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Abstract

During the operation of an active clearance control (ACC) system of a turbine, the aerodynamic performance of the intake grille indirectly influences its control. To improve the performance, an aerodynamic optimization method is proposed, consisting of parameterization, an optimization algorithm, and a fitness evaluation. During parameterization, its geometry is represented by seven geometric variables. A modified social spider algorithm is used as the optimization algorithm. To evaluate the aerodynamic performance of the grille, a special fitness function is adopted, obtained using an adaptive topological multi-layer feedforward artificial neural network. To verify the feasibility of this method, experiments and numerical calculations are carried out on the original and optimized intake grilles. The results show that the average intake flow rate and average total pressure recovery coefficient of the optimized grille have increased by 17.3% and 4.9%, respectively.

Keywords

ACC intake grille aerodynamic performance optimization MSSA MLANN

Introduction

The gas-turbine is a core component of civil aero engines, converting the energy of a high-temperature high-pressure gas into mechanical energy. However, its efficiency, performance, and service life are easily affected by its blade-tip clearance.^1,2 Active clearance control (ACC) technology tightly controls blade-tip clearance during engine operation, so as to improve engine performance, reduce fuel consumption, and increase service life.³ Among all feasible ACC methods, the thermal control ACC method is the most widely used. The working principle of the ACC of a high-pressure turbine (HPT) is to control the thermal deformation of the turbine casing.^4,5 Turbine casing is usually cooled or heated by intake airflow at various temperatures under the action of a high-pressure compressor or external bypass, resulting in radial deformation of the casing.⁶

Generally, the character of a thermal control ACC system exists in two main aspects: the outlet cooling character and the inlet intake character.^7,8 Various studies have focused on the outlet cooling characters of the ACC. The numerical simulations have been validated to be available,⁹ for example, have been used to examine the outflow characteristics and heat transfer characteristics of impinging holes of a cooling pipe with circular section for the ACC of a low-pressure turbine.¹⁰ The insulating coverage effect of a perforated impact-jet on the surface of casing with circular cross-section flat cooling air pipe has been explored.¹¹ Work has also been conducted on the influence of flow mode on the internal heat transfer characteristics of ACC casing in a high-pressure turbine.¹² However, very little research has been conducted on the inlet intake character of an ACC.

With the fast development of computational technology, various optimization theories have been investigated and some applied to the aerodynamic performance and stability optimization of machines.^13,14 Optimization is usually divided into three procedures: shape parameterization, deriving the optimization algorithm, and fitness value evaluation.¹⁵ Honing these procedures should result in the effectiveness and efficiency of optimization.¹⁶ During optimization, the dimensions of its design variables are completely determined using parameterization methods.¹⁷ Good parameterization methods have things in common, such as enough flexibility to cover the whole search space, few design parameters, and the ability to avoid curvature discontinuities at a junction.¹⁸

Among all the optimization theories, intelligent algorithms are the most popular. These are divided into two main categories: evolution algorithms and population algorithms.¹⁹ Population algorithms have a smaller control coefficient but a faster search speed. The particle swarm optimization algorithm (PSO), a traditional swarm algorithm, is widely used in the optimization of machines. However, when dealing with multidimensional optimization problems, the algorithm readily produces unwanted, local extreme values and does not converge, therefore, to a global optimal value.²⁰ To mitigate this problem, the social spider algorithm (SSA), a new population algorithm, was proposed by Yu and Li.¹⁹ Yu and Li²¹ showed that the SSA could deal with various high-dimension optimization problems. Unlike other population intelligence algorithms, the SSA could move toward the target using the current position of other spiders on the spider web and its own previous position. Although it exhibited a considerable searching ability for multiple extreme optimization problems, the convergence velocity and precision of the algorithm were weakened.

As well as the parameterization and the optimization algorithm, the fitness value evaluation is also very important. The fitness value is usually calculated using the aerodynamic performance parameters of the flow field. However, the computational consumption of the flow field calculation could not be stood in every case.²² To overcome this problem, a surrogate model has been proposed, in which the complex nonlinear relationship between the geometric parameters and aerodynamic performance are taken into account.^23–25 Owing to its significant approximation capability, a multilayer feedforward artificial neural network (MLFANN)²⁶ is often used as a surrogate model. An MLFANN’s approximation capability is mainly influenced by its topology structure.²⁷ To optimize the topology structure of an MLFANN, various global intelligent optimization methods have been adopted, such as genetic algorithms (GAs),²⁸ simulated annealing (SA),²⁹ and particle swarm optimization (PSO).³⁰ Although these methods can improve the approximation capability of an MLFANN by optimizing parts of its topology structure, there is still no way to determine a high-efficiency topology.

The intake grille is an effective solution for better aerodynamic performance.³¹ To improve the intake performance of an ACC intake grille, an improved aerodynamic performance optimization method is proposed in this work. This method is divided into three parts: parameterization, the optimization algorithm, and fitness value evaluation. Using the commercial software NX 12.0, the intake grille is parameterized using seven key variables. To improve the convergence velocity and precision of the original SSA, a modified SSA is proposed. In this modified algorithm, an adaptive factor and an adaptive mutation operator are added to the original SSA to increase its search velocity and decrease its loss in population diversity. In order to calculate the fitness value of the grille quickly, an MLFANN is used as a surrogate model. The topology structure of the MLFANN can be determined adaptively using an improved hybrid intelligent optimization method. To verify the feasibility of the aerodynamic optimization method, a practical intake grille is optimized.

Aerodynamic optimization methodology

Parameterization of intake grille

The intake airflow of the ACC is shown in Figure 1. The bypass bottom is arc-shaped, and the top surface of the intake grille is coplanar with the inner bottom surface. Figure 1(a) shows that a proportion of the cooling airflow in the bypass is imported to the ACC by the intake grille. Figure 1(b) shows a geometric model of the intake grille, which has four arc-shaped blades. The intake performance of grille is affected by its geometry.³² To realize the geometric deformation of the grille, it is parameterized using key geometrical parameters with the commercial software NX 12.0. Figure 2 shows the seven geometric parameters selected to describe the geometry of the intake grille: the inlet angle, outlet angle, chord, interval 1, interval 2, interval 3, and interval 4.

Figure 1.

ACC intake: (a) intake process and (b) intake grille model.

Figure 2.

Parameterization of the intake grille.

Modified social spider algorithm (MSSA)

For high-dimension problems, the SSA,^19,21 a swarm intelligent algorithm, has been proposed. The spider web is the solution space of the optimization problem, and each spider on the web represents a feasible solution. To find the position of a food source, spiders share the information through vibration. Then, using its own historical vibration intensity and other spiders’ current vibration intensities, a spider can choose its direction of movement to perform a random walk. The algorithm is divided into four main parts: initialization, vibration generation, random walk, and constraints control. The first is to set up control parameters and randomly assign positions to spiders. Then, their fitness values are evaluated. During the vibration generation, its intensity of vibration in its current position is defined using equation (1), and its vibration attenuation is calculated using equation (2). Based on the vibration, the random walk of a spider is described by equation (3). Before a spider’s random walk, the constraint in equation (4) is used to prevent spiders from moving out of the search space.

I (P_{s}, P_{s}, t) = \ln (\frac{1}{f (P_{s}) - C} + 1)

(1)

I (P_{a}, P_{b}, t) = I (P_{a}, P_{a}, t) \cdot \exp (- \frac{‖ P_{a} - P_{b} ‖}{\bar{σ} γ_{a}})

(2)

P_{s} (t + 1) = P_{s} + (P_{s} - P_{s} (t - 1)) \times r + (P_{s}^{0} - P_{s}) ⊙ R

(3)

P_{s, i} (t + 1) = {\begin{matrix} ({\bar{x}}_{i} - P_{s, i}) \cdot r, if P_{s, i} (t + 1) > {\bar{x}}_{i} \\ (P_{s, i} - x_{i}') \cdot r, if P_{s, i} (t + 1) < x_{i}' \end{matrix}

(4)

where $I (P_{s}, P_{s}, t)$ is the vibration value generated by a spider at position $P_{s}$ and time iteration number $t$ ; $f$ is the fitness value of the spider in its current position; $I (P_{a}, P_{b}, t)$ is the attraction intensity of spider a to spider b at time $t$ ; $C$ is a constant, satisfying $f (P_{s}) - C > 0$ ; $\bar{σ}$ is the standard deviation of each dimension of the spider population; $γ_{a}$ is the attenuation rate of the vibration intensity; $r$ is a random value from zero to one; $t$ is the current iteration number; $T_{\max}$ is the maximum iteration number; ${\bar{x}}_{i}$ is the upper bound of the $i$ dimension of the search space; and $x^{'}$ the lower bound of the $i$ dimension of the search space.

However, in the original SSA, there are two disadvantages that affect global optimization. One is that, during optimization, an individual spider uses its own information to randomly move toward the position of maximal vibration, where it can feel. This movement reduces the convergence speed of the algorithm, limits its global optimization ability, and reduces the adaptability of the iterative process. The other disadvantage is that individuals aggregate in multiple regions and in multiple solution spaces after combining their own experiences and information from the population. To provide solutions to these two problems, an adaptive factor and an adaptive mutation operator are adopted in the SSA. The equation for the adaptive factor $R^{*}$ is given by equation (5) and used to replace $R$ in equation (3). The use of the adaptive factor gives an individual spider a large search step to quickly search the whole space at the early stage of an iteration, and to have a smaller search step to quickly converge the optimal at the late stage of iteration. Once the position of the strongest vibration falls into the local optimum, there is a rapid decrease in population diversity, causing the multiple regions aggregation of spiders. To overcome this problem, the adaptive mutation operator in equation (6) has been proposed to disturb the position of strongest vibration. It is a combination of Gaussian mutation³³ (which has a strong local development ability and a fast convergence speed in the later stage of evolution) and Cauchy mutation³⁴ (which has a strong global search ability and can effectively restrain the loss of population diversity). To make the mutation own adaptivity, an adaptive factor is used in this process. In the early and late stages, Cauchy mutation and Gaussian mutation, respectively, dominate. Details of the modified social spider algorithm (MSSA) are shown in Figure 3.

\frac{R^{*} = 1}{1 + e^{(\frac{14 t}{T_{\max}} ())}}

(5)

P_{gbest}^{*} = P_{gbest} + ζ \cdot [R^{*} \cdot C (0, 1) + (1 - R^{*}) \cdot G (0, 1)]

(6)

where $t$ is the current iteration value, $T_{\max}$ is the maximum iteration value, $R^{*}$ is the adaptive factor, $C (0, 1)$ is the Cauchy mutation, $G (0, 1)$ is the Gaussian mutation, $P_{gbest}$ is the position of strongest vibration, and $ζ$ is the coefficient related to the boundary of the search space.

Figure 3.

The process of MSSA.

Surrogate model

Fitness function

In this work, aerodynamic optimization searches the intake grille with optimal aerodynamic performance. Two coefficients are used in the assessment of the aerodynamic performance of the intake grille, the total pressure recovery coefficient $C p_{t o t a l} = \frac{P_{o u t 2}}{P_{i n}}$ and the intake airflow coefficient $η_{m} = \frac{Q_{om 2}}{Q_{im}}$ . Using these two coefficients and various geometric limiting boundaries, the fitness function of the intake grille is derived (equation (7)). A larger fitness value yields a higher total pressure recovery coefficient and a higher intake airflow capability. On account of the control variables and the corresponding fitness values, the samples of the aerodynamic performance of intake grille are constructed.

The law of conservation of energy must be satisfied in a flow system involving heat exchange. The law can be expressed as the increase rate of energy in the microelements, which is equal to the net heat flow into the microelements plus the work done by physical and surface forces on the microelements. In other words, the first law of thermodynamics is obtained:

\begin{matrix} f = c_{1} (\frac{C p_{total} - {(\frac{P_{out 2}}{P_{in}})}_{ref}}{{(\frac{P_{out 2}}{P_{in}})}_{ref}}) + c_{2} (\frac{η_{m} - {(\frac{Q_{om 2}}{Q_{im}})}_{ref}}{{(\frac{Q_{om 2}}{Q_{im}})}_{ref}}) \\ + c_{3} | \min (0, \frac{| α_{1} - α_{1}' |}{α_{1}'} - 0.25) | \\ + c_{4} | \min (0, \frac{| α_{2} - α_{2}' |}{α_{2}'} - 0.25) | \\ + c_{5} | \min (0, \frac{| L - L' |}{L'} - 0.25) | \\ + c_{6} | \min (0, \frac{| I_{1} - I_{1}' |}{I_{1}'} - 0.25) | \\ + c_{7} | \min (0, \frac{| I_{2} - I_{2}' |}{I_{2}'} - 0.25) | \\ + c_{8} | \min (0, \frac{| I_{2} - I_{2}' |}{I_{2}'} - 0.25) | \\ + c_{9} | \min (0, \frac{| I_{4} - I_{4}' |}{I_{4}'} - 0.25) | \end{matrix}

(7)

where $c_{1}$ , $c_{2}$ , $c_{3}$ , $c_{4}$ , $c_{5}$ , $c_{6}$ , $c_{7}$ , $c_{8}$ , $c_{9}$ are weight factors, $P_{out 2}$ is the intake outlet total pressure, $P_{in}$ is the inlet total pressure, $Q_{om 2}$ is the intake outlet airflow rate, $Q_{im}$ is the inlet airflow rate, and the parameters $α_{1}, α_{2}, α_{1}', α_{2}'$ are described in equation (18).

Surrogate model

In this work, an MLFANN is adopted as a surrogate model to approximate the complex nonlinear relations of practical problems. In the MLFANN, the data are trained using the back propagation method,³⁵ the dataflow is controlled by equation (8), the linear function as the activation function for the input and output layers, the sigmoid function (shown in equation (9)) is used for the hidden layer, and the error between the practical output value and the ideal output value is calculated using equation (10).

u_{j} = \sum_{i = 1}^{m} ω_{i} x_{i} + θ_{j}

(8)

y_{j} = f (u_{j}) = \frac{1}{(1 + \exp (- u_{j}))}

(9)

E_{AV} = \frac{1}{N} \sum_{n = 1}^{N} E (n) = \frac{1}{2 N} \sum_{n = 1}^{N} \sum_{j = 1}^{T} {(d_{j} (n) - y_{j} (n))}^{2}

(10)

where $x$ is the input data; $u$ is the data of the hidden-layer neuron; $ω$ a weight coefficient; $θ$ is the bias value; $y$ and $d$ are, respectively, the actual output value and the ideal output value; $E_{AV}$ is the total average error; and T and N are the total number of output elements and the number of iterations, respectively.

The approximation precision of the MLFANN is easily affected by its topology structure. However, no solid theory exists that determines MLFANN topology accurately and effectively. To obtain MLFANN topology automatically, a hybrid intelligent algorithm is proposed. This algorithm combines the high-dimensional optimization of the MSSA, the quick-search capability of discrete particle swarm optimization (DPSO), and the ability to quickly jump out the local optimum of a modified very fast simulated annealing algorithm (MVFSA).³⁶ The MSSA was discussed above. The update velocity and the location of DPSO are given by equations (11) and (12), respectively. In the MVFSA, the initial particle is distributed by equation (13), and the random probability of acceptance $P_{r}$ is defined using the Boltzmann–Gibbs distribution in equation (14). Considering the high-dimensional and discrete character of the topology structure for MLFANN, the topological optimization is divided into two parts: the weights and biases of the neural network are optimized using the MSSA, and the number of hidden layers and the number of neurons per hidden layer are optimized using DPSO-MVFSA.³⁷ The evaluation of particles in the DPSO-MVFSA is conducted using the optimization result of the MSSA. To ensure the effectiveness of MLFANN, 80% of the sample data are used to train the MLFANN, and 20% are used to verify the prediction accuracy of the MLFANN. Four evaluation criteria, $SSE$ , $RMSE$ , $R^{2}$ , and $R^{2} - adj$ , are often used to validate the feasibility of ANN. Values of $SSE$ and $RMSE$ close to zero result in a small random error and a useful prediction. Values of $R^{2}$ and $R^{2} - adj$ close to one result in a significant proportion of variance and a better fit.³⁸ The fitness function is defined using the evaluation criteria (equation (15)).

\begin{matrix} v_{i, j} = (\frac{2}{| 2 - C - \sqrt{C^{2} - 4 C} |}) \\ [\begin{matrix} ((μ 3 minma x_{\min} ()_{i, j}^{0})] c_{1} γ_{1} (P_{i, j} - x_{i, j}^{0}) + c_{2} γ_{2} (P_{g, j} - x_{i, j}^{0})) \end{matrix}] \end{matrix}

(11)

\begin{array}{l} x_{i, j} = {\begin{matrix} 0 & r a n d () \leq 1 - \frac{2}{1 + e x p (- v_{i, j})} \\ x_{i, j} & r a n d () > 1 - \frac{2}{1 + e x p (- v_{i, j})} \end{matrix} . (v_{i, j} < 0) o r \\ {\begin{matrix} 1 & r a n d () \leq \frac{2}{1 + e x p (- v_{i, j})} - 1 \\ x_{i, j} & r a n d () > \frac{2}{1 + e x p (- v_{i, j})} - 1 \end{matrix} (v_{i, j} > 0) \end{array}

(12)

\begin{matrix} M_{ij}' = M_{ij} + α \cdot B \cdot \\ (Texp (- c k^{\frac{1}{N_{1}}}) sgn (u - 0.5) {[(1 + \frac{1}{Texp {(- c k^{\frac{1}{N_{1}}})}_{\max}} {()}^{| 2 u - 1 |}) []]}_{\max}) \end{matrix}

(13)

\begin{matrix} P_{r} = \\ [1 - (1 - h) \cdot (E (M_{ij}') - E (M_{ij})) / (Texp {(- c k^{\frac{1}{N_{1}}})}_{\max} ()) {[]}^{1 / (1 - h)}] \end{matrix}

(14)

F = C_{1} \cdot | R^{2} - 1 | + C_{2} \cdot | R^{2} - adj - 1 | + C_{3} \cdot SSE + C_{4} \cdot RMSE

(15)

where $v$ is the particle velocity; $i, j$ are the particle sequence and the particle vector, respectively; $C$ is modified factors; $C = c_{1} + c_{2}$ and $c_{1}, c_{2}$ are learning factors; $γ_{1}, γ_{2}$ are random numbers distributed uniformly in $(0, 1)$ ; $P_{i, j}$ is the best position in its flight history; $P_{g, j}$ is the best position in the particle swarm; $μ, σ$ are the average stochastic inertial weight and the variance of stochastic inertial weight, respectively; $N (0, 1)$ is a random number of the standard normal distribution; $x_{i, j}$ is the corresponding location; $M_{ij}^{'}$ is the disturbed variable; $M_{ij}$ is the undisturbed variable; $α$ is a positive value less than $1$ ; $B$ is the limitation value of a variable; $y_{ij}$ is the Cauchy distribution; $h$ is a real number (set to $h = 0.5$ ); $E$ is the particle current energy; $u$ is a random number ranging from 0 to 1; $T_{\max}$ is the initial temperature; $N_{1}$ is the syllogism coefficient; $k$ is the number marked annealing stage; $c$ is a constant and the random probability of acceptance of MVFSA $P_{r}$ ; and $C_{i} (i = 1, 2, 3, 4)$ is the weight of the evaluation parameters.

Aerodynamic optimization of intake grille

Based on the optimization order, the whole aerodynamic optimization of the intake grille is further divided into three parts (shown in Figure 4): the sample database, the surrogate model, and the aerodynamic optimization. During the construction of the sample database, the geometry of the intake grille is parameterized using the commercial software NX 12.0. In order to make the samples distributed in the whole search space uniformly, the Latin hypercube sampling design method³⁹ is adopted. To calculate the aerodynamic performance of a sample, the commercial meshing software ICEM and the commercial CFD software CFX are used. Using the sample database and the back propagation method, the MLFANN is trained. During training, weights and biases are optimized using MSSA, and the number of hidden layers and the number of neurons are optimized using DPSO-MVFSA. The trained MLFANN is selected as the surrogate model. In aerodynamic optimization, the sample database is as the initial position of spiders, and the fitness values of the moving spiders are calculated quickly using the surrogate model. The optimization is stopped until the maximum iteration is reached.

Figure 4.

Illustration of aerodynamic optimization.

MSSA validation

To validate the improvement of the MSSA and the feasibility of the surrogate model, two multimodal functions are used as test functions, the Griewank function (equation (16)) and the Rastrigin function (equation (17)).

f (x) = \sum_{i = 1}^{N} \frac{x_{i}^{2}}{4000} - Π_{i = 1}^{N} \cos (\frac{x_{i}}{\sqrt{i}}) + 1

(16)

f (x) = \sum_{i = 1}^{D} [x_{i}^{2} - 10 \cos (2 π x_{i}) + 10]

(17)

Figure 5 shows two-dimensional forms of these two functions, with many local minima but only one global minimum [0, 0]. These are usually used as benchmark functions to verify optimization methods. They are also used to describe the complex nonlinear relations. In this work, the dimensions of these two functions are set to 10, and the value ranges are set to [−10, 10].

Figure 5.

Two multimodal functions in two-dimensional form: (a) Griewank function and (b) Rastrigin function.

To verify the performance of our MSSA, two other social spider algorithms are selected as benchmark algorithms: the original SSA and an improved SSA (ISSA). Using convergence curves, optimal convergence results, and average convergence results, the optimization performances of the three algorithms are compared.

Figure 6 shows convergence curves for the three algorithms. It is seen that for the optimization of two functions, the iteration number of MSSA is the minimal among these three algorithms. Which means that the iteration velocity of MSSA is the fastest. The convergence velocity of the MSSA is significantly better than that of the SSA and ISSA in the early stage. This indicates that the adaptive factor increases the convergence velocity in the early stage of iteration and balances the global and local convergence capabilities during optimization. Table 1 shows that the MSSA is better than the SSA and ISSA in terms of convergence accuracy for both the optimal convergence results and the average convergence results. This means that the algorithm proposed in this paper delivers a good performance, which makes the algorithm jump out of the local optimal value effectively and approach the global optimal value, so as to obtain more accurate optimization results.

Figure 6.

Convergence maps of the three SSA algorithms using the (a) Griewank function and (b) Rastrigin function.

Table 1.

Comparison of results for the three optimization algorithms.

Function	Algorithm	Optimal convergence results	Average convergence results
Griewank	MSSA	2.3e-11	4.3e-11
	SSA	5.8e-08	6.7e-08
	ISSA	4.3e-09	7.2e-09
Rastrigin	MSSA	3.8e-14	6.5e-14
	SSA	5.4e-10	8.5e-10
	ISSA	6.9e-12	9.4e-12

To prove the feasibility of the surrogate model in approximating nonlinear relations, these two functions are approximated by this model. Firstly, two groups of the sample database are constructed using the Latin hypercube sampling design method, and the number of samples for both groups is set at 500. The MLFANN is then trained and optimized using the sample database. The final MLFANN constitutes the surrogate model. During training, four evaluation parameters, $SSE$ , $RMSE$ , $R^{2}$ , and $R^{2} - adj$ are calculated (Table 2). Table 2 shows that $SSE$ and $RMSE$ are close to 0, and that $R^{2}$ and $R^{2} - adj$ are close to 1. Based on the evaluation criteria of ANN, it is believed that surrogate models based on MFLANN can predict some nonlinear relations.

Table 2.

Evaluation results.

Nonlinear relation	Model	$SSE$	$RMSE$	$R^{2}$	$R^{2} - adj$
Griewank	MLFANN¹	0.065	0.023	0.968	0.920
Rastrigin	MLFANN²	0.009	0.008	0.971	0.933

Aerodynamic optimization of intake grille

Experiment

The intake grille of the ACC of a particular type of turbine has been designed. However, its aerodynamic performance is not up to design requirements. Therefore, an aerodynamic optimization method for the intake grille is proposed. Aerodynamic performance experiments on the original intake grille and on an optimized intake grille were done in Key Lab. for Power Machinery and Engineering of SJTU. Figure 7 shows the test apparatus, in which three total pressure probes and two static pressure probes are mounted. One flow sensor is mounted at the inlet and another at the outlet of the main pipe. Using these measured values, the total pressure recovery and the intake airflow coefficients can be calculated.

Figure 7.

Test apparatus of the intake grille: (a) practical test apparatus and (b) simple test model.

In assessing the aerodynamic performance of the two grilles, five special inlet operating conditions were selected, 0.1, 0.25, 0.3, 0.4, and 0.5 Ma. The opening static pressure was used as the outlet condition of the main pipe and the intake device. For each operating condition, the outlet Reynolds number of the intake device was adopted as the collection criterion, and the numbers were, respectively, 4.14e+04, 71.52e+05, 1.49e+05, 1.7e+05, 5.32e+04. Before the experiments, the inlet and outlet boundaries were required to be converted to mass flow. In each experiment, adjustments in the static pressure of the outlet were realized by controlling the outlet sections of the main pipe and the intake device. When the mass flow of the outlet of the intake device reached the mass flow corresponding to the special Reynolds number, the pressure at each measuring point was collected. Using these collected data, the intake airflow coefficient and the total pressure recovery coefficient were calculated.

Simulation and validation

During the aerodynamic optimization of the intake grille, a simulation calculation and validation were necessary for the construction of a sample database. Using the geometry of the original intake grille, a test apparatus model was built using NX 12.0 software. The model was meshed using unstructured grids in ICEM 19.2 software (Figure 8). Before calculation of the airflow field using CFX, the boundary was set such as the flow model is steady, mass flow selected as the inlet condition, opening pressure as the outlet condition, and the turbulence model, $k - ω$ used to close the 3-D Reynolds Average Navier-Stokes (RANS) equations. Then, the grid independence of the model is analyzed and the total pressure recovery coefficient is used to be an evaluation parameter. The following figures show the parameter curve corresponding to different grids. As can be seen from the Figure 9, when the number of grids reaches 24 million, the parameter no longer increases with the number of grids, so it can be considered that the influence of more than 24 million grids on flow field parameters can be ignored.

Figure 8.

Mesh model: (a) mesh of test apparatus and (b) mesh of intake grille.

Figure 9.

Total pressure recovery coefficient curve.

To compare the differences in aerodynamic performance between the experiment on and simulation of the intake grille, two coefficients, the intake airflow and the total pressure recovery coefficients, were used. Five working conditions were chosen to be analyzed.

Figure 10 shows small gaps between the two groups of coefficients obtained from experiment and CFD: the maximum relative error of the intake airflow efficiency is 3.4%, and the maximum relative error of the total pressure recovery coefficient 1.5%. The curve trends of these two coefficients obtained from CFD are similar to those of the experiment. Thus, the CFX simulation predicts the aerodynamic performance of the intake grille.

Figure 10.

Aerodynamic performance validation: (a) intake airflow coefficient and (b) total pressure recovery coefficient.

Aerodynamic optimization

Aerodynamic optimization process

Using the geometry of the initial intake grille, parameterization is conducted. Seven geometric control parameters, $α_{1}, α_{2}, L, I_{1}, I_{2}, I_{3}, I_{4}$ , were selected to optimize the aerodynamic performance of the intake grille. The value ranges of these parameters are shown in equation (18), in which $α_{1}^{'}, α_{2}^{'}, L^{'}, I_{1}^{'}, I_{2}^{'}, I_{3}^{'}, I_{4}^{'}$ are the original values of the intake grille; these are used to determine the searching space for the geometry of the intake grille. To construct the uniformly distributed sample of aerodynamic performance, 300 samples were chosen using the Latin hypercube sampling design method. The aerodynamic performance of a sample was calculated using the validated CFD method. Then, the fitness value corresponding to a sample was determined using equation (7). Using the aerodynamic samples, an MLFANN was trained via back propagation and its topology structure optimized using the improved intelligent hybrid algorithm. The final trained and optimized MLFANN was used as a surrogate model to predict the fitness value of the intake grille. This model consists of three hidden layers, each with a neuron count of 7, 5, and 8. After a series of aerodynamic optimization iterations using the MSSA, the intake grille with the best aerodynamic performance was found.

[\begin{matrix} α_{1} \\ α_{2} \\ L \\ I_{1} \\ I_{2} \\ I_{3} \\ I_{4} \end{matrix}] \in [\begin{matrix} [0.75, 1.35] \cdot α_{1}' \\ [0.75, 1.35] \cdot α_{2}' \\ [0.75, 1.35] \cdot L' \\ [0.75, 1.35] \cdot I_{1}' \\ [0.75, 1.35] \cdot I_{2}' \\ [0.75, 1.35] \cdot I_{3}' \\ [0.75, 1.35] \cdot I_{4}' \end{matrix}]

(18)

Aerodynamic performance analysis

By running the iterative optimization many times, the grille with the best aerodynamic performance was found. The geometries of the original and the optimal grilles are shown in Figure 11. Using the geometric structure of the original grille as a baseline, the inlet and the outlet angles of the optimized grille have increased, the chord length has decreased, and the interval has increased.

Figure 11.

Geometry of a grille: (a) original grille and (b) optimal grille.

To compare the practical differences in aerodynamic performance, we used the above mentioned five operating conditions for the two kinds of grilles. Two aerodynamic performance coefficients for an optimized grille were calculated. The coefficients of the two grilles are displayed in Figure 12. The average intake airflow and the average total pressure recovery coefficients of the optimized grille have, respectively, increased by 17.3% and 4.9%, using the initial grille as a baseline.

Figure 12.

Aerodynamic performance coefficient comparisons: (a) intake airflow coefficient and (b) total pressure recovery coefficient.

To analyze the effect of geometry on the airflow field of the intake grille, two operating conditions, 0.3 and 0.4 Ma, were adopted and simulated using the validated CFD method. In this work, the velocity contours are used as an analytical tool (Figure 13). The separation loss of the initial grille is bigger than that of the optimized grille. Combined with Figures 11 and 12 shows that the increase in inlet angle and outlet angle results in a decrease in the airflow separation of the lower surface of each blade. There is a concomitant decrease in chord length and increase in, all resulting in the interval can enhance the intake airflow. Therefore, the aerodynamic performance of the intake grille can be improved by this aerodynamic optimization method.

Figure 13.

Velocity contours: (a) original grille (0.3 Ma), (b) optimized grille (0.3 Ma), (c) original grille (0.4 Ma), and (d) optimized grille (0.4 Ma).

Conclusions

In this work, an improved aerodynamic optimization method for an intake grille was proposed. The geometry of the intake grille was parameterized using NX 12.0. A special fitness function was proposed related to its geometric parameters and aerodynamic performance. An MSSA was used as the aerodynamic optimization algorithm. To evaluate the fitness value quickly, an MLFANN was selected as a surrogate model. Two multimodal functions were used to verify improvements in the SSA and the feasibility of the surrogate model. Lastly, a practical intake grille was optimized using this method. This paper ran as follows.

The geometry of the intake grille was parameterized using seven geometrical parameters, and the shape transformation can be achieved by fewer control parameters in comparison of the traditional repetitive design. This method had enough flexibility to cover the important search space.

An MSSA was proposed and used as the optimization algorithm. Compared with other SSA methods, an adaptive factor and an adaptive mutation operator were added to the MSSA. The former not only increased the convergence velocity at an early stage, but also balanced the global and the local convergence capabilities during optimization. The latter can make the strongest vibration jump out of the local extremum.

To obtain the fitness value of the intake grille quickly during optimization, an improved MLFANN was used as a surrogate model. Its topology structure was optimized using an adaptive intelligent algorithm. Using this model, the effect of human factors on the training of an MLFANN was decreased, and the approximation capability of the surrogate model was increased.

Analyzing the results for the practical intake grille showed that the average intake airflow and the average total pressure recovery coefficients of the optimized grille increased, respectively, by 17.3% and 4.9%, in comparisons of those of the initial grille and the separation loss of the optimized grille is smaller than that of the original. Thus, this aerodynamic optimization could be used to optimize the aerodynamic performance of an intake grille.

Future research will focus on the applicability of this model, including further increasing optimization scenarios, such as how to further validate and enhance the accuracy of the proxy model after increasing control parameters.

Footnotes

Handling Editor: Sharmili Pandian

Author contributions

Hong Xie: conceptualization (equal), data curation (equal), and writing (original draft). Shuyi Zhang: formal analysis (equal), investigation (equal), and methodology (equal). Xiangfeng Shi: software (equal) and validation (equal). Bo Yang: writing (review and editing (equal)), Chunrong Wang: data curation (equal).

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The authors are grateful for the financial support from the National Fund Cultivation Program of Sanming University (Grant No. PYT2203), the funding for the Sanming University’s Introduction of High Level Talents Research Initiation Project (Grant No. 22YG08), the Special Project of Central Government Guiding Local Science and Technology Development (Grant No. 2021L3029), and the Fujian Natural Science Foundation (Grant No. 2023J011033).

ORCID iD

Hong Xie

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Aerodynamic optimization method of intake grille of active clearance control system for turbines based on modified social spider algorithm (MSSA)

Abstract

Keywords

Introduction

Aerodynamic optimization methodology

Parameterization of intake grille

Modified social spider algorithm (MSSA)

Surrogate model

Fitness function

Surrogate model

Aerodynamic optimization of intake grille

MSSA validation

Aerodynamic optimization of intake grille

Experiment

Simulation and validation

Aerodynamic optimization

Aerodynamic optimization process

Aerodynamic performance analysis

Conclusions

Footnotes

Author contributions

Declaration of conflicting interests

Funding

ORCID iD

References