Machine learning-based optimization of a pitching airfoil performance in dynamic stall conditions using a suction controller

A

amplitude of motion (deg)

c

chord length

$C d$

drag coefficient

$C l$

lift coefficient

$C μ$

jet momentum coefficient

$e$

strain

$E$

error function

f

frequency of pitching airfoil

G

global model

k

reduce frequency (⁠ $π f c U ∞$⁠)

$k$

turbulence kinetic energy

$L D$

aerodynamic performance

$N s$

number of constructing points

$P$

pressure

r(.)

Gaussian correlation function

Re

Reynolds number

$R c$

correlation matrix

S

area

SC

shape coefficient

$S jet$

jet suction area

$U jet$

jet velocity amplitude

$U τ$

Frictional velocity

$U ∞$

free stream velocity

$w$

neuron weight

$y j$

guessed data

$y Kriging$

guessed data of Kriging method

$y RBF$

guessed data of BRF method

$y j actual$

original data

$y j ¯ actual$

average of the original data

Z

stochastic Gaussian process determining

$α$

angle of attack

$α 0$

initial angle of attack

$θ$

the jet angle with respect to chord line

$β$

the jet angle with respect to airfoil surface

$γ$

coefficient vector of the RBF

$Γ$

learning rate

$δ$

error gradient

$η$

efficiency of a control jet

$μ t$

eddy viscosity

$ρ$

density

$σ 2$

process variance

$τ$

surface stress

$τ w$

wall shear stress

$υ$

kinematic viscosity

$ϕ$

radial basis function

$ω$

turbulence frequency length scale

Dynamic stall is a nonlinear phenomenon that arises in an oscillating airfoil. It results from rapid changes in the angle of attack and influences wind turbines, helicopter blades, and flapping wings. Numerous papers have examined the dynamic stall of a pitching airfoil for laminar flows,^1,2 laminar to turbulent transition,^3–8 and fully turbulent flows.^9–14 Researchers have explored how dynamic stall develops and what phenomena occur during it. Ham and Garelick¹⁵ experimentally investigated the effect of the pitching rate and pitching axis location. They demonstrated that dynamic stall occurs at a higher angle of attack than static stall. McCroskey et al.¹⁶ studied unsteady boundary-layer separation. They discovered that the abrupt separation of the turbulent boundary layer generated the initial vorticity of the shed vortex. Visbal and Shang¹ conducted a numerical study for an oscillating National Advisory Committee for Aeronautics (NACA) 0015 airfoil. They inferred that separation propagation, vortex shedding into the wake, formation of the leading-edge vortex, and shear layer vortex defined the flow structures. Lee and Gerontakos¹⁷ experimentally analyzed oscillating NACA0012 at the Reynolds number of $1.35× 10 5$⁠. They concentrated on the effect of oscillation frequency and amplitude on flow. They reported the locations of the boundary-layer transition, separation, reattachment, and relaminarization points. They observed that the onset of the various boundary-layer events and the peak values of aerodynamic coefficients were significantly affected by the reduced frequency of the oscillation. Sheng et al.¹⁸ enhanced the Leishman–Beddoes model for dynamic stalls by proposing a new stall-onset indication. Martinat et al.⁹ assessed the influence of different turbulence models on the numerical simulation of the dynamic stall of an airfoil. They realized that the results of the SST model were closer to the experimental results than the classical unsteady Reynolds-averaged Navier–Stokes (URANS) model. They showed that 2D simulation could capture significant parts of coherent structures, although 3D simulation improved the downstroke phase. Wang et al.⁴ compared the results of two URANS models for the dynamic stall of a NACA0012 airfoil. They showed that the transient SST k–ω model captured the experimental data in a significant part of the flow, while the standard k–ω model could not. The dissipative prediction of adverse pressure gradient near the suction side prevented the standard k–ω model from capturing the characteristic of dynamic stall well. Wang et al.⁵ simulated an oscillating NACA0012 airfoil at a Reynolds number of $1.35× 10 5$⁠. They used several turbulence models and showed that SST k–ω was better than the others. They found that 3D simulation did not improve the prediction of lift coefficients. During stall development, Melius et al.¹⁹ experimentally determined a wind turbine blade's vortex size, separation angle, and separation point. They used time-resolved particle image velocimetry for visualization. They showed that the angles of attack for the transition from attached flow to full stall were higher than those for the transition from full stall to attached flow. Benton and Visbal¹¹ simulated a ramp-type pitching NACA0012. They used a wall-resolved large-eddy simulation and showed that the dynamic stall process was susceptible to the Reynolds number. They explained the formation of the leading-edge and dynamic stall vortex (DSV) in detail. Li et al.²⁰ experimentally studied the detachment mechanism for pitching and plunging flat plates. They focused on the effect of startup, cyclic motions, and maximum effective angle of attack on leading-edge development. Their results showed that the start cycle did not affect leading-edge development. However, the maximum effective angle of attack determined the different secondary structures that caused the detachment of the leading-edge vortex. Ansell and Mulleners²¹ examined the multiscale development of dynamic stalls and linked them to physically significant phenomena. The dynamic stall phenomenon often occurs in Vertical Axis Wind Turbines (VAWTs). Therefore, many authors have explored this phenomenon with a wind turbine approach to investigate the influence of operating parameters,⁷ startup process,²² optimization of blade aerodynamics,²³ application of micro‐plasma actuator,²⁴ high reduced frequencies,⁸ sensitivity analysis,²⁵ surface grooves,²⁶ and ice accretion shapes²⁷ on wind turbines. Because two-dimensional simulation of URANS gives acceptable results for dynamic stalls and can capture the main flow structure and vortices, researchers^7,8,25 studied this phenomenon with two-dimensional simulation.

In an oscillating airfoil, the drag force increases significantly when the angle of attack approaches the angle of dynamic stall. As the drag force increases, more power is needed to overcome flow resistance. Aerodynamic performance can be improved by applying flow control methods. Flow control methods are divided into two general categories: active and passive. Passive methods try to control the flow without consuming energy, while active methods require energy consumption. Wingtip devices, roughness, and vortex generators are commonly used examples of passive flow control, while examples of active flow devices include blowing and suction jets and synthetic jets. Much research has been done on dynamic stall flow control. Their main activities have been in movable trailing-edge,³ cylindrical disturbance generators,²⁸ passive vortex,¹² steady-blowing jets,^29–31 synthetic jets,³² and plasma actuators.^13,14 In the following, some studies on this topic are described. In their experimental study, Müller-Vahl et al.³¹ applied an adaptive blowing jet to eliminate lift oscillations of a pitching NACA0018. In their work, the jet momentum coefficient of the blowing jet was changed during the oscillating period. Zhang et al.³³ controlled the separation of a NACA0012 airfoil at low Reynolds at several angles of attack. Suction coefficient, location, angle, slot width, and porosity were control parameters considered for the suction jet. Tadjfar and Asgari¹⁰ utilized a continuous blowing jet to control the separation of an oscillating NACA0012 airfoil at a Reynolds number of $10 6$⁠. They investigated the impact of slot location and velocity ratio on the flow. They found that jet performance improved when placed within a minimal location upstream of the leading-edge vortex. Tadjfar and Asgari³⁴ investigated the impact of a tangential synthetic jet in controlling the dynamic stall. They studied the role of the phase difference between airfoil oscillation and jet actuation frequency and found that this difference significantly affected aerodynamic performance. Williams et al.³⁵ designed a feed-forward controller for a NACA0018 airfoil in dynamic stall conditions and evaluated the controller's ability to keep the lift coefficient constant during the oscillation period. Samara and Johnson³⁶ used a trailing edge flap and studied its influence on the dynamic stall of a pitching S833 cambered airfoil. They used different phase lags for flap motion and concluded that the flap only reduced the leading-edge vortex magnitude and could not prevent its formation.

The determination of optimal values for flow control parameters has been neglected in most studies of dynamic stalls. All studies related to flow control optimization have paid attention to the static stall.^37–40 Some studies have focused on finding optimal parameters for a static airfoil using different methods. Here are some examples of such studies. Kamari et al.⁴¹ used continuous blowing and suction jets to control the separation of a Selig–Donovan 7003 airfoil and applied a control mechanism in both cross-boundary layer and tangential-boundary layer configurations. They optimized the injection angle, velocity, orifice location, and opening length using a genetic algorithm (GA) coupled with two neural networks. They found that the suction jet had more control effect than the blowing jet in optimal conditions. Nair et al.⁴² proposed a feedback controller based on a cluster for separation control of a NACA0012 airfoil at a Reynolds number of 23 000. Their actuator allowed them to suck and blow the jet. They minimized power consumption with the simplex search algorithm, obtained the optimal control law, and reduced the mean drag. Yildizeli and Cadirci⁴³ optimized impinging jet arrays for a pneumatic contactless levitation system. They used a multi-objective genetic algorithm and chose levitation capacity and energy consumption as objective functions. The distance between sequential jet nozzles, aspect ratio of cylindrical pillars, and jet Reynolds number were considered as design parameters. Yuhui and Yufei⁴⁴ introduced a new internally blown flap configuration based on a multi-element airfoil. They obtained the optimal configuration and enhanced the lift coefficient, lift-to-drag ratio, and low angle of attack performance. They also examined the sensitivity of each design parameter by parametric analysis. Tadjfar and Kamari⁴⁵ numerically studied the effect of synthetic jets in tangential-boundary layer and cross-boundary layer configurations on an SD7003 airfoil at a Reynolds number of 60 000. They selected location, opening length, injection velocity amplitude, injection angle, and non-dimensional frequency as variable parameters. They used a genetic algorithm coupled with two neural networks for their optimizations. They showed that the synthetic jet in the tangential-boundary layer configuration improved aerodynamic characteristics more than the cross-boundary layer configuration.

The authors discussed the effect of selecting functional parameters of blowing and suction jets in other studies^10,46,47 on pitching airfoils in dynamic stall conditions. Finding the optimal parameters for flow control is crucial for achieving maximum efficiency. Optimization algorithms are used to find the optimal shape of the airfoil,⁴⁸ the vortex control strategy for plunging airfoil,⁴⁹ the propulsion performance of flapping wings,⁵⁰ and synthetic jets on two tandem airfoils.⁵¹ The authors studied the optimal suction jet⁴¹ and synthetic jet⁴⁵ for static stall airfoils. All previous research focused on static stalls, but few studies investigated the optimal jet parameters for oscillating airfoils and dynamic stalls. The authors⁵² recently optimized the performance of a blowing jet for an airfoil under dynamic stall conditions. The literature survey on pitching airfoils under deep dynamic stall conditions shows the need to determine the optimal suction jet parameters to reach maximum efficiency. This paper used a continuous suction jet to prevent the dynamic stall of an oscillating NACA0012 airfoil over the surface at a Reynolds number of $1.35× 10 5$⁠. A genetic algorithm (GA) approach was employed to find the optimal values of the jet control parameters: jet angle, jet velocity, location of the orifice, and orifice width. Due to the high computational cost of optimization, artificial neural networks (ANNs) were applied to reduce the number of computational fluid dynamics (CFD) simulations. The optimal configuration was obtained with about 150 CFD simulations using this methodology.

The outline of the current study is organized as follows: Sec. II presents the governing equations for this study. Section III consists of four subsections. The first subsection presents the physical model and motion model of the airfoil. Section III B introduces the computational domain. Section III C expresses time step and grid resolution verification. Section III D provides validation of this solver for the simulation of dynamic stalls. The optimization methodology is discussed in Sec. IV. This section includes four subsections. Section IV A states design variables and objective function. Section IV B describes the general optimization process. Sections IV C and IV D explain the genetic algorithm and neural network used in the optimization process. Section V presents the convergence of the optimization process and optimization results using a proposed framework, and the suction jet's effect in different operational parameters is fully analyzed. Finally, obtained main conclusions are summarized in Sec. VI.

The two-dimensional explicit finite volume method was used to simulate a pitching NACA0012 airfoil at the Reynolds number of $1.35× 10 5$⁠. The SIMPLE algorithm was utilized for pressure-velocity coupling, and pressure, momentum, and turbulence transport equations were discretized with second order in space. Flow turbulence properties were modeled with the transient k–ω SST model because of its compatibility with previous studies of similar flows. Wang et al.^4,5 showed that this model accurately predicted flow separation and reattachments. The base of this model is the coupling of k–ω SST that requires two transport equations, one for intermittency and the other for the laminar to turbulent boundary layer transition onset criteria in terms of the momentum-thickness Reynolds number. Free stream velocity at the far field was considered inlet velocity and no-slip condition over the walls was set for computational domain boundaries.

A structured O-type grid was employed to solve the governing equations. An interface boundary was created to distinguish the moving and stationary zones. The computational domain was extended 19 chord lengths upstream and downstream of the airfoil. Tadjfar and Asgari¹⁰ employed 15 and 21 chord lengths upstream and downstream for the same airfoil at a higher Reynolds number. Chapin and Bernard⁵⁶ employed 15 chord lengths on both sides. The computational domain and dynamic zone are depicted in Fig. 2. The grids near the leading edge and trailing edge are illustrated in Fig. 3. The no-slip wall and inlet velocity were imposed as boundary conditions for the sample without a jet. For the sample with a jet, the part of the airfoil with the suction jet was isolated from other parts, and the inlet velocity condition was imposed on it. The dynamic mesh technique was utilized to impose the pitching condition. The airfoil oscillated around its quarter chord, and zones 2 and 3 oscillated with it to preserve grid quality. Free flow velocity was imposed on the outer boundary of zone 1, far from the airfoil. Wang et al.,⁴ Gharali and Johnson,⁶ and Asgari and Tadjfar¹⁰ imposed similar conditions.

Two time scales were employed as the initial guesses for the time step: the time for a shed vortex to traverse the airfoil $(c/ U ∞)$ and the time for the oscillating motion $(1/f)$⁠. By choosing a smaller timescale $[ t *= min c / U ∞ , 1 / f]$⁠, we realized that the time step size of $Δt= t */110$ would be suitable for our simulation. The simulation was repeated with $Δt= t */220$ to verify the time step independence of the aerodynamic coefficients, and no significant difference was observed in the results. The comparison of the aerodynamic coefficients is presented in Fig. 4. Wang et al.⁵ employed $Δt=0.0125c/ U ∞$ for a grid with 180 000 cells and similar oscillating motion and Reynolds number as ours.

The grid independence study was conducted to verify the adequacy of the final grid and the aerodynamic coefficients' independence from the grid. The result indicated that 203 000 and 210 000 cells were sufficient for the uncontrolled and active flow control (AFC) cases, respectively. The effect of grid accuracy on the aerodynamic coefficients in the desired flow is depicted in Fig. 5. The grid size boundaries are obtained from numerical simulations of Asgari and Tajfar¹⁰ and Wang et al.⁴ Asgari and Tajfar employed 100 000 cells for the case without jet and 150 000 cells for the AFC case, and our simulation employed more cells than both. In most areas, the value of $Δ y +$ is below 1.0 and is limited to a maximum of 3, and the maximum value of $Δ x +$ is 40 on the airfoil surface. The $Δ y +=y u τ/υ$ and $Δ x +=x u τ/υ$ are the non-dimensional distances (based on the local cell fluid velocity) from the wall to the first mesh node and the distance between two nodes of the wall in the streamwise direction, respectively. In these definitions, $u τ= τ w / ρ$ is the frictional velocity determined based on wall shear stress. All presented results are obtained after at least two time cycles from the beginning and by reaching a repetitive pattern in time for aerodynamic coefficients. The lift and drag coefficient variations during flow time and their periodicity for the uncontrolled case are shown in Fig. 6.

We compared the results of the present numerical simulation with the experimental data of Lee et al.,¹⁷ the URANS results of Wang et al.⁵ and Geng et al.,²⁵ and the LES results of Boye and Xie.⁸ These comparisons in the aerodynamic coefficients (⁠ $C l$ and $C d$⁠) have been presented in Fig. 7.

As shown in Fig. 7, our numerical simulation adapts well in the upstroke motion with experimental data but predicts stall earlier and with approaching the dynamic stall angle, the difference increases. Also, the general trend is maintained in the downstroke motion. However, our results, similar to Wang et al.'s and Geng et al.'s data that used URANS methods, have fluctuations in these areas that are not observed in the experimental data. What makes simulation difficult near dynamic stall angle and move downstroke is the existence of a complex flow including separation bubble bursting, transition, multiple vortex shedding, and flow reattachment that occurs after separation of dynamic stall vortex and the onset of vortex shedding. The simulation has captured the main structures and is suitable for the investigation of general behavior and general parameters. Also, we believe that the smoothness of the experimental data is because of averaging over eddies. Asgari and Tajfar¹⁰ also mentioned this issue. Previous numerical works have had similar differences with experimental results, so our results are acceptable both quantitatively and qualitatively levels. An example of that is the numerical results of Wang et al.,^4,5 which confirm the accuracy of our simulation (see Fig. 7 of Wang et al.⁴ or Figs. 7 and 8 of Wang et al.⁵). Other researchers have also reported such discrepancies in their result simulation with experimental data in dynamic stall conditions; for example, see Fig. 4 of Gharali and Johnson⁶ or Figs. 9 and 10 of Tadjfar and Asgari.¹⁰ By comparing different turbulence models, Wang et al.⁵ showed that in the URANS approach, the k–ω SST model is the best turbulence model for simulating airfoil dynamic stall and matching the experimental data. These differences in k–ω SST model results with experimental data are attributed to the addition of SST limiter, which causes an abrupt decrease in eddy viscosity.⁹ Even large eddy simulation approaches do not perform well when the airfoil experiences dynamic stall conditions and have similar disparities that can be seen in the LES research of Boyi and Xie⁸ in Fig. 7. Moreover, other LES studies, for instance, Fig. 6 of Rajasekharan and Farhat⁵⁷ or Fig. 16 of Kasibhotla and Tafti,⁵⁸ have similar disparities. The reason is that DNS simulation can only capture physics detail and effective complex structures. Due to the computational cost of DNS simulation, it cannot be generalized for industrial applications and optimization usages.⁸ See Kasmaiee et al.⁵² and Tadjfar and Asgari¹⁰ for more details about simulation validation. Therefore, the difference between the present calculation and experimental results is reasonable and present results are acceptable both quantitatively and qualitatively levels.

Design parameters are entities that affect the final answer of the algorithm. They play an essential role in the optimization problem, and their proper selection plays a key role in the algorithm reaching the optimal solution faster. The smaller the number of these parameters, the smaller the problem state space. As a result, the costs required to find the optimal values are reduced, but the effect of the removed parameters is not seen in the results. Based on previous research,^{10,41,45,59–62} we extracted the acceptable and applicable range for the momentum coefficient. After that, the design parameter ranges were determined for optimization. In our problem, these parameters are the same as jet control parameters that were selected according to Kamari's study:⁴¹ jet location (at 0.1%–60% of chord length), jet-opening length (0.05%–0.3% of chord length), suction velocity magnitude (0 to $5 U ∞$⁠), and suction angle with respect to chord line (0°–180°). The design parameter ranges are summarized in Table I. The results were also analyzed by the suction angle relative to the airfoil surface. These two angles are shown in Fig. 1(b).

Initially, a database of 50 samples with randomly selected design parameters was prepared. Then, two neural networks were trained using this database, one for the drag coefficient and the other for the lift coefficient. Due to the nature of neural networks, their training and performance depend on the value of weights, particularly their initial values, which are chosen randomly. To ensure that the networks were properly trained and provided accurate estimates of lift and drag coefficients, the regression of the total data (training, validation, and test data), which measures the network's error in training and prediction, was evaluated. If it was below the limit, the training process was repeated until the desired condition was achieved. This limit is determined by trial and error (in this research, it was considered 90% for the drag coefficient and 95% for the lift coefficient). By doing this, the correct performance of the networks was ensured. A genetic algorithm (GA) was applied to find the optimal values of the design parameters. GA evaluated individuals in each generation using the objective function, whose required quantities were obtained from neural network outputs. After determining the optimal values using GA, a simulation was performed using these parameters. The convergence condition was checked to ensure that GA had accurately determined the optimal values of the design parameters. Changes of less than 5% for design parameters in the last ten cases in the database or no improvement in average aerodynamic performance compared to the last ten cases in the database were considered as convergence criteria. If the convergence condition was met, the optimization process was finished. Otherwise, CFD simulation results were added to the database. This process was repeated until convergence was established. The overall flow chart of optimization is presented in Fig. 8.

A genetic algorithm (GA) is a metaheuristic algorithm based on natural selection and belongs to the class of evolutionary algorithms (EA).⁶³ GA is commonly used in optimization problems due to its success in finding optimal or near-optimal values and its ability to escape local optima. It has three biologically inspired operators: mutation, crossover, and selection. This population-based algorithm requires an initial population for the first generation. The fitness value (the reciprocal of the objective function value) is used as the criterion for evaluating individuals. Selection is used to determine both parents and survivors. Parents generate new offspring through crossover, while mutation alters some of the genes in the offspring. Mutation allows for escape from local optima but decreases the convergence rate, so its occurrence probability should not be high. The algorithm flow chart is shown in Fig. 9.

Due to the wide range of design parameters, the number of generations and the population size of each generation should be large enough to cover the state space. Therefore, we considered 150 generations and a population size of 800 for each generation. Two-point crossover and uniform mutation were used with probabilities of 0.8 and 0.01, respectively. The best 5% of each generation was passed on to the next generation. Stochastic uniform and ranking were chosen as the selection method and fitness scaling, respectively.

The purpose of using a neural network in this study was to reduce the number of CFD simulations. In other words, we used machine learning to obtain the value of the objective function without the need for simulation. Two MLPs were trained, one to predict the mean lift coefficient and the other to predict the mean drag coefficient. The number of hidden layers was set to four for both MLPs, according to the number of design parameters. After several investigations, it was found that four neurons in each hidden layer were sufficient for both MLPs. The network topology used is shown in Fig. 10. The activation function of output layer neurons and hidden layer neurons was considered linear and sigmoid functions, respectively. Initially, MLPs were trained with 50 data points in the database, and as data were added, the database was updated and the MLPs were retrained using an online learning methodology. Levenberg–Marquardt and mean squared error were chosen as the training algorithm and measure of network performance, respectively. The number of epochs was set to 240, and 1 × 10⁻¹⁵ was selected as the network convergence criterion. The data were divided into three categories during the training and evaluation of the networks: training, validation, and testing. Training data were used to train the network and correct weights based on its error. Validation data were used during the training phase to complete the learning process, prevent overfitting, and halt the process when generalization stopped improving. Testing data examined the network's performance on non-training data. In other words, the first two groups were used during the training phase, while the third group was used after learning was completed. In this study, 70%, 15%, and 15% of the available data in the database were considered training, validation, and test data, respectively. The data were randomly divided into these three categories during each new training process, so it was shuffled and classified randomly.

In this relation, $y j actual$ is the original data and $y j$ is the guessed data and $y j ¯ actual$ is the average of the original data. The maximum network error was 24% for the drag coefficient and 26% for the lift coefficient. However, it is crucial for us to recognize all the data and general trends, which it can predict with high accuracy. Therefore, these networks can predict trends correctly and can be used in optimization. Finally, after optimization, we can simulate the optimal case.

The trained neural networks were made as efficient as possible. Any function can be approximated with a single hidden layer and infinite neurons. However, adding more hidden layers allows greater flexibility and reduces the required neurons. Thus, networks with multiple hidden layers can learn faster and more efficiently. Four hidden layers were chosen because there are four independent physical parameters. Four-layer networks with different numbers of neurons in each layer were tested, and the results are shown in Table II. As shown, when the number of neurons in each hidden layer is 3, the training accuracy is low, indicating that the network is underfitting and cannot capture the complexity of the training data for the two neural networks that predict the drag and lift coefficients. When the number of neurons in each layer is 5, the training accuracy is high, but the test accuracy is low, indicating that the network has too many free parameters and is overfitting. When the number of neurons in each layer is 4, the training and test accuracies are similar and reasonable, implying that the network has sufficient neurons.

The value of the fitness function in terms of the number of generations for the last optimization loop is presented in Fig. 13 to investigate the convergence of the genetic algorithm. According to the diagram, it is clear that the problem state space was well-searched. The algorithm was stopped before reaching a certain number of generations due to the lack of improvement in the best available individual in several consecutive generations. The logical difference between the value of the best individual fitness function and its average for individuals in the initial generation and reaching almost the same value at the end of the algorithm indicates that early convergence and getting stuck in the local extremum did not occur.

After the implementation and convergence of the optimization algorithm, the design parameters for the optimal suction jet were obtained, and their values are presented in Table IV. The optimal jet significantly improved aerodynamic performance, especially the average drag coefficient, which was reduced by more than 20 times. Convergence and variations in the design parameters during the optimization algorithm in terms of optimization loop iterations are shown in Fig. 14. The initial database contains 50 data points whose design parameter values were obtained using a random search in the problem state space. It should be noted that each simulation case using an Intel^® Core™ i7-4790 CPU series system took about one day, and the entire optimization process (145 cases) took about 145 days.

Figures 14(a) and 14(b) show that the jet velocity and orifice opening tend toward their maximum values in the optimal case, resulting in the jet momentum coefficient approaching its maximum value. The orifice opening size tends to be minimal in the initial iterations due to the search for optimal values of other parameters. However, it can modify itself during the algorithm. Due to the density of points in Fig. 14(c), the optimal jet location is an interval between 3% and 6% of the chord, which is around the formation of the leading edge vortex. Angle and location changes are related so that small changes in jet angle can compensate for small location changes. This effect occurs for small changes close to optimal conditions. Kamari et al.⁴¹ also presented an interval for the optimal jet location to prevent static stall. In the present study, the flow situation becomes much more complicated, and a more extensive change interval can provide optimization conditions. In the present study, the flow situation becomes much more complicated because of the airfoil pitching motion, and a more extensive change interval for the jet location can provide optimization conditions with changing jet suction angle.

The variation in the jet angle with respect to the chord line during the optimization algorithm is presented in Fig. 14(d). As can be seen, although the algorithm converges and the other parameters converge after 110 iterations, the suction angle fluctuates. The acceptable range for the optimal angle is between 90° and 120°. The high and close values of average aerodynamic performance related to these angles make it difficult for the algorithm to converge to a specific value. This issue is presented in Fig. 14(e), which shows the mean aerodynamic performance in terms of jet angle. This figure specifies the range of 105°–115° as the optimal jet angle range. The best jet performance among the samples in the database is related to a jet with a suction angle of 107.531° and an average $( L / D ) ave=48.796$⁠. For a closer look, the suction jet angle relative to the airfoil surface in terms of the number of optimization loop iterations and the average aerodynamic performance in terms of this angle are shown in Figs. 14(f) and 14(g), respectively. According to these figures, the optimal value of the jet angle with respect to the airfoil surface (β) is between 80° and 100°, with 90° being the best angle. In the definition of the jet momentum coefficient, the sine of this angle is used, and since the sine changes for this range are less than 2%, this quantity tends toward its maximum for the optimal jet.

Figure 15(a) shows the average performance vs jet location. There is a sizable functional difference in the optimal location range. Three jets with approximately similar locations but different performance coefficients were selected to investigate this issue and are shown in the figure. The jet's momentum coefficient difference has caused the jets not to act similarly. The parameters of these jets and their momentum coefficients are summarized in Table V. The velocity contours and streamlines around the airfoil in the upstroke motion at an angle of attack of 25° for these cases are shown in Fig. 16. As shown in Fig. 16, the uncontrolled case and case C had significant separations and formed several vortices. In both the uncontrolled case and case C, the dynamic stall vortex (DSV) formed from the leading edge vortex (LEV) and the turbulent separation vortex (TSV) and moved downward after separation. In the uncontrolled case, both the DSV and trailing edge vortex (TEV) can be seen, but in the case of C, the jet causes the two vortices to combine to form an enormous vortex. Cases A and B have no dynamic stall due to a lack of DSV formation. Both cases have trailing edge separation behind the airfoil due to the high angle of attack, but flow separation is less in case A. Less separation in case A is due to its high output jet momentum coefficient. Increasing the jet momentum coefficient increases its effect on flow and performs better in controlling flow separation. It should be noted that the three vortices of LEV, TSV, and DSV move clockwise, while TEV is a counterclockwise vortex.

Figure 15(b) presents the average performance vs the jet momentum coefficient. Despite having a high momentum coefficient in some jets, the average performance has not improved. As shown in the figure, two jets with almost identical momentum coefficients but different performances were selected to investigate this. The difference in jet location has caused this significant difference. Information on jet parameters is given in Table V. The velocity contours and flow streamlines of these two cases are shown in Fig. 17. According to Fig. 17, in case D, no vortex is formed at the leading edge, and only the flow near the trailing edge is separated due to the high angle of attack. In case E, several vortices are seen, but a large vortex results from the combination of DSV and TEV. Due to the distance of the jet from the LEV, it could not prevent the formation of DSV and only caused the vortices to combine.

Figure 18 shows the lift and drag duty cycles (hysteresis curves) for three jets with different average performances compared to the uncontrolled case. The jet with the highest value of average performance (optimal case) and two other jets were evaluated. The parameters of the other two jets are summarized in Table VI.

As seen in Fig. 18, the behavior of case II is very similar to the uncontrolled case due to the jet being too close to the leading edge, causing it to move away from the LEV formation site. Therefore, it has not a significant effect on the flow. On the other hand, due to the sharp increase in the angle of the airfoil surface as it approaches the leading edge, the jet angle should increase sharply to maintain the jet momentum coefficient high. However, the jet angle of this sample has not changed much compared to other samples. As a result, this jet is less efficient than the other two jets. In contrast, the optimal case and case I do not resemble the model without a jet, and the jet's effect can be seen in them. The graphs of the two cases are similar during the upstroke motion, with only the optimized jet performing slightly better. However, the two jets are very different during the downstroke motion. In the optimal jet, flow is controlled so that the lift coefficient is higher than during the upstroke motion, and for the drag coefficient, we see a significant decrease in these areas and drag is negative at most angles. This phenomenon is not seen in case I which is less effective than the optimal jet due to its low value of jet momentum coefficient. This low momentum is due to its small jet-opening length.

For a more detailed examination, the vorticity contours for these jets are shown in Fig. 19. It is clear that at the beginning of the movement and AOA of 10° during the upstroke motion, all jets behave similarly to the sample without a jet, and there is not much difference in their vorticity contours. With increasing AOA, the effect of jets becomes more apparent. At AOA of 20° during the upstroke motion in the uncontrolled case, DSV is formed from the expansion of LEV and combination with TSV and moves downstream. DSV is forming and growing in case II, but in the optimal case and case I, there is no DSV formation and most flow remains attached. At AOA of 25° during the upstroke motion for the uncontrolled case, and case II, the dynamic stall occurs and DSV separates and TEV forms, but this does not happen for the two other cases. In case I, more flow separates than in the optimal case, and this issue causes us to see more separation during the downstroke motion. While for uncontrolled and case II at AOA of 23° during the downstroke motion, we see severe separation and a vortex with different sign at the trailing edge but with decreasing AOA flow begins to reattach and return to initial conditions.

Dynamic stall conditions in oscillating airfoils, blades, and wings reduce efficiency. Preventing dynamic stall can significantly improve performance, so determining the optimal conditions to avoid it is particularly important. In this paper, we optimized the variable parameters of a suction jet for a pitching NACA0012 airfoil in deep dynamic stall conditions at a Reynolds number of $1.35× 10 5$⁠. These parameters include the suction jet angle, output jet velocity, orifice length, and jet location. We used the genetic algorithm as the optimization algorithm and employed machine learning to reduce the number of CFD simulations due to the population-based nature of this algorithm. Since the evaluation criterion in the genetic algorithm depends on the average lift and drag coefficients, we trained two artificial neural networks, one for the mean lift coefficient and the other for the mean drag coefficient. The neural networks are multilayer perceptrons, and we used an online learning methodology to train them. We examined the total data regression and changes in the fitness function to check the accuracy of network performance and genetic algorithm, respectively.

The results demonstrate the efficiency of optimization based on machine learning. This method can enhance optimization efficiency and the magnitude of improvement achieved. The optimized jet with this method can control flow well and increase the performance coefficient (average lift to drag ratio during a period) by about 24 times. Additionally, this method can decrease the number of simulations needed for optimization, thus reducing computational time. The results also show that orifice length and jet velocity reach their optimal value by approaching their maximum value, which increases the jet momentum coefficient. Locations between 3% and 6% of chord length are best for installing the jet. This area is close to the leading-edge vortex formation site, and changes in jet angle can compensate for slight changes in location. The jet angle relative to the chord direction also has an optimal range between 90° and 120°, with 107.088° being the value the algorithm converged to in the optimal sample and 107.531° being the value for the best sample available in the database. Furthermore, by investigating the angle of the jet relative to the airfoil surface, we find that its optimal value tends toward 90° but still has an optimal range between 80° and 100°. Studies indicate that in our cases, two independent parameters are effective for a suction jet: the jet location and the jet momentum coefficient, but the momentum coefficient varies with three design parameters: jet angle, orifice size, and jet velocity. The optimal jet prevents the formation of leading edge vortex (LEV), turbulent separation vortex (TSV), and trailing edge vortex (TEV) and precludes the appearance of dynamic stall vortex (DSV). As a result, it significantly improves the value of lift and drag coefficients especially in the downstroke motion. Using this jet makes values of lift coefficient for moving downward higher than those moving upward, and we observe negative drag coefficient in most AOAs of moving downward. It also smooths out lift and drag duty cycles (hysteresis diagrams). The optimum jet can increase the average performance coefficient (average ratio of lift to drag during a period) by about 24 times with a significant part of this improvement related to the reduction of drag force. By using this jet average lift coefficient increases from about 0.58 to about 0.92, while the average drag coefficient decreases from about 0.23 to about 0.02.

To summarize, key findings include:

• This paper presents a novel optimization method for a suction jet control of a pitching NACA0012 airfoil in deep dynamic stall regime. The method achieves a remarkable improvement in the performance coefficient (average ratio of lift to drag) of the airfoil, which is increased by about 24 times. The method decreases the number of CFD simulations needed for the optimization, thus saving computational time and resources.

• The results show that the optimal values of the parameters are: jet location between 3% and 6% of the chord, jet angle between 90° and 120°, orifice length and jet velocity close to their maximum values.

• The optimal jet prevents the formation of the leading edge vortex (LEV), turbulent separation vortex (TSV), and trailing edge vortex (TEV). Therefore, it precludes the appearance of the dynamic stall vortex (DSV), resulting in higher lift and lower drag coefficients, especially in the downstroke motion.

The authors have no conflicts to disclose.

Saman Kasmaiee: Conceptualization (lead); Data curation (lead); Software (lead); Writing – review & editing (equal). Mehran Tadjfar: Resources (equal); Software (equal); Supervision (equal); Writing – review & editing (equal). Siroos Kasmaiee: Conceptualization (equal); Software (equal); Validation (equal); Writing – original draft (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.