Aircraft Thrust Vector Control Using Variable Vanes; Numerical Simulation and Optimization

Among the various thrust vector control methods, this study investigates the utilization of mechanical deviation of the nozzle by incorporating a flow deflector path in the flow emanating from the nozzle outlet which is applied in the aviation industry. The researchers conducted simulations to assess the deviation of the thrust vector using exit vanes within the nozzle. Two parameters, namely the thrust force ratio and the axial force ratio, were calculated to regulate the thrust vector. The results demonstrated that employing vanes successfully deviated the thrust vector, resulting in a reduction in force coefficients. Notably, vanes with lower angles proved more suitable for thrust vector deviation. To forecast the performance of the vanes based on their angle and nozzle pressure ratio (NPR), optimization and prediction were conducted. The prediction outcomes exhibited favorable agreement with the simulation results, confirming the accuracy of the prediction method. This research provides valuable insights into the thrust vector control mechanism employing mechanical deviation of the nozzle and underscores the significance of predicting vane performance for optimal thrust vector selection.


Introduction
The propulsion system is one of the most important components of a jet aircraft, and one of the important subcomponents of the propulsion system is the nozzle system.Changing the direction of movement according to the previous plan and changing the state of the jet plane during the flight, with the engine on, could be the main reasons for using the thrust vector control system [1].Among the thrust vector control methods are the injection of a secondary fluid in the divergent part of the nozzle, mechanical deviation of the nozzle, placing a flow deflector device at the nozzle outlet, side nozzles producing thrust, and controlling the thrust vector using different nozzles with variable flow rates [2].One of the thrust vector control methods commonly used today in the aviation industry is the mechanical deviation of the nozzle from the divergent area.This mechanism deviates the nozzle output flow more than the nozzle deviation due to the oblique shock and expansion waves.In this article, this trust vector control method has been studied.Jessen and Peters [1] compared two vector control methods with a moving nozzle.In both methods, the nozzle is moved mechanically with the difference that in the first case, the nozzle rotates from upstream of the throat, and in the second case, the nozzle rotates from downstream.
They found that the structure's weight becomes heavier in the second case, and the control torque increases by fifty percent.They mentioned that in this case, because the nozzle is rotated from the supersonic part of the flow, the efficiency of the nozzle is slightly reduced, and the wear increases in this area.Of course, this method has an outstanding advantage because a magnification factor greater than one is accessible in this case.Tiokal Company [2], after conducting various research and tests of thrust vector control, proposed the nozzle with a cut in the ultrasonic part as the second stage nozzle for the MLV SAT-1B-5A engine to NASA and expressed its advantages.Their method has a significant deviation of the thrust vector and also creates fewer loads on the nozzle.
Ikazah [3], during his research on thrust vector control methods in airplanes, concluded that the most optimal way to control the thrust vector is the mechanical deviation of the divergent part of the nozzle.His results showed that the fighter has a better angle of attack at a certain point of flight and a it has a certain load in the flights without changing altitude, which will reduce drag and thus reduce fuel consumption.Also, due to the deviation of the thrust vector, the take-off and landing distance of the plane could be reduced.
Eli and Bank [4] utilized numerical analysis to obtain the characteristics of thrust vector control and compared thrust vector control using a flexible nozzle and secondary injection.Using the SST k- turbulence model, they observed that the flexible nozzle provides more precise control in flight and reduces thrust, while one of the advantages of thrust vector control using secondary fluid injection is reducing the weight of the vector control system.Kiuchi et al. [5] analyzed the three-dimensional numerical analysis of the two-phase flow inside diverting nozzle whose converging part moves inside the engine compartment.Their two-phase analysis was based on the Euler-Lagrange method, and they used the  −  turbulence model.
Hoi Man et al. [6] investigated the transient flow inside a three-dimensional moving nozzle using Jameson's finite volume method.Their intended nozzle deviates from the initial angle of 0 degrees to the final angle of 20 degrees, only in the direction of the screw, with a constant speed of 90 degrees per second.They compared the results obtained from the flow field and the performance parameters of the nozzle in the transient and steady states at the same deviation angles.They concluded that the results obtained from the transient analysis are very similar to those obtained from the steady state analysis.Strom [7] at the Air Force Research Center successfully concluded that the moving-type nozzles with cutting in the supersonic region have a force magnification factor greater than one by conducting many experiments.Their experiments showed that the deflection angle of the thrust vector is greater than the deflection angle of the nozzle.
Thrust vector controlling is one way of guidance and control for applications which has recieved the attention in recent decades.These methods are divided into four main types: methods with multiple nozzles, moving nozzles, interference methods, and secondary injection methods.
Hot gas injection is one of the suitable methods of thrust vector control due to its small volume and high efficiency.In the past, this method was not used much due to the mechanical corrosion of the materials of the valves on the road, but today, with the advancement of metallurgy, this method is of great interest.Weather first proposed secondary fluid injection for thrust vector control in 1949, and its implementation was completed in 1952 [20].
In this field, analytical studies have more history than other research, and the basis of thrust vector control with the help of secondary injection is established.However, simplifying assumptions in this category leads to achieving unrealistic answers.The analytical model of the last wave was presented by Broadwell in 1963 [21].This model deals with the flow in a two-dimensional and non-viscous manner.This study considers the injection jet movement size a general parameter.In 1964, Sandor and Walker combined the linearized theory with the assumption of a low flow deviation angle to investigate the flow behavior with one-dimensional analysis [22].In 1964, Zakowsky and Spade [23] proposed a model under the title of a thick body.It has been observed that the injection of secondary flow in the supersonic flow produces a flow field resembling a thick body (like a step).
Richard Gauss conducted detailed studies on thrust vector control with secondary injection [24].This study showed that increasing the static pressure of the secondary flow, by keeping the rest of the parameters constant, transfers the separated shock further upstream and increases the produced thrust.In the field of experimental activities, the lack and sometimes lack of required hardware, the need for expensive hardware, problems caused by vibrations, the formation of strong shocks, and blockage of flow during execution are among the factors that have made this research difficult.
The ever-increasing development of computational methods and the remarkable progress made in this field have paved the way for numerical research.Many studies conducted in recent years have dealt with numerical work and validated their results with laboratory studies.From the first measures taken for numerical study was the investigation of this system by Ballou about the solving Euler's equations in three dimensions in 1991 [25].This research has been done by studying parameters such as the secondary jet amplification factor and added axial thrust.The ratio of lateral force to axial force is also one of the results of this research, which is in good agreement with the experimental results and has a maximum value.
In 2002, Hyun and Sapp Yun numerically investigated the efficiency of secondary gas injection into a conical nozzle [26].This research has produced a three-dimensional, viscous network with the organization and considered two Baldwin-Lomax and A-K algebraic turbulence models.The effects of the angle of the divergent part of the nozzle and the injection location on the system's efficiency have been investigated [27][28][29][30].
The responses of two-dimensional and three-dimensional flow and the efficiency of an ultrasonic nozzle with secondary injection are investigated [31].In these studies, which were carried out numerically, physical parameters such as the length of the separation region were investigated.Some integral parameters of the system efficiency were compared with the experimental and numerical results, which showed a good match between the results of the numerical program.
Salimi.et al. [32] focused on the numerical investigation of the impact of secondary injection geometry on the performance of a fuel-injected planar dual throat thrust-vectoring nozzle.
Different injector cross-sections (slot and circular) and four different fuels were considered, along with variations in the center-to-center distance of injection holes for circular injectors.
Their results showed that slot injection performs better in terms of discharge coefficient, thrustvector angle, and thrust-vectoring efficiency, while circular injection provided a higher thrust ratio.
The study conducted by Hamedi et al. has numerically modeled the double-port nozzle [33].
This research introduces the k- SST turbulence model as a suitable model for predicting flow details.The effect of the injection gap length on the nozzle vectoring has been investigated, which increases with the injection level and the deviation angle of the thrust vector.
Recently, the use of artificial intelligence in various applied problems is becoming common [34][35][36][37][38].However, the use of artificial intelligence in thrust vectoring has been overlooked.
Therefore, in the present study, we present some models to predict the output parameters using artificial neural networks (ANNs).In this research, the mechanical method (flow deflector) is employed to deviate the thrust vector, and the angle of the vanes is considered for different states.Simulations are conducted to study and analyze the effects of these angles.Additionally, artificial intelligence techniques are utilized to predict the thrust vector through simulation.

Governing equation
If the equations of conservation of mass, conservation of momentum, conservation of energy, and the equation of state of the perfect gas are used in the steady state and time averaging is used, these equations will be expressed as follows [39]: (3) where   is Kroniker's delta, E is the total energy,   =  +   is the effective thermal conductivity coefficient and (  )  is the deviatoric stress tensor, which is defined by the following relationship [39]: Using Bozinsk's assumption is a common method to establish a relationship between Reynolds stresses and average velocity gradients [39]: where   is the average strain rate tensor and is expressed by the following equation [39]:

RNGk-ε Turbulence model
Yakhut et al. [39] have presented a model of the model, whose performance features and features are optimized in streams with rapid strain rates and rotational currents compared to the standard model.It also performs better than the standard model in the currents with low Reynolds.Transfer equations in RNGK-ε are as follows [39]: The parameters of equations 8 and 9 are defined as follows [39]: Equation 11 indicates that it reflects the equilibrium characteristics of the disturbed flow field.
The ideal gross thrust is calculated from the following equation [40].
The ideal weight flow is also measured using equations 15 and 16.Equation 15 is for unchocked nozzle flow, while equation 16 is for chocked flow nozzle [40].

Geometry
In order to generate the flow for the present problem, a single-engine propulsion system was designed, and different thrust-vectoring nozzle configurations were tested.The air pressure provided by this system is 90 psi in the nozzle.Also, the temperature of the dry air was about 26.85 ºC.The nozzle studied during this research is symmetrical convergence.Figure 1   The three-dimensional view of this nozzle is presented in Figure 2. From this figure, it is clear that the vanes are 120 degrees apart from each other.Also, it is observed that deflecting these vanes could affect the outlet flow angle to control the thrust vector.The blockage caused by the vane angles could negatively or positively affect the nozzle exit flow.Also, the study is carried out in various NPRs, ranging from 2 to 6.09.

Boundary conditions and grid independency
The walls are considered adiabatic with no-slip conditions in the present simulation.As can be seen in Table 1, the simulations have been carried out for different mesh sizes.Due to three dimensional geometry of the control vanes the numerical simulation mus be done in three dimensions.Therefor, a three dimentional mesh is needed to generate.To compare the results and the effect of mesh size, / and   /, are presented, where   is the resultant force and  is the axial thrust force.The selected case for the mesh independency study is NPR=6.09.
The results show that the error of mesh size changes is negligible.Therefore, the selected case for the simulations is 356355.The error between the results obtained from thenumerical simulations using two mesh size of 356355 and the 383245 is less than 1%.Therefore, choosing the second case to run the rest of the simulations is logical.Figure 3 shows the generated mesh for the present investigation.It is clear that in the regions where the boundary layer is formed, the mesh size is so small.This would help to capture the dynamics of the flow with a higher accuracy.

Validation of the simulation results
The presented numerical simulation method is carried out for the three dimensional gemometry of the nozzle is shown in Figure 1, without any control vane variation.The results of this three dimensional simulation case are used for the validation of the present numerical simulation methodology.In order to validate the results of the present three dimensional numerical simulation, a comparative study of the results with the experimental results of Berrier and Mason [40] is presented in Figure 4.The results of the validation case are related to the vane with 0 angles.Figure 4 shows the axial force in different NPRs.The results show how the numerical simulations closely match the experimental results.The results have been carried out for a range of NPRs (2-6.09).The small differences between the numerical results and experimental data are due to simulation errors.Also, the results of the resultant force of the present simulations have been compared with the experimental results of Berreir and Mason [40] for the NPR of 2-6.09, as shown in Figure 6.
The results prove that the present three dimensional numerical simulation of the present study has a close agreement with the experimental measurements in Ref. [40].

Results and discussion
After the simulation metodology is verified as describe in privious section, now, the control vane angle variation are considered.The effects of vane angles on the velocity profile and streamlines are investigated for various NPRs.In addition, the results of the axial and resultant forces are presented.These results are studied and presented for different vane angles and NPRs.As mentioned, due to three dimensional geometry of the control vanes and their locations so, the numerical simulation must be done in three dimensions.
Figure 7 shows the contour of velocity for different NPRs.It should be noted that the vane angles of these cases are fixed at zero.It is clear that by increasing the NPR, the velocity of the outlet would increase considerably.This increase in the velocity would increase the axial and resultant thrust force.Since the vane angle in all the vanes is equal to zero, the axial thrust force is equal to the resultant thrust vector.Also, the resultant vector angle is 0 because there is no deviation in the flow as can be seen in Figure 7.  this is because there is no deviation form the horizontal axis, so the highest velocity is there.As can be seen from Figure 10, the velocity increases with the inccrease of NPR.The difference between this case with Figure 9 is since there are vane angles, the highest velocity is no longer at the horizontal axis.Instead, the vane angles cause a disturbance in the outlet flow.Thus, there are some wakes downstream of the outlet.As can be seen in Figures 11a and 11b, two different force components, axial and resultant thrust vectoring, are presented.Also, in Figures 11c and 11d, the thrust vectoring coefficients are compared for the resultant and axial forces.Figure 11a shows the resultant thrust vectoring.It is clear that in this configuration, only the third vane angle is changing, ranging from 0 to 25. Figure 11a shows that by increasing the third vane's angle, the resultant thrust vectoring decreases.
A similar trend is observed in Figure 11b for the axial thrust vector.It is observed that by increasing the vane angle, the axial thrust vectoring decreases.
Also, it is presented that increasing the NPR increases the reduction in the resultant and axial thrust forces compared to the conditions related to zero vane angle.This is because the drag force increases as the vane angles are increased.Also, as the vane angles augment, a friction force is introduced to the vanes that contribute to reducing the thrust vectoring.As the NPR increases, up to NPR=4, the coefficients get closer to each other in all cases.After NPR=4, thrust vectoring coefficients start to drift apart from each other.The maximum of value thrust vectoring coefficients among all cases occurs in NPR=5.The differences percentage in NPR=2 in two vane configurations of (0,25,5) and (0,0,0) is 8.77%.When the NPR=4, this difference is 10.87%; when the NPR=6.09,this parameter is 10.47%. Figure 11d also shows the ratio of axial thrust force to isentropic force in different NPRs.As can be seen in Figure 11d, the ratio of these forces increases with the increase of NPR in all modes of vane configuration.This process is similar to the case seen in Figure 11c until NPR=4.Approaching a growth slope tends to be zero, and in the case of NPR=5, the value of this coefficient shows the maximum, and after that, it starts to decrease with a negative slope.This ratio of axial thrust force to isentropic force shows the thrust force's efficiency, and the system performance is improved as it goes towards one.In the case of NPR=5, more efficiency is observed in the system.It is also observed that the best efficiency occurs in the case of vanes with a zero angle, and as the angle of the vanes increases inward (+), the values of this ratio decrease.The lowest value of this ratio was observed for vanes with (0, 25,25), which compared to vanes with zero angles at NPR=2, 4, and 6.09, respectively 10.31%, 12.64%, and 12.45% decrease.12b show axial and resultant thrust vector forces in different NPRs.It can be seen that with the increase of NPR, the values of thrust and thrust forces have increased.Also, by increasing the angle of the variable vane, the reduction of the thrust force has increased.For example, in the case of NPR=6.09 and with a vane angle of 25 degrees, the vertical and axial thrust vector values decreased by 21% and 25%, respectively.As can be seen in Figures 12c and 12d also show the ratio of resultant and axial force to isentropic force.As the NPR is increased up to 3, the growth slope of the force ratios is sharp, and when reaching NPR=4, the growth slope of the force ratios tends to zero.At NPR=5, for the ratio of output force and axial force ratio, a decrease of 20.40% and 24.64% is observed in the vane angle of 0, 25, 0 compared to the mode of zero vane angle.can be optimized to achieve enhanced performance metrics such as increased maneuverability, reduced fuel consumption, and improved stability.This data-driven approach allows for the utilization of large datasets to train models and make accurate predictions, reducing reliance on complex analytical models.Additionally, ANNs and machine learning algorithms can adapt in real-time, making them suitable for dynamic thrust vectoring applications that require on-the-fly adjustments based on sensor inputs.Lastly, these technologies provide robustness and fault tolerance by learning from diverse datasets that encompass different operating conditions, sensor noise, and uncertainties, enabling thrust vectoring systems to effectively handle variations and disturbances.Overall, the use of ANNs and machine learning algorithms in thrust vectoring offers the potential for improved control performance, adaptability, and optimization, making them valuable tools in this field.

Artificial neural network
The artificial neural network is a deep learning method designed based on the human neural system.It consists of an input layer, a group of hidden layers, and an output layer through which the feed-forward process occurs.Each layer comprises a number of nodes called artificial neurons, the basic computational unit in the neural network.As is shown in Figure 14a, each neuron receives several signals (called "input"), carries out some calculations on the input signal and their weights, adds the bias to the summation of input signals, passes the whole summation to an activation function, and generate one output signal.There are lots of activation functions such as rectified linear unit (ReLU), Sigmoid, Linear, Softmax, and hyperbolic tangent functions.Figure 14b depicts the most common activation functions in regression problems.The most popular choice of activation function seems to be ReLU, as it does not suffer from the vanishing gradient problem [41].To understand how well the model works, the backpropagation technique is used.In this process, after each epoch (i.e., when the feed-forward process completes one pass through the whole training dataset), the loss function is calculated, and then the weights and biases are updated using a gradient descent optimization in order to minimize the loss function.
The model selection process is probably the most important section of a neural network because the parameters investigated and the decisions made in this section would directly affect the final output of the model.To discover the ideal model, we should investigate and optimize different architectures and hyperparameters, including the number of input parameters, number of neurons, number of hidden layers, activation, and loss functions [36].It should be noted that the deepest and the largest network possible would not necessarily be the best choice.For example, a neural network with a few numbers of hidden layers cannot perform sufficiently well.On the other hand, by excessively increasing the hidden layers, the model would fall into the over-fitting problem.
Prior to constructing the ANN, the preprocessing of data is done.The input parameters are normalized in a range of zero to one.The dataset is split into training and test data points with a 70%-30% split ratio.
Table 2 shows the optimized ANN's hyperparameters used in the present study.The learning rate is set to 0.001.Of course, larger values could be chosen to achieve convergence faster, but this could simultaneously cause disturbance near the optimum point.The Adam optimizer [42] is selected as the gradient descent solver in this study.Finally, the exponential first and secondmoment vector is set to be 0.9 and 0.999, respectively.Mean squared error We use the mean squared error (MSE) as the loss function, which is calculated as: The model accuracy is based on the mean absolute error (MAE), which is measured as: In Equations 17 and 18, n is the number of data points,   is the real value of the target quantity, and  ̂ is the predicted value of the target quantity by the ANN.
We use the coefficient of determination (R-squared or  2 ) to show how well the model is capable of predicting the target quantity.If  ̅ is the mean of the data,  2 is defined as: The  2 value ranges between 0 and 1, where zero is the case when the model is unable to predict the target quantities.On the contrary, the R-squared of one is equivalent to the best model, which predicts all the cases correctly.

The ANN results
In the present study, the ANN model is utilized to predict four key parameters of the thrust vectoring problem.These four parameters are: /  ,   /  , , and   .Prior to the presentation of the results, the hyperparameter tuning of each model is indicated.Hyperparameter tuning is the study of the key factors in the ANN model and finding the optimum state.Hyperparameters are the number of neurons, number of hidden layers, activation functions, number of epochs, batch size, and learning rate.In this work, we have used hyperparameter tuning to optimize our models and get the best predictive capabilities.In the following, the hyperparameter tuning of   is presented.

The ANN model selection procedure
After the preprocessing stage, the local wall temperature dataset consists of 60 numerical results.The dataset is split into training and test data points with a 70%-30% split ratio, resulting in 40 training data points and 20 test data points.The next step is to determine the number of hidden layers.The widely used doubling sequence is utilized for the number of hidden layers, starting with 32 neurons.To begin with the process, the number of neurons and the number of hidden layers must be investigated as they are the principal components of the neural network.Table 3 presents the results for different numbers of neurons and hidden layers, using ReLU as the activation function of all layers.It is obvious that by increasing the number of neurons and the number of hidden layers, the model becomes more accurate; however, this improvement is stopped, and an opposite pattern is observed when overfitting occurs.Therefore, the selected model is (32,64,128,256,128,64,32).After selecting the number of hidden layers, the impact of different activation functions for the output layer is investigated in Table 4. Apparently, using both Linear and ReLU results in the same MAE.However, for the sake of consistency, the ReLU activation function is utilized for the local wall temperature.
Another contributing element in the model architecture is the batch size, i.e., the number of data points going through the feed-forward process before the backpropagation begins.In other words, the batch size is the number of data points that are processed in the neural network before the weights and biases are updated.According to Table 5, the mean absolute error of both batch sizes of 8 and 16 are the same.However, the selected batch size is 16 because this choice is less time-consuming.Finally, the number of epochs is compared with each other, as shown in Table 6.The best choice is the model with 20,000 epochs.20,000 is picked over 30,000 or higher epochs despite the same MAEs because the computational expenses in the former are way less than the later ones.It is evident that the best choice for hyperparameters is not necessarily the deepest ANN or the slowest model, so it is logical that a search for the best model architecture should be done prior to model selection.This selection procedure is carried out to identify the best model architecture for every single target quantity in this study.The final selected models for all of the quantities of interest are presented in Table 7.These models achieve the best performance among the examined cases.The results show the deviation of the numerical and the predicted parameters from the ideal case, which is  = .

Conclusion
In present research, the issue of thrust vector control for an aircraft engine has been investigated using three control vanes consept (vanes are installed at an angle of 120 degrees to each other).
Undoubtedly, in operational conditions, it is very important to have a predictive model of thrust vector as a function of input parameters such as Nozzle Pressure Ratio (NPR) and vane deflection angles.In this research, Artificial Neural Network (ANN) technique was used to achieve this predictive model.Thus, a three-dimensional model of the engine exhust nozzle along with three control vanes was prepared.Flow simulation was performed on the existing three-dimensional model, using Computational Fluid Dynamics (CFD).After validating the numerical results using the available experimental data, simulations performed in different values of the pressure ratio and varios vane angles.The results of the simulations were used to train the neural network model and the predictive function was extracted.The thrust vector function was implemented, and the prediction results exhibited a strong agreement with the simulation outcomes, affirming the accuracy of the prediction methodology.
is a plan that shows the geometry of the nozzle.The nozzle has an area of 3.145  2 , the ratio has an expansion of     = 1.482 ⁄ and the Nozzle Design Pressure Ratio (NPR) des = 6.09(the NPR for a fully expanded executor in the nozzle exit) based on the experimental study presented in Ref. [40].The thrust-vectoring process was tested with three vanes mounted downstream of the nozzle.The vane angles are changed in different simulations from -25 to 25degrees.The effects of these changes on the nozzle thrust vectoring performance are investigated in the present study.In the three-vane concept, which is studied in the present study, the vanes are mounted 120 degrees apart.The simulations showed the achievements of pure yaw for this configuration of vanes which requires the deflection of two vanes or even all of them.Using various vane angles, the resultant axial forces are studied in the following.

FigureFigure 1 .
Figure1ashows the schematic view of the convergent-divergent nozzle investigated in the present study

Figure 3 .
Figure 3.The three dimensional mesh generation of the geometry

Figure 4 .
Figure 4.The comparison of the axial force in the present numercal simulations with the experimental data of Ref. [40]

Figure 5 .
Figure 5.The comparison of the air flow rate in the present simulations with the experimental data of Ref. [40]

Figure 6 .
Figure 6.The comparison of the resultant force in the present simulations with the experimental data of Ref. [40]

Figure 13 .
Figure 13.The effect of NPR for different vane angles configurations on (a) resultant thrust force (  ), (b) axial thrust force (), (c) resultant thrust coefficient (  /  ), and (d) axial thrust coefficient (/  ) Artificial Neural Networks (ANNs) and machine learning algorithms are instrumental in the field of thrust vectoring for several reasons.Firstly, these technologies excel at modeling the complex relationships inherent in thrust vectoring systems, capturing the intricate interplay between nozzle geometry, flow conditions, and control inputs.Secondly, ANNs and machine learning algorithms are well-suited for handling the nonlinearity and dynamics associated with thrust vectoring, adapting and learning from data to identify and control nonlinear systems effectively.Moreover, by leveraging machine learning techniques, thrust vectoring systems

Figure 14 .
Figure 14.(a) A schematic model of an artificial neuron and (b) The most common activation functions for a regression problem

Figure 15 .5. 2 . 3 .
Figure 15.The results of the ANN model for   with the hidden layer structure 32,64,128,256,128,64,32 and input parameter of ,  1 ,  2 ,  3 5.2.3.The ANN model's axial thrust vector Figure 16 shows the ANN model for the axial thrust vector.The predictive model for the axial thrust vector comprises of 32,64,64,32 hidden layer structure.The model has obtained the MAE of 0.89% and an  2 of 0.99.Similarly, the input parameters are ,  1 ,  2 ,  3 .Also, the hyperparameters for this model have been optimized.The number of epochs for this ANN model is 30,000.The batch size of this model is 16.The activation function of the output layer is sigmoid, but all other activation functions for this predictive model are ReLU.

Figure 17 . 3 Figure 18
Figure 17.The results of the ANN model for   /  with the hidden layer structure 128,128,64,64,32,32 and input parameter of ,  1 ,  2 ,  3 Figure 18 shows the results of the predictive model for /  , and how the ANN model predicts the results.The hidden layer structure of this model is 512,256,128,64,32, which is able to have the MAE of 1.08% and the  2 of 0.99.The results of the model are so close to the ideal case,  = .This model had 40,000 epochs with a batch size of 16.Also, all the activation functions are ReLU.

Figure 18 .
Figure 18.The results of the ANN model for /  with the hidden layer structure 512,256,128,64,32 and input parameter of ,  1 ,  2 ,  3

Table 1 .
Mesh numbers in order to gain grid independency

Table 2 .
ANN model parameters chosen in the present study

Table 3 .
ANN model predictions for fixed input parameters and different combinations of hidden layers

Table 4 .
ANN model predictions for fixed ANN model and different activation functions for the output layer

Table 5 .
ANN model predictions for a fixed number of hidden layers and different batch size

Table 6 .
ANN model predictions for a fixed number of hidden layers and different numbers of epochs

Table 7 .
Finally selected modelsThe ANN model for the resultant thrust vector is presented in Figure15.It is clear that this model was able to capture the physics of the flow.Hence, the accuracy of the model is calculated.It seems that the model with a hidden layer structure of 32,64,128,256,128,64,32 was able to achieve the MAE of 0.18% and  2 of 1.This show how accurate the model is.The input parameters for this model are ,  1 ,  2 ,  3 .Also, the number of epochs is 20,000.The batch size of this model is 16, and all the activation functions of layers are ReLU.The 30% of the data which are unseen in the training of the model are used to evaluate the ANN model.