Michel Bergmann

Laurent Cordier

Jean-Pierre Brancher

On the power required to control the circular cylinder wake by rotary oscillations

Keywords: circular cylinder, rotary oscillations, power control, energetic efficiency

In this Brief Communication, we determine an approximate relation which gives the mean time power required to control the wake flow downstream from a circular cylinder. The control law is the sinusoidal tangential velocity imposed on whole or part of the cylinder surface. The mean control power thus depends on four parameters: the amplitude and the Strouhal number of forcing, the control angle which defines the controlled upstream part of the cylinder and the Reynolds number. This relation indicates that the control power grows like the square of the forcing amplitude, like the square root of the forcing Strouhal number, linearly with the control angle and varies like the inverse of the square root of the Reynolds number. We show that the values obtained with this approximate relation are in very good agreement with the corresponding values given numerically.

Finally, the energetic efficiency of the control is discussed. We claimed that the most energetically efficient control law corresponds a priori to low forcing amplitudes applied to a restricted upstream part of the cylinder for relatively high values of the Reynolds number.

The control of the cylinder wake flow by rotary oscillations has been intensively studied over the last decade. The majority of these studies were motivated by the experiments of Tokumaru and Dimotakis 1 where 80% of relative mean drag reduction was empirically found at Re = 15, 000. Due to the rapid progress achieved in computational fluid dynamics and the remarkable developments in modern control theory (optimal and robust control mainly), the active control of wake flows was intensively explored numerically [START_REF] He | Active control and drag optimization for flow past a circular cylinder. Part 1. Oscillatory cylinder rotation[END_REF][START_REF] Homescu | Suppression of vortex shedding for flow around a circular cylinder using optimal control[END_REF][START_REF] Bergmann | Optimal rotary control of the cylinder wake using POD Reduced Order Model[END_REF] . All these studies gave place to convincing results for the different values of Reynolds number considered.

However, in most of the cases, the cost of the control was not considered, except in Refs. 2,5.

Consequently, the energetic efficiency of a given technique of control is difficult to evaluate and comparing the various control methods is even more delicate. It is thus of interest to analyze the influence of the control parameters on the control power evolution. Hence, the objective of this Brief Communication is to determine, and validate numerically, an approximate relation giving the control power according to the different control parameters of the cylinder wake and the Reynolds number.

Here, the incompressible and viscous flow around a circular cylinder of diameter D (radius R) is considered in a two-dimensional domain Ω. The cylinder boundary is divided in two regions: an upstream part animated with an unsteady tangential velocity V T , and the remainder of the cylinder which is not controlled. Hereafter, the upstream controlled region is geometrically defined by -θ c < θ < θ c (see Fig. 1) where θ and θ c , initialized by convention at the front stagnation point of the cylinder, are respectively the curvilinear coordinates of a point on Γ c and the control angle. Equivalently, the controlled boundary of the cylinder is characterized by the control surface S c = 2θ c R ℓ z where ℓ z is the spanwise length of the cylinder (for two dimensional configurations, ℓ z = 1). As it is generally the case [START_REF] Tokumaru | Rotary oscillatory control of a cylinder wake[END_REF][START_REF] He | Active control and drag optimization for flow past a circular cylinder. Part 1. Oscillatory cylinder rotation[END_REF][START_REF] Homescu | Suppression of vortex shedding for flow around a circular cylinder using optimal control[END_REF][START_REF] Bergmann | Optimal rotary control of the cylinder wake using POD Reduced Order Model[END_REF][START_REF] Protas | Optimal rotary control of the cylinder wake in the laminar regime[END_REF][START_REF] Min | Suboptimal feedback control of vortex shedding at low Reynolds numbers[END_REF][START_REF] Protas | Drag force in the open-loop control of the cylinder wake in the laminar regime[END_REF][START_REF] Bergmann | On the generation of a reverse Von Kármán street for the controlled cylinder wake in the laminar regime[END_REF] , the control law is selected to be harmonic V T (t) = A 0 sin(ω t) where A 0 and ω re-spectively represent the dimensional amplitude and pulsation of forcing. For convenience, the dimensionless velocity γ(t) = V T (t)/U ∞ is introduced, where U ∞ denotes the inflow velocity. Then, the control law writes equivalently γ(t) = A sin (2πSt f t) with A = A 0 /U ∞ and St f = Rω/(πU ∞ ). Here, A and St f denote respectively the nondimensionalized forcing amplitude and Strouhal numbers. Moreover, since the control is applied only on a restricted part of the cylinder surface, the physical problem depends implicitly on four control parameters, namely A, St f , θ c and the Reynolds number Re = ρU ∞ D/µ, where ρ and µ are respectively the density and the dynamic viscosity of the fluid. Finally, for all the numerical simulations used for the validation, the Navier-Stokes equations are solved in the pressurevelocity formulation with a fractional step method in time and a finite element method in space [START_REF] Bergmann | Optimal rotary control of the cylinder wake using POD Reduced Order Model[END_REF] . Now, we examine the power budget related to flow control i.e. the sum of the work that has to be done against the drag force and the work needed to control the flow. By definition, the force which is acting on the cylinder can be expressed as F = -Γc σ n dΓ where σ is the stress tensor and Γ c is the cylinder boundary. Let F D be the total drag force obtained by projecting F on the direction e x , the power spending related to this effort is thus given by the relation

P D (t) = F D (t)U ∞ .
We then focus on the power used to control the cylinder flow. If we do not account for inertial effects related to rotating the cylinder, the control power can be evaluated by

P c (t) = M z (t) θ where θ = γ U ∞ /R
is the angular velocity of the cylinder and M z is the component on the direction e z = e x ∧ e y of the moment of forces, or torque M, applied to the obstacle. By definition, the moment of forces on a body placed in a flow is given by M = -Γc r ∧ σ n dΓ where r = xe x + ye y defines the coordinates of any point on Γ c . For reasons of simplicity, we introduce the coefficient of moment defined as

C Mz = Mz 1/2Mr
, where M r is a reference moment given by M r = F r × L r with F r and L r , a reference force and a reference length, respectively. The reference force is now determined as F r = p r ×S r where p r and S r are the pressure reference and surface reference, respectively. Usually p r = ρ U 2 ∞ and S r = 2R ℓ z . Here, the reference length L r is selected equal to the diameter of the cylinder

D = 2R.
After some algebraic manipulations, the instantaneous coefficient of moment can be evaluated numerically from:

C Mz (t) = 2 Re 2π 0 - ∂v ∂x x 2 -2 ∂v ∂y xy + 2 ∂u ∂x xy + ∂u ∂y y 2 - ∂u ∂y x 2 + ∂v ∂x y 2 dθ,
where all the variables are nondimensionalized with respect to the oncoming velocity U ∞ and the cylinder diameter D. Finally, since

M z = 1 2 M r C Mz with M r = F r L r = 4ρU 2 ∞ R 2 ℓ z ,
the instantaneous power useful for rotation merely writes as

P C (t) = ρU 3 ∞ Rℓ z 2γ(t)C Mz (t).
To simplify the future comparisons of the control power, let us define the time average over a finite horizon T , corresponding to a few cylinder oscillation periods, as

• T = 1 T t 0 +T t 0 • dt.
The analytic formulation of the mean control power is then

P C = P C T = ρU 3 ∞ Rℓ z 2γ(t)C Mz (t) T . (1)
Unfortunately this relation does not depend explicitly on the control parameters A, St f , θ c and Re. Hence, more precise developments seem necessary.

The mean time power necessary to ensure the oscillatory rotation of the cylinder can be approximated as follows:

P C = -S c τ w A 0 sin(ω t) T ,
where τ w is the wall shear stress. To determine an expression of τ w which depend on the control parameters and the Reynolds number, we now make the following assumptions:

H1 the oscillations of the cylinder do not affect the external flow, H2 the effects of curvature can be neglected ( δ R ≪ 1).

The H1 assumption is equivalent to saying that the oscillations do not diffuse outside the presumedly laminar boundary layer of characteristic thickness 9 δ = O νR U∞ . This assumption implies immediately that the Reynolds number must be sufficiently large (Re ≫ 1). In addition, the H1 assumption implies that we must satisfy the condition δ 0 δa ≪ 1 where and the condition δ 0 δa ≪ 1 writes:

δ 0 = O ν ω is
ψ = Im{f (z)} simply writes ψ(X, Y ) = U 2 ∞ Y - R 2 Y (R+X) 2
δ 0 δ a = O 2U ∞ Rω = O 2 πSt f ≪ 1. (2)
Since the velocity field is tangent to the cylinder surface near the front stagnation point, the velocity component in the direction e y is written as:

v(r * , t) = A 0 exp - ω 2ν r * sin ωt - ω 2ν r * ,
where r * = r -R (here, r represents the radial coordinate). One can then easily deduce the expression of the wall shear stress:

τ w = µ ∂v ∂r * r * =0 = -µA 0 ω 2ν [sin(ω t) + cos(ω t)] .
Since we have cos(ω t)sin(ω t) T = 0 and sin 2 (ω t) T = 1 2 for T = 2kπ/ω, k ∈ N, the mean control power becomes

P C = S c µ A 2 0 2
ω/2ν. Finally, after some manipulations, the expression of the power needed to apply the rotary oscillations to the cylinder can be simplified as

P C = ρ U 3 ∞ R ℓ z √ π θ c A 2 St f √ Re . (3)
This approximate expression now depends on the four control parameters. For the numerical validation, we chose to compare directly the mean control power coefficient obtained respectively from Eq. ( 1) and Eq. ( 3). Let P r = F r U ∞ = ρU 3 ∞ 2Rℓ z be the reference control power, by definition, the mean control power coefficient is C P = P T / 1 2 P r . Applying this relation to the drag power P D (t) = F D (t) U ∞ we can deduce an expression for the mean power drag coefficient:

C D P = C D T , (4)
where C D (t) = F D (t)/ρU 2 ∞ Rℓ z is the instantaneous drag coefficient, given for example in Ref. 8. Similarly, the expression of the mean control power coefficient based on Eq. ( 1) is

C C P = 2 C Mz (t) γ(t) T , (5)
and the corresponding expression for the mean control power coefficient obtained with the approximate relation ( 3) is written

C C P = √ π θ c A 2 St f √ Re . ( 6
)
The control is said energetically efficient (resp. inefficient), if the value of the total mean power coefficient C P = C D P +C C P (where C C P can be evaluated either by Eq. ( 5), or by Eq. ( 6))

is lower (resp. greater) than the value obtained in the uncontrolled case (where naturally

C C P = 0).
Now, we will compare the results given by the approximate relation (6) to the numerical ones given by Eq. ( 5). For that, we will successively represent the evolutions of the control powers according to one of the four control parameters of the flow, the others being fixed. For Figure 2 shows an excellent agreement between the values obtained numerically and those determined with the approximate relation. One can thus claim that the power required to control the cylinder wake grows effectively like the square of the forcing amplitude as the expression (3) predicted it. As it is noticeable in Fig. 3, the agreement between the numerical and approximate expressions of the control power according to the forcing Strouhal number is still very good, and that even if the Strouhal number is of order unit. The small differences observed for the smallest values of the Strouhal number can be explained by the assumptions made to derive the approximate relation, and more particularly that given by Eq. ( 2). In spite of that, the values obtained numerically exceed hardly the values found by the approximate relation. The same behavior is clearly visible in Fig. 4 where the linear dependence of the control power as a function of the angle θ c is quite well verified. As we can see in Fig. 5, the approximate relation does not suitably predict the variation, according to the Reynolds number, of the control power obtained numerically.

Indeed, in one hand, for Reynolds number lower than 200, the numerical values exceed those predicted by the approximate relation. This result can be explained easily by the assumption of laminar boundary layer (Re >> 1) carried out to determine the approximate relation.

In the other hand, for Reynolds number greater than 200, the numerical values are slightly lower than those determined with the approximate relation. This time, these differences can be explained by the fact that for a value of the Reynolds number close to 200, where a good agreement is observed, a three-dimensional Hopf bifurcation occurs [START_REF] Williamson | Vortex dynamics in the cylinder wake[END_REF] . As a normal consequence, our two-dimensional numerical simulation cannot capture accurately threedimensional dynamic phenomena. However, from a qualitative point of view, the control power seems to evolve like the reverse of the square root of the Reynolds number as it was predicted by Eq. ( 3).

In this Brief Communication, we determined an approximate relation which gives, according to the 4 parameters of controlled flow A, St f , θ c and Re, the mean power which is necessary to control by oscillatory rotation the cylinder wake. Even if all the assumptions used to determine this approximation are not verified exactly, we showed that this relation was in very good agreement with the numerically given corresponding values. Consequently, the power required for the cylinder oscillations is almost completely dissipated in the Stokes layer. Since the control power grows with the square of the forcing amplitude, it is nearly impossible to obtain an energetically efficient control for high amplitudes. Protas and Styczek 5 had numerically shown this result besides while considering as cost function the total power in an optimal control approach. In addition, the approximate relation shows that the control power is much less penalized by the Strouhal number than it is by the amplitude of forcing. It is also remarkable that the control power decreases linearly with the control angle θ c . Indeed, since a gain of reduction of drag is obtained by controlling only a well defined upstream part of the cylinder 8 (and not all the surface of the cylinder as it is usually the case), the maximum of energetic efficiency is awaited for θ c lower than 180

  the thickness of the diffusion Stokes layer induced by the oscillations and δ a is the thickness of the boundary layer at the front stagnation point. Using the H1 assumption a priori (what was checked a posteriori), we can consider that the front stagnation point does not move. One can thus calculate the flow in the vicinity of the stagnation point starting from the complex potential without circulation. In this case, the stream function

+Y 2 with

 2 X = x -R and Y = y. Near the stagnation point, i.e. for (X, Y ) → (0, 0), the stream function behaves like 2U∞ R XY . Therefore, δ a = O ν R 2U∞

  these comparisons, the parameters maintained fixed are selected among A = 5, St f = 1, θ c = 180 • and Re = 200. Figures 2, 3, 4 and 5 show respectively the evolutions of the mean control power coefficient versus the amplitude A, the Strouhal number St f , the control angle θ c and the Reynolds number.

FIG. 1 .FIG. 2 .FIG. 3 .FIG. 4 .A = 5 ,FIG. 5 .

 123455 FIG. 1. Sketch of the controlled flow configuration.

  • . Finally, the approximate relation indicates that the control power decreases when the Reynolds number increases. Moreover, Protas and Wesfreid 7 showed that the potential gain of drag increased with the Reynolds number. Consequently, the most energetically efficient controls should


Adrien Cassagne

Mathieu Leonardon

Romain Tajan

Camille Leroux

Christophe Jégo

Olivier Aumage

Denis Barthou

Mathieu Léonardon

A Flexible and Portable Real-time DVB-S2 Transceiver using Multicore and SIMD CPUs

Keywords: Real-time system, SDR, Multicore CPU, SIMD, DVB-S2 standard, Radio transceiver

Software implementation of digital communication systems is more and more used in different contexts. In the case of satellite communication standards, they are an appealing alternative in ground stations. The challenge is to push the performance of these digital communication systems to meet the real time constraints. In this paper, we propose an open source digital communication transceiver that enables to exploit the parallelism of general purpose processors (multicore, SIMD). It is also flexible, supporting several modulation and coding schemes. Finally, it is portable, being able to adapt to the level of parallelism of different CPU architectures (x86 and ARM).

I. INTRODUCTION

Software implementation of digital communication standards is a high-stakes research path for communication network players. It is promising from many points of view: hardware mutualization, energy savings, maintainability and scaling. It is also an attractive choice when the production volume is low and does not enable to compensate for nonrecursive engineering costs. In the case of the DVB-S2 standard, on which this article focuses, ground stations are good candidates. However, the existing solutions for implementing the DVB-S2 transceiver in software are not sufficient. They are either flexibility oriented, as for example by using the GNU radio software suite [START_REF] Economos | gr-dvbs2rx: GNU Radio Extensions for the DVB-S2 and DVB-T2 Standards[END_REF], but at the expense of performance, not suitable with real time processing. On the other hand, highly optimized solutions exist, but they sacrifice flexibility [START_REF] Grayver | Extreme Software Defined Radio -GHz in Real Time[END_REF]. Moreover, the code is not available, which limits its reusability and reproducibility. The main objective of this paper is to provide a clear, structured, open and efficient code base, for the implementation of all the elementary blocks of the DVB-S2 system, from transmission to reception through the signal acquisition and synchronization steps. The rest of the paper is organized as follows. Sec. II presents the communication system and its different parts. Sec. III details the real time constraints and the optimizations that have been implemented to achieve them, including multi-threading and SIMD parallelization. Finally, Sec. IV presents the related works.

II. DVB-S2 TRANSCEIVER

The second generation of Digital Video Broadcasting standard for Satellite (DVB-S2) [START_REF]EN 302 307 -Digital Video Broadcasting[END_REF] is a flexible standard designed for broadcast applications. DVB-S2 is typically used for the digital television (HDTV with H.264 source coding). The full DVB-S2 transmitter and receiver are implemented in a SDRcompliant system. Two Universal Software Radio Peripherals (USRPs) N320 have been used for the analog signal transmission and reception while all the digital processing of the system have been implemented on CPU.

A. Transmitter Software Implementation

Fig. 1 shows the DVB-S2 transmitter decomposition in tasks. The filled tasks are intrinsically sequential. The initial information bits are read from a binary file (t Tx 1 ). Then, the DVB-S2 coding scheme rests upon the serial concatenation of a BCH code (t Tx

3 ) and an LDPC code (t Tx 4 ). The selected modulation (t Tx 6) is a Phase-Shift Keying (PSK). The scrambler tasks (t Tx 2 and t Tx 8 ) apply predefined repeated sequences of xor to the frame in order to avoid overly long sequences of the same bit or symbol in the frames sent by the radio (t Tx 10 ). Depending on the DVB-S2 MODulation and CODing scheme (MODCOD), the frame can be interleaved (t Tx 5 ) after the encoding process. If there is no interleaver, the frame is just copied. After the modulation, PayLoad Header (PLH) and pilots are inserted (t Tx 7 ). These extra data are used by the synchronization tasks at the receiver side. Before the radio transmission (t Tx 10 ), the signal bandwidth is rescaled by a shaping filter (t Tx 9 ). The DVB-S2 standard contains 32 different configurations or MODCODs. This work focuses on 3 typical MODCODs given in Tab. I. Depending on the MODCOD, the PSK modulation and the LDPC code rate R vary. In MODCOD 1 and 2 there is no interleaver, MODCOD 3 includes a column/row interleaver. K is the number of information bits and the input size of the BCH encoder. K LDPC is the output size of the BCH encoder and the input size of the LDPC encoder. The LDPC codeword size is constant for the 3 MODCODs (N LDPC = 16200).

B. Receiver Software Implementation

Fig. 2 and Fig. 3 describe the tasks decomposition of the DVB-S2 receiver software implementation with five distinct phases. The first one is called the waiting phase (see Fig. 2). It consists in waiting until a transmitter starts to transmit. The Synchronizer Frame task (t Rx 8 ) possesses a frame detection criterion. When a signal is detected, the acquisition (acq.) phase 1 (see Fig. 2) is executed during 150 frames. After that, the acq. phase 2 (see Fig. 2) is also executed during 150 frames. After the acq. phase 1 and 2, tasks have to be rebound for the acq. phase 3 (see Fig. 3). This last acq. phase is applied over 200 frames. After the 500 frames of these acq. phases, the final transmission phase is established (see Fig. 3).

In a real life transmission systems, radios internal clocks can drift slightly. A specific processing has to be added in order to be resilient. This is achieved by the Synchronizer Timing tasks (t Rx 5 and t Rx 6 ). Similarly, the radio transmitter carrier frequency does not perfectly match the receiver carrier frequency. So, the Synchronizer Frequency tasks (t Rx 3 , t Rx 10 and t Rx 11 ) aims at estimating and compensating the carrier frequency offset and the phase offset. Finally, LDPC FEC is a block coding scheme that requires to know precisely the first and last bits of a codeword. The Synchronizer Frame task (t Rx 8 ) uses the PLH and pilot symbols inserted by the transmitter (t Tx 7 ) to recover the first and last symbols. One can notice that the Synchronizer Timing module is composed by two separated tasks (synchronize or t Rx 5 and extract or t Rx 6 ). This behavior is different from the other Synchronizer modules. The synchronize task (t Rx 5 or t Rx 3,4,5 ) has two outputs: one for the regular data and another one for a mask. The regular data and the mask are then used by the extract task (t Rx 6 ) to screen which data that is selected or not for the next task. This specific implementation has been retained for two reasons. Firstly, the Synchronizer Timing tasks (t Rx 5 and t Rx 6 ) have a high latency compared to the others tasks, thus splitting the treatment in two tasks is a way to increase the throughput of the pipeline (this is further discussed below). Secondly, the extract task (t Rx 6 ) introduces a new possible behavior. In some cases the task does not have enough samples to produce a frame. In such cases, the extract task raises an exception. When this exception is caught, the chain restarts from the first task (t Rx 1 ). This implies to manage a buffer of samples in the extract task (t Rx 6 ). If the buffer contains more than one frame then the next task (t Rx 7 ) is executed, otherwise the chain is restarted. Fig. 4 shows the FER decoding performance results of the 3 selected MODCODs. The shapes represent the channel conditions: squares stand for a standard simulated AWGN channel, triangles are also a simulated AWGN channel in which frequency shift, phase shift and symbol delay have been taken into account, circles are the real conditions measured performances with the two USRPs. One can notice a 0.2 dB inaccuracy in the noise estimated by the t Rx 13 task. It is symbolized by the extra horizontal bars over the circles. For each MODCOD, the LDPC decoder is based on the belief propagation algorithm with horizontal layered scheduling (10 ite.) and with the min-sum node update rules. Each DVB-S2 configuration has a well-separated SNR predilection zone.

C. Open Source Integration with AFF3CT Toolbox

The proposed software implementation of the DVB-S2 digital transceiver is open source and available online1 . It is described with the help of the AFF3CT toolbox [START_REF] Cassagne | AFF3CT: A Fast Forward Error Correction Toolbox![END_REF]. AFF3CT is a library dedicated to the digital communication systems and more specifically to the channel decoding algorithms. At the time of the writing, the project focuses more on functional simulations/evaluations than on real-time digital communication systems. In this paper, we extend AFF3CT to the SDR use case while keeping the interoperability, reproducibility and maintainability philosophy initiated in the toolbox.

Some components are directly used from the AFF3CT library (black dashed-dotted tasks in Fig. 1 and Fig. 3). Thus, the software implementations are optimized for efficiency and the features have been checked. For instance, knowing that the LDPC decoding is one of the most compute intensive task, an existing high performance SIMD implementation is used. This implementation decodes multiple frames in parallel to reach very high throughputs. The Decoder LDPC task (t Rx 16 ) is the only one in the receiver that takes advantage of the multiple frames to fill the SIMD registers (inter-frame SIMD strategy). The other tasks simply process multiple frames sequentially. This implementation choice negatively affects the latency of these tasks by a factor equal to the number of frames. But, in the targeted video streaming app., the latency is not important.

New tasks have been implemented specifically for this project (blue plain boxes in Fig. 1, 2 and 3). They have been described according to the AFF3CT interfaces. Thus, allowing to take advantage of instrumentation for fine-grained throughput and latency measurements, as well as tools to facilitate debugging. The proposed implementations are mainly focusing on two missing aspects in AFF3CT: End of the acquisition phase 3 all of these software implementations, that are optimized for real-time processing, are reusable for the community.

III. REAL TIME IMPLEMENTATION FEATURES

A. Reliable Sample Collection and Delivery

The processing rate of the whole system is determined by the rate (in MS/s) at which USRP modules are set to run. The transmitter part has to provide samples at a sufficient rate, whereas the receiver part must consume them. In our study, the real time constraint was imposed by our industrial partner whose goal was that the complete chain reach several tens of MB/s. Furthermore, there should be a very limited jitter in the samples delivering and gathering rates at both sides. Indeed, the inner buffers of the USRP modules are limited. Precautionary measures are necessary to avoid temporary rate drops, that would cause underflows at the transmitter side, or overflows at the receiver side. Firstly, the corresponding threads are pinned to a dedicated core (in t Tx 10 and in t Rx 1 ). Then, preemption of these threads has to be avoided. To this purpose, we chose to isolate the CPU core to which these threads are pinned, using the isolcpus Linux kernel option. Provided with sufficient buffers on the host side, these threads efficiently handle the data exchange between the host and the USRP modules. Accessing real time processing is now just a matter of getting sufficient throughput for other tasks.

The transmitter is not the most resources consuming part that is why delivering high data rate is not as challenging as the receiver counterpart. The DVB-S2 transmitter software implementation has been split into 3 pipeline stages. Stages 1 and 3 are sequential, stage 2 is parallel. An Intel ® Core ™ i7 CPU with 4 cores (SMT was switched on) has been selected (Haswell architecture). One entire core has been reserved for the radio thread (stage 3), one hardware thread has been assigned to the source (stage 1) and the five remaining hardware threads have been dedicated to the stage 2. The pipeline implementation is a copy-less and lockfree producer/consumer using a passive waiting strategy.

B. SIMD Implementations of Synchronization and Filter Steps

The synchronization and filter tasks extensively rely on vectorized implementations. SIMD instructions are used to speedup the processing inside a frame (intra-frame SIMD strategy). The synchronization tasks are working on complex numbers. Actually, we have chosen to represent these numbers as an Array of Structures (AoS) in memory. To guarantee the portability and the flexibility of the code, the MIPP SIMD library [START_REF]MIPP: A Portable C++ SIMD Wrapper and its use for Error Correction Coding in 5G Standard[END_REF] is used. Additional reordering operations (mipp::deinterleave and mipp::interleave) were necessary to manage the AoS representation. The filter task (t Rx 4 ) mainly perform linear algebra operations. Thus, the proposed implementation extensively rely on efficient Fused Multiply-Add (FMA) instructions (mipp::fmadd).

C. High Throughput Receiver

This section details the receiver part of the system in which achieving high data rates is clearly challenging. The presented results have been obtained on two high-end NUMA machines. The first one is composed by two Intel ® Xeon ™ Platinum 8168 CPUs (denoted as x86). The frequency of the CPUs is 2.70 GHz (24 cores, 128 GB of RAM) and the Turbo Boost mode has been disabled for the reproducibility of the experiment results. Each core is powered by AVX-512F SIMD ISA. The second architecture is composed by two Cavium ThunderX2 ® CN9975 v2.1 CPUs (denoted as ARM). The frequency of the CPUs is 2.00 GHz (28 cores, 256 GB of RAM). Each core is powered by NEON SIMD ISA. In both targets the SMT was switched off. In the proposed implementation, the data are represented by 32-bit numbers. Thus, the data parallelism level is 16 for AVX-512F ISA and 4 for NEON ISA.

Tab. II presents the task throughputs and latencies measured from a sequential execution of the MODCOD 2 in the transmission phase and on the x86 target. The tasks been regrouped per stage in order to introduce the future decomposition when the parallelism is applied. The throughputs have been normalized to the number of information bits (K). This enables the comparison among all the reported throughputs.

The stage 7 takes 76% of the time with especially the Decoder BCH task (t Rx 17 ) that takes 59% of the time. t Rx 17 should not take so much time compared to the other tasks.

However, we chose to not spend time in optimizing the BCH decoding process as the stage 7. Indeed, throughput can easily be increased by running the tasks on multiple threads. The second slower stage is the stage 3. This stage is the main hotspot of the implemented receiver. The stage 3 contains only one synchronization task (t Rx 5 ). In the current implementation, this task cannot be duplicated (or parallelized) because there is an internal data dependency with the previous frame (state-full task). The stage 3 is the real limiting factor of the receiver. For information, considering a CPU with an infinite number of cores, the maximum reachable throughput is 55.42 Mb/s.

We did not try to parallelize the waiting and the acq. phases. We measured that the whole acq. phase (1, 2 and 3) takes about one second (on x86). During the acq. phase, the receiver is not fast enough to process the received samples in real time. To fix this problem, the samples are buffered in the Radio -receive task (t Rx 1 ). Once the acq. phase is done, the transmission phase is parallelized. Thus, the receiver becomes fast enough to absorb the radio buffer and samples in real time. During the transmission phase, the receiver is split into 8 consecutive pipeline stages as presented in Fig. 3. This decomposition has been motivated by the nature of the tasks (sequential or parallel) and by the sequential measured throughput. The number of stages has been minimized in order to limit the pipeline overhead. Consequently, sequential and parallel tasks have been regrouped in common stages. The slowest sequential task (t Rx 5 ) has been isolated in the dedicated stage 3. The other sequential stages have been formed to always have a higher normalized throughput than the stage 3. The sequential throughput of the stage 7 (5.35 Mb/s) is lower than the throughput of the stage 3 (55.42 Mb/s). This is why we duplicated this stage to run over 28 threads. This looks overkill but the machine was dedicated to the DVB-S2 receiver and the throughput of the Decoder LDPC task (t Rx 16 ) varies depending on the SNR. One can notice that an early termination criterion was activated. When the signal quality is very good, the Decoder LDPC task runs fast and the threads can spend a lot of time in waiting. In Tab. II, the presented Decoder LDPC task throughputs and latencies are optimistic because we are in a SNR error-free zone. All the threads are pinned to a single core with the hwloc [START_REF] Broquedis | hwloc: A Generic Framework for Managing Hardware Affinities in HPC Applications[END_REF] library. The 28 threads of the stage 7 are pinned in round-robin between the CPU sockets. By this way, the memory bandwidth is maximized thanks to the two NUMA memory banks. During the duplication process, the thread pinning is known and the data are copied into the right memory bank (first touch policy). All the other pipeline stages (1, 2, 3, 4, 5, 6 and 8) are running on a single thread. Because of the synchronizations between the pipeline stages, the threads have been pinned on the same socket. The idea is to minimize the pipeline stage latencies in maximizing the CPU cache performance. It avoids the extra-cost of moving the cache data between the sockets. On the ARM target, the pipeline has been decomposed in 12 sequential stages and 1 parallel stage of 40 threads (stage 7).

The receiver program assigned around 1.3 GB of the global memory when running in sequential while it assigned around Tab. III summarizes the obtained throughputs for the 3 MODCODs presented in Tab. I. Each time, sequential and parallel throughput are given. To measure the maximum achievable throughput, the USRP modules have been replaced by pre-registered samples. This is because the pipeline stages are naturally adapting to the slowest one. It means that in a real communication, the throughput of the radio is always configured according to the slowest stage. Indeed, it is necessary for real time communication otherwise the radio task has to indefinitely buffer the samples while the amount of available memory in the machine is finite. The throughput value is the final useful information throughput (K bits) for the user. Between the MODCOD 1 and 2, only the LDPC code rate varies (R = 3/5 and R = 8/9 resp.). It has a direct impact on the information throughputs. Between the MODCOD 2 and 3, the modulation varies (QPSK and 8-PSK resp.) and the frames have to be deinterleaved (column/row). High order modulation reduces the amount of samples processed by the Synchronizer Timing task (t Rx 5 ). This results in higher throughput in the slowest stage 3 (80 Mb/s for the 8-PSK on the x86 target). In the parallel implementations, the pipeline stage throughputs are adapting to the slowest stage 3. It results in an important speedup. In the sequential implementations, we observe a slowdown. This is mainly due to the demodulate task (t Rx 14 ) which takes a higher amount of time. This additional demodulation cost is absorbed by the 28 threads of the stage 7 when running in parallel. For MODCOD 2 in parallel, one can note that the achieved information throughput is 55 Mb/s on the x86 target. This is very close to the sequential throughput of t Rx 5 on the same target (55.42 Mb/s). Knowing that this task is the limiting factor in the pipeline, it demonstrates the efficiency of the proposed multi-threaded implementation.

The throughputs obtained on the ARM target are lower than on the x86 CPUs (by a factor of ≈ 2 when running in parallel). It can be explained by the limited mono-core performance of the ThunderX2 architecture: the frequency is lower (2.0 GHz versus 2.7 GHz) and the SIMD width is smaller (128-bit in NEON versus 512-bit in AVX-512F).

IV. RELATED WORKS Some other works are focusing on SDR implementations of a DVB-S2 transceiver. To the best of our knowledge, the competitive existing projects are discussed thereafter.

gr-dvbs2rx [START_REF] Economos | gr-dvbs2rx: GNU Radio Extensions for the DVB-S2 and DVB-T2 Standards[END_REF] is an open source extension to GNU Radio. The project sounds promising but lacks efficiency. Its main maintainer affirms that the receiver is not yet able to meet the satellite real time constraints on a Xeon ™ Gold/Platinum processor. The use of a dedicated GPU or an FPGA for the LDPC decoding is advised.

leansdr [START_REF] Pabr | leansdr: Lightweight, Portable Software-defined Radio[END_REF] is a standalone open source project. The project creation was motivated to reach higher receiver throughput than GNU Radio at the cost of decoding performance degradation. For instance, a low complexity LDPC bit-flipping decoder is chosen. At the time of the writing, the project does not support multi-threading.

Grayver and Utter recently published a paper [START_REF] Grayver | Extreme Software Defined Radio -GHz in Real Time[END_REF] in which they succeed to build a 10 Gb/s DVB-S2 receiver on a cluster of server-class CPUs. On a comparable CPU, their work is able to double or even triple the throughput of our implementation (approximately 185 Mb/s on 20 cores). This is mainly due to new algorithmic improvements in the synchronization tasks and the use of a very fast LDPC decoder [START_REF] Gal | High-Throughput Multi-Core LDPC Decoders Based on x86 Processor[END_REF], [START_REF] Grayver | Scaling the Fast x86 DVB-S2 Decoder to 1 Gbps[END_REF]. They are able to express more parallelism in the Synchronizer Timing task (t Rx 5 ). In our work, we could have split t Rx 5 in multiple pipeline stages to increase its throughput but we preferred to stop the optimization process since we met the industrial constraints. One may note that Grayver and Utter's work focuses on a single DVB-S2 MODCOD (8-PSK, N = 64800 and R = 1/2) and the achieved decoding performances are not reported.

V. CONCLUSION

This paper deals with the software implementation of realtime DVB-S2 transceiver. For this purpose, USRP modules were combined with multicore and SIMD CPUs. The implementations were done thanks to the AFF3CT toolbox. Thus, some components are directly from the AFF3CT library and others such as the synchronization functions have been described. Experiment results show the performance but also the flexibility and the portability of the transceiver. An obvious strength of this work is also its collaborative context and the code publicly released.

Fig. 3 .Fig. 4 .

 34 Fig. 3. DVB-S2 software receiver: acquisition phase 3 & transmission phase.

  1) signal synchronizations and filters, 2) real-time communications. Being open source,

		USRP					Synchronizer Pilot Feedback
			receive (t Rx 1 ) Radio	imultiply (t Rx 2 ) Multiplier AGC	synchronize Synchronizer Freq. Coarse	filter Filter Matched	synchronize Synchronizer extract (t Rx 6 ) Timing (Gardner)	imultiply (t Rx 7 ) Multiplier AGC	(t Rx 8 ) synchronize Synchronizer Frame
							synchronize (t Rx 3,4,5 )
			Fig. 2. DVB-S2 software receiver: waiting phase and acquisition phase 1 & 2.
	USRP
	Radio	Multiplier AGC	Synchronizer Freq. Coarse	Filter Matched		Synchronizer Timing (Gardner)		Multiplier AGC	Synchronizer Frame	Scrambler Symbol	Synchronizer Freq. Fine L&R	Synchronizer Freq. Fine P/F
	receive (t Rx 1 )	imultiply (t Rx 2 )	synchronize (t Rx 3 )	filter (t Rx 4 )	synchronize (t Rx 5 )	extract (t Rx 6 )	imultiply (t Rx 7 )	synchronize (t Rx 8 )	descramble (t Rx 9 )	synchronize (t Rx 10 )	synchronize (t Rx 11 )
	Stage 1		Stage 2				Stage 3			Stage 4		Stage 5
	Framer PLH	Noise Estimator	Modem PSK		Interleaver	Decoder LDPC		Decoder BCH	Scrambler Binary	Sink Binary File
	remove (t Rx 12 )	estimate (t Rx 13 )	demodulate (t Rx 14 )	deinterleave (t Rx 15 )	decode SIHO (t Rx 16 )	decode HIHO (t Rx 17 )	descramble (t Rx 18 )	send (t Rx 19 )
	Stage 6					Stage 7				Stage 8

TABLE II RECEIVER

 II SEQUENTIAL TASK CHARACTERISTICS ON THE X86 CPU. SEQUENTIAL TASKS ARE REPRESENTED BY BLUE ROWS. THE SLOWEST SEQ. STAGE IS IN RED WHILE THE SLOWEST OF ALL IS IN ORANGE .

	Stages and Tasks Throughput (Mb/s)	Latency (µs)	Time (%)
	Radio -receive (t Rx 1 ) Stage 1	431.83 431.83	527.32 527.32	0.94 0.94
	Multiplier AGC -imultiply (t Rx 2 ) Synch. Freq. Coarse -synchronize (t Rx 3 ) Filter Matched -filter (t Rx 4 ) Stage 2	367.45 841.32 116.41 80.00	619.71 270.66 1956.08 2846.45	1.11 0.48 3.49 5.08
	Synch. Timing -synchronize (t Rx 5 ) Stage 3	55.42 55.42	4108.52 4108.52	7.34 7.34
	Synch. Timing -extract (t Rx 6 ) Multiplier AGC -imultiply (t Rx 7 ) Synch. Frame -synchronize (t Rx 8 ) Stage 4	281.83 685.51 159.41 88.65	807.97 332.18 1428.51 2568.66	1.44 0.59 2.55 4.58
	Scrambler Symbol -descramble (t Rx 9 ) Synch. Freq. Fine L&R -synchronize (t Rx 10 ) Synch. Freq. Fine P/F -synchronize (t Rx 11 ) Stage 5	1682.89 1246.85 112.56 97.27	135.31 182.63 2022.98 2340.92	0.24 0.33 3.61 4.18
	Framer PLH -remove (t Rx 12 ) Noise Estimator -estimate (t Rx 13 ) Stage 6	1008.60 550.06 355.94	225.77 413.98 639.75	0.40 0.74 1.14
	Modem PSK -demodulate (t Rx 14 ) Interleaver -deinterleave (t Rx 15 ) Decoder LDPC -decode SIHO (t Rx 16 ) Decoder BCH -decode HIHO (t Rx 17 ) Scrambler Binary -descramble (t Rx 18 ) Stage 7	40.47 1347.25 164.21 6.92 32905.37 58.79 5626.34 10.05 169.02 0.30 1386.74 2.48 91.11 2499.41 4.47 5.35 42586.88 76.09
	Sink Binary File -send (t Rx 19 ) Stage 8	1838.31 1838.31	123.87 123.87	0.22 0.22
	Total	4.09 55742.37	99.57

TABLE III THROUGHPUTS

 III DEPENDING ON THE SELECTED DVB-S2 CONFIGURATION.

			Throughput (Mb/s)
		Sequential	Parallel	Latency (ms)
	Configuration x86 ARM x86 ARM x86	ARM
	MODCOD 1 MODCOD 2 MODCOD 3	3.4 4.1 4.0	1.0 1.4 1.1	37 55 80	19 28 42	-56 -	37 41 51
	30 GB in parallel. The memory usage increases because of the chain duplications in the stage 7. The duplication operation takes about 20 seconds on the x86 and ARM targets. It is made at the very beginning of the program. It is worth mentioning that the amount of memory was not a critical resource.

DVB-S2 digital transceiver repository: https://github.com/aff3ct/dvbs2.


Michel Bergmann

Laurent Cordier

Jean-Pierre Brancher

On the generation of a reverse Von Kármán street for the controlled cylinder wake in the laminar regime

Keywords: Partial rotary control, Cylinder wake, Drag minimization, reverse Kármán vortex street

In this Brief Communication we are interested in the maximum mean drag reduction that can be achieved under rotary sinusoidal control for the circular cylinder wake in the laminar regime.

For a Reynolds number equal to 200, we give numerical evidence that partial control restricted to an upstream part of the cylinder surface may increase considerably the effectiveness of the control. Indeed, a maximum value of relative mean drag reduction equal to 30% is obtained when applying a specific sinusoidal control to the whole cylinder, where up to 75% of reduction can be obtained when the same control law is applied only to a well selected upstream part of the cylinder. This result suggests that a mean flow correction field with negative drag is observable for this controlled flow configuration. The significant thrust force that is locally generated in the near wake corresponds to a reverse Kármán vortex street as commonly observed in fish-like locomotion or flapping wing flight.

Finally, the energetic efficiency of the control is quantified by examining the Power Saving Ratio: it is shown that our approach is energetically inefficient. However, it is also demonstrated that for this control scheme the improvement of the effectiveness goes generally with an improvement of the efficiency.

The control of the cylinder wake flow by sinusoidal rotation (Fig. 1) was intensively studied numerically 1-3 and experimentally 4 . Generally, a global rotation of the cylinder is considered, i.e. the whole cylinder oscillates around its principal axis. When the wake flow is controlled by rotary oscillation, a maximum value of relative mean drag reduction of about 30% could be obtained 5,6 for a Reynolds number equal to 200. In Ref. 7, Protas and Wesfreid argued that this mean drag reduction seems to be associated with control driving the mean flow toward the unstable symmetric state i.e. the basic flow. Therefore, as it was already pointed out by Protas and Wesfreid, is it possible for the cylinder wake flow to obtain under periodic forcing, a mean drag reduction lower than in the basic flow? The aim of this Brief Communication is to demonstrate numerically that a more effective drag reduction can be achieved by controlling only a limited part of the cylinder surface 8 . In our approach the cylinder boundary is divided into two regions: an upstream part on which a specific control law discussed below is applied, and the remainder of the cylinder which is not controlled.

Hereafter, the upstream controlled region is geometrically defined by -θ c < θ < +θ c (see Fig. 1) where θ and θ c are respectively the curvilinear coordinate of a point on Γ c and the control angle. By convention, these angles are initialized at the front stagnation point of the cylinder. Obviously the case θ c = 180 • corresponds to the rotation of the cylinder as a whole. The objective of this numerical experience is to determine the value of θ c such that the reduction of the mean drag coefficient is maximized.

Here, the incompressible and viscous flow around a circular cylinder of diameter D (radius R) is considered in a two-dimensional domain Ω (Fig. 1). The Reynolds number Re = U ∞ D/ν is equal to 200, where U ∞ is the uniform velocity of the incoming flow and ν is the kinematic viscosity of the fluid. The cylinder is animated with an unsteady tangential velocity V T , possibly equal to zero for the uncontrolled part of the cylinder surface. The 3 Navier-Stokes equations are solved using a fractional step method in time and a finite element method in space (see Ref. 5).

Let u be the velocity component in the x-direction, p be the pressure and n be the external normal vector to the boundary Γ c of the cylinder, the drag coefficient is given by:

C D (u, p) = Γc 2p n x dΓ - Γc 2 Re ∂u ∂x n x + ∂u ∂y n y dΓ,
where all the variables are non-dimensionalized with respect to the cylinder diameter D and the oncoming flow U ∞ . For a circular cylinder, the mean temporal drag coefficient estimated over a finite horizon T writes:

C D T = 2π 0 1 T T 0 2p n x dt R C p D (θ) dθ + 2π 0 1 T T 0 - 2 Re ∂u ∂x n x + ∂u ∂y n y dt R C ν D (θ) dθ. (1)
In a first stage, the cylinder is considered to rotate as a whole (θ c = 180 • ) with a non By comparison, the mean drag associated to the basic flow is 0.94. Therefore, in this case, as it was previously discussed in Ref. 7, the lower limit of the mean drag coefficient, even for a controlled flow, seems to be the drag associated with the basic flow. The wake flow generated using the control law γ min is symmetric and tends to the wake associated with the Moreover, the maximal mean drag reduction is obtained for θ c = 120 • . Figure 5 is equivalent to Fig. 3 but corresponds to the case where the rotary partial control characterized by θ c = 120 • is considered. Figure 5 clearly demonstrates that, for this particular control, almost all the surface cylinder contributes to the relative reduction of the term C D . In comparison with Fig. 3, it can be inferred that the major improvement is obtained near the rear stagnation point. In consequence, as we can see in Figs. 4 and6, the mean drag coefficient is then greatly reduced. Indeed, this coefficient decreases to a value equal to 0.34 corresponding to a relative mean drag reduction greater than 75%. Moreover, the drag coefficient becomes periodically negative expressing the generation of a propulsive effect (see Fig. 6). However, whatever the value of θ c , the control is not energetically efficient. The Power Saving Ratio (PSR) as defined in Ref. 7 is everywhere less than unity (see table I). In spite of that, it is remarkable that as soon as θ c is greater than 30 • , the efficiency is improved compared to the case where θ c = 180 • demonstrating, that for θ c contained between 73 • and 177 • , the improvement of the effectiveness goes with an improvement of the efficiency.

dimensional sinusoidal velocity γ(t) = V T (t)/U ∞ = A sin(2πSt f t).
Next in Fig. 7(a) the organization of the vorticity ω z in the near wake is presented.

Alternatively, positive (resp. negative) vortices are shed in a region where the net mean vorticity is of opposite sign. This phenomenon is physically similar to the one observed in fish-like locomotion 9 or flapping wing flight 10 . The upstream surface cylinder produces thrust as it oscillates developing a reverse Kármán vortex street that corresponds, at least locally, to a jet-like average vorticity profile. The moving walls on a part of the cylinder surface have the effect of increasing the flow momentum that may be large enough to overcome the adverse pressure gradient formed at the rear part of the cylinder and to delay the separation, resulting in the pressure drag reduction (see Fig. 4). As shown in Fig. 7(b), the Kármán vortex street tends to disappear in the far wake. Moreover, the wake flow is quasi-symmetric and its width is greatly reduced in comparison with the global controlled case (see Ref. 5).

In this Brief Communication, we demonstrated numerically that the effectiveness of a partial control restricted to an upstream part of the cylinder surface may be significantly higher that the one corresponding to the rotation as a whole. Indeed, while a relative mean drag reduction of about 30% is obtained when the control γ min is applied on the whole cylinder, this reduction becomes greater than 75% when a partial control characterized geometrically by θ c = 120 • is considered. In this case, the value of the mean drag coefficient is even lower than that obtained for the basic flow, suggesting that a mean flow correction field with negative drag is observable. Physically, a partial control gives rise in the near wake to a reverse Von Kármán street that generates locally a significant thrust force. This unsteady flow control mechanism is similar to the drag reduction phenomenon encountered in fish-like locomotion where fish and cetaceans employ instinctively their flapping tails and several fins to produce propulsive and maneuvering forces. * Electronic address: Michel.Bergmann@ensem.inpl-nancy.fr 1 J.-W. He, R. Glowinski, R. Metcalfe, A. Nordlander, and J. Périaux, "Active control and drag optimization for flow past a circular cylinder. Part 1. Oscillatory cylinder rotation," J. Comp.

Phys., 163, 83 (2000).

2 C. Homescu, I. Navon, and Z. Li, "Suppression of vortex shedding for flow around a circular cylinder using optimal control," Int. J. Numer. Meth. Fluids, 38, 43 (2002).

3 B. Protas and A. Styczek, "Optimal rotary control of the cylinder wake in the laminar regime,"

Phys. Fluids, 14, 2073Fluids, 14, (2002)).

4 P. Tokumaru and P. Dimotakis, "Rotary oscillatory control of a cylinder wake," J. Fluid Mech., 224, 77 (1991).

5 M. Bergmann, L. Cordier, and J.-P. Brancher, "Optimal rotary control of the cylinder wake using proper orthogonal decomposition reduced order model," Phys. Fluids, 17 (2005).

  Hereafter, the parameters A and St f are selected as the values which minimize the mean drag coefficient of the wake flow. In Ref. 5, a series of simulations with various amplitudes A varying from 0 to 6.5 per step of 0.5 and various forcing Strouhal numbers St f varying from 0 to 1 per step of 0.1 was carried out. The numerical results of this open loop control approach demonstrated (see Fig. 31 in Ref. 5) that the mean drag coefficient was minimized for a control law γ min characterized by the parameters (A min , St f min ) = (4.3, 0.74). The corresponding relative mean drag reduction amounts to more than 30% ( C D T varies from 1.39 to 0.99).

  unstable stationary basic flow (see Ref. 5). Moreover, the controlled flow exhibits a vortex street thinner than the one obtained for the uncontrolled flow. Now, it is of interest to examine the relative contributions of the different quantities C p D and C ν D to C D = C p D + C ν D (see Eq. (1)) with respect to the angle θ. Such representations are done in Fig. 2 for the uncontrolled flow and in Fig. 3 for the global controlled flow with γ min . The light grayed and dark grayed areas correspond to the angles for which the fluid exerts on the cylinder a drag force and a thrust force respectively. It is noticeable (compare Figs. 2 and 3) that the application of the control law γ min is effective only on the upstream part of the cylinder. Then, it is natural to expect that the effectiveness of the control law γ min increases as a controlled upstream part of the cylinder is coupled to an uncontrolled downstream part. Different numerical experiences have been successively considered for θ c varying from 0 • (uncontrolled flow) to 180 • (global controlled flow) by step equal to 30 • . In Fig. 4, wenotice that for angles θ c contained between 73 • and 177 • , the mean drag coefficient is lower than the minimal expected value corresponding to the basic flow 7 , suggesting that a mean flow correction field with negative drag can be obtained for this controlled configuration.

6

  M. Bergmann, L. Cordier, and J.-P. Brancher, "Control of the cylinder wake in the laminar regime by Trust-Region methods and POD Reduced Order Models," 44 th IEEE Conference on Decision and Control and European Control Conference, 12 -15 December 2005, Seville, Spain.

7

  B. Protas and J. Wesfreid, "Drag force in the open-loop control of the cylinder wake in the laminar regime," Phys. Fluids, 14, 810 (2002).8 B. Choi and H. Choi, "Drag reduction with a sliding wall in flow over a circular cylinder," AIAA

FIG. 1 .

 1 FIG. 1. Sketch of the controlled flow configuration.

TABLE I :

 I

	θ c	30 •	60 •	90 •	120 •	150 •	180 •
	PSR ×100	1.3	9.6	22.3	24.1	16.2	6.6
		Bergmann, Cordier and Brancher, Physics of Fluids

D. Barrett, M. Triantafyllou, D. Yue, M. Grosenbaugh, and M.J. Wolfgang, "Drag reduction in fish-like locomotion," J. FluidMech., 392, 183 (1999).

S. Ho, H. Nassef, N. Pornsinsirirak, Y.-C. Tai, and C.-H. Ho, "Unsteady aerodynamics and flow control for flapping wing flyers," Progress in Aerospace Sciences, 39, 635 (2003).