Delay of laminar–turbulent transition by counter-rotating cylindrical roughness elements in a laminar flat plate boundary layer

Delaying laminar–turbulent transition in boundary layers is of great interest since the skin-friction coefficient can be reduced by up to one order of magnitude. In this experimental research, it is shown that counter-rotating cylindrical roughness elements are able to delay transition under realistic flow conditions. Evidence is given by the intermittency, evaluated from hot-film measurements in a laminar water channel. An increase in rotation speed results in a delay of transition of up to 6.5%\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${6.5}{\%}$$\end{document} in the center of the plate. This trend can be explained by the streaks amplified by the rotating cylinders, resulting in a damping of the fluctuation amplitude in the boundary layer. The advantage of this method is that the transition delay can be actively controlled with conventional cylindrical roughness elements.


Introduction
Depending on the intensity of external disturbances, the laminar-turbulent transition process can take several paths. The primary scenario is triggered by the exponential growth of infinitesimal instability, i.e., Tollmien-Schlichting (TS) waves, followed by a secondary instability and breakdown into turbulence. With higher external disturbances, due to the non-normality of the governing stability operator, the instability is characterized by a transient algebraic increase in the perturbations (Chomaz 2005). With even larger disturbances, the boundary layer can transit directly to turbulence, bypassing the scenarios mentioned above.
Three-dimensional (3D) surface roughness elements have long been thought to promote the transition process. The induced flow is featured by a horseshoe vortex wrapping around the 3D roughness element and two elongated counter-rotating streamwise vortices. The alternating low-highlow velocity streaks are subjected to two inviscid inflectional instabilities, i.e., sinuous (anti-symmetric) and varicose (symmetric) instabilities (Andersson et al. 2001). The competition between the wall-normal and spanwise shear in a boundary layer determines which of the two instabilities will predominate. However, Fransson et al. (2006) showed that boundary layer streaks induced by cylindrical roughness elements are capable of controlling the TS instabilities up to a complete transition delay. The mechanism is attributed to a negative spanwise production in the perturbation kinetic energy (PKE) budget, brought by the spanwise modulated 1 3 42 Page 2 of 8 velocity streaks (Cossu and Brandt 2004). However, the cylinder array needs careful construction, and a specific flow configuration is required to avoid the fast-growing inflectional secondary instability which is inevitable in a streaky boundary layer. Moreover, the recirculation region behind a cylindrical roughness element is prone to self-sustained global instability (Loiseau et al. 2014). Concerning this, miniature vortex generators (MVGs) are preferable where both high-amplitude streaks and small recirculation regions can be generated (Shahinfar et al. 2012), making the transition delay control method with surface roughness elements more robust.
Similarly, the linear stability analysis performed by Wu et al. (2021) has shown that by rotating the cylinder elements, higher amplitude streaks and small recirculation regions can also be obtained, meanwhile, the growth rate of the TS-like mode is effectively reduced up to complete stabilization. In contrast to the classic TS mode, the TSlike mode contains a three-dimensional (3D) distortion. The mechanism is based on the reduction in wall-normal PKE production . However, the inviscid inflectional mode caused by the distortion of the cylinder in the boundary layer at medium rotation speed can premature transition. Even direct laminar-turbulent transition can occur at high rotational speeds as a result of a deceleration mechanism that takes place on the rotating cylinder's decelerated side .
The purpose of this experimental work is to demonstrate the proof-of-concept, that by actively rotating the cylinder pair at low rotation speed, the boundary layer primary instability can be stabilized and the corresponding laminar-turbulent transition can be delayed in a realistic flow configuration. Existing transition delay techniques with surface roughness elements can be categorized into passive control methods. The authors are not aware of any other studies that actively delay the laminar-turbulent transition utilizing cylindrical surface roughness elements except our own. Further, if the transition delay can be confirmed, it is of interest how far the transition point in the streamwise direction can be delayed.

Experimental setup
Experiments have been performed in the laminar water channel (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. The turbulence intensity of the facility is Tu = 0.05% between 0.1 and 10 Hz, as reported by Wiegand (1996) and validated in a recent measurement campaign by Puckert et al. (2017). A steady 2D laminar boundary layer forms on the flat plate, which has an elliptical nose at the leading edge. Measurements are carried out using a Dantec 55R15 hot-film probe, whose position can be set by a traverse system. The hot-film probe is connected to a Dantec Streamware bridge, which works according to the constant temperature anemometry principle. The output voltage of the bridge is recorded with a 16-bit National Instruments USB-6216 A/D converter and converted to velocity u through King's law. Unless otherwise noted, each measured point has a measurement duration of 120 s and a sampling rate of f = 100 Hz . The measurement uncertainties are discussed in Appendix 1.
Two cylinders, each with a height k = 7.4 mm and a diameter of d = k∕2 , are placed at a distance of x = 1400 mm behind the leading edge with a spanwise spacing of ∕k = 2 . Between the two cylinders at (x, y, z) = (1400, 0, 0) mm, the x k ∕y k ∕z k -coordinate system (non-dimensionalized by k) has its origin. The free-stream velocity is U e = 0.1 m s −1 , resulting in a displacement thickness at the streamwise cylinder position of 1 ∕k = 0.87 . The roughness Reynolds number Re kk = u(k)k∕ is 461.3, where u(k) is the undisturbed boundary layer velocity at roughness height k and is the kinematic viscosity. To prevent uncontrolled transition directly behind the roughness elements, Re kk is lower than the critical Reynolds number Re kk,crit = 610 for aspect ratio = d∕k = 0.5 (von Doenhoff and Braslow 1961;Wu et al. 2021). The cylinders rotate counterwise, inducing a high-momentum fluid between them. The rotation speed is given by Ω u = Ωd∕(2u(k)) , where Ω is the angular velocity. An overview of the experimental setup is shown in Fig. 1.
In order to compare the transition location over the flat plate with and without the rotating cylinders in a controlled boundary layer, a disturbance wire oscillating in the wall-normal direction is used to excite the TS wave. The disturbance must cross branch I of the Blasius stability diagram, which separates the stable region from the unstable region, in order to achieve amplification of the TS wave. Figure 2 shows the evaluated stability diagram for U e = 0.1 m s −1 . The wire, which has a diameter of 0.1 mm , is oscillating with f TS = 0.15 Hz and is located at x = 1700 mm (cross × in Fig. 2). This configuration Fig. 1 Schematic of experimental setup is slightly upstream of branch I, because there is not a fully settled TS mode immediately downstream of the TS generator in experiments, and thus, a relaxation distance must be considered (Wiegand 1996). To reduce the relaxation distance to a minimum, the wall-normal position y = 4 mm ( y k = 0.54 ) of the wire is located in the local critical layer. The critical layer is defined as the height in the boundary layer where the phase velocity c of the TS wave is equal to the local baseflow velocity u(y) (Schmid and Henningson 2001). This position is ideal for triggering the TS wave, as shown in experiments by Wiegand (1996).

Verification of the TS wave setup
The boundary layer without roughness elements, but disturbed by the TS generator, is measured and compared with LST to verify that the TS wave can be properly triggered in the current LaWaKa setup. The measured TS wave at x = 3000 mm ( x k = 216.2 ) is compared to the theoretical TS amplitude and TS phase in Fig. 3a and b, respectively, by bandpass filtering the measured velocity data in the physical frequency range f = 0.14-0.16 Hz ( f TS = 0.15 Hz ). Both figures show good agreement between measurement and LST. The maximum of the fluctuation velocity |u � |∕u � max is located at y k = 0.68 and agrees with the theory.
The amplitude ratio of the TS wave in the experiment is measured downstream of the TS generator at the local maximum position y k = 0.68 . According to Wiegand (1996), this comparison is the most accurate verification of whether LST and experiment are in agreement. In general, the evolution of the amplitude ratio can be calculated with (Mack 1984) being A(x) and A 0 the amplitude at position x and the initial station x 0 , respectively. The amplification rate in streamwise direction is donated as − i . From the experiment, the and A 0 are obtained by the root-mean-square (rms) of the velocity fluctuation u � rms (x) at the streamwise position x and at the streamwise position when passing the branch I, respectively. As can be observed from Fig. 2, the branch I is traversed at approximately x = 2000 mm . From Fig. 3c, it can be confirmed that the amplitude ratios in the experiment and LST are in good agreement up to x k = 340 . For 340 < x k < 380, the amplification rate in the experiment is higher than that predicted by the LST, indicating that nonlinear effects are present and a turbulent boundary layer develops. The transition location will be addressed in further detail in the next chapter. The small deviation between the experiment and LST in the linear stage x k < 340 can be explained by the changing position of the maximum TS amplitude in the wall-normal direction as the TS wave travels downstream, whereas the hot-film probe in Fig. 3c is traversed at constant y k = 0.68 position. The influence is small as the probe position is sufficiently close to the maximum amplitude u � max in the linear stage. The maximum amplitude at x k = 81 is at y k = 0.61 and moves to y k = 0.68 at x k = 216.2 . Thus, the amplitude ratio in the experiment is provided with sufficient accuracy in Fig. 3c.

Results
The counter-rotating cylinder pair is effective in generating high-amplitude velocity streaks (Wu et al. 2021). Here, the streak amplitude defined by Shahinfar et al. (2012) is evaluated and shown in Fig. 4. A st (y) is calculated with the maximum and minimum of ū along constant y position. The maximum streak amplitude is defined with A st,max = max y (A st (y)) . The error bars are defined in Appendix 1. Compared to the static case Ω u = 0 , the maximum streak amplitude at rotation speed Ω u = 0.265 is more than three times larger. The maximum streak amplitudes are given in Fig. 4. According to the hypothesis of Fransson et al. (2005), a higher streak amplitude is destined to stabilize the boundary layer.
To determine the transition location x k,tr , measurements are carried out to obtain N x = 21 equidistant measurement points from x k = 216.2 to x k = 486.5 . The position of the probe is traversed at y k = 0.68 . This position is located in the maximum TS wave amplitude at x k = 216.2 , as shown in Fig. 3. According to Matsubara et al. (1998), the transition location is independent of y k , as long as y k is lower than the boundary layer thickness. After the measurements are performed, the intermittency function as described by Zhang et al. (2013) is evaluated, where varies between 0 and 1. If = 0 , the flow is laminar, whereas = 1 indicates a fully turbulent flow. The transition point x k,tr is defined at the x k position, where = 0.5 . Figure 5 illustrates the intermittency function for different rotation speeds Ω u together with the TS wave disturbed boundary layer without roughness elements. The solid lines are obtained by a least-squares fit to where c 1 and c 2 are individual constants. Comparing the intermittency function of the static case Ω u = 0 with the TS disturbed boundary layer at z k = 0 , a slight delay of transition is noticeable. Similar to the experiments from Fransson et al. (2006), streaks were generated that have a stabilizing effect on the TS-like wave. However, the influence on transition is limited. As the rotation speed increases, the transition is tending to be postponed downstream. The trend ends with Ω u = 0.41 , where no fit has been applied in Fig. 5 (dashed line). The boundary layer is still laminar at this rotation speed up to x k = 351 , but it rapidly turns turbulent at x k = 365 . This behavior is comparable to a bypass mechanism. An explanation for this behavior may be related to the high total PKE of the inviscid inflectional mode (Wu et al. 2021). Therefore, rotation speeds higher than Ω u = 0.343 are not investigated further. The most effective rotational speed for delaying transition is Ω u = 0.343 , which allows the transition point to be delayed up to x k,tr = 381 . To allow a comparison of the transition delay in future work, the transition delay related to the TS wave with respect to the leading edge is given with Compared to the transition location of the TS disturbed boundary layer x k,tr (TS) = x k,tr,TS = 346 (x tr,TS = 3960.4 mm ), the transition delay gained by the rotating cylinders is 6.5% at the spanwise location z k = 0 . It will be shown that a larger transition delay can be obtained at other spanwise locations for Ω u = 0.343.
To give a more detailed comparison, the power spectral density (PSD) of the velocity signal at different x k positions from Fig. 5 is evaluated for the TS setup and the Ω u = 0.343 setup. Figure 6 shows the PSD as a function of the nondimensional angular frequency = 2 fk∕U e . The dashed (4) Δx tr x tr,TS = x tr,Ω u − x tr,TS x tr,TS . lines mark TS = 9.42 of the TS wave generator. Near the vibrating wire at x k = 121.6 , the PSD distribution of both setups is similar, with the highest peak giving the frequency of the TS and TS-like mode, respectively. One might expect an additional peak due to cylinder rotation, which is = 134.8 for Ω u = 0.343 and originates from the inviscid inflectional mode. However, this peak is only present in the close vicinity downstream of the cylinders (Wu et al. 2021) and is thus not present at the streamwise positions in Fig. 6. From x k = 148.6 to x k = 229.7, the PSD of the TS wave increased by three orders of magnitude, as predicted by LST in Fig. 3c. With cylinder rotation, the peak at TS is decreased between x k = 148.6 and x k = 229.7 , which clearly demonstrates the attenuation of the TS-like mode. The PSD distribution of both configurations grows at x k = 324.3 , however, for the TS setup, it grows to a considerably higher level. The higher-harmonic peaks at the TS setup indicate that a secondary instability is present as outlined by Klebanoff et al. (1962). Downstream at x k = 459.5 , both PSD distributions are at the same high level, implying that a turbulent boundary layer is present. In summary, Fig. 6 reveals that the TS wave is attenuated by the cylinder rotation. In the following, the influence of cylinder rotation in a y k ∕z k -plane at x k = 216.2 is analyzed. Figure 7 shows the fluctuation amplitude u � rms ∕U e (background color) and mean velocity isolines ū∕U e (solid white lines). Measurements are taken exemplarily for Ω u = 0 (Fig. 7a) and Ω u = 0.265 (Fig. 7b) at evenly spaced intervals of Δy k = 0.135 and Δz k = 0.27 to gain a measurement grid with N y × N z = 35 × 22 points. Each point is measured with a duration of 60 s , resulting in a total measurement duration of 25.67 hours for each case. Note that, due to the symmetry of the setup, measurements are only performed in a spanwise direction beginning in the center of the plate ( z k ≥ 0 ). The location of the cylinders is displayed as white dashed lines. First, it can be seen by the ū∕U e isolines that a rotation of the cylinders results in a stronger high-speed streak around z k = 0 . By comparing both figures with respect to the fluctuation amplitude u � rms ∕U e , it is clear that almost the entire boundary layer shows lower fluctuation amplitude values with rotation speed Ω u = 0.265 . To get a better impression of this, the difference in fluctuation amplitude between Ω u = 0.265 and Ω u = 0 is shown in Fig. 8 by the background color ( Δu � rms ∕U e ). Negative contour levels indicate a decrease in fluctuation amplitude when the cylinders rotate and vice versa. The lines in Fig. 8 mark the difference between the mean velocity Δū for Ω u = 0.265 and Ω u = 0 . Distinct decrease in fluctuation amplitude can be seen in Fig. 8 at regions y k , z k = (0.7, 2.2) and y k , z k = (1, 0) . The former location is in the amplified low-speed region (dashed lines), while the latter location is in the amplified high-speed region (solid lines). At around y k , z k = (2.1, 2.1) , a slight increase in the fluctuation amplitude is visible. At this location, the increase in fluctuation amplitude is Δu � rms ∕U e = 0.4 × 10 −2 , which is significantly lower than the decrease in fluctuation amplitude Δu � rms ∕U e = 1.4 × 10 −2 at y k , z k = (0.7, 2.2) and can be regarded as a side effect by the cylinder rotation. This effect is probably caused by an inflectional instability, as reported in Wu et al. (2021). To sum up, the dampening effect of rotating cylinders extends over the entire span and is not just present at the high-speed streak regions. An additional investigation with = 1 and ∕k = 4 was also performed. A similar decrease in fluctuation amplitude up to rotation speed Ω u = 0.19 was observed. It should be noted that the stabilization effect is due to the reduction in the wall-normal PKE production brought in by the velocity streak modulation (Wu et al. 2021), where both positive and negative patches coexist in the PKE production. They cancel each other out in the integral sense, which results in an overall stabilization effect. Therefore, if further counter-rotating cylinder arrays were repeated in the spanwise direction, the stabilization effect will not be changed qualitatively.
To verify that the transition delay is not only present in the center of the high-speed streak ( z k = 0 ), measurements at the spanwise positions z k = 1 (directly behind a cylinder) and z k = 2 are also performed up to Ω u = 0.343 . In Fig. 9, the difference between the transition positions x k,tr (Ω u ) and x k,tr,TS is shown, by evaluating the intermittency function in the same way as in Fig. 5. Positive values indicate a transition delay by the cylinders and can be confirmed for all z k positions in Fig. 9. For the static case Ω u = 0 and the low rotation speed case Ω u = 0.095 , the intermittency functions are almost identical for z k = 0 and z k = 1 . With higher rotation speeds, a slightly postponed transition point x k,tr at z k = 0 compared to z k = 1 is noticeable. At z k = 2 , this trend is even stronger and is probably caused by the strong decay of the fluctuation amplitude at this position (compare Fig. 8). The slightly different behavior of the transition delay in the spanwise direction can be explained by the streak variation in the streamwise direction, as can be observed from the graphical abstract. The results confirm that transition delay is not only present in the high-speed streak region, but also in the low-speed streak region.

Conclusion
Experiments have been performed with a counter-rotating cylinder pair placed in a Blasius boundary layer. The present work has provided direct evidence that rotating cylinders have a controllable positive effect on boundary layer stability. Up to Ω u = 0.343 , the intermittency function shows that the transition can be delayed further as the rotation speed is higher. The best effect on the transition between the cylinders is gained with rotation speed Ω u = 0.343 , where transition can be delayed by 6.5% compared to a TS wave disturbed boundary layer. If the rotational speed is increased further, it is observed that the boundary layer rapidly turns into a turbulent state. The mechanism for the transition delay is given by the amplified streaks induced by the rotating cylinders, which are able to attenuate the TS-like mode (Wu et al. 2021). Confirmation is given by a PSD analysis and the fluctuation amplitude downstream of the cylinders, which is lower when the cylinders rotate. Compared to passive transition delay mechanisms, e.g., static roughness elements (Fransson et al. 2006;Shahinfar et al. 2012), the advantage of the here presented active transition delay mechanism is that a control of the laminar-turbulent transition is possible by adjusting the rotation speed of the cylinders.

Appendix 1: Measurement uncertainty
Measurement uncertainties consist of two errors: systemic errors and random errors (Tropea et al. 2007). Systematic errors are a systematic underestimation or overestimation of, e.g., Re kk caused by an inaccurate calibration, deviating kinematic viscosity and manufacturing errors of the cylinder. Random errors are the difference between scattered samples and the mean of all scattered samples. They can be detected by repeating the measurements, which is not possible for systematic errors. In this work, the measurement uncertainties are determined as demonstrated in Appendix 2 from Puckert et al. (2020). To obtain the total systematic error, the authors compared the mean velocity measured by the hot-film probe and the mean velocity identified by a hydrogen-bubble visualization in the LaWaKa. Moreover, they give the systematic error resulting from using a mercury thermometer to determine the kinematic viscosity , and they provide an appropriate systematic error for the cylinder height k. The total systematic error from the current LaWaKa measurement setup is given with (Puckert et al. 2020): sys = 2.6e − 3 m s −1 ( sys ∕U e = 2.6% ). The random error cannot be taken from Puckert et al. (2020) because it has to be determined individually for each measurement. The random error is determined with a free-stream measurement and validated by several measurements at different free-stream positions. This assumption is necessary because the random error in the boundary layer is distorted by the TS wave and the rotating cylinders. The random error is two orders of magnitude smaller than the systematic error: ran = 8.04e − 5 m s −1 ≪ sys = 2.6e − 3 m s −1 . Therefore, the random error can be neglected here, and the resulting error A st = sys is shown in Fig. 4. The resulting error of the free-stream velocity is U e = 1.3e − 3 m s −1 ( U e ∕U e = 1.3% ) and is derived from the systematic error of the free-stream velocity sys,U e = 1.3e − 3 m s −1 from Puckert et al. (2020).