Vincent Samy

Karim Bouyarmane

Abderrahmane Kheddar

Active shock absorber control in humanoid robot falls with nonlinear optimization on a reduced dynamics model

We propose a reduced-model-based analytic approach to active shock absorption by a falling humanoid robot at impact. We model the segment between the contact point at impact and the impacting limb extremity attached to the torso or waist of the humanoid robot as a one degree-offreedom linear mass-spring-damper system. By mapping the joint angle limits and torque limits of the joints in the impacting limb to a corresponding position limit and force limit in the reduced model, we formulate a nonlinear optimization problem to find an admissible pair of parameters of the reduced model (stiffness, damping) that prevents the model from violating the constraints before reaching a steady state rest. The nonlinear constraints are analytically derived using symbolic computation tools and then numerically solved with off-the-shelf nonlinear optimization solver. The reduced model trajectories are then mapped back on the full body of the humanoid robot and illustrated on the HRP-4 robot in simulation.

I. INTRODUCTION

Reduced-dynamics models, such as the center-of-mass (CoM) and multi-dimensional mass-spring-dampers, proved to be extremely efficient in capturing the dominant behavior of complex dynamical systems, among which humanoid robots. For example, the CoM and related centroidal dynamics are used to design various strategies for dynamic walking, see [START_REF] Orin | Centroidal dynamics of a humanoid robot[END_REF], [START_REF] Wieber | Modeling and control of legged robots[END_REF]. The linear inverted pendulum (LIP) model can be cited as one of the most popular in the humanoid robotics research community since many years [START_REF] Kajita | Biped walking pattern generation by using preview control of zero-moment point[END_REF]. The CoM model enhanced by a spring along the massless CoM/Center of Pressure point system (spring-loaded inverted pendulum, or SLIP) is used to generate behaviors beyond walking, e.g. jumping and running [START_REF] Geyer | Compliant leg behaviour explains basic dynamics of walking and running[END_REF], [START_REF] Mordatch | Robust physics-based locomotion using low-dimensional planning[END_REF], [START_REF] Wensing | 3D-SLIP steering for high-speed humanoid turns[END_REF]. Reduced CoM models were extended to generate multi-contact motions, e.g. [START_REF] Mordatch | Discovery of complex behaviors through contact-invariant optimization[END_REF], [START_REF] Audren | Model preview control in multi-contact motionapplication to a humanoid robot[END_REF]. Mass-spring-damper models have also been used as nominal desired impedance or admittance at the contact space [START_REF] Hogan | Impedance control: An approach to manipulation: Part i-theory[END_REF]. There are of course many other variants of reduced-dynamics models and their usages in robotics.

In our previous work, we have addressed humanoid falling using locally linearized control to comply with post-impact dynamics using task-space quadratic programming (QP) control formalism [START_REF] Vaillant | Multi-character physical and behavioral interactions controller[END_REF], [START_REF] Bouyarmane | On weight-prioritized multi-task control of humanoid robots[END_REF]. The idea [START_REF] Samy | Falls control using posture reshaping and active compliance[END_REF] is to reshape the humanoid posture so as to meet the impacts at contact positions that maximize compliance by active change of the actuators Proportional-Derivative (PD) gains. Recently in [START_REF] Samy | QP-based adaptive-gains control to lower damage in humanoid falls[END_REF], we showed that the actuator gain parameters can also be integrated as decision variables in the QP controller, 1 CNRS-University of Montpellier LIRMM, Interactive Digital Human group, France.

2 Université de Lorraine, Inria, CNRS, Loria UMR 7503, Larsen team, France.

3 CNRS-AIST Joint Robotics Laboratory, UMI3218/RL, Japan.

and can thus be computed together with the state acceleration and contact forces. However, a problem that still needed to be solved was the computation of motor PD gains using a prediction of the system over a horizon of time (preview control) to ensure that the structural limits of the robot (torque limits, joint limits) are not reached during the whole post-impact phase, until the system comes to a rest.

In this paper, we propose to study a mass-spring-damper reduced-dynamics model applied to the situation of falling for a humanoid robot, to be used in a two-stage approach followed by whole-body QP control, similarly to what is well established in dynamic walking. The idea of the reduceddynamics model is to emulate a shock absorber on the impacting limbs of the robot (see Fig. 1).

(a) Landing on arms.

(b) Landing on legs.

Fig. 1: Shock-absorber reduced-dynamics-model representation in two different landing configurations.

To this end, we present a detailed analysis of the parameter tuning of a one degree-of-freedom (1 dof) mass-springdamper system to serve as a model-preview controller for post-impact trajectory generation of humanoid whole-body motion. We analytically derive the constraints on the stiffness and damping parameters of the reduced model and formulate a nonlinear optimization problem to solve for them. Once the parameters of the model are obtained, the corresponding trajectory is tracked by the robot in the post-impact phase using a whole-body controller.

The rest of the paper is organized as follows. Section II presents the reduced model and the analytical study that leads to the nonlinear optimization problem on its stiffness and damping parameters. Section III shows results obtained in simulation on full humanoid robot falling in different Fig. 2: Mass-spring-damper system. configurations. Finally, Section IV concludes the paper with a discussion on limitations, possible improvement, and future work suggestions.

II. ANALYTICAL RESOLUTION

Let us consider the system in Fig. 2. The analytical solution for the motion of this system comes from the standard resolution of the well-studied second order linear differential equation:

mz(t) -Ke(t) -B ė(t) + mg = 0 , ⇔ mz(t) -K(z ref -z(t)) -B( żref -ż(t)) + mg = 0 , ( 1
)
where m is the mass of the system, z(t) is the position of the mass at time t, g the gravity and K, B the gains that we want to compute to satisfy the constraints that will be mapped from the torque limit and joint angle limits of the full system. This equation can also be interpreted as a PD controller of a linear motor supporting a mass m under the gravity g. In such case, z ref is the desired position which is set to the position just before impact q -(0) and żref is always 0.

It is well known that when solving for z(t), three cases are distinguished. An under-damped solution occurs if B 2 < 4mK. The system keeps oscillating at its natural frequency. The critically damped solution occurs when B 2 = 4mK. It is the fastest way to reach the z ref position without oscillating. The last solution is the over-damped one when B 2 > 4mK. This also leads to reaching z ref without oscillating, but over a longer time. Fig. 3: Phases of landing and impact of the reduced model. The left figure represents the system just before impact at time t = 0 -, the velocity of the base is v -and the joint velocity is 0 (velocity of the mass relative to the base). The middle figure represents the system just after impact at t = 0 + . The right figure represents the system at the time of maximum amplitude t = t ma .

Given a solution of Eq. ( 1) that depends on the parameters K and B, our objective is to find admissible values of these parameters such that:

• joint limit constraint: a position limit constraint is satisfied on z(t) (e.g. preventing the mass from colliding with the ground after the impact, or equivalently reaching a maximum compression limit), • torque limit constraint: a force limit constraint is satisfied on K(z refz(t)) + B( żrefż(t)) (the impact force generated by the spring recoil should be kept below a safety threshold when possible), • the system converges to zero velocity.

The above constraints capture in the reduced model the full-body constraints on both joints angle and torques limits.

Based on these constraints, we choose to search for overdamped solutions. The reasons behind this choice are that: 1) we want to avoid unnecessary oscillations of the reduced model which, if mapped to the whole-body, will reproduce an undesired shaking behavior of the system after impact; and 2) We are not interested in critically converging to z ref , as long as we reach a rest steady state over time. Hence, we solve Eq. ( 1) assuming B 2 > 4mK. We will see later that this is not a limiting assumption in terms of finding a solution whenever the solution set is nonempty.

Note that we do not consider an explicit velocity constraint in the formulation, as during the entire post-impact phase, the system is only decelerating (slowing down) due to the dissipation of energy through the damping, so the constraint would only be effective if written on the initial velocity. However, the latter is an input to the problem over which we have no direct control. Therefore, the post-impact initial velocity might or might not cross the velocity limit if it was to be set. For these reasons, we do not explicitly formulate a velocity constraint.

Solving the differential equation Eq. ( 1) for z as a function .25]). The red axis is the stiffness K in N.m -1 , the green axis is the damping B in N.s.m -1 and the blue axis is the position z in m. of t, with K and B as parameters, leads to:

z(t, K, B) = e -1 2m (B- √ γ)t (Bg + 2K ż(0) + √ γg)m 2 √ γK - e -1 2m (B+ √ γ)t (Bg -2K ż(0) - √ γg)m 2 √ γK + Kz(0) -mg K , (2)
with γ = B 2 -4Km.

In the following, for the sake of readability and in order to reduce the size of the symbolic expressions, but without loss of generality, we set example numerical values to the initial conditions and to the mass of the system (all the developments can be kept with their symbolic expressions):

         m = 1 , g = 9.81 , z(0) = 0 , ż(0) = -5 . (3)
The derivations and reasoning to follow are still valid for any other numerical values corresponding to the specific robot and falling conditions at hand. Initializing the velocity to a value different from zero simulates a falling system state just after an impact. Suppose the system is impacting with a net velocity v -= -0.5 ms -1 (external floating-base velocity) and ż-(0) = 0 ms -1 (internal joint velocity) and that the impact is inelastic (no bounce), the remaining velocity in the joint post-impact is then ż+ (0) = -5 ms -1 , see Fig. 3.

With the numerical values (3), z(t, K, B) in (2) becomes: .25]). The red axis is the stiffness K in N.m -1 , the green axis is the damping B in N.s.m -1 and the blue axis is the velocity ż in m.s -1 .

z(t, K, B) = e -1 2 (B- √ γ)t (-10K + 9.81(B + √ γ) 2 √ γK - e -1 2 (B+ √ γ)t (10K + 9.81(B - √ γ) 2 √ γK - 9.81 K , (4)
with γ = B 2 -4K (we recall that γ > 0).

A 3D time-evolution representation of z as function of K and B can be represented, this is illustrated in Fig. 4.

Taking the time-derivative of z(t, K, B) gives us the joint velocity:

ż(t, K, B) = (-B + √ γ)e -1 2 (B- √ γ)t (-10K + 9.81(B + √ γ)) 4 √ γK - (-B - √ γ)e -1 2 (B+ √ γ)t (10K + 9.81(B - √ γ)) 4 √ γK . (5)
The 3D time-evolution of ż as a function of K and B can also be visualized (Fig. 5).

Lastly, from z and ż, we can compute the joint force (spring force) with respect to time and as a function of K and B (Eq. ( 6) and Fig. 6). .25]). The red axis is the stiffness K in N.m -1 , the green axis is the damping B in N.s.m -1 and the blue axis is the force f in N. The joint limit and force limit constraints can be respectively written from the expressions (4) and [START_REF] Wensing | 3D-SLIP steering for high-speed humanoid turns[END_REF] as

f (t, K, B) = K(z ref -z(t, K, B)) -B ż(t, K, B) = - (2.5B 2 -5K -4.905B + (4.905 -2.5B) √ γ)e -1 2 (B- √ γ)t √ γ - (-2.5B 2 + 5K + 4.905B + (4.905 -2.5B) √ γ)e -1 2 (B+ √ γ)t √ γ + 9.81 . (6)
∀t > 0 z ≤ z(t, K, B) ≤ z , ( 7
) ∀t > 0 f ≤ f (t, K, B) ≤ f , (8)
where z, z, f , f are the lower and upper bounds of the reduced system position and force mapped from the joint limit constraints and torque limits constraints of the considered robot impacting limb (see Section III).

The projection of the joint limit constraint and the torque limit constraint on the K-B plane through time is visualized in Fig. 7. The joint limit visualization is obtained by intersecting the 3D time-evolution plot of z(t, K, B) with the plane z = z, while the force limit constraint visualization is obtained by intersecting the 3D time-evolution plot of f (t, K, B) with the plane at f = f (the reason why only one side of these two constraints is relevant is explained below). By analyzing these time visualizations, we can substitute the time-continuous constraints ( 7) and ( 8) with fixed time constraints.

First, we can define the maximum amplitude time t ma as the time at which the velocity reaches 0. Solving for Eq. (5) = 0 with K and B as parameters using a symbolic computation software (e.g. Maple), we get:

t ma = ln - 4905B-2500K-9623.61 1250B 2 -1250B √ γ-2500K-4905B+4905 √ γ+9623.61
√ γ .

(9) This equation is stiffness-dependant and damping-dependant. The joint limit constraint can then be captured by:

z ≤ z(t = t ma , K, B) ≤ z. (10)
Secondly, we observe that the force limit can be reached in three main ways: i) high damping force when having high velocity; ii) high stiffness force when having high position error; iii) a mix of the two. In this study, we are interested in impacts involving a relatively short post-impact distance traveled between initial position and the limit, and high postimpact velocity coming from the fall. Placing ourselves in these conditions, the force maximum value is more likely to come from the damping of the velocity and not from the stiffness related to the distance. And since we will only decelerate the system, we consider that the maximum force is reached at t = 0 + (impact force). The torque limit constraint (here a force limit constraint) thus writes:

f ≤ f (t = 0 + , K, B) ≤ f . (11)
Of course, it is necessary to confirm this latter assumption once the t → f (t) trajectory is derived with the computed values of K and B. This is the case in Fig. 7b. As only the over-damped behavior is taken into account, a third constraint, the over-damped system constraint, is added to the above joint limit and torque limit constraints.

4K -B 2 < 0. ( 12
)
The problem we have formulated up to this point thus reduces to the following system of inequalities:

find K, B (13)
such that

     z ≤ z(t = t ma , K, B) ≤ z f ≤ f (t = 0 + , K, B) ≤ f 4K -B 2 < 0 (14)
This is a system of nonlinear inequalities, as the constraints ( 10)-( 12) are nonlinear inequalities in K and B, of which we have derived the analytical expressions throughout this section using symbolic computation tools. To solve our system of nonlinear inequalities, we formulate it as the system of constraints of a nonlinear constrained optimization problem in K and B, with an arbitrary convex cost function, e.g. minimizing

K 2 + B 2 : min K,B K 2 + B 2 (15) such that      z ≤ z(t = t ma , K, B) ≤ z f ≤ f (t = 0 + , K, B) ≤ f 4K -B 2 < 0
that finds the minimum norm solution for (K, B) satisfying the system of nonlinear inequality constraints.

We use the Matlab software to compute the solution of this optimization problem (The use of other non-linear optimization frameworks that are robotics-applications optimized, such as [START_REF] Moulard | Roboptim: an optimization framework for robotics[END_REF], is possible). The result is shown in Fig. 8. In this example, the impulse happens at the time t 0 = 0 such that at t - 0 the velocity is zero and at t + 0 the velocity is -5m.s -1 and z(0) = 0, z = -0.5, f = 60. 12), the green curve Eq. ( 11) and the orange curve Eq. [START_REF] Vaillant | Multi-character physical and behavioral interactions controller[END_REF]. The set of admissible solutions of the system of nonlinear inequalities is the white region. The red dot is the particular solution of the the nonlinear optimization problem. When the joint limit is not too close to the current position (Fig. 8a), we are able to find a set of solutions that satisfy all the constraints (Fig. 9). However, if the joint limit is set to 0.3 instead of 0.5 (much closer to the current position) as in Fig. 8b, no solution can be found, as the solution set is empty. The latter situation can also occur in falling conditions from which it is impossible for the robot to safely recover (e.g. dropping from a high altitude, impacting with very high velocity).

III. RESULTS

The accompanying video shows simulations that were realized based on the presented approach, first with lowdimensional systems (matching the studied model) and on a full humanoid robot HRP-4.

Fig. 10a is a trivial application of the method on a 1-dof Fig. 11: Effective mass computation.

system, in this case the reduced-model corresponds to the full model. Fig. 10b shows an application to a 2-dof system with two prismatic joints. The maximum force f is the sum of the maximum forces of the two linear actuators, and the minimum position z is obtained when the two joints are put at their position limits.

In both examples, we obtain a perfect realization of the desired behavior in simulation, after both systems reach the ground with an impact velocity of v -= -5ms -1 .

Figs. 12 and 13 show falling motions of the HRP-4 robot, controlled based on the reduced-model approach illustrated in Fig. 1. The reduced-model analysis is used to generate trajectories for the limb extremity attached to torso (arms) or to the waist (legs), and these trajectories are tracked in the post impact phase with a whole-body controller.

In these two cases, the limit position of the reduced model z is obtained from forward kinematics of the limb of interest when all the joints of that limb are put in their respective joint limits: z = FK limb (q limb limit ) ,

where FK limb denotes the forward kinematics function of the limb and q limb limit the configuration of the limb at its limits (here limb ∈ {arm, leg}). However, it is only an approximation, since FK limb is generally a nonlinear mapping. This example shows that obtaining the exact bounds on the reduced model range of motion (i.e. the compression limit of reduced system) from the robot joint limits and the kinematics is not trivial. We discuss other possible methods in the discussion section (Section IV). For the limit force of the reduced model, it can be obtained in first approximation from the pseudo inverse of the Jacobian transpose applied on the torque limits of the limb, or obtained exactly by using the force polytope analysis detailed in [START_REF] Samy | Adaptive Compliance in Post-Impact Humanoid Falls Using Preview Control of a Reduce Model[END_REF], that accounts for both torque limits in the limb and friction limits at the contact.

As for the effective mass of the reduced model in case of landing on the arms, we use the reasoning illustrated in Fig. 11. Modeling the part of the robot between the knee and the arm as an inverted pendulum rod, the effective mass can be defined as the point mass localized at the extremity of the rod that has the same moment around the center-ofpressure (CoP) of the rod as the total gravity force applied at the CoM, i.e.

M gl c cos(θ) = m eff gl r cos(θ), where M is the mass of the rod (mass of the robot between the knee and the arm), l c the distance of the CoM of the rod from its CoP at the knee, l r the total length of the rod (distance between the effective mass and the CoP), and θ the angle between the rod and and ground. Therefore, we have:

m eff = M l c l r . (18)
The impact velocities for the arm landing and leg landing falls were respectively measured at v -= -3ms -1 and v -= -1.47ms -1 . In Fig. 12, the particular posture that the robot adopts just before it starts falling is generated by applying the method presented in our previous work on nonsingular falls and optimal falling postures [START_REF] Samy | Falls control using posture reshaping and active compliance[END_REF], [START_REF] Samy | QP-based adaptive-gains control to lower damage in humanoid falls[END_REF].

IV. CONCLUSION AND DISCUSSION

We presented a study on a reduced model to capture the dynamics of active post-impact shock absorption when a humanoid robot falls and collides with the environment.

The analysis of the system allowed us to find the stiffness and damping parameters (K, B) of the reduced model that satisfy, when possible, the structural limits that are the joint limits and torque limits mapped onto a position limit and force limit on the reduced model. This reduced-model approach results in trajectories that are subsequently tracked by a whole-body controller, simulating the behavior needed by the humanoid to realize active impact compliance following the model.

As discussed in Section III, a general exact mapping from the limbs joint limits of a humanoid robot to the bounds on the range of motion of the corresponding reduced models can be difficult to obtain. A good approximation and practical method consisted in using forward kinematics and reachable space computation of the point corresponding to the attachment of the mass considering the contact constraint. This is the method we used in our examples. An alternative method would be to represent the kinematic tree structure of the humanoid as a deformable structure (through a rigidity matrix [START_REF] Bouyarmane | Fem-based static posture planning for a humanoid robot on deformable contact support[END_REF]) with repulsive potential fields at the joint limits, and use computer graphics animation techniques (such as a virtual force pulling the reduced-model's attachment point along the gravity until equilibrium is reached) to compute the desired bounds iteratively. This can be obtained with very fast simple computations. We plan to investigate this technique in future work.

Our analysis highlighted situations in which no solution exists (when the solution set on (K, B) is empty), due to extreme falling conditions for example (high impact velocity) or very limiting bounds. In these situations, the proposed approach is unsuccessful and the model appears to be no longer valid to propose a practical damage-reduction solution. Future work will investigate what alternative models can be used to deal with these extreme situations.

Moreover, other limiting situations in which a constant (K, B) solution does not fit in the constraint region throughout the time can also occur. It can be overcome in future work by seeking for time-varying (K(t), B(t)). We can optimize for those with trajectory optimization techniques using basisfunctions parameterization (e.g. splines).

The approach in this work was specifically designed for position-controlled robots, and applied in simulation by tracking the desired trajectories by a whole-body controller with high motor PD gains. We plan in the future to combine this work with our previous work on low-level motor PD gain adaptation [START_REF] Samy | QP-based adaptive-gains control to lower damage in humanoid falls[END_REF] to realize the tracking of the trajectories with lower adaptive motor PD gains.

Finally, we plan to extend the analysis proposed here to directly tune the low-level motor PD gains of the joints by using a multi-dof system modeling the compliance with a torsional spring-damper behavior at each joint.

Fig. 4 :

 4 Fig. 4: Evolution of the position z for a given set (K ∈ [0..50], B ∈ [0..25]). The red axis is the stiffness K in N.m -1 , the green axis is the damping B in N.s.m -1 and the blue axis is the position z in m.

  t = 200ms 5: Evolution of the velocity ż for a given set (K ∈ [0..50], B ∈ [0.

Fig. 6 :

 6 Fig. 6: Evolution of the force f for a given set (K ∈ [0..50], B ∈ [0..25]). The red axis is the stiffness K in N.m -1 , the green axis is the damping B in N.s.m -1 and the blue axis is the force f in N.

  (a) Joint limit projection.

  (b) Force limit projection.

Fig. 7 :

 7 Fig.7: Projection of the joint limit and the force limit on the K-B plane for different time with z = -0.3m and f = 60N.

- 1 )Fig. 8 :

 18 Fig. 8: Computation of the solution region depending on the problem constraints in the K-B domain. The blue curve represents Eq. (12), the green curve Eq. (11) and the orange curve Eq.[START_REF] Vaillant | Multi-character physical and behavioral interactions controller[END_REF]. The set of admissible solutions of the system of nonlinear inequalities is the white region. The red dot is the particular solution of the the nonlinear optimization problem.

  Evolution of z(t).

1 )

 1 (b) Evolution of ż(t).

  Evolution of f (t).

Fig. 9 :

 9 Fig.9: Evolution of the different joint parameters through time. The system reaches the joint limit without crossing it and after around 0.5s it is completely stopped.

(a) 1 -

 1 dof system simulation. (b) 2-dof system simulation.

Fig. 10 :

 10 Fig. 10: low-dimensional systems simulations.

Fig. 12 :

 12 Fig. 12: Front fall screenshots from the accompanying video.

Fig. 13 :

 13 Fig. 13: Leg fall screenshots from the accompanying video.

ACKNOWLEDGMENT

This work is supported in parts from grants of the CO-MANOID EU H2020 project www.comanoid.eu and of the Japan Society for Promotion of Sci-ence (JSPS) Grant-in-Aid for Scientific Research, Kakenhi B No. 16H02886


Kianoosh Yousefi
email: kianoosh_py@yahoo.com

Reza Saleh

Peyman Zahedi

Numerical study of blowing and suction slot geometry optimization on NACA 0012 airfoil †

Keywords: Blowing, Flow control, Jet width, Lift and drag coefficients, Slot geometry optimization, Suction ----------

The effects of jet width on blowing and suction flow control were evaluated for a NACA 0012 airfoil. RANS equations were employed in conjunction with a Menter's shear stress turbulent model. Tangential and perpendicular blowing at the trailing edge and perpendicular suction at the leading edge were applied on the airfoil upper surface. The jet widths were varied from 1.5% to 4% of the chord length, and the jet velocity was 0.3 and 0.5 of the free-stream velocity. Results of this study demonstrated that when the blowing jet width increases, the lift-to-drag ratio rises continuously in tangential blowing and decreases quasi-linearly in perpendicular blowing. The jet widths of 3.5% and 4% of the chord length are the most effective amounts for tangential blowing, and smaller jet widths are more effective for perpendicular blowing. The lift-to-drag ratio improves when the suction jet width increases and reaches its maximum value at 2.5% of the chord length.

Introduction

Although potential theory can be used to explain many aerodynamic phenomena, the boundary layer significantly alters theoretical predictions in some cases. An example of which is the flow passing an airfoil. At low angles of attack, the streamline pattern with such a shape is closed to the predictions of inviscid theory. However, a drag force unaccounted for by such a theory exists. This drag is large because of viscous shear forces and is called skin-friction drag. In the regions over the surface, where the boundary layer flow is laminar, the fluid mixing and viscous skin friction are low. However, such laminar flows are often unstable and develop into turbulent flows. Turbulent flows involve more rapid mixing than laminar flows, which produce higher skin-friction drag. The combined action of viscous forces and an adverse pressure gradient produces a reverse of the flow next to the surface, which causes separation of the adjacent flow from the surface. The presence of the boundary layer has produced many design problems in all areas of fluid mechanics. The most intensive investigations have been directed toward its effect on the lift and drag of wings. Techniques developed to manipulate the boundary layer, either to increase the lift or decrease the drag, are classified under the general heading of boundary layer control or flow control [START_REF] Hazen | Boundary layer control[END_REF]. In 1961, Flatt indicated that boundary layer control includes any mechanism or process which the boundary layer of a fluid flow is caused to behave differently than it normally would when the flow develops naturally along a smooth straight surface. Methods of flow control to achieve transition delay, separation postponement, lift enhancement, drag reduction, turbulence augmentation, and noise suppression have been considered [START_REF] Gad-El-Hak | Control flow: Passive, active and reactive flow management[END_REF].

Numerous studies have been conducted on flow control techniques. Prandtl [START_REF] Schlichting | Boundary layer theory[END_REF] was the first scientist who employed boundary layer suction on a cylindrical surface to delay boundary layer separation in 1904. The earliest known experimental works [START_REF] Richards | An airfoil designed to give laminar flow over the surface with boundary layer suction[END_REF][START_REF] Walker | Wind tunnel test on the 30 percent symmetrical griffth aerofoil with ejection of air[END_REF][START_REF] Braslow | A history of suction type laminar flow control with emphasis on flight research[END_REF] on boundary layer suction for wings, primarily in the wind tunnel, were conducted in the late 1930s and the 1940s. Suction and blowing approaches have emerged and have been studied in a variety of experiments [START_REF] Dannenberg | Section characteristics of a 10.5 percent thick airfoil with area suction as affected by chordwise distribution of permeability[END_REF][START_REF] Heugen | An experimental study of a symmetrical aerofoil with a rear suction slot and a retractable flap[END_REF][START_REF] Dannenberg | Section characteristics of an NACA0006 airfoil with area suction near the leading edge[END_REF][START_REF] Howe | An experimental evaluation of a low propulsive power discrete suction concept applied to an axisymmetric vehicle[END_REF][START_REF] Dirlik | Experimental evaluation of a 50-percent thick airfoil with blowing and suction boundary layer control[END_REF]. Such experiments have demonstrated that suction and blowing can modify the pressure distribution over an airfoil surface and have a substantial effect on lift and drag coefficients. Over the past few decades, various numerical works have been performed on the most common NACA airfoils to measure the lift and drag coefficients under different flow conditions [START_REF] Guowei | Control of unsteady vertical lift on an airfoil by leading-edge blowing suction[END_REF][START_REF] Wu | Post-stall flow control on an airfoil by local unsteady forcing[END_REF][START_REF] Nae | Synthetics jets influence on NACA0012 airfoil at high angle of attacks[END_REF][START_REF] Ravindran | Active control of flow separation over an airfoil[END_REF][START_REF] Rizzetta | Numerical investigation of synthetic jet flow fields[END_REF]. In these studies, the effects of zero net mass flux oscillatory jet (synthetic jet) and leading edge blowing/suction were considered on the vortex flow passing airfoils. Moreover, flow control methods, such as suction, blowing, and synthetic jets, have been investigated through analytical methods [START_REF] Glauert | The application of the exact method of aerofoil design[END_REF][START_REF] Preston | The theoretical estimation of power requirements for slot-suction aerofoils with numerical results for two thick griffith type sections[END_REF][START_REF] Abzalilov | The inverse boundary-value problem for an airfoil with a suction slot[END_REF] over thick and NACA airfoils.

Huang et al. [START_REF] Huang | Numerical study of blowing and suction control mechanism on NACA0012 airfoil[END_REF] studied the suction and blowing flow control techniques on a NACA 0012 airfoil. When jet location and angle of attack were combined, perpendicular suction at the leading edge increased lift coefficient better than other suction situations. The tangential blowing at downstream locations was found to lead to the maximum increase in the lift coefficient value. Rosas [START_REF] Rosas | Numerical simulation of flow separation control by oscillatort fluid injection[END_REF] numerically investigated flow separation control through oscillatory fluid injection, in which lift coefficient dramatically increased. Beliganur and Raymond [START_REF] Beliganur | Application of evolutionary algorithms to flow control optimization[END_REF] applied an evolutionary algorithm to optimize flow control. Results of their study showed that the use of two suction jets along with two blowing jets enhanced the lift-to-drag ratio for a NACA 0012 airfoil. Akcayoz and Tuncer [START_REF] Akcayoz | Numerical investigation of flow control over an airfoil using synthetic jets and its optimization[END_REF] examined the optimization of synthetic jet parameters on a NACA 0015 airfoil in different angles of attack to maximize the lift-to-drag ratio. Their results revealed that the optimum jet location moved toward the leading edge and the optimum jet angle incremented as the angle of attack increased. Jensch et al. [START_REF] Jensch | Numerical investigation of leading edge blowing and optimization of the slot geometry for a circulation control airfoil[END_REF] numerically conducted an aerodynamic analysis of a two-dimensional airfoil to improve the efficiency of a circulation control system. Varied slot heights at different flap deflection angles and leading edge blowing were investigated to optimize high-lift performance. The occurrence of a thin separation bubble near the leading edge was prevented through applying a second blowing slot at the leading edge.

With the development of computational facilities in recent years, computational fluid dynamics (CFD) has been increasingly used to investigate boundary layer control. Numerous flow control studies through CFD approaches [START_REF] You | Active control of flow separation over an airfoil using synthetic jets[END_REF][START_REF] Kim | Separation control on NACA23012 using synthetic jet[END_REF][START_REF] Genc | Performance of transition model for predicting low re aerofoil flows without/with single and simultaneous blowing and suction[END_REF][START_REF] Rumsey | Numerical study comparing RANS and LES approaches on a circulation control airfoil[END_REF][START_REF] Lee | Unsteady airfoil with a harmonically deflected trailing edge flap[END_REF][START_REF] Benini | Efficiency enhancement in transonic compressor rotor blades using synthetic jets: A numerical investigation[END_REF][START_REF] Yagiz | Drag minimization using active and passive flow control techniques[END_REF][START_REF] Yousefi | Numerical study of flow separation control by tangential and perpendicular blowing on the NACA 0012 airfoil[END_REF][START_REF] Yousefi | Numerical investigation of suction and length of suction jet on aerodynamic characteristics of the NACA 0012 airfoil[END_REF] have been conducted to investigate the effects of blowing, suction, and synthetic jets on the aerodynamic characteristics of airfoils. In the current study, the optimization of blowing and suction slot geometries, including suction and blowing jet widths, is numerically analyzed. Nearly 500 numerical simulations are performed with several parameters, such as jet amplitude, jet width, jet angle, and jet momentum coefficient at different angles of attack, on a NACA 0012 airfoil.

Governing equations

The fluid flow was modeled as a two-dimensional, unsteady, turbulent, and viscous incompressible flow with constant properties. The governing partial differential equations for mass and momentum conservation are as follows:

(1) [START_REF] Gad-El-Hak | Control flow: Passive, active and reactive flow management[END_REF] where ρ is the density, P is the mean pressure, ν is the kinematic viscosity, and u is the mean velocity. The term

' ' i j u u -
is the Reynolds stress tensor that incorporates the effects of turbulent fluctuations. The Reynolds stresses were modeled via the Boussinesq approximation [START_REF] Alfonsi | Reynolds-averaged Navier-Stokes equations for turbulence modeling[END_REF], in which the deviatoric part is considered proportional to the strain rate tensor through the turbulent viscosity. The incompressible form of the Boussinesq approximation is

(3) (4)
where ν t is the turbulent viscosity, k is the average kinetic energy of the velocity fluctuations, and δ ij is the Kronecker delta. To simulate the turbulent flow, eddy or turbulent viscosity distribution was employed rather than the Reynolds stress tensor through eddy viscosity turbulent models, including algebraic or zero-equation models, one-equation models, and two-equation models.

The turbulence model used in the present computation was the Menter's shear stress transport two-equation model (k-ω SST), which provides excellent predictive capability for flows with separation. This model includes k-ω and k-ε standard models, which improve the calculations of boundary layer flows with separation and remove the sensitivity of k-ω model in external flows. The transport equations in the Menter's shear stress model are as follows:

(5) [START_REF] Braslow | A history of suction type laminar flow control with emphasis on flight research[END_REF] where F 1 is the blending function, S is the invariant measure of the strain rate, β * is 0.09, and σ w2 is 0.856. The blending function is equal to zero when is away from the surface (k-ε model) and switches over to unity inside the boundary layer (k-ω model). A production limiter, k P % , was used in the SST model to prevent the build-up of turbulence in the stagnation regions. All constants were computed by a blend from the corresponding constant of the k-ε and k-ω models via α, σ k , σ ω , and so on [START_REF] Menter | Ten years of industrial experience with the SST turbulence model[END_REF].

Numerical simulation

Parameter selection

The commercial RANS-based code FLUENT, which follows a finite volume computational procedure, was used in this study. The first-and second-order upwind schemes were employed to discretize the convective terms in the momentum and turbulence equations. The central difference scheme was also used for the diffusive terms, and SIMPLE algorithm was applied for pressure-velocity coupling.

Calculations were performed over the NACA 0012 airfoil with 1 m chord length and a chord Reynolds number of 5 × 10 5 . The NACA 0012 profile, blowing and suction jet location (L j ), blowing and suction angle (θ), and jet width (h) are shown in Fig. 1. A previous study [START_REF] Huang | Numerical study of blowing and suction control mechanism on NACA0012 airfoil[END_REF], under similar conditions at Re = 5 × 10 5 and jet width of 2.5% of the chord length, has shown that the blowing at downstream locations, approximately 0.4 and 0.8, is better than other blowing situations for increasing lift. Moreover, recent investigations [21 and 23] have indicated that blowing at the trailing edge is more appropriate in various flow conditions than that at other locations. Therefore, the blowing slot was considered at 80% of the chord length from the leading edge in this work.

Meanwhile, suction at the leading edge, approximately 10% of the chord length, is better than other positions in improving the lift-to-drag ratio at different flow conditions over different airfoils [START_REF] Huang | Numerical study of blowing and suction control mechanism on NACA0012 airfoil[END_REF][START_REF] Rosas | Numerical simulation of flow separation control by oscillatort fluid injection[END_REF][START_REF] Akcayoz | Numerical investigation of flow control over an airfoil using synthetic jets and its optimization[END_REF][START_REF] You | Active control of flow separation over an airfoil using synthetic jets[END_REF]. In this study, the suction slot was therefore located at 10% of the chord length from the leading edge. The blowing and suction slot widths were varied from 1.5% to 4% of the chord length, and the jet amplitude (the jet velocity-to-free-stream velocity ratio) was 0.3 and 0.5. The angles of attack analyzed were 12, 14, 16, and 18°. The jet amplitude and jet entrance velocity are defined as follows:

(7) (8) (9)
where β is the angle between the free-stream velocity direction and the local jet surface, and θ is the angle between the local jet surface and the jet output velocity direction. Negative θ represents suction condition, and positive θ indicates blowing condition. For tangential blowing, perpendicular blowing, and perpendicular suction, θ is +90°, 0°, and -90°, respectively.

Finally, the jet momentum coefficient is as follows: [START_REF] Howe | An experimental evaluation of a low propulsive power discrete suction concept applied to an axisymmetric vehicle[END_REF] (11) [START_REF] Guowei | Control of unsteady vertical lift on an airfoil by leading-edge blowing suction[END_REF] Eq. [START_REF] Guowei | Control of unsteady vertical lift on an airfoil by leading-edge blowing suction[END_REF] indicates that the jet momentum coefficient depends on two non-dimensional factors, namely, jet amplitude (A) and jet width (H). A moving fluid exerts pressure and shear forces on the body surface. Both of these forces generally have components in the normal and tangential directions of flow; thus, the drag force is due to the combined effects of pressure (pressure drag) and wall shear forces (viscous drag) in the flow direction. Pressure and wall shear forces occur in the normal direction, and their sum (pressure lift and viscous lift) is called lift. The lift and drag forces are expressed as follows:

(13) [START_REF] Nae | Synthetics jets influence on NACA0012 airfoil at high angle of attacks[END_REF] Dimensionless numbers of drag and lift coefficients, which represent the drag and lift characteristics of the body, are more convenient to use than other numbers. They are defined as follows:

(15) [START_REF] Rizzetta | Numerical investigation of synthetic jet flow fields[END_REF] where the subscripts P and V refer to pressure and viscous, respectively. In the calculations of airfoils, A is taken to be the planform area. The pressure and viscous drag and the pressure and viscous lift were considered in all current simulations.

Numerical solution method

The first-and second-order upwind methods were employed to discretize the governing equations. In the simulations, the first-order upwind discretization in space was used. The resulting system of equations was solved through SIMPLE procedure until a convergence criterion of O( 5) reduction in all dependent residuals was satisfied. The secondorder upwind method was then applied to discretize the equations. The equations were resolved through SIMPLE method until a precise convergence was achieved at O( 7) in all dependent residuals. The results obtained from the first-order upwind method were used as an initial assumption for the second-order upwind method.

A C-type-structured grid with multizonal blocks , which is shown in Fig. 2, was generated as computational area. The computational area was large to prevent the outer boundary from affecting the near flow field around the airfoil. The inlet (left) and lower boundaries were fixed, with a uniform inlet velocity, u ∞ = 7.3 m/s. The upper and outer (right) boundary conditions were free-stream boundaries that satisfy the Neumann condition. Moreover, no-slip boundary condition was used at solid surfaces. A low free-stream turbulence level was used to match the wind tunnel characteristics, such that the stream turbulence intensity was selected as less than 0.1%. Different sized grids were used to ensure grid independence of the calculated results through the study of lift and drag coefficients at angles of attack of 10 and 16° for basic condition, without jets implemented on the airfoil. According to Table 1, the grid size giving a grid independent result with reasonable accuracy was selected to be 40,640 cells.

An interval size of 0.005 with 412 nodes was used on the upper and lower surfaces of airfoil, where the first-cell height was 1 × 10 4 . To simulate the boundary layer flow, the first layer grid near the wall satisfied the y-plus between 0.2 and 1.6. The mesh features and y-plus distribution over the airfoil are shown in Table 2 and Fig. 3, respectively. The interval size was reduced to 0.00125 for blowing and suction slots, as indicated in Fig. 4.

The residuals in all simulations were continued until the lift and drag coefficients reach a full convergence. The results were compared with the numerical simulation data of Huang et al. [START_REF] Huang | Numerical study of blowing and suction control mechanism on NACA0012 airfoil[END_REF] and the experimental values of Critzos et al. [START_REF] Critzos | Aerodynamics characteristics of NACA0012 airfoil section at angle of attacks from 0° to 180°[END_REF] and Jacobs et al. 3 (all experimental data have a Reynolds number of 5 × 10 5 ). Fig. 5 and Table 3 show that the experimental data from literature significantly varied, which implies a large amount of experimental uncertainty. This uncertainty is attributed to several factors, such as different flow regimes, angles of attack, and airfoil geometries.

Turbulence model selection has a significant effect on stall prediction and lift-to-drag ratio accuracy. The k-ε realizable model at the same condition, with changing stall angle from 14° to 16°, was considered. The k-ω SST model has better stall prediction capability than the k-ε realizable model. The prediction by k-ε realizable was good in the pre-stall region, but this model failed to predict the stall condition and poststall phenomena accurately. Moreover, the maximum errors for lift and drag coefficients in the k-ε realizable model at an angle of attack of 14° were 17% and 25%, respectively.

Given an apparent lack of experimental survey on suction and blowing flow control over a NACA 0012 airfoil at the current flow conditions, the computation results of airfoil with flow control were compared with numerical works. To validate the suction and blowing over NACA 0012 airfoil, the simulation data were compared with the numerical study of Huang et al. [START_REF] Huang | Numerical study of blowing and suction control mechanism on NACA0012 airfoil[END_REF] under a Reynolds number of 5 × 10 5 and a suction/blowing jet width of 2.5% of the chord length conditions. C L /C L, Base and C D /C D, Base ratios were compared for the present study and [START_REF] Huang | Numerical study of blowing and suction control mechanism on NACA0012 airfoil[END_REF] at a jet amplitude of 0.1 and an angle of attack of 18°, as shown in Table 4. In the study of Huang et al., the blowing and suction slots were located at 0.8 and 0.1 from the leading edge, respectively. Fig. 6 illustrates the comparison between the present study and the numerical work of Goodarzi et al. [START_REF] Goodarzi | Investigation of flow control over a NACA0012 airfoil by suction effect on aerodynamic characteristics[END_REF] under a Reynolds number of 5 × 10 5 , a jet width of 2.5% of the chord length, a jet location of 10% of the chord length from the leading edge, and a jet amplitude of 0.5 Fig. 5. Comparison among the current computational results, the numerical work result of Huang et al. [START_REF] Huang | Numerical study of blowing and suction control mechanism on NACA0012 airfoil[END_REF], and the experimental measurement results of Critzos et al. [START_REF] Critzos | Aerodynamics characteristics of NACA0012 airfoil section at angle of attacks from 0° to 180°[END_REF] and Jacobs et al. [START_REF] Jacobs | Airfoil section characteristics as affected by variations of the Reynolds number[END_REF].

Table 4. Comparison of computational results and numerical study [START_REF] Huang | Numerical study of blowing and suction control mechanism on NACA0012 airfoil[END_REF] for blowing and suction flow control. at different angles of attack. The simulation results agree strongly with the mentioned studies.

Slot geometry

Two slot geometries are available for blowing and suction simulations over a NACA 0012 airfoil; their slot types are shown in Figs. 7 and8. The perpendicular suction and per-pendicular blowing have similar results with the normal inlet and outlet flow directions. The result of perpendicular suction was used for perpendicular blowing. The streamlines and re-sults of these two slot types were compared in Fig. 9 and Ta-ble 5. The perpendicular suction has a jet amplitude of 0.5, a jet width of 2.5% of the chord length, and angles of attack of 16 and 18°. Fig. 9 illustrates streamlines around the airfoil and suction slot. The flow pattern around the airfoil for both suc-tion slot geometry types is closely resembled. In experimental investigations, the slot geometry of type 1 for suction can be constructed through porous media. This type of geometry is also widely used in previous studies. In the current study, we also used slot geometry of type 1 for perpendicular suction and perpendicular blowing.

The comparison of streamlines around the airfoil and blow-ing slot for tangential blowing at the trailing edge is shown in Fig. 10. The values of lift and drag coefficients are presented in Table 6, which indicates that the results for two slot geome-try types are strongly closed, whereas the vortexes passing the airfoil are different. The slot geometry of type 1 models the moving surface (wall motion) over the airfoils [START_REF] Modi | Effects of moving surfaces on the airfoil boundary-layer control[END_REF][START_REF] Yousefi | Numerical investigation of moving surface boundary-layer control on NACA 0012 airfoil and comparison with tangential and perpendicular blowing[END_REF] rather than the tangential blowing. Blowing is defined as a method of preventing separation to supply additional energy to the particles of fluid being retarded in the boundary layer [START_REF] Schlichting | Boundary layer theory[END_REF]. The set up of slot geometry of type 1 in experimental surveys is roughly unreachable, whereas the slot geometry of type 2 is a common approach. Therefore, the blowing slot geometry of type 2 was used in this investigation for tangential blowing at the airfoil trailing edge.

Results and discussion

Tangential blowing

The effects of jet amplitude and jet width on lift and drag coefficients were investigated for tangential steady blowing at the trailing edge. The effects of blowing and its strength on aerodynamic coefficients are illustrated in Fig. 11 under conditions of L j = 0.1, H = 3.5%, and A = 0.3, 0.5. The figure indicates that the changes of jet amplitude have negligible effect on the aerodynamic characteristics of airfoil. The lift and drag coefficients are roughly constant with varying jet amplitude from 0.3 to 0.5. By increasing blowing amplitude, lift coefficient increases progressively and drag coefficient decreases marginally. The maximum increase of lift-to-drag ratio occurred in a blowing amplitude of 0.5, which increased at approximately 16% at an angle of attack of 18°. The lift and drag coefficients had a 7% increase and decrease, respectively. When blowing amplitude increases, stall shows no alteration and occurs at the same angle of attack of 14°.

Figs. 12-14 present the lift, drag, and lift-to-drag ratio versus jet width with blowing jet amplitudes of 0.3 and 0.5 for tangential blowing at the trailing edge, respectively. Increasing the jet width leads to lift coefficient improvement and drag coefficient reduction slightly. Under an angle of attack of 14° and a blowing amplitude of 0.5, the lift coefficient increased to 1.157 and 1.163 for blowing jet widths of 1.5% and 4.0% of the chord length, respectively. Therefore, the lift coefficient increased approximately by 1%, whereas the drag coefficient remained nearly constant. The least improvement of the lift coefficient arises in jet widths of 3.5% to 4%, and the lift coef- ficient value generally increases with a constant slope until a jet width of 3.5% of the chord length and then declines steadily. A similar trend was also found for the lift-to-drag ratio. By contrast, the drag coefficient reduces linearly as jet width increases. Moreover, the drag coefficient remains roughly constant in different jet amplitudes. In the conducted studies on tangential blowing, no maximum or minimum values were found in lift, drag, and lift-to-drag ratio. The vortexes around the blowing slot move toward downstream because of the tangential blowing at the airfoil trailing edge. Considerable vortexes are transferred to downstream by increasing the blowing jet width, as shown in Fig. 15.

The lift-to-drag ratio increases continually up to jet widths of 3.5% to 4% of the chord length along with jet width and then decreases. Hence, the blowing jet widths of 3.5% to 4% of the chord length are extremely effective. Generally, the variations of tangential blowing parameters have a remarkable effect on the lift coefficient and a marginal effect on the drag coefficient. As mentioned earlier, the blowing jet amplitude has an insignificant effect on aerodynamic coefficients and poses only a 2% increase in the lift-to-drag ratio. Consequently, the lift-to drag ratio increased by 17% under a jet amplitude of 0.5, a jet width of 4% of the chord length, and an angle of attack of 18°. The streamline patterns of different jet widths passing an airfoil are plotted in Fig. 15 at a jet amplitude of 0.5 and an angle of attack of 16°. The streamlines when jet widths are 3.5% and 4% clearly demonstrate smaller vortex around the airfoil particularly closed to the blowing slot than other cases.

Perpendicular blowing

Fig. 16 mainly focuses on the effects of jet amplitude variations with angle of attack for perpendicular blowing. In this subsection, the jet location was fixed at 0.8C, the jet width was considered as 1.5%C, the jet amplitude was limited to 0.3 and 0.5, and the jet angle was +90°. Smaller jet amplitude is more appropriate in perpendicular blowing, contrary to tangential blowing. The drag coefficient increased by approximately 8% and 5% for jet amplitudes of 0.3 and 0.5, respectively, at an angle of attack of 14°. On the contrary, the lift coefficient declined by approximately 8.5% and 14.5%. Generally, perpendicular blowing decreases the lift-to-drag ratio before stall angle intensively and increases the aerodynamic characteristics after stall, for instance, perpendicular blowing increased the lift-to-drag ratio by 17.5% at an angle of attack of 18°. Drag is usually an undesirable effect, and we do our Author's Proof best to minimize it. The reduction of drag is closely associated with the reduction of fuel consumption, noise, and vibration. However, drag produces a beneficial effect in some cases, which we try to maximize, for example, in the brakes of airplanes.

Figs. 17-19 illustrate the changes of lift coefficient, drag coefficient, and lift-to-drag ratio with jet width and jet amplitude for perpendicular blowing at the airfoil trailing edge, respectively. Unlike in tangential blowing, the aerodynamic coefficients decrease continuously as blowing jet width increases in perpendicular blowing. When the jet width varied from 1.5% to 4% of the chord length, the lift and drag coefficients decreased by approximately 23% and 16%, respectively, under A = 0.5 and an angle of attack of 14°. In the similar conditions, the lift-to-drag ratio decreased by 7% and 3.5% at angles of attack of 14° and 16°, respectively. In all studied cases for perpendicular blowing, the decrease of lift and drag coefficients is almost linear with a constant slope. In Fig. 20, streamlines passing an airfoil for tangential and perpendicular blowing were compared at a blowing amplitude of 0.5 and an angle of attack of 16°. The perpendicular blowing in the trailing edge makes larger vortexes in comparison with tangential blowing. In addition, the velocity vectors over the airfoil surface were compared for tangential and perpendicular blowing, as shown in Fig. 21. The velocity vectors were plotted under a blowing jet width of 4% of the chord length, a blowing amplitude of 0.5, and an angle of attack of 16°. The tangential blowing postpones separation more than perpendicular blowing. The separation in tangential blowing occurred in 0.535 of the chord length, whereas that in perpendicular blowing occurred in 0.460 of the chord length from the airfoil leading edge. Consequently, we indicate that smaller blowing jet widths provide more positive effects for using perpendicular blowing to minimize drag and larger jet widths are effective to maximize it. amplitudes of 0 and 0.5, respectively. A jet amplitude of 0 refers to no suction conditions. With the use of perpendicular suction, not only the lift-to-drag ratio increases dramatically but also the stall angle delays effectively.

Figs. [START_REF] Akcayoz | Numerical investigation of flow control over an airfoil using synthetic jets and its optimization[END_REF][START_REF] Jensch | Numerical investigation of leading edge blowing and optimization of the slot geometry for a circulation control airfoil[END_REF][START_REF] You | Active control of flow separation over an airfoil using synthetic jets[END_REF] show the lift, drag, and lift-to-drag ratio at different jet widths and angles of attack for perpendicular suction at the leading edge with a jet amplitude of 0.3, respectively. The variations in lift and drag coefficients with jet width are almost negligible at low angles of attack, although they have shown significant changes with the increment of angle of attack. With increasing jet width, the lift coefficient increases continuously until a jet width of 2.5% of the chord length and then insignificantly decreases. The drag coefficient has the similar trend. The lift coefficient increased by 6.3% and the drag coefficient reduced by 15% when the jet width varied from 1.5% to 2.5% of the chord length at an angle of attack of

Suction

The effects caused by the changes of suction amplitude were investigated in Fig. 22. The jet width and jet location were fixed at 2.5% and 10% of the chord length, respectively. As suction amplitude increased from 0.3 to 0.5, the lift coefficient improved and the drag coefficient declined. However, the lift coefficient increase and the drag coefficient decrease are neg-ligible for angles of attack smaller than 10°. The lift coefficient increased by 75%, and the drag coefficient decreased by 56%, with a suction amplitude of 0.5 at an angle of attack of 18°. Moreover, increasing suction jet amplitude leads to stall angle improvement, which increased from 14° to 22° for jet 18°. The jet width then increased from 2.5% to 3% of chord length, leading to trivial changes in the lift and drag coefficients. The lift-to-drag ratio also conforms to the abovementioned trend and increased by 24% for jet width variation from 1.5% to 2.5%, whereas the lift-to-drag ratio increased only by 2% for jet width variation from 2.5% to 3% of chord length. Thus, the optimum jet width considered was 2.5% of the chord length for perpendicular suction at the leading edge.

Danenberg and Weiberg [START_REF] Dannenberg | Section characteristics of a 10.5 percent thick airfoil with area suction as affected by chordwise distribution of permeability[END_REF] experimentally evaluated the effects of suction jet width on a symmetrical airfoil (NACA airfoil 0010.51). They considered two slot widths, 2.3% and 6.3% of the chord length, for suction on the upper surface of airfoil and demonstrated that the optimum suction jet width was 2.3% of the chord length. The maximum lift increased approximately from 1.3 to 1.8 by an area suction of 2.3% of the chord length. The results of [START_REF] Dannenberg | Section characteristics of a 10.5 percent thick airfoil with area suction as affected by chordwise distribution of permeability[END_REF] confirm the obtained data of the present investigation.

Figs. 26-28 present the effects of suction jet amplitude, which changes from 0.3 to 0.5 for perpendicular suction at the airfoil leading edge. The lift coefficient increases until a jet width of 2.5% and then declines slightly particularly at high angles of attack. Increasing suction jet amplitude from 0.3 to 0.5 increases lift coefficient and reduces drag coefficient. The jet width increase from 2.5% to 3% led to a 1.4% reduction in the lift coefficient, a 1% improvement in the drag coefficient, and a 2.5% increase in the lift-to-drag ratio at angle of attack of 18°. In this case, the alterations of lift, drag, and lift-to-drag ratio were marginal at lower angles of attack.

Suction control on changing flow patterns at different jet widths is illustrated in Fig. 29. The results for suction jet widths of 1.5%, 2%, 2.5%, and 3% of the chord length are plotted and compared in the figure. These cases all have a jet

Conclusion

In this study, we evaluated the effects of blowing and suc-tion flow controls on a NACA 0012 airfoil at a Reynolds number of 5 × 10 5 and different angles of attack. When several parameters, including jet width, jet amplitude, and jet angle, were changed, interesting and valuable results were analyzed. The following results were obtained.

The computational studies were conducted with the blowing flow control approach. Downstream tangential blowing improves the lift and drag characteristics, whereas perpendicular blowing has often adverse effects in comparison with the desirable objectives of flow control results, particularly at angles of attack smaller than stall angle. However, smaller amplitudes are better than larger amplitudes for perpendicular blowing. The lift and drag coefficients are almost constant with variations of jet amplitude for tangential blowing. Increasing blowing jet width improves the lift-to-drag ratio continuously for tangential blowing and reduces it constantly and quasilinearly for perpendicular blowing. The gradient of the lift-todrag ratio declines from a jet width of 3.5% of the chord length in tangential blowing. Jet widths of 3.5% and 4% of the chord length appear to be the most effective choice for tangential blowing at the airfoil trailing edge. However, smaller jet widths pose more beneficial effects of using perpendicular blowing to minimize drag, whereas larger jet widths are highly efficient in maximizing drag. The lift-to-drag ratio increased by approximately 17% for tangential blowing under H = 4%, A = 0.5, and Lj = 0.8C from the leading edge, with an angle of attack of 18°.

The effects of suction control flow on the aerodynamic characteristics of NACA 0012 airfoil were subsequently investigated. In suction, larger amplitude results in larger effect on the flow field around the airfoil, particularly at high angles of attack. Owing to the use of perpendicular suction, not only the lift-to-drag ratio increases dramatically but also the stall angle improves effectively from 14° to 22°. The suction jet width improvement leads to a significant augmentation in the lift-todrag ratio, and separation effectively travels toward downstream. The lift-to-drag ratio increases continuously until a jet width of 2.5% of the chord length and then decreases. By employing the suction control flow technique, the lift coefficient increased by approximately 75% and the drag coefficient decreased by 56% under H = 2.5%, A = 0.5, and Lj = 0.1C from the leading edge, with an angle of attack of 18°. The most effective jet widths for achieving all desirable effects are 2.5% to 3% of the chord length for suction at the airfoil leading edge.

Further works are needed to investigate some essential suction/blowing parameters, including the number of slots, slot arrangements, and synthetic jet parameters. Laboratory surveys on suction and blowing parameters are highly recommended. -----------------------------------------------------------------------A : Jet amplitude C : Airfoil chord length location of 10% of the chord length from the leading edge, a jet amplitude of 0.3 for perpendicular suction, and θ = -90°.

Nomenclature-

When the jet width is increased, the separation bubble is effectively delayed; hence, the separation bubbles and vortices are almost entirely eliminated in a jet width of 2.5% of the chord length. Therefore, a suction jet width of approximately 2.5% to 3% of the chord length is the most effective width for the suction flow control approach to manipulate the boundary layer and increase the lift-to-drag ratio. An increase of suction area beyond of 3% of the chord length will not increase the aerodynamic characteristics significantly.

C

Fig. 1 .

 1 Fig. 1. NACA 0012 airfoil with blowing and suction slots: (a) blowing slot; (b) suction slot.

  [START_REF] Jacobs | Airfoil section characteristics as affected by variations of the Reynolds number[END_REF].Huang et al. investigated the suction and blowing flow control over a NACA 0012 airfoil under a Reynolds number of 5 × 10 5 and an angle of attack of 18°. Moreover, they examined several parameters, including jet location, jet amplitude, and jet angle, through the CFD GHOST code. Critzos et al. experimentally studied the aerodynamic characteristics of a NACA 0012 airfoil, with Reynolds numbers of 0.5 × 10 6 and 1.8 × 10 6 and angles of attack varied from 0° to 180°. E. Jacobs et al. investigated symmetrical NACA airfoils in wind tunnel over a wide range of Rey-

Fig. 2 .Fig. 3 .

 23 Fig. 2. C-type-structured mesh with multizonal blocks.

Fig. 4 .

 4 Fig. 4. Structured grid of NACA 0012 airfoil with jet slot.

Fig. 6 .Fig. 7 .Fig. 8 .

 678 Fig.6. Comparison of computational results and numerical study[START_REF] Goodarzi | Investigation of flow control over a NACA0012 airfoil by suction effect on aerodynamic characteristics[END_REF] for suction flow control.

Fig. 9 .Fig. 10 .

 910 Fig. 9. Streamlines around the airfoil with a jet amplitude of 0.5, a jet width of 2.5%C, and an angle of attack of 18°: (a) suction slot geometry of type 1; (b) suction slot geometry of type 2.

Fig. 11 .

 11 Fig. 11. Effects of blowing amplitude on lift and drag coefficients for tangential blowing at the airfoil trailing edge.

Fig. 12 .

 12 Fig. 12. Changes of lift coefficient with blowing jet width for jet amplitudes of 0.3 and 0.5 for tangential blowing at the trailing edge.

Fig. 13 .

 13 Fig. 13. Changes of drag coefficient with blowing jet width for jet amplitudes of 0.3 and 0.5 for tangential blowing at the trailing edge.

Fig. 14 .Fig. 15 .

 1415 Fig. 14. Changes of lift-to-drag ratio with blowing jet width for jet amplitudes of 0.3 and 0.5 for tangential blowing at the trailing edge.(a)

Fig. 16 .

 16 Fig.[START_REF] Rizzetta | Numerical investigation of synthetic jet flow fields[END_REF]. Effects of blowing amplitude on lift and drag coefficients for perpendicular blowing at the trilling edge.

Fig. 17 .

 17 Fig. 17. Changes of lift coefficient with blowing jet width for jet amplitudes of 0.3 and 0.5 for perpendicular blowing at the trailing edge.

Fig. 18 .

 18 Fig. 18. Changes of drag coefficient with blowing jet width for jet amplitudes of 0.3 and 0.5 for perpendicular blowing at the trailing edge.

Fig. 19 .

 19 Fig. 19. Changes of lift-to-drag ratio with blowing jet width for jet amplitudes of 0.3 and 0.5 for perpendicular blowing at the trailing edge.

Fig. 20 .

 20 Fig. 20. Comparison between tangential and perpendicular blowing at a jet amplitude of 0.5, an angle of attack of 16 degrees, and jet widths of 2.5 %C and 4 %C.

Fig. 21 .

 21 Fig. 21. Comparison between the velocity vectors of tangential and perpendicular uniform blowing, with a jet width of 4 %C, a blowing amplitude of 0.5, and an angle of attack of 16°.

Fig. 22 .

 22 Fig. 22. Effects of suction amplitude on lift and drag coefficients for perpendicular suction.

Fig. 23 .

 23 Fig. 23. Changes of lift coefficient with suction jet width for a jet amplitude of 0.3 for perpendicular suction at the leading edge.

Fig. 24 .

 24 Fig. 24. Changes of drag coefficient with suction jet width for a jet amplitude of 0.3 for perpendicular suction at the leading edge.

Fig. 25 .

 25 Fig. 25. Changes of lift-to-drag ratio with suction jet width for a jet amplitude of 0.3 for perpendicular suction at the leading edge.

Fig. 26 .

 26 Fig. 26. Changes of lift coefficient with suction jet width for a jet amplitude of 0.5 for perpendicular suction at the leading edge.

Fig. 27 .

 27 Fig. 27. Changes of drag coefficient with suction jet width for a jet amplitude of 0.5 for perpendicular suction at the leading edge.

Fig. 28 .

 28 Fig. 28. Changes of lift-to-drag ratio with suction jet width for a jet amplitude of 0.5 for perpendicular suction at the leading edge.

Fig. 29 .

 29 Fig. 29. Effect of suction jet width on eddies behind the airfoil at 16° angle of attack.

Table 1 .

 1 Grid independence study for NACA 0012 airfoil at a Reynolds number of 5 × 10 5 and an angle of attack of 16°.

	Number of	Angle of attack of 10°	Angle of attack of 16°
	cells	Lift	Drag	Lift	Drag
		coefficient	coefficient	coefficient	coefficient
	8096	0.82201	0.07858	0.64594	0.20889
	17160	0.91364	0.05898	1.05134	0.12544
	24480	0.90865	0.05525	1.09073	0.11567
	40640	0.90873	0.05332	1.12352	0.10938
	58080	0.90878	0.05394	1.12319	0.11187

Table 3 .

 3 Comparison of computational lift coefficient and experimental values at angles of attack less than 10°.

			Experimental	Experimental	Experimental
	Angle of	Computational	result of	result of	result of
	attack	results	Critzos et al.	Jacobs et al.	Sheldahl et al.
			[36]	[37]	[38]
	0°	0.0021	0.0	0.0	0.0
	2°	0.1853	0.2053	0.1807	0.22
	5°	0.4715	0.5855	0.4511	0.55
	10°	0.9087	0.9542	0.9019	1.003

Table 5 .

 5 Comparison of lift and drag coefficients for different suction slot geometries.

	Angle of	Suction slot geometry of type 1	Suction slot geometry of type 2
	attack	CL	CD	CL	CD
	12°	1.1124	0.04474	1.1008	0.07025
	14°	1.2846	0.05414	1.2707	0.08271
	16°	1.4424	0.06487	1.4298	0.09193
	18°	1.5778	0.07713	1.5605	0.10533

Table 6 .

 6 Comparison of lift and drag coefficients for different suction slot geometries.

	Angle of attack	Blowing slot geometry of type 1	Blowing slot geometry of type 2
		CL	CD	CL	CD
	12°	1.0551	0.06994	1.0217	0.06918
	14°	1.1596	0.08685	1.1388	0.08635
	16°	1.1548	0.11015	1.1525	0.11074
	18°	0.9659	0.16387	0.9749	0.16673

Author's Proof