# Large-eddy simulations of control of a separated flow over a 2D bump by means of pulsed jets

Jean-Philippe Laval, Caroline Braud, Guillaume Fournier, Michel Stanislas

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Jean-Philippe Laval, Caroline Braud, Guillaume Fournier, Michel Stanislas. Large-eddy simulations of control of a separated flow over a 2D bump by means of pulsed jets. Journal of Turbulence, 2010, 11, pp.52. 10.1080/14685248.2010.522579. hal-05510169

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
Large-eddy simulations of control of separated flow over a 2D
bump by means of pulsed-jets

J.P. Laval*(1,3), C. Braud(1,3), G. Fournier(1,3), M. Stanislas(1,2)

(1) CNRS, UMR 8107, F-59650 Villeneuve d'Ascq, France

(2) ECLille, LML, F-59650 Villeneuve d'Ascq, France

(3) Univ Lille Nord de France, F-59000 Lille, France

(Received 00 Month 200x; final version received 00 Month 200x)

A numerical study of active flow control of a separated converging-diverging channel flow is presented. Several geometrical and operating parameters of the continuous and pulsed jets are investigated using Large Eddy Simulations (LES) which are highly resolved in the actuation region. The LES without control is validated with a Direct Numerical Simulation (DNS) of the same flow. Configurations with both counter-rotating and co-rotating jets are tested. The main statistical results are in general agreement with the literature. Thanks to the unsteady highly resolved simulation, both temporal and spatial organization of the generated vortices are investigated. An optimal frequency associated to the convective time as well as an optimal duty cycle are obtained, leading to a control efficiency comparable to the cases with continuous jets and the same operating velocity and leading consiquently to a significant saving in mass flow rate.

**Keywords: turbulence, active control, pulsed-jets**

# 1. Introduction

Delaying or preventing Turbulent Boundary Layer (TBL) separation in aeronautic applications (during take off/landing or manoeuvre of an aircraft for example) leads to enhance the lift to drag ratio and thus to fuel saving and diminution of the emitted pollution. For these reasons, flow separation control has been the subject of strong attention in the last 50 years. In particular, many types of actuator were explored in the last decades, over many different flow configurations (flat plates, ramps, bumps, ducts, airfoils, wind turbines, ...) [8].

Among the variety of actuators used for that purpose, the Vortex Generators (VGs) were found efficient to reduce and even sometimes suppress the separated region [18]. These devices generate a streamwise vortex structures which can entrain high-momentum fluid towards the wall, hence energizing the turbulent boundary layer, increasing quantities such as wall shear stress, turbulence intensities, momentum transfer, etc ... and delaying separation. The optimal device would produce streamwise vortices just strong enough to overcome the separation without unnecessarily persisting within the boundary layer once the flow control objective is reached. It can be either passive or active. The resulting streamwise vortices has been the subject of several studies which helped to characterize the optimal actuator parameters and the resulting mean and turbulent organization (see for instance [1, 4, 5, 9, 10, 14, 15, 18, 19, 22, 24, 27, 28]).

<footnote>*Corresponding author. Email: jean-philippe.laval@univ-lille1.fr</footnote>
Passive devices were rapidly replaced by active ones which can be turned off when not necessary in order to avoid additional drag (i.e. during cruise flight for instance). Also, active devices allows to implement different control strategies to adapt the actuation to the baseline flow situations. Many active device types were thus developed and used in continuous and pulsed operating modes.

For continuous jets, the origin of the streamwise vortical structures in the con-
tinuous jet case comes from the high shear from the jet edges which induces a
roll-up of the boundary layer flow [25, 29]. When the jet is inclined to the wall
and/or to the incoming baseline flow (α skew and/or β pitch angle respectively),
two counter-rotating vortices are initially created just downstream of the device
which evolves rapidly into an enhanced single coherent vortex of one sign accom-
panied by a much smaller and weaker region of circulation of the opposite sign
near the wall [28]. When there exist a skew and pitch angle, the skin friction
increases which make it more robust for flow control separation [10]. The opti-
mal pitch and skew angle is found to be around α = β = 45° [5, 10]. Several
studies were performed with a single jet arrangement in a ZPG configuration to
analyze the streamwise evolution of the vortices (among others [13, 28]). However
this does not allow to take into account the interaction between the streamwise
vortices which is found of major importance [10]. Therefore, many studies were
performed with a spanwise distribution of the jets with a co-rotating (CoR) or a
counter-rotating (CtR) arrangement [4, 10, 14, 15, 21, 29]. Distances between de-
vices (defined in Fig. 6) of λ/Φ = 6 for CoR jets was found to significantly increase
the skin friction for a bump configuration [10] (where Φ is the diameter of round
jets). Also, counter-rotating arrangements (CtR) were found more efficients than
co-rotating arrangements (CoR) [10]. The counter-rotating arrangement generates
stable counter-rotating streamwise vortices which stay coherent and individualized
far downstream, contrary to co-rotating jets which merged rapidly to form a large
vortical region [4]. At last, increasing the Velocity Ratio (VR) increases the shear
from the jet edges which has been demonstrated to produce stronger streamwise
vortices, i.e. a greater circulation [25, 28, 29]. However, too high exit velocity of
the jets can blow the produced vortices out of the boundary layer which reduces
the control efficiency [28].

In the pulsed mode, the operating parameters of the resulting jet excitation (frequency bandwidth, amplitude or VR, duty cycle or phase ...) are usually used as inputs for different control strategies. Depending on the main flow configuration (i.e. attached or separated TBL over a ramp, bump, ...) and the actuator-jet strength, the pulsating frequency may interact favorably with the baseline flow. These strategies can be classified in three categories : the robust control (high excitation amplitude and adapted frequencies), the control of the baseline instabilities or turbulent structures in turbulent flows (lower excitation amplitudes and adapted frequencies), the control of turbulent structures at a Kolmogorov scale (very low excitation and high frequencies). The main goal of all these strategies is, for the same control efficiency, to reduce the energy expense of the control which is of major importance for realistic applications. Using pulsed jets with round orifices and a high excitation amplitude (VR > 1), [14, 28] have reported that the pulsating frequency and the Duty Cycle (DC) can be tuned so that successive streamwise vortex segments can merged downstream, producing quasi-continuous streamwise vortices with less mass flow rate than in the steady case. [14, 23] have recently reported that the opening and closing of the valve create an enhanced mixing mechanism which significantly increases the skin friction gain and is not observed for a continuous jet configuration. [3, 14] associate the opening and closing effects to a wave propagation in the tubes behind the valves. Using zero net mass flux
actuators in a slot jet configuration at low jet exit amplitude, preferential excita-
tion frequencies are systematically reported [2, 6, 11, 26]. This is associated by the
authors to a resonance between the produced structures and the structures from
the Kelvin-Helmholtz instability of the separated baseline flow. However, turbulent
separated flows are sensitive to a wide range of frequencies which are not yet fully
understood. In studies using active vortex generator actuators only few authors
have reported such a frequency dependence and only for a complex geometry and
a relatively high jet exit amplitude (VR = 2) [21].

The present numerical study (LES) aims at extending the knowledge of the
influence of active vortex generator actuators for separation control purpose. From
the knowledge of the authors, this is the first LES study on active vortex generators.
Most of the numerical studies were performed for zero-net mass flux actuators [2, 6,
26] and only one study on passive devices was recently performed [7]. Parametric
investigations of the influence of geometrical parameters are less numerous for
active devices than for passive devices as it is experimentally difficult to modify
them. Moreover, it is experimentally difficult to isolate a single excitation parameter
[3, 25].

In the present study the control over a bump similar to [10] at a lower Reynolds number is performed. Firstly, the influence of the geometrical parameters are analyzed for a low authority actuation VR < 1. Two streamwise positions, expected to be sensitive ones, are tested. Then the influence of the jet orientation is explored by varying the skew (α = 45°, 90°, 135°) and pitch (β = 45°, 70°, 90°) angles. Secondly, the geometry is fixed and operating parameters are explored. The influence of the jet exit amplitude is analyzed for continuous jets. Using one amplitude of the jet exit, a large range of the different pulsating frequency is explored together with the DC parameter. At last, for a fixed configuration, the streamwise evolution of the Reynolds stress components is analyzed and the momentum balance is extracted.

2. Numerical Simulations

**2.1. Numerical method**

The LES and the reference DNS of converging-diverging channel flow were per-
formed using the MFLOPS3D code developed at LML for simulation of wall
bounded flow with a curved wall. Instead of writing the Navier-Stokes system in
curvilinear coordinates, the wall curvature is obtained by a mathematical mapping
of the partial differential operators from physical coordinates to Cartesian ones (see
[20] for more details on the numerical algorithm). The spatial discretisation uses
Fourier modes in the spanwise direction, a Chebyshev collocation is used in the
normal direction and 4^th^ order explicit finite differences are used in the streamwise
direction. The main interest of this code is to keep a direct resolution method in
addition to high order spatial discretisations. The three-dimensional incompressible
Navier-Stokes equations are made dimensionless using half the channel width h as
reference length. The maximum velocity at the inlet U_o is the reference velocity
and hence the Reynolds number is Re = hU_o / v = 12600. The Reynolds number is
often based also on the wall friction velocity u_T : Re_T = h u_o / v. In the present case,
Re_T = 617 at the inlet.

A detailed analysis of the reference DNS of the uncontrolled case can be found
in [16] at the same Reynolds number and in [20] at lower Reynolds number (Reτ =
395). The uncontrolled flow slightly separates at the lower wall and is close to
separation at the upper wall. However, in the present study, only the lower wall
will be investigated in detail. As a large number of parameters will be considered, it is necessary to use LES instead of DNS to limit the computational time expenses.

## 2.2. LES validation

A complete LES study of the uncontrolled flow has been performed in [17]. The WALE model was found to give results comparable to the dynamic Smagorinsky model and the LES is able to reproduce the right separation when the streamwise spatial resolution is high enough. All the LES of the present study are performed with the required resolution of 1280($x$) × 129($y$) × 256($z$), $x$, $y$ and $z$ being the streamwise, wall-normal and spanwise coordinates respectively. As a comparison, the reference DNS case was performed on a 2304 × 384 × 576 mesh for the same size of the domain (12h × 2h × $\pi h$). The mesh is strongly stretched in the streamwise direction (see Fig. 1) in order to have a better resolution of the actuated region which is almost fully resolved. The mesh size in wall units is $\Delta x^+ = 11$, $\Delta z^+ = 7.5$ at the inlet and $\Delta x^+ = 5.7$, $\Delta z^+ = 10$ at the actuation location ($x = 0$). Due to the use of Chebyshev collocation, the normal direction is over-resolved near the wall as 4 grid points are located below 2 wall units (at $x = 0$). The streamwise resolution $\Delta x$ is of the same order or even lower than the spanwise resolution $\Delta z$ which is related to the lower order of the scheme discretisation in the spanwise direction.

In order to obtain converged statistics, all the LES (with and without control) are initialized with a DNS velocity and pressure field, and are integrated over ten convective times (based on the inlet maximum velocity ($U_o = 1$) and half the channel height ($h = 1$)). Only the last six convective times are retained to compute statistics. The inlet flow is interpolated from the same inlet data (recorded from a precursor channel DNS without bump) that was used for the DNS.

In order to validate the LES, the friction coefficient ($C_f$) and the pressure coefficient ($C_p$) are compared with the DNS (Fig. 2). The two coefficients are defined by:

$$C_p = \frac{P - P_o}{\frac{1}{2} \rho U_o^2}, \qquad C_f = \frac{\tau_w}{\frac{1}{2} \rho U_o^2}, \qquad (1)$$

where $\tau_w$ is the wall shear stress and $P_o$ and $U_o$ are respectively the pressure and the streamwise velocity at the inlet ($y = 1$) and $\rho = 1$. The comparison with the DNS is rather accurate except within the separation region, for negative values of the skin friction, where the intensity of the shear stress is slightly over-estimated by the LES. However, the position of the separation as well as the re-attachment point are predicted accurately. The comparison of the mean streamwise velocity is shown in Fig. 3 in the diverging part of the flow. The agreement between DNS and LES is very good, even in the separation region.

In the following $C_f$ and $C_p$ coefficients from the LES with actuation will be systematically compared to the same quantities from the LES without actuation to highlight the benefit of the control.

## 2.3. Baseline flow

The baseline flow is a converging-diverging channel flow which can be described from the friction and pressure coefficient distributions. The curved lower wall creates a converging part. This, until the summit at $x = 0$, induces an acceleration of the flow and thus a favorable pressure gradient (see Fig. 2). In that converging part, the boundary layer thickness decreases until the point of maximum skin friction
coefficient at $x = -0.66$. From this position, the boundary layer begins to grow. Also, for $-0.66 < x < 0$ the skin friction decreases while the pressure gradient is still favorable. After the summit, a pressure recovery occurs in two steps due to a flow separation. The flow separates between $x = 0.5$ and $x = 1.5$ and induces a negative value of the skin friction coefficient. At the minimum of the skin friction coefficient ($x = 1.2$), the pressure gradient is almost zero. Before and after the flow separation, the adverse pressure gradient is of the same order of magnitude. Therefore, two positions are expected to be sensitive to the actuation. The first one, at the maximum of skin friction $x = -0.75$ where the produced streamwise vortices may take advantage of the growing boundary layer thickness. The second one, at the summit of the bump, where the pressure gradient begins to be adverse.

## 2.4. Actuation method

In all the controlled cases, the actuation is generated by setting a bleeding velocity profile at the wall. The general shape of the velocity profile is given by:

$$V^p(x, z, t) = \begin{cases} \frac{1}{4}G(t)V \left[ \cos \left( 2\pi \frac{|x-x_a^p|}{\delta x} \right) + 1 \right] \left[ \cos \left( 2\pi \frac{|z-z_a^p|}{\delta z} \right) + 1 \right] & \text{if } \begin{cases} |x-x_a^p| < \frac{\delta x}{2} \\ \text{and} \\ |z-z_a^p| < \frac{\delta z}{2} \end{cases} \\ 0 & \text{otherwise} \end{cases}$$

where $V^p(x, z, t) = |\vec{V}^p(x, z, t)|$ is the modulus of the velocity of actuator $p$, $x_a^p$, $z_a^p$ and $\delta x = \delta z$ are the position of the center and the size of actuator $p$ and $G(t)$ is the time evolution of the jet amplitude. An example of $V^p(x, z, t = t_o)$ is given in Fig. 4 for a size $\delta x = \delta z = 0.08$. Note that, because of the shape of the outlet profile, the effective diameter of the actuator is estimated to be lower than $\delta x = \delta z = 0.08$ by at least 15%. The time evolution $G(t)$ of the actuation is set to unity for continuous jets. Two types of function are used for pulsed jets. The first type is a smoothed squared wave function described as follows:

$$G(t) = \left[ \text{mod}(C \times F \times t, C) - \frac{C}{2} \right]^8 \quad (2)$$

where $C$ is a constant to adjust the duty cycle. A constant $C = 4$ was used for $DC = 0.47$, $C = 10$ for $DC = 0.19$ and $C = 6.5$ for $DC = 0.28$. The second type of function is a linear ramp. The two types of function are given in Fig. 5 for two different frequencies ($F = 2$ and $F = 10$).

For both space and time smoothed functions are preferred to a sharp ones for numerical stability reasons.

## 2.5. Control parameters

Due to the computational effort induced by each simulation, only few parameters are investigated in the present study. The investigated range for each parameters is based on previous experimental studies of Godard et al [10] on the same geometry but at much larger Reynolds number. First, several geometrical parameters, such as the streamwise location of the jet $x_a^p$, the jet arrangement (co-rotating jets (CoR) or counter-rotating jets (CtR)), the spanwise distance between jets $\lambda$ and $L$) and the jet orientation ($\alpha$ and $\beta$), were briefly investigated (see Fig. 6). Then, based on the best geometrical parameters, the influence of the operating parameters ($V$, $F$ and $DC$) were analyzed. All the tested configurations are listed in Table 1. The
<table><caption>Table 1. List of all controlled cases with their corresponding parameters. The geometrical parameters ($x_a$, $\lambda$, $L$, $\alpha$, $\beta$) are defined in Fig. 6. The other parameters are: the mass Flux of the actuation in percentage of the mass flux of the boundary layer at the position of the actuation ($Fl$), the Frequency ($F$) used in the time function $G(t)$ defined by Eq. (2) except for the case indicated by a * symbol which use the ramp function plotted in Fig. 5.</caption><thead><tr><th>Case</th><th>x<sub>a</sub></th><th>Nb×Type</th><th>λ/L</th><th>V</th><th>Fl.</th><th>α</th><th>β</th><th>F</th><th>DC</th></tr></thead><tbody><tr><td>C1</td><td>-0.664</td><td>8 CtR</td><td>4</td><td>1.0</td><td>6.1</td><td>90</td><td>45</td><td>0</td><td>1.0</td></tr><tr><td>C2</td><td>+0.000</td><td>8 CoR</td><td>1</td><td>1.0</td><td>6.1</td><td>90</td><td>45</td><td>0</td><td>1.0</td></tr><tr><td>C3</td><td>+0.000</td><td>8 CtR</td><td>4</td><td>1.0</td><td>6.1</td><td>90</td><td>45</td><td>0</td><td>1.0</td></tr><tr><td>C4</td><td>+0.000</td><td>8 CtR</td><td>4</td><td>0.5</td><td>3.0</td><td>90</td><td>45</td><td>0</td><td>1.0</td></tr><tr><td>C5</td><td>+0.000</td><td>8 CtR</td><td>4</td><td>1.5</td><td>9.2</td><td>90</td><td>45</td><td>0</td><td>1.0</td></tr><tr><td>C6</td><td>+0.000</td><td>8 CtR</td><td>4</td><td>2.5</td><td>15.3</td><td>90</td><td>45</td><td>0</td><td>1.0</td></tr><tr><td>C7</td><td>+0.000</td><td>8 CtR</td><td>4</td><td>0.5</td><td>3.0</td><td>90</td><td>70</td><td>0</td><td>1.0</td></tr><tr><td>C8</td><td>+0.000</td><td>8 CtR</td><td>4</td><td>0.5</td><td>3.0</td><td>90</td><td>90</td><td>0</td><td>1.0</td></tr><tr><td>C9</td><td>+0.000</td><td>8 CtR</td><td>4</td><td>1.0</td><td>6.1</td><td>45</td><td>45</td><td>0</td><td>1.0</td></tr><tr><td>C10</td><td>+0.000</td><td>8 CtR</td><td>4</td><td>1.0</td><td>6.1</td><td>135</td><td>45</td><td>0</td><td>1.0</td></tr><tr><td>P1</td><td>+0.000</td><td>8 CtR</td><td>4</td><td>1.0</td><td>2.8</td><td>90</td><td>45</td><td>1</td><td>0.47</td></tr><tr><td>P2</td><td>+0.000</td><td>8 CtR</td><td>4</td><td>1.0</td><td>2.8</td><td>90</td><td>45</td><td>2</td><td>0.47</td></tr><tr><td>P3*</td><td>+0.000</td><td>8 CtR</td><td>4</td><td>1.0</td><td>2.8</td><td>90</td><td>45</td><td>2</td><td>0.47</td></tr><tr><td>P4</td><td>+0.000</td><td>8 CtR</td><td>4</td><td>1.0</td><td>2.8</td><td>90</td><td>45</td><td>5</td><td>0.47</td></tr><tr><td>P5</td><td>+0.000</td><td>8 CtR</td><td>4</td><td>1.0</td><td>2.8</td><td>90</td><td>45</td><td>10</td><td>0.47</td></tr><tr><td>P6</td><td>+0.000</td><td>8 CtR</td><td>4</td><td>1.0</td><td>2.8</td><td>90</td><td>45</td><td>20</td><td>0.47</td></tr><tr><td>P7</td><td>+0.000</td><td>8 CtR</td><td>4</td><td>1.0</td><td>1.1</td><td>90</td><td>45</td><td>2</td><td>0.19</td></tr><tr><td>P8</td><td>+0.000</td><td>8 CtR</td><td>4</td><td>1.0</td><td>1.7</td><td>90</td><td>45</td><td>2</td><td>0.28</td></tr><tr><td>P9</td><td>+0.000</td><td>8 CtR</td><td>4</td><td>1.0</td><td>1.7</td><td>90</td><td>45</td><td>3</td><td>0.28</td></tr><tr><td>P10</td><td>+0.000</td><td>8 CtR</td><td>4</td><td>1.0</td><td>1.1</td><td>90</td><td>45</td><td>10/3</td><td>0.19</td></tr></tbody></table>

size of the jets, which is identical in all cases, is $\delta x = \delta z = 0.08$. Therefore the size of the jet is approximately $1.33 \delta_o$ where $\delta_o \simeq 0.06$ is the boundary layer thickness at the top of the bump. The actuation is distributed in the full spanwise domain size.

### 3. Results

The aim of the present study is not to find the optimal parameters as too many configurations would be required for a single bump geometry. The targeted objective is rather to analyze the influence of different geometrical (the streamwise position of the actuators, the jet arrangement and orientation) and operating parameters (amplitude, pulsating frequency, duty cycle of actuation) in order to get a better physical understanding of the control action.

#### 3.1. Influence of geometrical parameters

##### 3.1.1. Influence of the streamwise position

Two positions of the actuation were investigated as they are expected to be sensitive. The first one was chosen close to the maximum of the friction velocity ($x \simeq -0.66$) which corresponds to the position of the minimum boundary layer thickness. The second position is located at the summit of the bump, close to the minimum of the pressure coefficient. The comparison of the friction velocity is shown in Fig. 7. For the first streamwise position ($x \simeq -0.66$) the skin friction coefficient is slightly negative around $x = 1$ which indicates that a very small separation remains. On the other hand, for the second position ($x = 0$), the separation is totally suppressed with a significantly higher level of the skin friction. Therefore, the structures generated by jets/free stream interaction are not able to take advantage of the growing boundary layer thickness as expected. This can be explained in the light of the analysis performed in section 3.3. Indeed, for the first position, the shear stress coefficient, which was found to be the dominant quantity to balance
the adverse pressure gradient and then the necessary quantity to maintain the flow attached, presents a peak which does not persist at the position of separation (not shown here). This indicates that the lifetime of the produced vortices from the first position is not long enough to reach the location of the flow separation, it seems that they have already grown and have been dissipated. Moreover, the shear stress peak has already moved away from the wall. On the another hand, the actuators located at the summit of the bump affect significantly the Reynolds stresses close to the wall down to $x = 1.0$. Therefore, between the two streamwise positions, the one located at the summit of the bump is the most efficient and is chosen for the investigation of the other parameters.

### 3.1.2. Influence of the jet arrangement

Jets in Counter-rotating (CtR) (see Fig. 6) or co-rotating (CoR) arrangements have been found to produce a different organization of the streamwise vortices [4, 10]. From a CoR arrangement, streamwise vortices merge rapidly to form a large vortical region which moves transversely by self-induction. In the CtR arrangement, vortices stay coherent and individualized far downstream. Also, for the same bump configuration, Godard et al (2006c) [10] found that the CtR arrangement is more efficient than the CoR.

In the present study the two configurations are compared with a constant mass flux and the same number of jets (C1 and C2 in Table 1). For the initial spanwise spacing of $L = 5\delta_z$ in both cases, no differences in the friction coefficient was found between the two configurations (not shown here) although the CoR spacing corresponds to the optimal one found by [10]. These results indicates that coherent vortices created by each jet from both arrangements do not interact significantly. Indeed, the advantage of the CtR is usually linked to the ability of the neighboring vortices to enhance the circulation while in the CoR arrangement they merged rapidly with a decrease of circulation. In the present case the CoR jets do no interact before $x = 1$. The spacing of the CoR arrangement was decreased down to $L = 2.5\delta_z$ which gives exactly the same evolution of skin friction (see Fig. 8). This could be attributed to the Reynolds number differences ($Re_θ = 27610$ in [10]), the different streamwise positions of the jets relatively to the minimum of skin friction and also to the low amplitude of the jet. This last parameter is supposed to be the dominant one as the jet amplitude of the present study is at least three time lower than the one of [10] which induces much smaller streamwise vortices. Indeed, in section 3.2.1 where the jet amplitude is investigated, higher jet amplitudes are able to modify the size of the produced streamwise vortices and thus the interaction between them (see Fig. 17). In the following only the counter-rotating arrangement is explored.

### 3.1.3. Influence of the jet orientation

From [25], the origin of the counter-rotating vortex pair from the jet/free stream interaction when a single jet is perpendicular to the flow is explained by the high shear from the jet edges which induces a roll-up of the boundary layer flow. Depending on the jet orientation $α$ and $β$ (see Fig. 6), one vortex of the counter-rotating pair may dominates the other which leads to a single coherent vortex of one sign accompanied by a much smaller and weaker region of circulation of the opposite sign near the wall [28]. When this configuration occurs, the produced vortex is stronger and the control efficiency is improved. Therefore, the orientation of the jet is a parameter able to modify the induced vortices, and thus the intensity of these vortices which have the ability to sustain for a longer convective time. In the present study, three pitch angles and three skew angles were tested for a constant $V$. The results are shown in figure 9 and figure
10. Keeping the skew angle to 90 degrees (perpendicular to the flow), the lowest pitch angle $\beta = 45^{\circ}$ leads to a skin friction which is significantly larger behind the actuators down to $x = 1.2$ and almost identical further downstream. The two other pitch angles ($\beta = 70^{\circ}$ and $\beta = 90^{\circ}$) give very similar results. This analysis confirms the improvement of the control efficiency when the jets have a pitch angle. Then, keeping the most efficient pitch angle ($\beta = 45^{\circ}$), the most efficient skew angle is found to be $\alpha = 135^{\circ}$ which corresponds to a jet blowing upstream while the case with $\alpha = 45^{\circ}$ (blowing downstream) is the less efficient. Difference in the fluidic obstacles generated by the two configurations is clearly highlighted in Fig. 11. In comparison with jets with a pitch angle $\beta = 45^{\circ}$, actuators which are flowing upstream ($\alpha = 135^{\circ}$) creates a stronger fluidic obstacle at $x = 0$ (see Fig. 12), which penetrates further in the boundary layer for the same mass flow rate. Consequently, the shear generated by the jet is higher for $\alpha = 135^{\circ}$ and induces stronger streamwise vortices further downstream (Fig. 13). Therefore, blowing upstream can be viewed as an increase of the velocity of the jet with respect to the local mean velocity. As a matter of fact the effect on the friction coefficient and on the pressure coefficient (not shown here) is similar to what has been observed by increasing the velocity (see section 3.2.1, Figs. 14 and 15). In the following the orientation of the jets is arbitrarily fixed to $\beta = 45^{\circ}$ and $\alpha = 90^{\circ}$.

## 3.2. Influence of the operating parameters

Starting from the analysis of geometrical parameters for continuous jets, the influence of the operating parameters (the jet amplitude, the pulsating frequency and the duty cycle) is now investigated.

### 3.2.1. Influence of the jet amplitude

This parameter is usually defined by the velocity ratio $VR = V/U_{\infty}$ with V the jet amplitude and $U_{\infty}$ the free stream velocity. In our case, the indication of the jet amplitude V was preferred to the velocity ratio as the free-stream velocity is not clearly defined. However, the maximum of the streamwise velocity profile and the streamwise velocity at the estimated boundary layer thickness $\delta_o$ are respectively $U_{max} = 1.45$ and $U_{\delta_o} = 1.43$ at the top of the bump. Consequently, the velocity ratio can be estimated by dividing the parameter V in Table 1 by one of these two definitions of $U_{\infty}$. The range of velocity ratios investigated is approximately from 1/3 for $V = 0.5$ to 1.6 for $V = 2.5$. These values are in the low range of VR when compared to what has already been tested (see for instance [5, 10, 14, 29]). This range of velocity ratio corresponds to an actuation mass flux of approximately 3% to 9% of the mass flux integrated over the boundary layer thickness and 0.1% to 0.3% of the total channel mass flux at the inlet. The comparison of the friction and pressure coefficients are given in Fig. 14 and 15 respectively. All the tested configurations are able to remove the negative range of the skin friction coefficient and the zero pressure gradient which are associated to the flow separation. The general evolution of the skin friction coefficient is similar for all the tested values of V except for $V = 2.5$. Behind the jets, at around $x = 0.05$, the skin friction and pressure coefficients are both locally strongly modified and increase almost linearly with V. This is directly linked to the jets which are bent and create a local fluidic obstacle over which a region of favorable pressure gradient ($-0.1 < x < 0.1$) followed by a lower adverse pressure gradient is naturally created. At this location, increasing the jet amplitude V increases the shear induced by the jets (see Fig.
16) which leads, as expected, to stronger vortices and then to a higher friction at the wall (not shown here). However the skin friction is not homogeneous (see Fig. 17). Increasing the jet amplitude from $V = 1$ to $V = 1.5$ only slightly extends the spanwise gain of $C_f$ while a large extend of negative friction (i.e. flow separation) remains in between jets from $x = 0.2$ to $x = 1$. Actually, the minimum of $C_f$ at $x = 0.9$ on figure 14, is associated to the end of the region of negative skin friction between jets (see the two top Figs. 17). For $V = 2.5$, the skin friction distribution at the wall shows that stronger vortices are able to produce a more homogeneous control efficiency, however this affects the skin friction after $x = 1$ (see bottom Fig. 17). The loss of skin friction after this location is existing for all the velocity ratios as shown by the skin friction coefficient for $1.2 < x < 4$. Therefore, for a spanwise average gain in the skin friction, a better compromise in the jet distribution might exist. For the same mass flow rate, a configuration with small jets and low $V$, but close enough to each others to avoid the separation between jets, could be more efficient. However, this configuration was out of reach of our computing capabilities. Moreover, the lifetime of the vortices would be reduced and the control would be efficient over a smaller streamwise extend.

### 3.2.2. Influence of frequency

Only few studies have reported a frequency dependence of the controlled flow using fluidic actuators (see e.g. [21]) while it is always reported when slot jets are used (see e.g. [6, 11, 26]). Some authors have reported that pulsed round jets contrary to continuous round jets have a starting vortex at the early opening stage of the valve (see [12, 14, 23] which could be at the origin of the differences in control efficiency between the two cases. However, Braud et al.[3] and Kostas et al.[14] indicate that this starting vortex is linked to the experimental arrangement of the valves used. It is then of interest to analyze the effect of the frequencies from the present LES studies.

The time evolution of the jet velocity is defined by Eq. (2). As the a-priory choice of an optimal frequency is difficult to argue, a large range of frequency was investigated starting from $F = 1$ to $F = 20$ which can be more or less related to the convective and turbulent time scales respectively. The control efficiency, observed with the friction coefficient distribution along the streamwise direction, is shown on Fig. 18 and 19 for a high frequencies range (from $F = 5$ to $F = 20$) and a low frequencies range (from $F = 1$ to $F = 5$) respectively, using fixed DC and mass flux. The continuous case with the same mass flux (with $V = 0.5$ instead of $V = 1.0$ for pulsed jets) is given as a reference. The first conclusion is that the benefit for the high frequency range is small as compared to continuous jets and even less efficient for $F = 10$ (see Fig. 18). The case with the highest frequency ($F = 20$) presents only slight differences with the continuous case. For $F = 5$ a significant increase of $C_f$ is observed in the vicinity of the jets (from $x = 0.2$ to $x = 0.9$) as compared to continuous jets. However, this benefit is significantly reduced from $x = 1$ to $x = 2$ and then almost identical. The benefit of pulsed jets at low frequencies is much more significant. From the four frequencies investigated, the optimal one in terms of global minimum of friction velocity seems to be $F = 2$. Compared to continuous jets, the friction coefficient is significantly increased from $x = 0.2$ to $x = 1.2$. The $C_f$ distributions represent a time averaged vision of the control efficiency for the pulsed jet cases. Phase average measurements are then performed at four instants of the actuation cycle, indicated with black circles on Fig. 5. The two first instants ($t_o, t_o + 0.1$) are taken when the maximal amplitude of the jet velocity is reached, at the beginning and at the end of the actuation stage respectively. The two remaining instants ($t_o + 0.2, t_o + 0.3$) are chosen in the cycle for a null exit velocity. The phase average of $C_f$ is presented in Fig. 20. For
$t = t_o$, when the jet is activated, the first peak occurs at the actuator location $x = 0$. Then a second peak, which corresponds to the structures from the previous actuating period $t = t_o - 0.5$, appears at $x = 0.25$. This second peak is broader and noisier. At $t = t_o + 0.1$ the peak is stronger and has not moved from the actuation location. This seems to indicate that the generated structures remain aggregated during the blowing action. For $t = t_o + 0.2$, two peaks are present around $x = 0.1$ and $x = 0.4$. They corresponds to the structures formed by two successive actuation cycles which have been convected in the streamwise direction. Hence the distance from the first peak to the second peak ($\Delta x \simeq 0.3$) and the time interval for one actuation cycle ($\Delta t \simeq 0.5$) give an evaluation of the convection velocity of the structures $\Delta x/\Delta t = 0.6$. The convective velocity corresponds to the mean streamwise velocity at $1/4$ of the boundary layer thickness at $x \simeq 0.1$. The second peak has a lower energy and is broader in space than the first peak. This can be explained by the fact that the structures have grown and mixed with the turbulent boundary layer [4]. For $t_o + 0.3$ the first peak at $x = 0.9$ corresponds to the peak found for $t = t_o + 0.2$ which has moved downstream with the same convecting velocity $\Delta x/\Delta t \simeq 0.06/0.1 = 0.6$. Again a second broader peak is present (near $x = 0.45$) but is less intense. From this analysis, it seems that there exist an optimum blowing duration under which the structure cannot be formed. The evaluation of the optimal time duration will be performed in the following through the investigation of the influence of the duty cycle. The improvement of the control for $F = 2$ is then due to an average of the positive and negative effects from the traveling coherent structures produced by pulsed jets while the control for $F = 20$ or $F = 0$ generate vortices which are less regular (see Fig. 18).

### 3.2.3. Influence of the duty cycle

When the jet amplitude is fixed, the effect of DC can be associated to a mass flux effect. In the previous investigation the DC was fixed. Hence, departing from the optimal pulsating frequency $F = 2$ and fixed values of the exit velocity ($V = 1$), the DC is investigated with the objective to reduce the mass flux without affecting the control efficiency. The duty cycle can be defined as follows $DC = T_{on} * F = T_{on}/(T_{on} + T_{off})$ where $T_{on}$ is the period of blowing (i.e. when $G(t) = 1$) and $T_{off}$ the period without actuation (i.e. when $G(t) = 0$).

Firstly, the DC can be varied using a fixed value of the frequency $F = 2$ and modifying both the period of blowing $T_{on}$ and the period without actuation $T_{off}$. The friction coefficients are compared in Fig. 21 for DC = 0.19, DC = 0.28 and DC = 0.47. The results for the three DCs investigated show that the minimum of $C_f$ located near $x = 0.9$ decreases almost linearly with the duty cycle. Therefore, for this frequency, it is not possible to reduce the mass flux (or DC) without decreasing the control efficiency. Then, keeping a constant period of actuation $T_{on}$, the influence of the rest period $T_{off}$ is investigated. In this case both the frequency and the duty cycle are modified. The resulting friction velocity is shown in Fig. 23 and the corresponding time function $G(t)$ is shown in Fig. 22. This figure indicates that the rest time is increasing from case P4 to case P7, decreasing DC. For all the tested cases, the control is less efficient than the continuous jet case. However, from P4 to P9, the control efficiency is maintained for $0 < x < 0.9$ and even slightly increased for $0.9 < x < 1.8$ while the DC is decreased by 40% (Fig. 22). Increasing further the rest time does not allow to maintain the control efficiency (see the P7 case). Therefore, for this range of frequency, there exist a minimum of the rest period $T_{off}$ for the structures to be convected. At last, departing from a constant $T_{off}$, $T_{on}$ is decreased to investigate the effect of the blowing duration. Again, in this case, both the frequency and the duty cycle are modified. The results are shown in Fig. 24 and the corresponding time functions $G(t)$ are shown in Fig. 25. Very similar
results are found for case P2 and P9 while the duty cycle is decreased by around 40% (Fig. 24). However, decreasing further $T_{on}$ decreases the control efficiency (from P9 to P10). Therefore, there exist a minimum of the blowing duration $T_{on}$ for the structures to be formed while longer actuation periods corresponding to a larger mass flux are useless in term of efficiency. From these last two investigations on the DC effect, we can conclude that the benefit of choosing a lower DC is neither strongly linked to the period of actuation nor to the period of rest, both have an important role in the formation of the structures and need to be optimized to save mass flux. Note that, in both cases, the investigation end up with the same efficient parameters $F = 3$ and $DC = 0.28$ which leads to a mass flux gain of 40% as compared to a DC of 0.47 used for the frequency analysis. The optimal frequency, which is refined in the present investigation to $F = 3$, is actually linked to the optimal duration for the structure to be formed and convected downstream before the following one. Of course the optimal DC parameter, which was determined with a constant $V = 1$, depends on the jet amplitude, on all the other geometrical parameters and on the baseline flow configuration such as the turbulent boundary layer characteristics or the pressure gradient.

### 3.3. Statistical analysis

#### 3.3.1. Reynolds stress components

The main components of the Reynolds stress tensor are presented in Fig. 26 in the local coordinate system $(s, n)$ ($s$ and $n$ being the local streamwise and normal components respectively). The results are shown at four streamwise locations, for two cases with continuous jets (C3 and C4), and compared to the uncontrolled case. For the first streamwise position at $x = -0.05$, located just upstream of the actuators, the streamwise and spanwise components of the Reynolds stress are the most affected. Also, increasing the jet amplitude mainly increases the level of the Reynolds stress components. At last, for this position, the peak of the spanwise component is located closer to the wall. The increase of the streamwise component is probably due to the non-zero spanwise velocity component of the jet. The shear stress is not affected yet. At $x = 0.25$, corresponding to approximately $4\delta_o$ downstream of the actuators, all the Reynolds stress components are largely affected. The dominant term is the streamwise stress $\langle u'_s u'_s \rangle$. The peak is larger than the peak from the uncontrolled case and located slightly above ($nu_{\tau o}/\nu \approx 30$, corresponding to $n \approx \delta_o/2$). Also, at this location, increasing the jet amplitude slightly moves the peak away from the wall, but does not increase its level. On the contrary, the peak from the other components, which appears closer to the wall than in the uncontrolled case, increases with the jet amplitude. At $x = 0.50$, a second peak of $\langle u'_s u'_s \rangle$ is growing closer to the wall for the two controlled cases when the peak from the uncontrolled case at $nu_{\tau o}/\nu \approx 30$ does not move significantly. The peak of the shear stress located near $nu_{\tau o}/\nu \approx 20$ has grown significantly and it will be shown in section 3.3.2 that the larger negative contribution to the Reynolds stress gradient comes from the normal derivative of the shear stress. At the last location ($x = 1.25$), where the flow separates, all the Reynolds stresses are lower than for the uncontrolled case. This seems to indicate that the separation is delayed as highlighted on the distribution of friction coefficient which is clearly dependent on $V$.
### 3.3.2. Momentum balance

The benefit of the control is linked to its ability to modify the near wall shape of the boundary layer profile and especially its normal derivative at the wall $\partial\langle u_s \rangle/\partial n$. In all the present cases with continuous jets, the flow stay attached at any streamwise position. Therefore, it is interesting to determine which terms of the mean streamwise velocity equation are affected under this control action. This equation reads in the local coordinate system $(s, n)$:

$$ \underbrace{\langle u_s \rangle \frac{\partial \langle u_s \rangle}{\partial s}}_{I} + \underbrace{\langle u_n \rangle \frac{\partial \langle u_n \rangle}{\partial n}}_{II} = - \underbrace{\frac{\partial \langle p \rangle}{\partial s}}_{III} - \underbrace{\frac{\partial \langle u'_s u'_s \rangle}{\partial s}}_{IV} - \underbrace{\frac{\partial \langle u'_s u'_n \rangle}{\partial n}}_{V} + \underbrace{\nu \frac{\partial^2 \langle u_s \rangle}{\partial^2 s}}_{VI} + \underbrace{\nu \frac{\partial^2 \langle u_n \rangle}{\partial^2 n}}_{VII} \quad (3) $$

The streamwise evolution of terms I to VII of Eq. (3) is shown in Fig. 27 for the uncontrolled case and the continuous case C3 ($V = 1, \alpha = 90^\circ$ and $\beta = 45^\circ$). The terms are normalized with the characteristics of the boundary layer at the summit of the bump ($u_{\tau o}, \delta_o, U_{\delta o}$). At the wall, up to $0.01 \delta_o$ (corresponding to approximately one wall unit), equation (3) can be reduced to an equilibrium between the pressure gradient (III) and the viscous stress (VI + VII) similarly to the uncontrolled case, as they are dominant for the four streamwise positions. At a wall distance $0.01 \delta_o < n < 0.2 \delta_o$, the balance evolves with the streamwise position as follows. In the vicinity of the actuation ($x = 0.1$), the dominant term is the Reynolds stress gradient (IV and V) which is balanced mostly by the viscous stresses (VI and VII) and partly by the pressure gradient (III). The situation is different at the three other downstream positions, $x = 0.2, 0.5$ and $0.9$, where the Reynolds stress gradient is compensated almost completely by a larger pressure gradient. For $0.2 \delta_o < n < \delta_o$, viscous terms are negligible and the balance of the convection terms is done by the pressure gradient and the Reynolds stress gradient. From the above analysis, the negative contribution of the Reynolds stress gradient is found to be an important ingredient of an efficient control. Moreover, from the analysis of each independent term of the Reynolds stress gradient, the shear stress gradient is the most dominant (Fig. 28). Hence, actuators which are able to induce shear stresses are the most effective ones to delay separation. However, producing high level of shear stress is not straightforward. Indeed the spatial and time evolution of the vortices generated by continuous jets are, in the present configuration, more complex than the one described in the literature (see for instance [25]). The mean coherent structures created by the counter-rotating jets are highlighted in Fig. 29. It shows the negative and positive iso-values of the streamwise component of the vorticity computed with the time average velocity for two pairs of counter-rotating jets from case C3. This figure confirms the complexity of the structures generated by each jet. The main vortices visible down to $x = 1$ are the results of several vortices of each sign induced by the shear generated by the jets. The two main elongated vortices from a pair of jets slightly moves away from each other, but they are not intense enough to stay alive after $x = 1$ which corresponds to approximately $15\delta_o$. The weakness of the induced vortices is probably due to the low value of the outlet velocity of the jets ($V = 1$). The statistics of the time averaged vorticity are important as it shows the spatial distribution of negative and positive vorticity. However, the coherent structures generated by the jets are unstable and an instantaneous analysis shows that the vortices are in fact much smaller and highly intermittent even for actuation with continuous jets.
## 4. Conclusions

In the present study, a LES investigation of separation control in a converging-diverging channel by means of active pulsed jets was performed. The lower wall of the channel is similar to the bump shape of [10] but at a lower Reynolds number ($Re_τ = 395$). The influence of the geometrical and operating parameters were explored for a low authority control $V < 2.5$ (i.e. $VR < 2$) with a systematic investigation of the skin friction coefficient gain and the pressure coefficient distribution. Two streamwise positions of the actuation were evaluated. The produced vortices were not able to take advantage of the growing boundary layer thickness at the maximum of the skin friction coefficient as the second position at the summit of the bump was more efficient. In agreement with the literature, jets with a pitch angle of $β = 45°$ (i.e. angle to the wall) are more efficient. Moreover, this study demonstrates that jets oriented in the opposite direction to the main flow (i.e. $α = 135°$) are found more efficient than co-flowing jets. Co-rotating arrangements are found to give the same efficiency as counter-rotating jets. This result is attributed, in the present case, to the low jet velocity ($V = 0.5$). In the continuous mode, increasing the jet velocity increases almost linearly the control efficiency similarly to [15]. However, the skin friction distribution at the wall indicates that this improvement is rather due to a local increase of the control efficiency behind the jets while regions of flow separation remains between the produced streamwise vortices. For a fixed mass flow rate and a low amplitude $V = 1$ (i.e. $VR < 1$), an optimal pulsating frequency is found. Only the low range of pulsating frequencies are found to lead to an improvement of the control efficiency while some high frequencies were found to have negative effects. The gain for this optimal frequency is associated to the average value of the skin friction from structures traveling in the streamwise direction at a convective velocity which corresponds to the mean streamwise velocity at $1/4$ of the boundary layer thickness near the jet exit. For the optimal low frequency range, an optimal duty cycle is found which enables to reduce the mass flow rate consumption by almost 72% with the same control efficiency than continuous jets. At last, a statistical analysis, for one continuous jet case, highlights that an efficient control is accompanied with an important negative shear stress gradient. However, producing mainly this quantity is not straightforward.

The production of vortices from the active vortex generators embedded in the turbulent boundary layer are not yet fully understood and the present LES simulations highlights the complex instantaneous organization and evolution of the resulting jet/baseline flow interaction, even for continuous jets. Future work will aim at understanding this organization.

## Acknowledgments

This work was supported by CISIT (International Campus on Safety and Intermodality in Transport) and WALLTURB (A European synergy for the assessment of wall turbulence) which is funded by the EC under the 6th framework program (CONTRACT: AST4-CT-2005-516008). The DNS was performed through two successive DEISA Extreme Computing Initiatives (DEISA is a Distributed European Infrastructure for Supercomputing Applications). The other calculations were performed with the computing facilities of Lille University (CRI) and GENCI (French National Computing Facilities).
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List of Figures

* Figure 1: 2D grid used for LES of uncontrolled and controlled flow (1/4 of grid points are plotted in each direction).

* Figure 2: Comparison of the friction coefficient $C_f$ (left) and pressure coefficient (right) between the DNS (black curves) and the LES (red curves) of uncontrolled flow. The position of the bump is indicated in yellow.

* Figure 3: Comparison of the mean streamwise velocity between the DNS (black) and the LES (red) of uncontrolled converging-diverging flow.

* Figure 4: Amplitude of the velocity profile of one actuator $V^p(x, z, t = t_o)$ as a function of space. The streamwise and the spanwise sizes of the actuator $\delta x = \delta z = 0.08$ are indicated by red lines.

* Figure 5: Example of the two types of function $G(t)$ (see eq. 2) for two frequencies: $F=2$ (black and blue) and $F=10$ (red).

* Figure 6: Geometrical parameters of a pair of counter-rotating jets. $\alpha$ is the skew angle, $\beta$ is the pitch angle, $L$ is the distance between the two jets of the same pair and $\lambda$ is the distance between two pairs.

* Figure 7: Friction coefficients $C_f$ defined by Eq. (1) for simulation C1 and C3 of table 1.

* Figure 8: Friction coefficients $C_f$ defined by Eq. (1) for simulation C2 and C3.

* Figure 9: Friction coefficients $C_f$ defined by Eq. (1) for simulation C4, C7 and C8.

* Figure 10: Friction coefficients $C_f$ defined by Eq. (1) for simulation C3, C9 and C10.

* Figure 11: Mean streamwise velocity in a plane parallel to the flow (aligned with a jet) for the two cases C9 and C10.

* Figure 12: Mean velocity and mean streamwise vorticity (in color) in a plane perpendicular to the flow at $x = 0$ for the two cases C9 and C10.

* Figure 13: Mean velocity and mean streamwise vorticity (in color) in a plane perpendicular to the flow at $x = 0.10$ (case C10) and $x = 0.15$ (case C9).

* Figure 14: Friction coefficients $C_f$ defined by Eq. (1) for simulation C3, C4, C5 and C6.

* Figure 15: Pressure coefficients $C_p$ defined by Eq. (1) for simulation C3, C4, C5 and C6.

* Figure 16: Mean velocity and mean streamwise vorticity (in color) in a plane perpendicular to the flow at $x = 0$ for the three cases C4, C5 and C6.

* Figure 17: Time averaged distribution of the friction coefficient $C_f$ at the lower wall behind the actuators for the three cases C3, C5 and C6.

* Figure 18: Friction coefficients $C_f$ defined by Eq. (1) for the cases with pulsating jets P4, P5 and P6 ($V = 1.0$) and the continuous case C4 ($V = 0.5$) with the same mass flux.

* Figure 19: Friction coefficients $C_f$ defined by Eq. (1) for the cases with pulsating jets P1, P2 and P4 ($V = 1.0$) and the continuous case C4 ($V = 0.5$) with the same mass flux.

* Figure 20: Phase averaged of the friction coefficients $C_f$ defined by Eq. (1) for simulation P2. The phases corresponding to the statistics are indicated with black dots in Fig. 5.

* Figure 21: Friction coefficients $C_f$ defined by Eq. (1) for simulation P7, P8 and P2.

* Figure 22: Time evolution $G(t)$ of pulsed jet outlet velocity defined by Eq. (2) for three duty cycles: DC=0.47 (P4, purple), DC=0.28 (P9, blue) and DC=0.19 (P7, green).
* Figure 23: Friction coefficients $C_f$ defined by Eq. (1) for simulation P4, P7 and P9.

* Figure 24: Friction coefficients $C_f$ defined by Eq. (1) for simulation P2, P9 and P10.

* Figure 25: Time evolution $G(t)$ of pulsed jet outlet velocity defined by (Eq. 2) for three duty cycles: DC=0.47 (P2, purple), DC=0.28 (P9, blue) and DC=0.19 (P10, green).

* Figure 26: Comparison of the Reynolds Stresses at 4 streamwise locations between the uncontrolled case and the cases C3 and C4 with continuous jets ($V = 0.5$ and $V = 1$).

* Figure 27: Balance of the mean streamwise component of the momentum equation for the uncontrolled case and the case C3 controlled with continuous jet. The terms are normalized by the friction velocity at the position of the actuation (left and low axis) and with the external velocity $U_\delta$ at the same reference position (upper and right axis). The streamwise position of each statistics ($x = 0.1, 0.2, 0.5, 0.9$) are indicated at the lower right corner of each plot.

* Figure 28: Comparison of two components of the Reynolds Stress gradient (VI) and (VII) of Eq. (3) for the case C3 and the uncontrolled case.

* Figure 29: Iso-value of positive (blue) and negative (red) streamwise vorticity computed with the time averaged velocity (simulation C3). The spanwise domain is limited to the first two pairs of counter-rotating jets.
<figure><img src="image_2.png" /><figcaption>Figure 1. 2D grid used for LES of uncontrolled and controlled flow (1/4 of grid points are plotted in each direction).</figcaption></figure>
<figure><img src="image_3.png" /><figcaption>Figure 2. Comparison of the friction coefficient $C_f$ (left) and pressure coefficient (right) between the DNS (black curves) and the LES (red curves) of uncontrolled flow. The position of the bump is indicated in yellow.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Figure 3. Comparison of the mean streamwise velocity between the DNS (black) and the LES (red) of uncontrolled converging-diverging flow.</figcaption></figure>
<figure><img src="image_5.png" /><figcaption>Figure 4. Amplitude of the velocity profile of one actuator $V^p(x, z, t = t_o)$ as a function of space. The streamwise and the spanwise sizes of the actuator $\delta x = \delta z = 0.08$ are indicated by red lines.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Figure 5. Example of the two types of function G(t) (see eq. 2) for two frequencies: F=2 (black and blue) and F=10 (red).</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>Figure 6. Geometrical parameters of a pair of counter-rotating jets. α is the skew angle, β is the pitch angle, L is the distance between the two jets of the same pair and λ is the distance between two pairs.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>Figure 7. Friction coefficients $C_f$ defined by Eq. (1) for simulation C1 and C3 of table 1.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Figure 8. Friction coefficients $C_f$ defined by Eq. (1) for simulation C2 and C3.</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>Figure 9. Friction coefficients $C_f$ defined by Eq. (1) for simulation C4, C7 and C8.</figcaption></figure>
<figure><img src="image_11.png" /><figcaption>Figure 10. Friction coefficients $C_f$ defined by Eq. (1) for simulation C3, C9 and C10.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>Figure 11. Mean streamwise velocity in a plane parallel to the flow (aligned with a jet) for the two cases C9 and C10.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>Figure 12. Mean velocity and mean streamwise vorticity (in color) in a plane perpendicular to the flow at $x = 0$ for the two cases C9 and C10.</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>Figure 13. Mean velocity and mean streamwise vorticity (in color) in a plane perpendicular to the flow at $x = 0.10$ (case C10) and $x = 0.15$ (case C9).</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>Figure 14. Friction coefficients $C_f$ defined by Eq. (1) for simulation C3, C4, C5 and C6</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>Figure 15. Pressure coefficients $C_p$ defined by Eq. (1) for simulation C3, C4, C5 and C6.</figcaption></figure>
<figure><img src="image_17.png" /><figcaption>Figure 16. Mean velocity and mean streamwise vorticity (in color) in a plane perpendicular to the flow at $x = 0$ for the three cases C4, C5 and C6.</figcaption></figure>
<figure><img src="image_18.png" /><figcaption>Figure 17. Time averaged distribution of the friction coefficient $C_f$ at the lower wall behind the actuators for the three cases C3, C5 and C6</figcaption></figure>
<figure><img src="image_19.png" /><figcaption>Figure 18. Friction coefficients $C_f$ defined by Eq. (1) for the cases with pulsating jets P4, P5 and P6 ($V = 1.0$) and the continuous case C4 ($V = 0.5$) with the same mass flux.</figcaption></figure>
<figure><img src="image_20.png" /><figcaption>Figure 19. Friction coefficients $C_f$ defined by Eq. (1) for the cases with pulsating jets P1, P2 and P4 ($V = 1.0$) and the continuous case C4 ($V = 0.5$) with the same mass flux.</figcaption></figure>
<figure><img src="image_21.png" /><figcaption>Figure 20. Phase averaged of the friction coefficients $C_f$ defined by Eq. (1) for simulation P2. The phases corresponding to the statistics are indicated with black dots in Fig. 5.</figcaption></figure>
<figure><img src="image_22.png" /><figcaption>Figure 21. Friction coefficients $C_f$ defined by Eq. (1) for simulation P7, P8 and P2.</figcaption></figure>
<figure><img src="image_23.png" /><figcaption>Figure 22. Time evolution $G(t)$ of pulsed jet outlet velocity defined by Eq. (2) for three duty cycles: DC=0.47 (P4, purple), DC=0.28 (P9, blue) and DC=0.19 (P7, green).</figcaption></figure>
<figure><img src="image_24.png" /><figcaption>Figure 23. Friction coefficients $C_f$ defined by Eq. (1) for simulation P4, P7 and P9.</figcaption></figure>
<figure><img src="image_25.png" /><figcaption>Figure 24. Friction coefficients $C_f$ defined by Eq. (1) for simulation P2, P9 and P10.</figcaption></figure>
<figure><img src="image_26.png" /><figcaption>Figure 25. Time evolution $G(t)$ of pulsed jet outlet velocity defined by (Eq. 2) for three duty cycles: DC=0.47 (P2, purple), DC=0.28 (P9, blue) and DC=0.19 (P10, green).</figcaption></figure>
<figure><img src="image_27.png" /><figcaption>Figure 26. Comparison of the Reynolds Stresses at 4 streamwise locations between the uncontrolled case and the cases C3 and C4 with continuous jets ($V = 0.5$ and $V = 1$).</figcaption></figure>
<figure><img src="image_28.png" /><figcaption>Figure 27. Balance of the mean streamwise component of the momentum equation for the uncontrolled case and the case C3 controlled with continuous jet. The terms are normalized by the friction velocity at the position of the actuation (left and low axis) and with the external velocity $U_\delta$ at the same reference position (upper and right axis). The streamwise position of each statistics ($x = 0.1, 0.2, 0.5, 0.9$) are indicated at the lower right corner of each plot.</figcaption></figure>
<figure><img src="image_29.png" /><figcaption>Figure 28. Comparison of two components of the Reynolds Stress gradient (VI) and (VII) of Eq. (3) for the case C3 and the uncontrolled case.</figcaption></figure>
<figure><img src="image_30.png" /><figcaption>Figure 29. Iso-value of positive (blue) and negative (red) streamwise vorticity computed with the time averaged velocity (simulation C3). The spanwise domain is limited to the first two pairs of counter-rotating jets.</figcaption></figure>