# Flexible Gear Assembly With Visual Servoing and Force Feedback

Junjie Ming, Daniel Bargmann, Hongpeng Cao, and Marco Caccamo Junjie Ming and
Daniel Bargmann are with the Fraunhofer Institute for Manufacturing
Engineering and Automation, Stuttgart, Germany<EMAIL_ADDRESS><EMAIL_ADDRESS>Cao is with the School of Engineering and
Design, Technical University of Munich (TUM), Munich, Germany.
<EMAIL_ADDRESS>Caccamo is with the School of Engineering and Design,
Technical University of Munich (TUM), Munich, Germany and Munich Institute of
Robotics and Machine Intelligence, Technical University of Munich (TUM),
Munich, Germany<EMAIL_ADDRESS>video showing the gear assembly process with
the proposed approach is available at
https://www.youtube.com/watch?v=5BDMzEspp_M&ab_channel=HongpengCao.

###### Abstract

Gear assembly is an essential but challenging task in industrial automation.
This paper presents a novel two-stage approach for achieving high-precision
and flexible gear assembly. The proposed approach integrates YOLO to coarsely
localize the workpiece in a searching phase and deep reinforcement learning
(DRL) to complete the insertion. Specifically, DRL addresses the challenge of
partial visibility when the on-wrist camera is too close to the workpiece.
Additionally, force feedback is used to smoothly transit the process from the
first phase to the second phase. To reduce the data collection effort for
training deep neural networks, we use synthetic RGB images for training YOLO
and construct an offline interaction environment leveraging sampled real-world
data for training DRL agents. We evaluate the proposed approach in a gear
assembly experiment with a precision tolerance of 0.3 mm. The results show
that our method can robustly and efficiently complete searching and insertion
from arbitrary positions within an average of 15 seconds.

##  I INTRODUCTION

Figure 1: The figure provides an overview of the gear assembly task, with the
objective of grasping a gear from the placement tray, observing the
environment using the on-wrist RGBD camera, and inserting the gear held by the
robot onto the peg using visual servoing to complete the assembly.

(a)

(b)

Figure 2: The figure shows the assembly status after localization from the
first stage.

High-precision and flexible gear assembly have been essential in industrial
automation [1]. Typically, a gear assembly process involves searching and
insertion [2]. To complete these two procedures, conventional gear assembly
relies on tedious position teaching, which has the limitations of time cost
and low flexibility [3]. Recent work leverages visual information [4, 5, 6, 7]
and force/torque (FT) feedback [8, 9, 10, 11, 12, 13] in robotic assembly to
avoid the reliance on tedious position teaching, achieving promising
performance.

In this work, we aim to develop an efficient solution to realize the part
searching and insertion for high precision and flexible gear assembly task, as
shown in Figure 1. The assembly scenario consists of a placement tray with an
initial gear, a Frank Emika Panda Robot, and the target platform mounted with
a gear near the peg to be assembled. The robot is equipped with an RGBD camera
on the wrist to observe the environment and a force sensor on the gripper to
detect contact. This assembly process aims to grasp a gear from the placement
tray and insert it onto the peg to finish the assembly with the gear on the
platform under the precision tolerance of $0.3\text{\,}\mathrm{mm}$. We
simplify the grasping by initializing the gear with a known position on the
placement tray and pre-program a fixed motion for grasping, as the assembly is
the main focus of this work.

We propose a vision-based two-stage approach with force feedback to solve the
above gear assembly problem. Specifically, in the first stage, we use YOLO
[14] with depth information to roughly localize the object. With the coarse
position, we can efficiently drive the robot using position control to reach
the neighbor of the peg. We use force feedback to determine whether the gear
touches the peg or not for deciding the transition between the two stages.

As shown in 2(a), after the first stage, the centers of the gear and peg are
not aligned perfectly due to the perception error. Meanwhile, the camera loses
the direct view of the peg, which imposes the challenges of predicting or
measuring the relative position of the peg. We propose a novel solution to
this challenge by training deep reinforcement (DRL) agents to learn the
underlying relative positions from partially visible target platforms in RGB
images, as shown in 2(b). Through interaction with the environment, DRL learns
to output movement setpoints for position control to complete the insertion.
We explore DQN [15] algorithm with discretized action space and PPO [16] with
continuous action space for the insertion to study the potential of the
proposed method.

To reduce the data collection effort for training deep neural networks, we use
synthetic RGB images for training YOLO and constructing an offline interaction
environment leveraging sampled real-world data for training DRL agents. The
real-world assembly experiment shows that the proposed approach can achieve
high robustness and efficiency when tested in the gear assembly task from 100
different starting points.

To summarize our contributions, we propose a novel two-stage approach by
integrating YOLO and deep reinforcement learning with force feedback for
achieving high-precision and flexible gear assembly. We simplify the training
of YOLO using synthetic images and ease the training of DRL agents with a
real-images constructed offline training environment. For the insertion, we
explore DQN and PPO to effectively address the challenge of partial visibility
when the on-wrist camera is too close to the target. At last, we evaluate the
proposed approaches in a real-world assembly experiment under varying
positional conditions.

This work is structured as follows: Section I describes the background for the
gear assembly task and an overview of the proposed solution. Section II
summarizes the related work. The proposed approach is introduced in detail in
Section III. The experimental setup and the evaluation of the proposed
approach are discussed in Section IV. Section V concludes the work and gives a
brief outlook on future work.

##  II Related work

Robotic assembly is a challenging task in industrial automation. Related work
leverages visual information [4, 5, 6, 7] and force/torque (FT) feedback [8,
9, 10, 11, 12, 13] in robotic assembly to avoid the reliance on tedious
position teaching, improving assembling reliability and flexibility.

Vision-based approaches focus on pose detection for part searching, and recent
work mainly aims to improve the detection accuracy [4, 5, 7]. In [4], a CNN-
based algorithm is proposed to estimate the poses of the part from depth
images and apply pose refinement to improve the estimation accuracy. [6]
detects the pose of the part from the 3D point cloud and gets more accurate
pose estimation through refinement using the iterative close point (ICP)
algorithm. The pose detection can also be simplified by detecting a
checkerboard attached near the part [4].

Once the poses of the parts are available, the assembly motion can be
generated using a motion planer based on kinematics and geometry prior-
knowledge [17]. The movement can be realized using position control. However,
the quality of the vision-based approaches is highly dependent on the
detection accuracy. Furthermore, it can be affected by the uncertainties
presented in the insertion phase due to contact between different parts [1].
This challenge motivates using force/torque (FT) sensors to provide contact
information for the robot to deal with the uncertainties in the assembly
process.

Directly deriving relative positions from F/T signals is challenging, as the
contact dynamics model is complex [13]. Deep reinforcement learning (DRL)
holds the promise of completing the insertion by learning from interactions
with the environment, where the dynamic model is not needed. Recent work [8,
9, 10, 11, 12] train DRL agents to map the F/T signals with other
observations, for instance, the pose of the part, directly to the actions that
drive the robot, achieving promising performance. In contrast to outputting
action, [13] proposes utilizing DRL to select stiffness matrix in admittance
control to induce the motion that can complete insertion.

Our work integrates YOLO to coarsely localize the target in the parts
searching phase and deep reinforcement learning to align parts with finishing
the insertion. Different from the related work that trains DRL to interpret
relative positions from F/T signals for insertion, the novelty of our work
lies in learning the underlying relative positions from partially observed
target platforms in RGB images.

##  III Methods

Figure 3: The plot illustrates the Convolutional Neural Network (CNN) utilized
for extracting features from image input and outputting actions for the DQN
and PPO agents.

In this section, we introduce a vision-based two-stage approach with the
assistance of force feedback to address the gear assembly problem.
Specifically, in the first stage, we use YOLO [14] with depth information to
roughly localize the object. In the second stage, we train deep reinforcement
learning agents to mitigate the inaccuracies from the first stage and finish
the gear assembly by utilizing force feedback.

###  III-A Target localization

The first stage of our assembly task involves localizing the target platform
and bringing the end-effector to the center of the peg. We utilize an RGBD
camera to obtain RGB and depth images of the assembly environment and apply
the YOLO algorithm [14] on the RGB image to detect the 2D bounding box of the
target platform. The depth image is then used to estimate the 3D coordinates
$\\{x_{pf},~{}y_{pf},~{}z_{pf}\\}$ of the center of the platform with respect
to the camera center, as in [18]. Specifically, we convert the pixel
coordinates of the bounding box to 3D coordinates using the camera’s intrinsic
parameters and utilize the depth values along with the converted 3D
coordinates to obtain the 3D position of the target platform. The 3D
coordinates of the targeting peg $\\{x_{p},~{}y_{p},~{}z_{p}\\}$ are
calculated by subtracting the relative distance from the detected center of
the platform $\\{x_{pf},~{}y_{pf},~{}z_{pf}\\}$. Using position control, we
then navigate the end-effector to $\\{x_{p},~{}y_{p},~{}z_{p}\\}$ and move the
gripper downward along the z-axis. The first stage is deemed complete if the
force detected in the z-axis exceeds the predetermined threshold $f_{z}$.

It should be noted that the localization method in the first stage may not
align the center of the peg and gear perfectly due to the inaccuracies of the
detected bounding box, resulting in an alignment error, as shown in Fig. 2(a).
In the next section, we introduce training reinforcement learning agents to
address the alignment error and complete the assembly task.

###  III-B Gear assembly

To address the alignment error, we train DRL agents to generate $\\{x,~{}y\\}$
set-points and use the position control to navigate the end-effector.
Meanwhile, we apply a minor $z$ position control to attempt downward
insertion. Once the centers of the gear and peg are aligned, the gear can drop
onto the peg, and the alignment process is completed. The gear is then further
lowered using only $z$ control until it reaches the specified threshold
$h_{t}$. Finally, a rotation along the $z$ axis is applied to complete the
assembly process, resulting in the gear being successfully assembled with
another gear mounted on the platform.

####  III-B1 Centers alignment using DRL

After the first stage, the camera is close to the peg, losing the peg’s direct
view, as shown in Figure 2(b). This imposes a challenge when aligning the
centers of the gear and peg, as the center of the peg is not directly
observable. We train DRL agents to infer the underlying relative positions to
the peg’s center by observing its neighboring images to address this partial
observable problem.

The process can be formulated as a Partially Observable Markov Decision
Process (POMDP) with
$\mathcal{M}=\\{\mathcal{S},\mathcal{A},P,\Omega,\mathcal{T},R,\gamma\\}$. In
the POMDP, $\mathcal{S}$ represents a set of states, $\mathcal{A}$ a set of
actions, $P:\mathcal{S}\times\mathcal{A}\times\mathcal{S}\mapsto\mathbb{R}$
the state-transition probability function indicating the probability of a
state-action pair leading to a specific next state. $\Omega$ stands for the
set of possible observations that the agent can receive, and
$\mathcal{T}:\Omega\times\mathcal{A}\times\Omega\mapsto\mathbb{R}$ is the
probability of receiving observation $\Omega$ given that the agent takes
action $a$ in state $s$. The reward function
$R:\mathcal{S}\times\mathcal{A}\times\mathcal{S}\mapsto\mathbb{R}$ maps a
state-action-state triple to a real-valued reward. The discount factor
$\gamma\in[0,1]$ controls the relative importance of immediate and future
rewards. The goal in DRL is to find a policy
$\pi:\mathcal{S}\mapsto\mathcal{A}$, mapping a state to an action that
maximizes the expected return from step $t$

$G_{t}=\sum_{i=t}^{\infty}{\gamma^{i-t}R(s_{i},~{}a_{i},~{}s_{i+1})}.$ (1)

Specifically, in the POMDP, the observation of the agent at time $t$ is the
neighbor RGB image of the platform, as shown in Figure 2(b), and action is a
two-dimensional offset $\\{x_{\delta},~{}y_{\delta}\\}$ to the end-effector’s
position in $\\{x,~{}y\\}$ plane. To find a policy that maximizes the return,
we explore DQN [15] algorithm for discretized action output and PPO [16] for
continuous action output. We design a convolutional neural network (CNN), as
shown in Figure 3, to extract features from RGB images and output the movement
offset $\\{x_{\delta},~{}y_{\delta}\\}$ actions for DQN or PPO.

DQN is a value-based approach, and CNN stands for the $Q$ value network. The
action is selected by applying the $\arg\max$ function on Q values output on
different actions $a_{i}=\arg\max(Q(s_{i},A))$, where $A$ is the movement in
$\\{x,~{}y\\}$ direction associated with discretized step size
$\\{-1,~{}+1,~{}-5,~{}+5\\}$ $\mathrm{mm}$. In contrast, PPO is a policy-based
approach, and the CNN represents the policy network that directly maps the
image observation into continuous $\\{x_{\delta},~{}y_{\delta}\\}$ output in
$[-5,~{}+5]$ $\mathrm{mm}$. For more details about DQN and PPO, we refer
readers to the literature [15] and [16].

(a)

(b)

Figure 4: The figure displays the 3D mesh of the platform and the synthetic
image for training YOLO.

####  III-B2 Training

The training of deep neural networks is non-trivial due to the high demand for
training data. To ease the data collection effort for training YOLO, we use a
3D scanner to obtain the 3D mesh of the target platform, shown in Figure 4(a),
and import it into the Blender to generate synthetic images, shown in Figure
4(b), with annotations by leveraging the pipeline proposed in [19]. After
training, the pre-trained YOLO is directly deployed in the first stage for
coarse localization without further training.

The training of DRL agents involves extensive interaction with the
environment, which imposes a challenge when training in the real world due to
the sampling complexity [20]. To address this problem, we discretize the
$\\{x,~{}y\\}$ plane around the peg as a grid world. We then drive the robot
to take pictures at each grid point and record its coordinates, which will be
used to construct an environment proximity to interact with the DRL agents for
training. As shown in 5, the grid is a $1\text{\,}\mathrm{mm}$ x
$1\text{\,}\mathrm{mm}$ square, and the entire map has 30 rows and 35 columns,
covering a $3\text{\,}\mathrm{cm}$ x $3.5\text{\,}\mathrm{cm}$ space. This
sampling process can be automated by programming the movement of the robot
without measuring the position of the peg, and it can be done efficiently
within 30 minutes.

Figure 5: The plot displays a grid world consisting of cells with a length of
1mm, where the center of the peg is marked in red and green grid points
indicating the position for capturing real images using the on-wrist camera.

The grid world enables us to derive a dense reward function to guide the
agents to learn more efficiently. As shown in Equation 2, The first term of
the reward function is the normalized Manhattan distance between the current
grid point and the center of the peg, marked as red, where $S_{px}$ and
$S_{py}$ are the x, y coordinate of the grid point, $n_{Y}$ and $n_{X}$ are
the total number of rows and columns in the sampling map. The second term is
the penalty term to penalize the robot moving out of the
$3\text{\,}\mathrm{cm}$ x $3.5\text{\,}\mathrm{cm}$ space. The agent receives
the highest reward, $r=1$, when the center of the gear reaches the target and
receives an additional penalty, $\beta=10$, when the robot moves out of the
boundary, as shown in Figure 6.

$\displaystyle
r=-(\frac{\left|S_{px}-\frac{n_{X}}{2}\right|}{n_{X}}+\frac{\left|S_{py}-\frac{n_{Y}}{2}\right|}{n_{Y}})/2-\beta,$
(2)

Now we are ready to introduce the pre-training process. The training episode
starts from $T_{0}$ at a random point $S_{px_{0},~{}py_{0}}$ in the grid
world, as shown in Figure 6. The observation for the agent is the recorded
real-world image from the on-wrist camera. The agents take the image as input
and output the movement offset to the current position of the end-effector. By
applying the movement offset, the training environment will evolve to the next
state at $T_{1}$ at $S_{px_{1},~{}py_{1}}$. The interaction continues until
the episode reaches the maximum length or the episode terminates when the
robot moves out of the grid world. To note, PPO algorithm outputs continuous
action which might not lead the next state reaching the exact grid point on
the map. In this case, we take the nearest grid point to the center of the
end-effector as the next state. This approximation will introduce a gap
between the pre-training environment and real world, which will be mitigated
through fine-tuning in the real world.

Figure 6: The 3D diagram shows the rewards associated with the training
environment, with a red dot marking the center of the peg. The offline
training process is visualized using green dots and arrows in the
$\\{x,~{}y\\}$ plane.

After pre-training, we transfer the learned agents to the real world and
further train DRL agents with sparse reward $[0,1]$ to compensate for the
inaccuracy of the pre-training environment to reach optimal.

##  IV Experiments and results

Figure 7: The figure depicts the workflow of gear assembly, consisting of
three steps: 1) driving the robot to grasp the gear on the placement tray; 2)
returning the robot to the initial position while holding the gear; and 3)
completing the assembly process using the proposed two-stage approaches.
Figure 8: The plot illustrates the run time performance of various approaches
in gear assembly experiments with 100 different initial starting points, where
the confidence interval is $95\text{\,}\mathrm{\char 37\relax}$.

(a)

(b)

(c)

(d)

Figure 9: The plot demonstrates the robustness performance of different
approaches in gear assembly experiments with 100 various initial starting
points, where blue dots represent successful assemblies, purple dots denote
failure cases and the green circle corresponds to the shape of the peg.

In this section, we introduce the experimental setup of the assembly task and
the evaluation of the proposed methods.

###  IV-A Hardware setup

The assembly scenario consists of three main components: a placement tray, a
Frank Emika Panda Robot, and the target platform, as shown in Figure 7. The
placement tray is a C-shaped bracket with a blue platform on top for initial
gear placement. The target platform has an aluminum peg at the center and a
gear mounted for gear assembly with a tolerance of $0.3\text{\,}\mathrm{mm}$
111The two gears used in this work are GEABP1.0-40-10-B-10 and
GEABP1.0-20-10-B-10 respectively.. The robot is embedded with a Realsense D415
RGBD camera on the wrist to observe the environment and a force sensor on the
gripper to detect contact. Moreover, the robot’s gripper fingers are
customized to be a pair of crabs to increase grasping reliability.

As the initialization, the gear is placed at the placement tray horizontally
at a pre-configured known position to reduce the grasping uncertainty. The
target platform is mounted horizontally on the workbench. The robot arm is
initialized to view the target platform at any position of the $\\{x,y\\}$
plane and above a certain height. As depicted in Figure 7, The workflow of the
whole pipeline includes 1) the robot is programmed to grasp the gear on the
placement tray; 2) the robot returns to the initial position with the gear in
hand; 3) the assembly starts using our proposed two-stage approach.

###  IV-B Software setup

In the proposed two-stage assembly approach, we use the Darknet implementation
of YOLO as in [14] for object detection and further integrate it with ROS
framework as in [21] for efficient data communication. We leverage the off-
the-shell implementation for PPO and DQN from [22] to address the center
alignment problem at the second stage. The robot’s motion control and contact
detection are performed by pitasc [23], a modular robot operating system.
Furthermore, We use ROS as a communication bridge between different modules.

###  IV-C Evaluation

The target localization using YOLO in the first stage can efficiently and
reliably bring the in-hand gear to the neighbor of the peg, leaving the center
alignment in the second stage as an essential factor for successful assembly.
Therefore, we mainly evaluate the performance of the second stage, in which we
initialize the second stage at randomly sampled 100 points in the neighbor of
the peg and study the robustness and running efficiency of PPO and DQN.
Furthermore, we implement a heuristic spiral search approach as a comparison
baseline.

####  IV-C1 Robustness performance

In the robustness test, we are interested in answering how reliable different
approaches can achieve in assembly tasks with different starting points. We
set the maximum run time budget as 50 seconds, and the experiment that can
finish assembly within the budget will be treated as a success. We repeat the
experiment 100 times from different randomly sampled starting points and
report a success rate (SR) for spiral search and offline-trained DQN and PPO
approaches. We furthermore report the SR of DQN and PPO after real-world fine-
tuning.

As shown in Figure 9, the offline-trained PPO achieves the best performance
with 100% SR, whereas the offline-trained DQN fails in many cases, as marked
as purple, resulting in a 90% SR. The reason is that the relative pose, i.e.,
a relative rotation or translation offset, of the target platform to the end-
effector is not perfectly calibrated before sampling, which introduces a
mismatch regarding the origin and axis in the constructed offline training
environment. This mismatch imposes a big challenge for DQN, as DQN outputs
discretized action along the ${x,y}$ axis. The policy learned in the offline
environment might not have a valid solution to align the centers in the real
world finely. However, PPO outputs continuous actions, which has more
compliance that could compensate for planning errors in the real world. Spiral
searching only has a 19% SR, showing its sensitivity to the distance to the
peg, which also implies the demand for the precision of the target
localization.

We further fine-tune the learned DQN and PPO in the real world to study the
gap between the offline training environment and the real world. The real-
world fine-tuning takes approximately 10 minutes, after which DQN can achieve
a 100% SR.

(a)

(b)

(c)

Figure 10: The figure displays the trajectories of gear assembly from various
initial starting points using different methods, where PPO achieves the most
efficient performance with an average traveling score (ATS) of 1.3.

####  IV-C2 Efficiency performance

In the efficiency test, we aim to study how fast different approaches can
complete the assembly task. We additionally record the time consumed for each
experiment performed in the robustness test, where failed experiments consume
a maximum time of 50 seconds.

As shown in Figure 8, the real-world tuned PPO is the most efficient algorithm
that takes approximately 6 seconds to finish the second stage, whereas DQN, on
average, takes 14 seconds. Spiral searching fails in cases where starting
points are far away from the center of the peg, resulting in the most
inefficient solution. Real-world tuning is generally beneficial to improve
performance consistency and reduce the average run time, especially for DQN.
However, we also notice that the policies learned from the offline training
environment have comparable performance to the real-world tuned policies. This
indicates that learning only in the offline environment can achieve
sufficiently good efficiency, showing the potential of training policies
solely in the offline-environment constructed with sampling points under
tolerable accuracy.

To analyze the behaviors of different approaches, we test spiral search, real-
world tuned DQN and PPO from 12 different starting points across the
$\\{x,y\\}$ plane, and their trajectories are visualized in Figure 10. The
Figure shows that the trajectories of the PPO are closer to the optimal
trajectories, i.e., straight line between two points. In contrast, the DQN
performs in Manhattan geometry due to the discretized action space. To
quantitatively study the difference, we compute the traveling score
$T_{score}$ for each trajectory as

$\hfill T_{score}=\frac{L_{traj}}{|P_{target}-P_{start}|}\hfill,$ (3)

where $L_{traj}$ is the actual length of the trajectory,
$|P_{target}-P_{start}|$ stands for the euclidean distance between the start
and end point. The optimal trajectory would lead to a score of 1. As shown in
Figure 10, PPO achieves the average traveling score (ATS) of 1.3 among 12
trajectories, and DQN achieves 2.0. Spiral searching only succeeds in the four
nearest starting points, and the ATS for these four trajectories is 29.9.
Although PPO acts close to optimal solutions, it can perform a curved
trajectory starting from the bottom right corner, as shown in Figure 10(c).
This is possibly due to the space near the bottom right corner is not well
explored during training. This can be mitigated by resetting the training from
this region or using more exploration-efficient off-policy algorithms.

With the most efficient real-world tuned PPO, the whole assembly pipeline can
be completed within approximately 15 seconds, in which the first stage takes
roughly 9 seconds to finish object searching and approaching.

##  V CONCLUSIONS

In conclusion, we presented a novel two-stage approach for autonomous assembly
that integrates YOLO for target coarse localization and deep reinforcement
learning for fine alignment. Our proposed approach exhibits impressive
robustness and running efficiency when evaluated in an industrial gear
assembly task with 0.3mm precision tolerance. This has been possible thanks to
the use of deep reinforcement learning that can effectively address the
problem of partial visibility when the on-wrist camera is too close to the
target and the force feedback to simplify the transitions between two stages.
We also introduced a novel offline training strategy using real-world data to
pre-train DRL agents.

The limitation of this work is that it requires visually distinguishable
spatial features on the part that can signify its relative distance from the
background. In other words, our approach would struggle to learn from a broad
plane of uniform pattern and color, where every observation at different
positions looks identical. To overcome this limitation, one can, for example,
place printed irregular pattern near the part to enhance spatial information.
The current work is tested only in the 3D space with simplified grasping.
Future work will extend to complete the whole pipeline by combining the
grasping and assembly in 6D space.

## ACKNOWLEDGMENT

Marco Caccamo was supported by an Alexander von Humboldt Professorship endowed
by the German Federal Ministry of Education and Research.

The authors would like to thank Lukas Dirnberger for his assistance in
synthetic data generation.

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# An experimental realisation of steady spanwise forcing for turbulent drag
reduction

Max W. Knoop1<EMAIL_ADDRESS>Friso H. Hartog1,2  Ferdinand F. J.
Schrijer1  Olaf W. G. van Campenhout1,2  Michiel van Nesselrooij1,2  Bas W.
van Oudheusden1  1Faculty of Aerospace Engineering, Delft University of
Technology, 2629HS Delft, The Netherlands 2Dimple Aerospace B.V., 2629HS
Delft, The Netherlands

###### Abstract

We present an experimental realisation of spatial spanwise forcing in a
turbulent boundary layer flow, aimed at reducing the frictional drag. The
forcing is achieved by a series of spanwise running belts, running in
alternating spanwise direction, thereby generating a steady spatial square-
wave forcing. Stereoscopic particle image velocimetry is used to investigate
the impact of actuation on the flow in terms of turbulence statistics,
performance characteristics, and spanwise velocity profiles, for a waveform of
$\lambda_{x}^{+}=401$. An extension of the classical spatial Stokes layer
theory is proposed based on the linear superposition of Fourier modes to
describe the non-sinusoidal boundary condition. The experimentally obtained
spanwise profiles show good agreement with the extended theoretical model. In
line with reported numerical studies, we confirm that a significant flow
control effect can be realised with this type of forcing. The results reveal a
maximum drag reduction of 26% and a maximum net power savings of 8%. In view
of the limited spatial extent of the actuation surface in the current setup,
the drag reduction is expected to increase further as a result of its
streamwise transient. The second-order turbulence statistics are attenuated up
to a wall-normal height of $y^{+}\approx 100$, with a maximum streamwise
stress reduction of 44% and a reduction of integral turbulence kinetic energy
production of 39%.

###### keywords:

##  1 Introduction

Turbulent drag reduction is a crucial research area in fluid dynamics, as
reducing drag can lead to substantial energy savings and emission reductions
in a variety of industrial and transportation applications. Spanwise forcing
of the turbulent boundary layer through transverse wall motion has shown
significant potential for turbulent drag reduction (DR), with reported DR
values reaching over 40% (Ricco et al., 2021). To account for the energy
expenditure of the forcing, the net power saving (NPS) is defined as the
difference between the DR and the idealised power input to drive the spanwise
velocity profile.

Early work on spanwise forcing focused on oscillating wall (OW) motion, where
a wall section is subjected to a temporal oscillation. The strategy was first
studied using direct numerical simulations (DNS) by Jung et al. (1992); a DR
of 40% was found at a spanwise velocity amplitude of $A^{+}=12$ and an optimum
period of $T^{+}=100$. The characteristic parameters are commonly scaled in
viscous units, with the kinematic viscosity $\nu$ and the friction velocity
$u_{\tau}\equiv\sqrt{\tau_{w}/\rho}$; $\tau_{w}$ and $\rho$ are the wall-shear
stress and fluid density. Furthermore, we distinguish two scaling approaches,
using either the reference $u_{\tau 0}$ from the non-actuated case or the
actual $u_{\tau}$ of the drag-reduced flow case; these are denoted by the
superscript ‘+’ and ‘*’, respectively. Laadhari et al. (1994) conducted
experiments which corroborated the results of Jung et al. (1992). They
reported a DR of 36% using the same actuation parameters. Following their
work, OW forcing has been extensively researched over several decades, in both
experimental and numerical studies (Ricco et al., 2021). Another form of
spanwise forcing was introduced by Quadrio et al. (2009), by imposing a
spatio-temporal oscillation, which results in a travelling wave (TW) boundary
condition. A maximum DR of 48% was observed at $A^{+}=12$. Inspired by their
work, several experimental realisations of TW forcing have been made. Examples
are the deformable kagome lattice of Bird et al. (2018), and a more recent
contribution of Marusic et al. (2021), who employed a series of synchronised
spanwise oscillatory elements. The mechanism responsible for the DR lies in
part in the oscillatory nature of the resulting spanwise velocity profile.
This profile predominantly resides within the viscous region of the boundary
layer, thereby showing a close match to the laminar solution of the second
Stokes problem, or temporal Stokes layer (TSL), in the case of OW forcing. In
spatio-temporal form, the solution is described by the generalised Stokes
layer (GSL), as introduced by Quadrio & Ricco (2011). Regarding the working
mechanism, Ricco (2004) presents evidence that the Stokes profile breaks up
the interaction between the near-wall streaks and the quasi-streamwise
vortices by laterally displacing the two features. Their disrupted spatial
coherence decreases the strength of the near-wall cycle resulting in a drag-
reduced state.

A third actuation scheme is a pure spatial forcing, which has been the focus
in a small number of numerical studies (e.g Viotti et al., 2009; Yakeno et
al., 2009; Skote, 2013). Forcing is imposed in this case as a steady wall
velocity in the form of a standing wave (SW):

$W_{w}(x)=A\sin\left(\frac{2\pi}{\lambda_{x}}x\right),$ (1)

in which $A$ is the spanwise velocity amplitude and $\lambda_{x}$ the
streamwise wavelength of actuation. In this work, the $x$, $y$, and $z$
coordinates denote the streamwise, wall-normal, and spanwise directions,
respectively, which correspond to instantaneous velocity components $u$, $v$,
and $w$. Applying Reynolds decomposition, an overbar and prime indicate the
mean and fluctuating component, respectively. The oscillatory spanwise forcing
of the flow is in this concept achieved convectively, as the flow travels over
the regions of alternating forcing direction. SW forcing was first introduced
and numerically studied by Viotti et al. (2009), and an optimum DR of 45% was
found around $\lambda_{x}^{+}=1,000-1,250$ at $A^{+}=12$. Their results were
compared to OW simulations of Quadrio & Ricco (2004), under a convective
transformation. The same trends could be observed; quantitatively, the DR
potential was found to be about 20-30% higher. Under SW forcing, the GSL
reduces to the spatial Stokes layer (SSL). The work of Auteri et al. (2010)
realised TW forcing in their pipe flow setup, where the wave was generated by
means of multiple azimuthally rotating pipe segments. Although SW forcing was
not the primary focus of their studies, under steady conditions, spatial
forcing was obtained. Besides this study, no experimental realisations of SW
forcing have been reported.

To address the sparse attention devoted to SW forcing so far, especially
experimentally, we present a proof of concept for an experimental realisation
of such forcing in an external boundary layer flow. The setup consists of an
array of spanwise running belts, so as to generate a steady spatial square-
wave wall boundary condition. A theoretical model for the SSL for an arbitrary
waveform is proposed, by using linear superposition of Fourier modes that
describe the non-sinusoidal boundary condition. The resultant modified SSL is
compared to the experimentally obtained spanwise profiles. The flow modulation
effects and its performance characteristics are assessed using stereoscopic
particle image velocimetry (SPIV).

##  2 Experimental methodology

Experiments were performed in the M-tunnel of the Delft University of
Technology at a freestream velocity of $U_{\infty}\approx 6.7$ m/s and a
freestream turbulence intensity of the order of 0.7%. The open-return tunnel
has a test section with a cross-section of 0.4 m × 0.4 m. A clean turbulent
boundary layer was generated on a horizontally oriented flat plate with an
elliptical leading edge and tripped using carborundum roughness (24 grit),
achieving a development length of 3 m. The experimental setup was placed in an
open-jet configuration, behind and flush with the test section. A temperature
sensor and Pitot-static tube were used to monitor ambient conditions and
freestream velocity. In view of the limited field of view of the SPIV
experiment, the reference boundary layer under non-actuated conditions was
characterised over its full height using hot-wire anemometry. The hot-wire
probe was positioned 3 mm downstream of the actuation surface. The
characteristics for the current experimental conditions are $\delta=67$ mm
($\delta$ denotes the boundary layer thickness), $u_{\tau 0}=0.27$ m/s,
yielding $\Rey_{\tau}=1230$, where the friction Reynolds number is defined as
$\Rey_{\tau}\equiv\delta u_{\tau 0}/\nu$.

###  2.1 Steady spanwise excitation setup

The experimental setup, referred to as the steady spanwise excitation setup
(SSES), was designed to realise SW forcing experimentally in an external
boundary layer flow. Four streamwise-spaced, spanwise running belts, run in
alternating direction so as to generate a spatial square actuation waveform.
The simplification of using a square waveform instead of a sinusoidal waveform
is justified by the studies of Cimarelli et al. (2013) and Mishra & Skote
(2015), who showed that substantial DR and NPS could be obtained using non-
sinusoidal forcing. A schematic representation of the setup and its
constituting components are presented in figure 1.

Figure 1: Schematic representation of the SSES mechanism, and its placement
with respect to the wind tunnel exit. Arrows on the belts indicate their
respective rotation directions.

The SSES has four 9 mm wide neoprene belts, extending over a spanwise extent
of 256 mm. The streamwise separation of 2 mm between the belts is kept to a
minimum considering the mechanical feasibility of the design. We define the
start of the waveform, and the origin of the coordinate system, at 1 mm (i.e.
half the streamwise separation) upstream of the first belt. The belts are
embedded in an aluminium surface plate so as to run flush with the wall. In
addition, the belts are guided by a PTFE strip to constrain the belts to the
surface grooves. The grooves are precisely machined to strict tolerances, with
a maximum gap and step size of 50 µm. The belts are driven by a non-slip
pulley system, which allows for individually controlling the rotation
direction of the belts. The SSES is powered by two iHSV60 integrated AC servo
motors that drive the belts in positive and negative spanwise direction. The
motors have an adjustable feedback control loop for a fixed belt velocity
output, with a maximum spanwise velocity of 6 m/s. An advantage of such a
setup is the independent control over $A$ and $\lambda_{x}$, which are usually
coupled in OW and TW forcing which employ oscillating elements with a fixed
displacement amplitude.

###  2.2 Stereoscopic particle image velocimetry

SPIV was used to obtain the three velocity components in the streamwise-wall-
normal ($x-y$) plane. Each measurement consists of 1000 uncorrelated velocity
fields, obtained at an acquisition frequency of 10 Hz. Two digital LaVision
sCMOS cameras (2,560 × 2,160 pixels, 6.5 µm pixel size) were placed on either
side of the laser sheet in back-scatter configuration. The field of view (FOV)
was centred on the middle two belts of the SSES, spanning 24 mm × 14.5 mm in
the $x$ and $y$ direction, respectively. The corresponding spatial resolution
is 113 pixels/mm. Imaging was performed using Nikkor AF-S 200 mm lenses, with
Kenko Teleplus PRO 300 $2\times$ teleconverters, at an aperture of f/11. The
cameras were oriented at $\pm 30^{\circ}$ with respect to the laser sheet
normal. Scheimpflug adapters were used to ensure uniform focus across the FOV.
Illumination was provided by a Quantel Evergreen 200 laser (double-pulsed
ND:Yag, 532 nm), with a 100 mJ/pulse power setting. A combination of
cylindrical and spherical lenses was used to create a laser sheet with an
approximate thickness of 1 mm. Seeding was injected at the wind tunnel inlet
by a SAFEX Fog 2010+ fog generator to create 1 µm tracer particles from a
water-glycol mixture.

Stereo calibration was performed to transform the images to physical
coordinates. Additionally, self-calibration was performed to reduce the
calibration fit error to below 0.25 pixel. For each camera, the two velocity
component fields were obtained using cross-correlation, with 16 × 16 pixels
circular interrogation windows at 75% overlap. The resulting vector pitch was
0.035 mm or $0.64\nu/u_{\tau 0}$. The resultant velocity fields of the two
cameras were combined to obtain the 2D3C velocity field by a least squares
fitting approach. The average stereo reconstruction error was below 0.1 pixel.
Wall-normal profiles of the turbulence statistics were obtained by ensemble
averaging (denoted by $\langle\rangle$) in time and streamwise direction,
spanning $0.5\leq x/\delta\leq 1.5$. This method allows us to assess the
integral effect of actuation over one streamwise actuation wavelength. The
uncertainty was quantified using the uncorrelated samples, i.e. every fourth
element in the streamwise direction to account for the 75% overlap. The
uncertainties for the non-actuated case, at $y^{+}=15$, are 0.069% and 0.36%
for the mean streamwise velocity and streamwise Reynolds stress, respectively.

###  2.3 Boundary layer fitting and estimation of the drag reduction

The reference friction velocity $u_{\tau 0}$ was obtained from a fitting the
streamwise velocity profile to the inner-layer profile of Chauhan et al.
(2009) using a sequential quadratic programming iterative method. To constrain
the fitting procedure in order to obtain consistent values, the log-layer
parameters are kept constant, where standard values of $\kappa=0.384$ and
$B=4.17$ have been used (Nagib & Chauhan, 2008). Under drag-reduced
conditions, the procedure as outlined cannot be applied, due to the deviation
from the canonical profile. To obtain the viscous scaling under drag-reduced
conditions, the measured velocity data in the region of $3\leq y^{+}\leq 5.5$
(encompassing four data points) is fitted to the theoretical relation
$u^{*}=y^{*}$. Note that although the fitting domain is obtained from the non-
actuated case, the linear relation is still valid under drag-reduced flow
conditions as the viscous length scale increases. The maximum standard error
of $u_{\tau}$ was 0.78%. The DR is defined as the reduction in mean wall-shear
stress, which can be expressed in terms of $u_{\tau}$ as:

$\text{DR}=1-\frac{\tau_{w}}{\tau_{w0}}=1-\left(\frac{u_{\tau}}{u_{\tau
0}}\right)^{2}.$ (2)

##  3 Laminar solution of the spatial Stokes layer for an arbitrary waveform

An extension of the laminar SSL model is required to allow for the
representation of an arbitrary waveform of the spanwise wall velocity. The
classical Stokes layer solution is derived from the $z$-momentum equation of
the laminar boundary layer equations, under the assumption that the
characteristic wall-normal length scale ($\delta_{x}$) is small compared to
the outer-scale. Full details on the derivation can be found in Viotti et al.
(2009). The solution corresponding to a harmonic waveform reads as follows:

$w(x,y)=AC_{x}\Real\left[e^{\mathrm{i}k_{x}x}\Ai\left(-\frac{\mathrm{i}y}{\delta_{x}}e^{-\mathrm{i}4\upi/3}\right)\right],$
(3)

with

$\delta_{x}=\left(\frac{\nu}{k_{x}u_{y,0}}\right)^{1/3},$ (4)

where $C_{x}=\Ai(0)^{-1}$ is a normalisation constant, $\mathrm{i}$ is the
imaginary unit, $k_{x}$ is the streamwise wavenumber and $u_{y,0}$ is the
slope of the streamwise velocity profile at the wall. We now adapt the SSL
solution to account for an arbitrary waveform, following the approach taken by
Cimarelli et al. (2013) for the TSL. The linear nature of the governing
equation allows for a superposition of various harmonics to make up the
desired waveform. In line with their approach, the waveform describing the
wall motion is expressed in terms of a Fourier series:

$W_{w}(x)=A\sum_{n=-\infty}^{+\infty}B_{n}e^{\mathrm{i}k_{x}nx},$ (5)

where $B_{n}$ is the complex coefficient and $k_{x}n$ the wavenumber
associated with the nth Fourier mode. The corresponding penetration depth is
defined as:

$\delta_{x,n}=\left(\frac{\nu}{k_{x}nu_{y,0}}\right)^{1/3}=\delta_{x}n^{-1/3}.$
(6)

Substituting this and superimposing the elementary solutions, the SSL for an
arbitrary waveform is then found as follows:

$w(x,y)=AC_{x}\sum_{n=-\infty}^{+\infty}\Real\left[B_{n}e^{ik_{x}nx}Ai\left(-\frac{iy}{\delta_{x}n^{-1/3}}e^{-i4\upi/3}\right)\right].$
(7)

##  4 Experimental results and discussion

The experimental setup was used in a configuration where the belts move in
alternating spanwise direction, resulting in the sequence of two square
waveforms of $\lambda_{x}=22$ mm, and a viscous wavelength
of$\lambda_{x}^{+}=401$. Six cases are discussed at varying spanwise amplitude
from $A^{+}\approx 2$ to $12$.

###  4.1 Comparison to the modified spatial Stokes layer

To address how the experimentally realised forcing relates to the theoretical
SSL, we compare the measured spanwise profiles to the model of the modified
SSL derived in § 3. The periodic waveform, consisting of two 9 mm regions
actuated at $\pm A$, is prescribed by ten single-sided Fourier coefficients.
The 2 mm transition between belts is approximated by a linear increase from
$-A$ to $A$ to prevent Gibbs phenomena. We investigate the case at
$A^{+}=6.3$. The half-phase ($\lambda_{x}/2$) is used to normalise the
streamwise location. Consequently, the same value after the decimal point
indicates the same location on the half-phase. The experimental profiles are
is subjected to streamwise averaging across the $x\pm 0.25$ mm range.

The spanwise profiles at three locations for each half-phase (i.e. belt) are
presented in figure 2(a), where $\tilde{w}$ denotes the phase-wise component
as a result of the forcing. We observe that the profiles resemble a clear
Stokes-like layer, with a steep spanwise velocity gradient at the wall,
followed by an overshoot around $y^{+}\approx 10$, after which it decays to
zero at $y^{+}\approx 25$. The behaviour is consistent with the SSL model,
subject to equation 1, at moderate wavelengths (Quadrio & Ricco, 2011; Viotti
et al., 2009). Some small degree of asymmetry between the two half-phases can
be observed. This may be attributed to a spatial transient of the SSL, which
only sees actuation over one half-phase upstream of the FOV.

Figure 2: Spanwise velocity profiles at $\lambda_{x}^{+}=401$ and $A^{+}=6.3$,
where the colour denotes the location on the half-phase. Experimental data
plotted with solid lines, modified SSL displayed for the same colour indicated
with a dashed-dotted line. (a) Profiles over the belts at three locations per
half-phase, (b) a detailed view of the profiles over the first half-phase, (c)
five linearly spaced profiles in the transition region between belts.

To make a comparison to the theoretical model for the modified SSL, the first
half-phase is highlighted in more detail in figure 2(b). The experimental
profiles match the model well, with an almost exact match in the region
$y^{+}\leq 10$. The slope of the spanwise velocity profile is in line with the
model and decreases when moving downstream over the belt. Hence, the model can
be used to estimate the power input required to drive the spanwise flow, which
is calculated from the integral spanwise wall shear stress (i.e.
$\propto\left.(\mathrm{d}\tilde{w}/\mathrm{d}y)\right|_{y=0}$). We utilised
the model to estimate the input power, followed by the calculation of NPS,
further discussion on these outcomes is presented in § 4.2. Beyond $y^{+}=10$
it can be observed that the experimental data starts to divert from the
theory. The overshoot of the experimental profile is larger in magnitude,
present at higher wall-normal locations, and penetrates deeper into the
boundary layer. These differences may be attributed to turbulent flow further
away from the wall, in contrast to the laminar flow assumption of the SSL
model.

The idealised model furthermore deviates from the practical realisation by
linearly transitioning between half-phases, instead of the abrupt changes and
the zero spanwise wall velocity in the transition region. To assess the effect
of this choice, five linearly spaced profiles spanning $0.9\leq
x/(\lambda_{x}/2)\leq 1.1$ in the transition region are studied, as depicted
in figure 2(c). A good match between the model and the experiment can be
observed, showing only slight deviations in the region $y^{+}\leq 5$. The
penetration depth is defined as the wall-normal location where the phase-wise
standard deviation of $\tilde{w}$ reduces to $Ae^{-1}$. The model and
experimental values are $\delta_{s,mod}^{+}=5.0$ and $\delta_{s,exp}^{+}=4.8$,
respectively. At first glance this result appears conflicting, since the
experimental profile, specifically the overshoot, perpetrates deeper into the
boundary layer. However, following the above definition, the penetration depth
is confined to the viscous sublayer. In this region the model shows a slightly
higher variance, explaining the small difference. Since there are only minor
variations, we conclude that the current way of modelling can be applied
without significantly altering the response of the Stokes profiles.

###  4.2 Mean velocity and performance characteristics

Figure 3: (a-b) Streamwise velocity profiles for the non-actuated case and
cases with $\lambda_{x}^{+}=401$, (a) scaling with $u_{\tau 0}$, (b) scaling
with $u_{\tau}$. (c) DR (solid) and NPS (dashed) vs $A^{+}$ compared to the DR
of Viotti et al. (2009) for $\lambda_{x}^{+}=312$ (dotted), error bars
indicate the 95% confidence interval of DR calculated using Student’s
t-distribution.

The mean streamwise velocity profiles are shown in figure 3(a,b), in scaling
with the reference and actual friction velocity, respectively. Scaling with
$u_{\tau 0}$ reveals a reduction of mean velocity below $y^{+}\leq 15$, which
is correlated with $A^{+}$. This reflects a reduction of the near-wall
velocity gradient, indicative of a drag-reduced state. The behaviour is
consistent with previous studies on spanwise forcing (Ricco & Wu, 2004; Gatti
& Quadrio, 2016). To further emphasise this, scaling with $u_{\tau}$ is
applied. The profiles now show similarity in the viscous sublayer with an
upward shift $\Delta B$ of the logarithmic region under actuation, consistent
with the reduction in skin friction. The profiles, specifically noticeable in
figure 3(a), reveal an overshoot of the velocity profile in the region $20\leq
y^{+}\leq 40$ compared to the non-actuated case. These results are consistent
with the conceptual model of Choi (2002), which suggests that this
modification of the streamwise velocity profile is due to an induced negative
spanwise vorticity.

The performance characteristics are quantified in figure 3(c), where for
reference the data of DR from the DNS of Viotti et al. (2009) is added,
subject to equation 1 with $\lambda_{x}^{+}=312$ at $\Rey_{\tau}=200$. The DR
increases monotonically at a decreasing rate with $A^{+}$, to a maximum value
of 26%. Qualitatively this behaviour compares well to the DNS reference. In
absolute terms, the DR is lower, especially at higher values of $A^{+}$. This
is primarily attributed to the expected streamwise transient of DR from the
onset of actuation. As shown by Ricco & Wu (2004), the DR rises to a maximum
over a streamwise extent of $3\delta$, whereas our measurement domain extends
over only $0.16\leq x/\delta\leq 0.50$. Secondly, a slight drop in performance
is expected from the higher Reynolds number in the experimental conditions
(Gatti & Quadrio, 2016). Lastly, the experimental realisation of the SW
boundary condition may not perform to the same degree as the idealised
conditions in the numerical studies. The NPS exhibits a maximum of 8% in the
range of $A^{+}=2.1-4.2$, whereas Viotti et al. (2009) achieved an optimum of
23% at $A^{+}=6$. The maximum is followed by a sharp decline to large negative
values (not depicted for clarity), reaching a minimum of -51% for
$A^{+}=12.6$. This behaviour results from the balance between DR and input
power ($\propto A^{+}$). We believe the difference in optimum is a consequence
of the higher DR towards the lower $A^{+}$ values, and the higher power
requirements for square-wave forcing.

###  4.3 Higher-order turbulence statistics

The higher-order turbulence statistics in terms of the variance profiles for
the three velocity components, the Reynolds shear stress, and the turbulence
kinetic energy (TKE) production are depicted in figure 4. Reference scaling
with $u_{\tau 0}$ is applied to highlight the absolute changes with respect to
the non-actuated case.

---
Figure 4: (a,b) Streamwise and wall-normal Reynolds stresses and (c) Reynolds
shear stress. Spanwise stress is decomposed in (d) total spanwise stress and
(e) the stochastic component. (f) Premultiplied production of turbulence
kinetic energy. Legend is depicted in figure 3.

Assessing figure 4(a), we observe that upon actuation the streamwise stress is
attenuated in the region up to $y^{+}\approx 100$, and that the near-wall peak
is shifted to higher wall-normal positions. In line with the DR behaviour, the
effect is correlated with $A^{+}$. The most significant reduction is found in
the near-wall stress peak, which is centred around $y^{+}=15$ for the non-
actuated case. The near-wall peak shows a $44\%$ reduction in magnitude for
the maximum $A^{+}$. This response is consistent with earlier experimental
observations of drag-reduced flow under spanwise forcing (Laadhari et al.,
1994; Ricco & Wu, 2004). The stresses $\langle v^{\prime 2}\rangle^{+}$ and
$-\langle u^{\prime}v^{\prime}\rangle^{+}$, in figure 4(b,c), reflect similar
behaviour, where the profiles shift to higher wall-normal locations and
attenuation is most significant close to the wall.

Under periodic forcing, the total spanwise stress is decomposed into a
stochastic component and a phase-averaged component, according to $\langle
w^{\prime 2}\rangle^{+}=\langle w^{\prime\prime
2}\rangle^{+}+\langle\tilde{w}^{2}\rangle^{+}$. The total spanwise stress, in
figure 4(d), shows a strong peak in the near-wall region (note the log-log
scale), as a direct consequence of the phase-wise fluctuating component
$\langle\tilde{w}^{2}\rangle^{+}$. This component corresponds to the
streamwise variance of the Stokes profiles, hence, the peak’s magnitude is
proportional to $A^{+2}$. When the phase-averaged component is removed, the
stochastic stress is obtained, depicted in figure 4(e). It can be observed
that the stochastic fluctuations are also significantly increased in the
region $y^{+}\leq 10$. In agreement with the other stresses, we observe that
under actuation, the $\langle w^{\prime\prime 2}\rangle^{+}$ profiles are
shifted in the wall-normal direction and show a slight energy reduction up to
a height of about $y^{+}\approx 100$.

Lastly, we discuss the TKE production which is approximated by $P=-\langle
u^{\prime}v^{\prime}\rangle\mathrm{d}\langle\overline{u}\rangle/\mathrm{d}y$.
Figure 4(f) depicts the term in the premultiplied form ($y^{+}P^{+}$), such
that an equal area under the curve highlights an equal contribution to the TKE
production. Actuation significantly reduces the peak in the buffer layer, and
again, the profiles are shifted to a higher wall-normal position, similar to
the streamwise Reynolds stress. For $A^{+}=12.6$, a reduction of integral TKE
production, over the considered wall-normal region, of 39% is found.

##  5 Concluding remarks

A proof of concept was presented for an experimental realisation of spatial
spanwise forcing in an external boundary layer flow. An extension to the SSL
model was proposed, by linearly superimposing Fourier modes to account for the
non-sinusoidal boundary condition. General agreement between the model and the
experimental profiles was found, showing an almost exact match in the region
$y^{+}\leq 10$. Furthermore, the penetration depths of the two are in the same
order at $\delta_{s,mod}^{+}=5.0$ and $\delta_{s,exp}^{+}=4.8$. This leads us
to conclude that the model can be used to estimate the theoretical power input
associated with the spanwise forcing.

We confirm that a significant flow control effect can be realised with such
forcing. Qualitatively, the DR trends are in line with the expected trends
from the literature, with a maximum of 26% within the parameter space
investigated. In view of its spatial transient, the full DR potential is
expected to be underestimated. An optimum NPS of 8% was found for the
amplitudes ranging $A^{+}=2.1-4.2$. The turbulence statistics further confirm
a strong flow modulation effect. The Reynolds stress profiles are attenuated
up to a height of $y^{+}\approx 100$, and are shifted to higher wall-normal
locations. Specifically, the streamwise stress peak shows a reduction of 44%
at the maximum spanwise velocity amplitude. Furthermore, for this case, the
integral TKE production is reduced by 39%.

These findings offer perspective for future research into the working
mechanisms responsible for the DR. A fundamental understanding of the
underlying physics can advance energy efficiency in practical applications,
e.g through its application in passive form (Ricco et al., 2021). Extending
the streamwise extent of the actuation surface is foreseen as a valuable next
step. In such a setup the spatial transient of DR and its steady-state
response can be quantified. Wider belts are also advised to realise a larger
viscous wavelength of the order of the optimum value. Furthermore, the
modified SSL theory may hold potential for a predictive model for DR and NPS,
similar to those of Choi et al. (2002) and Cimarelli et al. (2013).

[Acknowledgements]This work was financially supported by the Netherlands
Enterprise Agency under grant number TSH21002. The authors wish to give
special thanks to BerkelaarMRT B.V. for the mechanical design and realisation
of the experimental apparatus.

[Declaration of interests]The authors report no conflict of interest.

[Author contributions]Max W. Knoop: conceptualisation, experiments, formal
analyses, original draft. Friso H. Hartog: conceptualisation, discussion &
analysis, review. Ferdinand F.J. Schrijer: supervision, discussion & analysis,
review. Olaf W.G. van Campenhout: funding acquisition, review. Michiel van
Nesselrooij: funding acquisition, review. Bas W. van Oudheusden: supervision,
discussion & analysis, review.

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# Fourier series representation of Ferrers function $\mathsf{P}$

Hans Volkmer  Hans Volkmer
Department of Mathematical Sciences
University of Wisconsin - Milwaukee
<EMAIL_ADDRESS>
In [1, §6] we considered a trigonometric expansion of the Ferrers function
$\mathsf{Q}$ of the second kind [2, §14.13.2]. In this note we consider the
analogous expansion of the Ferrers function $\mathsf{P}$ of the first kind.

######  Theorem 1.

Let $x\in\mathbb{C}-((-\infty,-1]\cup[1,\infty))$ and $\nu,\mu\in\mathbb{C}$
such that $\nu+\mu\not\in-\mathbb{N}$. Then

$\displaystyle\mathsf{P}_{\nu}^{\mu}(x)$ $\displaystyle=$
$\displaystyle\frac{2^{\mu}}{i\sqrt{\pi}}(1-x^{2})^{\frac{1}{2}\mu}\frac{\Gamma(\nu+\mu+1)}{\Gamma(\nu+\frac{3}{2})}$
$\displaystyle\times\left(u^{\nu+\mu+1}{}_{2}F_{1}\left(\genfrac{}{}{0.0pt}{}{\mu+\frac{1}{2},\nu+\mu+1}{\nu+\frac{3}{2}};\frac{u}{v}\right)-v^{\nu+\mu+1}{}_{2}F_{1}\left(\genfrac{}{}{0.0pt}{}{\mu+\frac{1}{2},\nu+\mu+1}{\nu+\frac{3}{2}};\frac{v}{u}\right)\right),$

where

$u=x+i\sqrt{1-x^{2}},\quad v=x-i\sqrt{1-x^{2}}=\frac{1}{u}.$

If we set $x=\cos\theta$, $\Re\theta\in(0,\pi)$, then Theorem 1 gives

(1) $\displaystyle\hskip
28.45274pt\mathsf{P}_{\nu}^{\mu}(\cos\theta)=\frac{2^{\mu}}{i\sqrt{\pi}}(\sin\theta)^{\mu}\frac{\Gamma(\nu+\mu+1)}{\Gamma(\nu+\frac{3}{2})}$
$\displaystyle\times\left({\rm
e}^{i(\nu+\mu+1)\theta}{}_{2}F_{1}\left(\genfrac{}{}{0.0pt}{}{\mu+\frac{1}{2},\nu+\mu+1}{\nu+\frac{3}{2}};{\rm
e}^{2i\theta}\right)-{\rm
e}^{-i(\nu+\mu+1)\theta}{}_{2}F_{1}\left(\genfrac{}{}{0.0pt}{}{\mu+\frac{1}{2},\nu+\mu+1}{\nu+\frac{3}{2}};{\rm
e}^{-2i\theta}\right)\right).$

If $\theta\in(0,\pi)$ then the arguments $w={\rm e}^{\pm 2i\theta}$ of the
hypergeometric function lie on the unit circle $|w|=1$. Provided the
hypergeometric series converges at ${\rm e}^{\pm 2i\theta}$ we obtain [2,
(14.13.1)]

(2)
$\mathsf{P}_{\nu}^{\mu}(\cos\theta)=\frac{2^{\mu+1}}{\sqrt{\pi}}(\sin\theta)^{\mu}\sum_{k=0}^{\infty}\frac{\Gamma(\nu+\mu+k+1)}{\Gamma(\nu+k+\frac{3}{2})}\frac{(\mu+\frac{1}{2})_{k}}{k!}\sin((\nu+\mu+2k+1)\theta).$

Regarding the convergence of the series in (2) we have the following result.

######  Theorem 2.

Let $\theta\in(0,\pi)$, $\nu,\mu\in\mathbb{C}$ such that
$\nu+\mu\in\mathbb{C}\setminus-\mathbb{N}$.
(a) If $\Re\mu<0$ then the series in (2) converges absolutely.
(b) If $0\leq\Re\mu<\frac{1}{2}$ then the series in (2) converges, but, if
$\theta\neq\frac{1}{2}\pi$, it does not converge absolutely.
(c) If $\Re\mu\geq\frac{1}{2}$ and $\theta\neq\frac{1}{2}\pi$, then the series
in (2) diverges.

###### Proof.

(a) It is known [2, §15.2(i)] that the Gauss hypergeometric series
${}_{2}F_{1}\left(\genfrac{}{}{0.0pt}{}{a,b}{c};w\right)$ converges absolutely
on the unit circle $|w|=1$ if $\Re(c-a-b)>0$. In our case $a=\mu+\frac{1}{2}$,
$b=\nu+\mu+1$, $c=\nu+\frac{3}{2}$ so $c-a-b=-2\mu$. If $\Re\mu<0$ it follows
that the series in (2) is the sum of two absolutely convergent series and so
is itself absolutely convergent.
(b) Suppose that $0\leq\Re\mu<\frac{1}{2}$. If $-1<\Re(c-a-b)\leq 0$ then the
Gauss hypergeometric series converges conditionally at $|w|=1$, $w\neq 1$ [2,
§15.2(i)]. It follows that the series in (2) is the sum of two convergent
series and so is itself convergent. However, it is not true that the sum of
two conditionally convergent series is conditionally convergent. We still have
to show that the series in (2) does not converge absolutely if
$\theta\neq\frac{1}{2}\pi$. According to [2, (5.11.12)],

$\frac{\Gamma(a+z)}{\Gamma(b+z)}\sim z^{a-b}$

as $z\to+\infty$. Therefore, as $k\to\infty$,

$\displaystyle\frac{\Gamma(\nu+\mu+k+1)}{\Gamma(\nu+k+\frac{3}{2})}\frac{(\mu+\frac{1}{2})_{k}}{k!}=$
$\displaystyle\frac{\Gamma(\nu+\mu+k+1)}{\Gamma(\nu+k+\frac{3}{2})}\frac{\Gamma(\mu+\frac{1}{2}+k)}{\Gamma(k+1)\Gamma(\mu+\frac{1}{2})}\sim\frac{k^{\mu-\frac{1}{2}}k^{\mu-\frac{1}{2}}}{\Gamma(\mu+\frac{1}{2})}=\frac{k^{2\mu-1}}{\Gamma(\mu+\frac{1}{2})}.$

Since $1/\Gamma(\mu+\frac{1}{2})\neq 0$, there are positive constants $\kappa$
and $K$ such that

$\left|\frac{\Gamma(\nu+\mu+k+1)}{\Gamma(\nu+k+\frac{3}{2})}\frac{(\mu+\frac{1}{2})_{k}}{k!}\right|\geq\frac{\kappa}{k},$

for $k\geq K$. The second part of statement (b) now follows from Lemma 3.
(c) Suppose that $\Re\mu\geq\frac{1}{2}$ and the series in (2) converges. Then
the terms of the series must converge to $0$. It follows that
$\sin((\nu+\mu+2k+1)\theta)\to 0$ as $k\to\infty$. By Lemma 4 this is
impossible unless $\theta=\frac{1}{2}\pi$. Therefore, the series in (2) will
diverge for $\theta\neq\frac{1}{2}\pi$. ∎

######  Lemma 3.

Let $a\in\mathbb{C}$ and $\theta\in(0,\pi)$. Then

$\sum_{k=1}^{\infty}\frac{1}{k}|\sin((a+2k)\theta)|=\infty$

unless $\sin a=0$ and $\theta=\frac{1}{2}\pi$.

###### Proof.

If $\theta=\frac{1}{2}\pi$ then $|\sin((a+2k)\theta)|$ is independent of $k$
and the assertion follows. Now suppose that $\theta\neq\frac{1}{2}\pi$. Since
$|\sin z|\geq|\sin(\Re z)|$ for all $z\in\mathbb{C}$ it is enough to consider
$a\in\mathbb{R}$. Then $|\sin((a+2k)\theta)|\geq\sin^{2}((a+2k)\theta)$.
Therefore, it is enough to show that

$\sum_{k=1}^{\infty}\frac{1}{k}\sin^{2}((a+2k)\theta)=\infty.$

Now

$\frac{1}{k}\sin^{2}((a+2k)\theta)=\frac{1}{2k}-\frac{1}{2k}\cos(2(a+2k)\theta).$

The series $\sum_{k=1}^{\infty}1/(2k)$ diverges. The series
$\sum_{k=1}^{\infty}\cos(2(a+2k)\theta)/(2k)$ converges by the Dirichlet test
provided the partial sums $\sum_{k=1}^{n}\cos(2(a+2k)\theta)$ form a bounded
sequence. This is true for $\theta\in(0,\pi)$, $\theta\neq\frac{\pi}{2}$. ∎

######  Lemma 4.

Let $a,b\in\mathbb{C}$ and $\sin b\neq 0$. Then the sequence $\sin(a+bn)$ does
not converge to $0$ as $n\to\infty$.

###### Proof.

Suppose that $\sin(a+bn)\to 0$ as $n\to\infty$. We have

$\sin(a+b(n+1))=\cos(a+bn)\sin b+\sin(a+bn)\cos b.$

If we let $n\to\infty$ and use $\sin b\neq 0$ we obtain that $\cos(a+bn)\to
0$. Since $\cos^{2}x+\sin^{2}x=1$ this is a contradiction. ∎

## References

  * [1] H. Cohl, J. Park and H. Volkmer, Gauss hypergeometric representation of the Ferrers function of the second kind, Sigma 17 (2021), 053, 33 pages.
  * [2] F.W. Olver, D.W. Lozier, R.F. Boisvert and C.W. Clark, NIST handbook of mathematical functions, Cambridge University Press, 2010.