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

STABILITY ANALYSIS IN BOUNDARY LAYER BEHIND A ROUGHNESSWITH FREE-STREAM TURBULENCE

Tristan M. Römer
Institute of Aerodynamics and Gas Dynamics
University of Stuttgart
Pfaffenwaldring 21, D-70569 Stuttgart, Germany
roemer@iag.uni-stuttgart.de

Ulrich Rist
Institute of Aerodynamics and Gas Dynamics
University of Stuttgart
Pfaffenwaldring 21, D-70569 Stuttgart, Germany
rist@iag.uni-stuttgart.deauthors focused on FST without roughness elements, Kumar
et al. (2015) investigated the combined influence of FST androughness elements on transition. By placing a grid at a specific streamwise wall-normal position, transition can be delayed. A numerical focus on the combination of FST androughness elements is done by Bucci et al. (2021). They wereable to derive predictions on instabilities that would occur asa function of the two variables. Their maximum investigated
FST is Tu = urms /Ue = 0.18%, with urms being the root-meansquare (rms) of velocity fluctuation and Ue the mean velocity of the free-stream. An experimental contribution to theroughness-induced transition with higher FST as in Bucci et al.
(2021) is provided by Puckert et al. (2021). These authors determined the transition Reynolds number Rek,tr for two higher
levels of FST. Rek is defined by Ue and k, with k being theheight of the roughness element. However, the influence of
FST in the streamwise direction is not considered, which - aswill be shown in this paper - has an effect on transition. Therefore, the experimental setup used in Puckert et al. (2021) isrevisited and extended to include detailed measurements withrespect to an extended FST range by two different types ofturbulence generators. Instead of a grid with vertical bars, acrossed grid is used here to minimize the wall-normal turbulence variation. The following questions will be answered:

ABSTRACTAn investigation of laminar-turbulent transition behind acylindrical roughness element in the boundary layer with controlled free-stream turbulence (FST) is performed using hotfilm measurements. Different levels of FST are generated by a
bubble generator and a grid. An equation for determining thetransition Reynolds number as a function of FST is presented.
FST is higher at the leading edge of the plate with grid, but
decays much faster in the streamwise direction. It is suggested
that the boundary layer with roughness is not mainly receptive
to FST at the leading edge of the flat plate, but rather at thecylinder and in its wake.
The results of a measurement campaign in the wake ofthe cylinder shows that FST lowers the entire instability range
to lower Reynolds number, explaining the lower transition
Reynolds number at higher FST. A power spectral density(PSD) analysis reveals that the shedding frequency of hairpin
vortices is independent of FST. However, the hairpin development starts and decays at lower Reynolds numbers with FST.
Above a certain FST level (here Tu = 1.95 %), the hairpin development becomes too unstable and hairpins decay immediately after their initial generation, should they develop at all.

INTRODUCTION• Which influence does a decreasing FST have on Rek,trcompared to an almost constant FST in streamwise direction?
• How does Rek,tr depend on Tu with constant FST in thestreamwise direction?
• How is the hairpin vortex development affected by FST?

Precise prediction of laminar-turbulent transition inboundary layers under given conditions could improve the design of an airfoil with respect to drag. However, the prediction of transition is not straightforward, since it depends on
disturbances like roughness and free-stream turbulence (FST).
The experimental results of roughness-induced transition withvarious aspect ratios of the roughness elements from different
authors is summarized by von Doenhoff & Braslow (1961).
It was found that despite the same measurement setup, there
is a variation in laminar-turbulent transition between the authors. This could be attributed to the varying measuring systems, which have slightly different levels of FST. Fransson& Shahinfar (2020) presented a transition semi-empirical prediction model that determines the transition as a function of
the FST level Tu and the integral length scale. Whereas these

Test Facility
Experiments have been performed in the laminar-waterchannel (Laminar-Wasser-Kanal, LaWaKa) at the Institute of
Aerodynamics and Gas Dynamics (IAG) at the University of
Stuttgart. The LaWaKa is a closed-loop water facility that provides a reproducible measurement environment for flat plate
laminar boundary layer studies. It has a turbulence intensity of
Tu = 0.05 % between 0.1 − 10 Hz (Wiegand, 1996). A steady1

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
two-dimensional (2D) laminar boundary layer forms on the flat
plate, where the leading edge has an elliptical nose. Wiegand(1996) showed that there is a high agreement between the theoretical and measured flow in the LaWaKa.
The measurements are carried out using a Dantec 55R15
hot-film probe, whose position can be set by a traverse system. A Dantec Streamware bridge is connected with the hotfilm probe and works according to the constant temperature
anemometry principle. The output voltage of the bridge isrecorded with a 16-bit National Instruments USB-6216 A/D
converter and converted to velocity u through King’s law.

Ue x

x

z

flat platefine/coarse gridbubble generator
Figure 1.
Table 1.

Experimental Setup
A cylindrical roughness element with height k = 0.01 m
and width d = k (aspect ratio η = d/k = 1) is placed 0.57 m
behind the leading edge in the center of the flat plate. The coordinate x represents the distance from the leading edge andy the wall-normal position. Unless otherwise noted, the coordinates x,y and z are nondimensionalized with respect to k.
According to this notation, the roughness element is placed at
x = (0.57 m)/k = (0.57 m)/(0.01 m) = 57.
To generate controlled FST, two different turbulence generators have been developed: a bubble generator and a coarse
grid. The bubble generator allows a continuous adjustment of
the FST level by modification of the release rate of rising air
bubbles from a strip upstream of the leading edge. A fine gridis placed between the bubble generator and the leading edge toallow for better control of the turbulence level and enable a homogeneous spanwise turbulence distribution. Three differentpressures p = 0.1 bar, p = 0.2 bar and p = 0.3 bar are considered, named setup B1 , B2 and B3 , respectively. The second
mechanism is based on a welded grid, with cylindrical wires
forming a lattice with square holes (setup G). The mesh widthis M = 25 mm having wire diameter g = 1.6 mm. The grid dimensions have been chosen such that Reg = Ue g/ν > 40. Thisrequirement was determined by Roach (1987). It indicates thatvortex shedding is present, leading to an increase of FST. A
lower Reg value would lead to a reduction of FST, as shownby e.g. Puckert et al. (2017). Moreover, the porosityg 2
β = 1−= 0.876
M

roughnessyrising air bubbles

Test facility with experimental setup.
Overview of setups and their parameters.

bubblegenerator(pressure)

setup nameRreference

B1

bubblesB2

bubblesB3

bubblesG

grid

× (0.1 bar)× (0.2 bar)× (0.3 bar)

fine
grid

coarse grid
M = 25 mm
g = 1.6 mm

×
×
×
×
×

boundary layer thickness δ99 (xmax = 236) = 2.9. Each data
point has a measurement time duration of t = 300 s and sampling rate of fs = 200 Hz. The data are only band-pass filteredby the Nyquist frequency condition, i.e. 2/t = 0.0067 Hz andfs /2 = 100 Hz. As expected, setup R shows very low Tu levels
along x. Therefore, setup R will be considered a reference casewith effectively no FST.
It can be seen from the grid-generated turbulence that the
turbulence level decays in the streamwise direction. Tennekes& Lumley (1972) predict that the turbulent energy decays as
x−1 , implying that the turbulence level Tu decays proportional
to x−1/2 . Note that different values are given for the exponent
in literature (Batchelor & Townsend, 1948; Pope, 2000; Tennekes & Lumley, 1972). For grid-generated turbulence, the decay can be fitted according to Batchelor & Townsend (1948):(1)should be larger than 0.55 to avoid large-scale mean velocity
variations, as pointed out by Kurian & Fransson (2009). In
order to compare the results, a reference setup R without turbulence generator is also considered. Only the fine grid is used
to reduce FST at low velocities as pointed out by Puckert et al.
(2017). Either the fine grid (setup B1,2,3 , R) or the coarse grid(setup G) is used and placed at x = −25. The complete measurement setup with both turbulence generators is illustrated in
Figure 1. The various investigated setups and their parameters
are summarized in Table 1.

s
Tu =

A



xg − xg,0
M

b

(2)

where xg is the distance downstream of the grid, xg,0 is a virtual
origin, A is an individual grid constant and b is the decay rate.
As shown by Kurian & Fransson (2009), the virtual origin canbe set to xg,0 = 0. For setup G, the nonlinear least-square fit
gives A = 0.0153 and b = −1.45. Equation (2) fits well for
setup G, as can be seen in Figure 2 (dashed curve G fit).
In contrast to the grid-generated FST, the bubblegenerated FST follows a relatively constant Tu-level in xdirection. This is remarkable given that no energy is input
and thus a decay of the turbulence is expected. The rising
air-bubbles not only cause a velocity fluctuation in the streamwise direction, but also a lateral fluctuation due to lateral quivering of the bubbles. Downstream, the lateral velocity fluctuation contributes to the spanwise fluctuation velocity. Thisphenomenon has also been reported by e.g. Kurian & Fransson (2009). Figure 2 also shows the proportional relationship

RESULTS
Free-Stream Turbulence
A major challenge of understanding the influence of FST
on transition is that an exact value of FST level along thestreamwise direction x can not be adjusted in experiments.
This is explained by the fact that turbulence decays without external input. Figure 2 shows the variation of Tu at several x positions for Ue ≈ 0.07 m/s in the free-stream. The wall-normal
position is y = 7, which is higher than the theoretical laminar2

12th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
0.06

between air-pressure and the FST, namely that higher bubble generator pressure yields a higher FST. Accordingly, thebubble setups can be ranked from low to high Tu as follows:
B1 < B2 < B3 . Setup G generates the most dominant FST at
x = 0. Downstream of x ≈ 34, this setup has a lower FST
compared to B1,2,3 . This difference will be interesting for thelaminar-turbulent transition, which is discussed in the next section. In summary, the setups R and B1,2,3 exhibit a relatively
constant Tu along x, whereas setup G shows a strong decay of
Tu in x.

u [m/s]

0.04
0.02
0.00
−0.02

R
B1

Tu [%]

3

B2
B3

G
G fit

50

100

150

30

40

50

section. However, the general assumption that the transition
region increases with higher FST cannot be concluded only by
investigating the FST. The integral length scale has an additional influence on the transition range, as also found withoutroughness by Fransson & Shahinfar (2020). From the present
observations, the integral length scale must play a key role in
the transition region and should be investigated in more detail.
The focus of this paper is on the influence of FST.
The R setup has the transition Reynolds number Rek,tr =670. With setup G, the transition Reynolds number is reduced
to Rek,tr = 654. The bubble-generated FST setups B1 , B2 andB3 reduce the transition Reynolds number to Rek,tr = 616,
Rek,tr = 575 and Rek,tr = 549, respectively. Although setup
B1 has a lower FST level than setup G at x = 0 (compare Figure 2), the transition Reynolds number is by far lower. This isastonishing at first, because it is known that Rek,tr decreases as
Tu increases. The reason for this can be attributed to the higher
turbulence level of setup B1,2,3 downstream of the roughness.
Thus, the first question of this paper can be addressed. Fromthe presently investigated FST levels, the roughness-induced
boundary layer is mainly receptive to FST at and downstreamof the roughness element rather than to FST upstream of theroughness element.

200

x

Figure 2.

20

Figure 3. Unfiltered (top) and filtered velocity (bottom) measured with setup G at Rek = 626, 0 s ≤ t ≤ 50 s < tmax = 600 s
and x = 157, y = 1, z = 0.

1

0

10

t [s]

2

0

0

FST in streamwise direction at y = 7, z = 0.

Laminar-Turbulent Transition
To determine the transition Reynolds number Rek,tr , measurements are carried out at the fixed position x = 157, y = 1
with increasing Rek by varying Ue . Each measurement has
a duration of 600 s and sampling rate of 200 Hz. The measurements are then evaluated by the intermittency function γ
as described by Zhang et al. (2013), where γ varies between
0 and 1. If γ = 0, the underlying flow is laminar, whereas
γ = 1 would indicate turbulent flow. The transition region
lies between 0.1 < γ < 0.9. In this region, Rek,tr is definedwhere γ = 0.5. Here, a slight modification compared to Zhang
et al. (2013) is performed. The velocity signal is band-pass filtered between 0.5 − 10 Hz before evaluating the intermittency,
which is similar to the procedure from Fransson et al. (2005).
In Figure 3, the top line shows the unfiltered velocity signal
and the bottom line represents the band-pass filtered velocity
signal from setup G for Rek = 626. Figure 3 makes clear that
turbulent spots can be recorded, but mean velocity deviationand signal noise are filtered out. This allows for the more accurate derivation of the intermittency function. After evaluating
the intermittency, the values are least-square fitted to:

1.0
0.8

γ

0.6

R
B1
B2
B3
G

0.4
0.2

γ(Rek ) =1
1 + e−c1 (Rek −c2 )

(3)

0.0

where c1 and c2 are constants.
Figure 4 shows the intermittency functions for all setups.
In general, the resulting Rek range for 0.1 < γ < 0.9 is wider
with FST and matches the experiments from Puckert et al.
(2021). This can be explained by the fact that, due to higher
FST, there is more turbulent energy in the free-stream. Disturbances at the roughness element are amplified by the freestream/boundary layer receptivity and have a higher amplitudewith FST than without FST. Details are provided in the next

500

Figure 4.

550

600
Rek

650

700

750

Evaluated intermittency function.

Given Rek,tr for the various setups, a correlation between
Rek,tr and Tu can be determined, i.e. Rek,tr (Tu). This requires
to assign Rek,tr of each setup to one Tu level. For the bubble generator B1,2,3 and the reference R setups, Tu is relatively
constant in the streamwise direction. Therefore, for these se-3

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

Rek,tr (Tu) = 690 − 58.7 · 102 Tu

fit
R

1000

B1
B2

B3
G

900

Rek

800

x = 161

x=0

700
600

von Doenhoff range

tups the Tu averages along x are calculated from Figure 2.
The values Tu = 0.27 % (R), Tu = 1.43 % (B1 ), Tu = 1.95 %(B2 ) and Tu = 2.32 % (B3 ) are assigned to their corresponding
Rek,tr from Figure 4 and depicted as a cross in Figure 5. Notethat the average is calculated for the range 0 ≤ x ≤ 161, sothat all measuring points from the intermittency measurement(Figure 4) are included, but the insignificant FST upstream ofthe leading edge is not taken into account. With setup B1,2,3
and R (constant FST setups) the linear fit

500

(4)

0.0

is calculated and is shown as a solid gray line in Figure 5.
The linear Equation (4) fits well and can be used to calculate
Rek,tr by a given Tu. It should be emphasized that Equation(4) assumes a constant Tu in the streamwise direction. It is notstraightforward to determine a specific Tu for setup G, since
the FST decays rapidly in the streamwise direction. This canbe comprehended by the error-bars in Figure 5, which indicate
the minimum and maximum Tu from Figure 2 in the range
0 ≤ x ≤ 161 for all setups. The minimum and maximum Tu
for setups B1,2,3 and R (solid lines) are closer together than in
setup G (dashed line), since they follow a relatively constant
FST level along x. For setup G, the minimum Tu is locatedat x = 161 and the maximum at x = 0, as shown in Figure 2.
An analogous Tu is calculated for setup G based on the linear
trend shown in Equation (4) and Figure 5. Given Rek,tr = 650
for setup G, the equivalent Tu can be found to be 0.61 %. Thismeans that setup G acts like a bubble-generator, which causes
a FST of Tu = 0.61 %. Or, to be more general, setup G acts
like a constant FST environment with Tu = 0.61 %.
For comparison, the gray background in Figure 5 shows
the lower and higher range of the von Doenhoff & Braslow(1961) transition diagram for η = 1. The range marks where
Rek,tr was found for different roughness element experimentsin the literature without additional FST. If Rek of an experiment is below the gray range, the flow downstream of theroughness remains laminar. Above the range, the flow has already passed transition. Note that the original von Doenhoff& Braslow diagram only displays Rek,tr points. The range in
Figure 5 is derived from Bucci et al. (2021). All presently investigated setups lie within this range. Setup R is clearly below
the upper limit, although the FST is very low. This reveals that
there is another influencing factor which explains the varying
Rek,tr in the von Doenhoff & Braslow diagram. One such factor is that experiments are included in the diagram with morethan one cylinder. Moreover, different facilities lead to a deviating base-flow. More detailed explanations have been reported by e.g. Bucci et al. (2021). Setup B3 with Tu = 2.32 %is located at the lower limit. This indicates that constant FSTlevels higher than Tu > 2.32 % were not present in experimentssummarized by von Doenhoff & Braslow (1961).

0.5

1.0

1.5
2.0
Tu [%]

2.5

3.0

Figure 5. Rek,tr for all setups and corresponding average Tu(marked as a cross, only setup B1,2,3 , R) from Figure 2. The
error bars show the minimum and maximum Tu level between
0 ≤ x ≤ 161. Gray background indicates lower and higher limitof Rek,tr according to von Doenhoff & Braslow (1961).

series results in 754 discrete measurements, where each discrete measurement has a duration of 60 s and a sampling rate of
100 Hz. This results in a total measurement duration of 50.27
hours. Band-pass filtering is applied between 0.1 and 10 Hz.
Figure 6 illustrates the distribution of u0rms /Ue as a function of Rek and x. In the region Rek < 570, no fluctuations arepresent for setup R. This region can be interpreted as steady or
quasi-steady (Puckert, 2019). Such a steady region can not beclearly identified with FST. However, looking at the various
setups at Rek < 670, it appears that the entire lower fluctuation
range moves to lower Rek as FST increases. To get a better understanding of this phenomenon, Figure 7 illustrates the lower
fluctuation range (u0rms /Ue = 0.03) for setups R, B1 and B3by dashed lines. For clarity, setup B2 is not shown, although
the conclusions drawn are equivalent. Thus, with an increaseof FST, disturbances are more amplified at lower Rek , whichleads to an unstable boundary layer at lower Rek . This explains
why an increase of Tu results in a decrease of Rek,tr . Setup B1
has a less steady contour along x compared to R, and the contour for B3 is even less constant. Hence, it is clear that FSTleads not only to a lower Rek,tr , but also to a larger transition
range along Rek (compare Figure 4).
For 60 < x < 70 at Rek > 550, the most dominant fluctuations appear for all setups. This region is marked by solid lines
in Figure 7, where u0rms /Ue = 0.09. Note that only 59 ≤ x ≤ 73
is plotted for the solid lines. Such a region can be associated with the development of hairpin vortices (Puckert, 2019).
Hairpin vortices periodically shed downstream of the roughness element, if a certain Rek is exceeded (Loiseau et al.,2014). As outlined in Puckert (2019) without FST, an increaseof Rek has almost no effect on the upstream front of this region,
which settles here at x = 61 for u0rms /Ue = 0.09. However, with
FST the front moves towards the roughness element (x = 60).
This effect can be explained by the higher ambient fluctuation.
The disturbances require a shorter distance to become amplified to the same amplitude level of lower FST. An increasein FST also stretches this region to lower Rek , indicating that
hairpins start to develop at lower Rek .
To get insights into the FST influence on the developmentof hairpins, the power spectral density (PSD) of u0 is plottedin Figure 8 at x = 63, which lies in the hairpin development
region (compare to Figure 7). Note that similar results are ob-

Velocity Fluctuation Analysis
A velocity fluctuation analysis with setups R1,2,3 is performed, which focuses on constant FST along x. Comparisonis enabled by reference setup R. Measurements are taken in
the center of the plate downstream of the roughness element
at fixed position y = 1 and z = 0 at evenly spaced intervalsof ∆x = 2 from x = 59 to x = 109. The interval lies between
the roughness element (x = 57) and the intermittency measurement from Figure 4 (x = 157). At each measurement position,
the free-stream velocity was uniformly increased to obtain 29
equally distributed Rek in ascending order. One measurement

412th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
0.00

0.03

0.06

u0rms /Ue
0.09

0.12

0.15

0.18

Rek

750

650

550

450

Rek

750

650

550

450

60

70

80

90

100

60

70

x

Figure 6.

100

Contour plots of u0rms /Ue . Top left: R, top right: B1 , bottom left: B2 , bottom right B3 .
hairpin vortex shedding. Looking at the dashed line in Figure
8, the slope and ω-intercept of the harmonic structure along
Rek is not affected by FST. Thus, the shedding frequency ofthe hairpin vortices behind the roughness element is independent of FST. However, three differences to the reference setup
R can be determined. First, the harmonic structure starts at alower Rek with FST. Second, as FST increases, the maximum
PSD tends to decrease and become noisier outside the fundamental frequency. Third, a higher harmonic structure in therange ω ≈ 2, Rek > 690 can be identified only for R. From theobservations, the following statements can be implied for thehairpin vortex development:

B1 : u0rms /Ue = 0.09, 59 ≤ x ≤ 73
B3 : u0rms /Ue = 0.09, 59 ≤ x ≤ 73
R: u0rms /Ue = 0.03

B1 : u0rms /Ue = 0.03
B3 : u0rms /Ue = 0.03

750

Rek

90
x

R: u0rms /Ue = 0.09, 59 ≤ x ≤ 73

650

• The hairpin vortex both develops and collapses with FST
at lower Rek .
• Above a certain FST the hairpin, if present at all, collapses
immediately.

550
450

80

60

70

80

90

The FST of the second statement is here set to Tu = 1.95 %(setup B2 ), which is chosen visually by Figure 8 and should
therefore be regarded as an estimate. Accordingly, bypasstransition may be present in B2 and B3 , but not in B1 andR. The two results clarify why in Figure 7 the solid lines(u0rms /Ue = 0.09) tend to be more elongated along Rek andmore narrow-banded in x compared to R.

100x

Figure 7.

Specific u0rms /Ue values for R, B1 and B3 .

tained in the area 62 < x < 66. The non-dimensional angularfrequency ω = 2π f k/Ue is calculated with the physical frequency f in Hz. In the reference setup R, a clear harmonic
structure starting at Rek = 650 can be observed. The fundamental frequency is ω ≈ 1 and grows slightly with Rek . Puckert (2019) has shown that the harmonic can be associated with

CONCLUSION
The influence of FST on stability behind a roughness
element has been investigated by two types of FST genera-

512th International Symposium on Turbulence and Shear Flow Phenomena (TSFP12)
Osaka, Japan (Online), July 19-22, 2022
PSD ×10−5 [m2 s−1 ]
0
5
10
15
20

ω

2

R

B1

B2

B3

ciety of London. Series A. Mathematical and Physical Sciences 193 (1035), 539–558.
Bucci, M. A., Cherubini, S., Loiseau, J.-Ch., Robinet, J.Ch., Loiseau, J.-C. & Robinet, J.-C. 2021 Influence of
freestream turbulence on the flow over a wall roughness.
Physical Review Fluids 6 (6).
von Doenhoff, A. E. & Braslow, A. L. 1961 The effect of distributed roughness on laminar flow. Boundary Layer Control 2, 657–681.
Fransson, J. H. M., Matsubara, M. & Alfredsson, P. H. 2005
Transition induced by free-stream turbulence. Journal of
Fluid Mechanics 527, 1–25.
Fransson, J. H. M. & Shahinfar, S. 2020 On the effect of freestream turbulence on boundary-layer transition. Journal of
Fluid Mechanics 899, A23.
Kumar, P. P., Mandal, A. C. & Dey, J. 2015 Effect of a meshon boundary layer transitions induced by free-stream turbulence and an isolated roughness element. Journal of Fluid
Mechanics 772, 445–477.
Kurian, T. & Fransson, J. H. M. 2009 Grid-generated turbulence revisited. Fluid Dynamics Research 41, 021403.
Loiseau, J.-C., Robinet, J.-C., Cherubini, S. & Leriche, E.
2014 Investigation of the roughness-induced transition:
global stability analyses and direct numerical simulations.
Journal of Fluid Mechanics 760, 175–211.
Pope, S. B. 2000 Turbulent flows, 1st edn. Cambridge: Cambridge Univ. Press.
Puckert, D. K. 2019 Experimental investigation on global
instability and critical reynolds number in roughnessinduced laminar-to-turbulent transition. PhD thesis, Universität Stuttgart, Stuttgart.
Puckert, D. K., Dieterle, M. & Rist, U. 2017 Reduction offreestream turbulence at low velocities. Experiments in Fluids 58 (5).
Puckert, D. K., Römer, T. M., Scibelli, G. & Rist, U. 2021
Experimental investigation on roughness-induced transition
under the influence of freestream turbulence. In New Results in Numerical and Experimental Fluid Mechanics XIII
(ed. Dillmann et al.), Notes on Numerical Fluid Mechanics and Multidisciplinary Design, vol. 151, pp. 205–214.
Cham: Springer International Publishing.
Roach, P. E. 1987 The generation of nearly isotropic turbulence by means of grids. International Journal of Heat andFluid Flow 8 (2), 82–92.
Tennekes, H. & Lumley, J. L. 1972 A first course in turbulence. Cambridge, Massachusetts and London, England:
MIT Press.
Wiegand, T. 1996 Experimentelle untersuchungen zum
laminar-turbulenten transitionsprozeß eines wellenzuges ineiner plattengrenzschicht. PhD thesis, Universität Stuttgart,
Stuttgart.
Zhang, J., Xu, M., Pollard, A. & Mi, J. 2013 Effects of externalintermittency and mean shear on the spectral inertial-range
exponent in a turbulent square jet. Physical review E 87 (5),053009.

1
0

ω

2
1
0
450

550

650
Rek

750

450

550

650
Rek

750

Figure 8. Spectral analysis by PSD at x = 63, y = 1, z = 0.
The dashed line is displayed for comparison.
tors: a bubble generator and a coarse grid. From the studies
conducted here, the general hypothesis can be confirmed that
Rek,tr is reciprocal proportional to Tu. As shown by the gridgenerated FST, a strong decay of Tu is usually present in various experiments in the literature and it is therefore not straightforward to assign Rek,tr to a specific Tu. With the given equation, Rek can now be determined as a function of Tu, assuming that Tu is relatively constant in streamwise direction. Thishas been made possible by the bubble-generator experiments,
where Tu varies only slightly in the investigated streamwise
range. From the results it is suggested that the boundary layer
with a roughness element is receptive to FST at or downstreamof the element rather than upstream. However, further measurements with other grids are needed to reinforce this statement.
The investigation of the fluctuation power behind theroughness element has shown that disturbances are more amplified at lower Rek with increased FST. The entire lower
fluctuation range is moved to lower Rek and is more unsteadyalong x, explaining the larger transition range along Rek andlower Rek,tr . In addition, a PSD analysis shows that the shedding frequency of the hairpin vortices (ω ≈ 1) is independentof FST. However, it is implied that hairpins develop and decay at lower Rek . If the FST is too high, the development of
hairpins is disturbed too significantly and thus they are directly
broken up. Here, this limit is set to Tu = 1.95 %, but should bevalidated by further experiments and visualizations.

ACKNOWLEDGMENT
The funding of the Deutsche Forschungsgemeinschaft(DFG) under grant number RI 680/39-1 is acknowledged.

REFERENCES
Batchelor, G. K. & Townsend, A. A. 1948 Decay of isotropic
turbulence in the initial period. Proceedings of the Royal So-

6