figure 9 (a) presents the instantaneous streamlines andℜ(û) (real part of û) contour fields downstream of the step at
a given phase of the TS cycle to gain insight into the flow features that give rise to the complex growth behavior observed
after the step. Examination of the production term is wellsuited to analyze the linear growth mechanisms that contributeto stabilize or destabilize the disturbances. The production
term from the energy balance of the fundamental mode (seemathematical development in Section C of Jin et al. (2021)) isexpressed asP̂ = −(û∗i û j + ûi û∗j )

∂Ui
,
∂xj

(2)

This last expression indicates that the increase or decrease on
production takes place in regions of high shear where û and v̂
are not in quadrature, i.e. (ϕu − ϕv ) 6= ±π/2.
In overall, it is observed from Figure 9 (a) and (b) thatstructures tilted against the shear (note stations −10 ≤ x ≤ 2
and 40 ≤ x ≤ 80) seem to correlate with regions of positive
production where the disturbance is amplified (Figure 6) andviceversa. Interestingly, in a region around x ≈ 35 mm, production becomes zero, the disturbances achieve very low amplitude and immediately downstream the direction of tilting ischanged. Further downstream the TS waves recover its uprightposition (i.e. (ϕu − ϕv ) ≈ ±π/2) and production goes back toclean case values.
Based on previous works, the growth of disturbances asa result of their tilting against the shear can be associated to
the Orr-mechanism. Åkervik et al. (2008) studied the stability of a Blasius boundary layer subject to 2-D disturbances bymeans of non-modal analysis and observed that the Orr mechanism is the most optimal condition to initiate TS waves. Theirresults show the evolution of a wavepacket tilted against theshear which effectively amplifies the disturbance linearly until
it forms a well-developed TS wave which switches to exponential growth downstream. The authors observe that when only
TS type modes are considered, the predicted maximum growthis much lower. Despite the differences with the present results,
where the incoming disturbance is already a well-developed
TS wave, Åkervik et al. (2008) findings manifest the importance of accounting for non-modal growth to correctly evaluate
the energy of the disturbance further downstream.

4

0
0

∂U
.
∂y

(1)

CONCLUSIONS AND FUTURE WORKThe stability of TS waves in the presence of an FFS has
been studied by means of experiments and DNS with focus on
the growth mechanisms and flow features emerging at the step
region. Additionally, the global effect of the step on transition
was analyzed from IR measurements. The present results showpoor correlation with the existing ∆N models and fits found inliterature, which in addition present large differences among
the works of different authors. These observations indicate thewhere the superscript ∗ denotes complex conjugate. Figure 9(b) shows the four different contributions to the production

2 Similar conclusions are derived when the N-factor is defined as

the integral of the different TS shape function components (|û| andRh|v̂|) along y, i.e. 0 DNS |ûi |2 dy

5

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

(a)

10

0.2
0

5

-0.2

0
(b)
0.02
0
-0.02
-20

-10

0

10

20

30

40

50

60

70

80

Figure 9: (a) Contours of ℜ(û)/Uref (%) and streamlines from DNS for the fundamental mode with frequency F = 90 at phase φ = πrad. (b) Streamwise evolution of the production term as defined in Equation 1 at reduced frequency F = 90. Full lines indicate step
case and dashed lines clean case. Different production terms are included in the step case: P̂11 (magenta), P̂12 (red), P̂22 (blue), P̂21
(green) and total production (black). Production is normalized with the disturbance kinetic energy integrated along y, Ẽ, and the TS
frequency, ω = 2π f .

difficulty of describing the effect of an FFS on transition witha few non-dimensional parameters.
Meanflow quantities, N-factors and shape functions of thefundamental mode from 2D-DNS and experiments show very
good agreement for two different forcing frequencies, F = 90
and F = 110. Upstream of the step the boundary layer thickness increases due to the adverse pressure gradient built by
the step and TS waves undergo large exponential growth. Immediately after the step edge, the boundary layer presents alarge curvature distortion as a combined effect of two quasisimultaneous but opposite pressure gradients. At this region
the mean shear profile exhibits two different maxima and the
TS waves manifest a new lobe at a near wall location where
no recirculation region is observed from DNS. Further downstream the disturbance depicts a complex growth behavior
characterized by highly distorted TS shape functions at locations where the meanflow has already recovered to clean caseconditions. Furthermore, the |û| and |v̂| maxima of the fundamental mode present different amplification trends, pointing towards non-modal growth mechanisms active in these regions. Examination of the DNS perturbation flowfield in thisarea exhibits the tilting direction of disturbances with respect
to the mean shear to be closely related to a change of signin the production term. These findings indicate that the Orrmechanism could play a role in the non-modal growth of disturbances after the step. Finally, immediately after the step
corner, a vortical structure is observed in the perturbation fieldcounterrotating the incoming TS waves. This near-wall featureemerges in regions of high mean shear, with no recirculating
flow, where both |û| and |v̂| maxima undergo a steep decay.
The mechanisms giving rise to this structure remain unclear.
Future numerical and experimental works should investigate the effect that changing the step height and incoming
TS wave amplitude has on the growth mechanisms dominant
in the vicinity of the step. This could help understand whichparameters result more detrimental for laminar-turbulent transition with FFS.

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schlichting instability. AIAA Journal 58 (7), 2943–2950.
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Drake, A, Bender, A, Korntheuer, A, Westphal, R, Rohe, W,
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the New Horizons Forum and Aerospace Exposition, p. 375.
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