Optimization Design of Rocket Deflector Channel Surface Based on Kriging Surrogate Model

This study aims to optimize the gas flow environment of a rocket’s deflector channel. Numerical calculations and analysis were conducted on the influencing factors of a certain type of single-sided deflector channel, with the injection coefficient used as the evaluation index for the flow clearance. The study identified the deflector channel’s impact angle, curvature diameter of the bending section, and distance from the top of the deflector surface to the bottom as important structural parameters. There exists a complex nonlinear relationship between these parameters and the gas flow guiding effect of the deflector channel. The kriging surrogate model was employed for the optimization of the deflector channel’s surface. The results showed that, while meeting the basic requirements of the deflector channel, the optimized design improved the gas flow guiding effect by 27.58%, significantly enhancing the gas flow environment. The optimization error was within 1%, confirming the effectiveness and accuracy of the optimization. This study provides a reference for the future design and optimization of gas flow deflector channels, with the potential to improve the performance and efficiency of rocket deflector channels.


Introduction
In recent years, with the continuous development of aerospace technology, China's aerospace field has achieved numerous breakthroughs, and rockets have become essential tools for space transportation in a series of missions.Launching rockets typically requires specific positions within the aerospace launch site, where rocket engine total pressure can reach tens of megapascals and total temperature can exceed 3000K.After ignition, the rocket engine produces high-temperature and high-pressure gas jets [1].The jets may also contain incompletely burned solid propellant particles, which must be adequately guided to prevent severe erosion on the launch equipment impacted by them.Erosion can reduce the equipment's lifespan and potentially affect the success of subsequent launches.Additionally, the gas itself can cause erosion and destabilize the rocket [2].Therefore, research and improvement of rocket deflector channel configurations are crucial for ensuring successful rocket launches and reducing the cost of multiple launches.
Experimental and simulation studies on gas jets and launch deflection problems have yielded numerous achievements, and the research techniques have become relatively mature.In 2010, Koseoglu [3] et al. used a three-dimensional low Reynolds number model to numerically simulate the flow fields of nine different jet exits, including circular, elliptical, and rectangular shapes.The simulation results indicated that increasing the aspect ratio of the jet exit enhanced the heat transfer in the recirculation region.In 2018, Khizar [4] et al. employed CFD software to investigate the flow characteristics of supersonic jets discharged in circular pipes.The results showed that, under given nozzle pressure ratio and Mach number, an optimal area could be selected to maximize the engine's thrust.In 2020, Lixuan Ma [5] et al. conducted numerical simulations of aerodynamic noise in gas jets under flashback conditions, integrating the DES turbulence model, finite-rate chemistry model, and FW-H equation-based acoustic analogy method.Debin Fu [6] et al. analyzed the influence of deflector shapes on gas flow field environments and the variations in inner and outer flow fields of concentric cylinders during the launch process using unsteady numerical computation methods.They obtained suitable deflector structures to effectively direct high-temperature gas expelled from the launch tube away from the tube, thus improving the missile launch environment.Nakai Y [7] et al. conducted research on factors influencing gas flow impact on inclined deflectors using pressure-sensitive coatings and visualization schlieren techniques, analyzing and comparing various operating conditions with different inclination angles, pressure ratios, and distances between the nozzle and the deflector plate.
Rocket deflector channels are generally divided into single-sided deflector channels and double-sided deflector channels, both of which can meet design requirements, but they differ in gas aerodynamic characteristics and engineering parameters.The deflection performance of the deflector channel is characterized by the injection coefficient γ, which is defined as the ratio of the fluid flow rate at the deflector channel exit to the fluid flow rate at the engine exit.This parameter intuitively reflects the flow clearance performance of the deflector channel.A larger injection coefficient indicates that more air is entrained into the deflector channel by the gas jet flow field, indicating smoother flow in the deflector channel.
This paper focuses on the single-sided rocket deflector channel and conducts numerical research on the variation of deflection effectiveness caused by changing three parameters.The kriging method is used to establish a surrogate model between the deflector channel surface and the injection coefficient, and a comparison is made.The multi-island genetic optimization algorithm is employed to optimize the surrogate model, and the results are validated through numerical calculations.

Fluid control equations
The governing equations of fluid flow are built upon the foundation of the three fundamental laws of physics, and all flow phenomena in nature adhere to these three basic principles.They are the conservation of mass, the principle of momentum, and the conservation of energy.Correspondingly, the governing equations consist of the continuity equation, the momentum equation, and the energy equation.The governing equations are as follows: Continuity equation Where, ρ, p, and E represent the density, pressure, and energy of the initial chamber mixed gas, respectively.v is the velocity of the initial chamber gas, and G represents the source term due to secondary combustion.t is the iteration step size, and τ is the viscous tensor.For specific meanings, please refer to references [8] [9].

Turbulence model
Gas jets generally exhibit extremely high velocities.As they move at high speeds, they entrain and mix with the surrounding air, leading to a highly unstable and turbulent flow field with a large Reynolds number.To accurately describe the gas jet flow field, it is necessary to introduce a turbulence model.The Realizable k   model is an improvement over the standard k   model and is widely used for gas jet problems.In the Realizable k   model, the k-equation and ε-equation are given as follows: Where, k represents turbulent kinetic energy, ε represents turbulent dissipation rate, and the meanings of other symbols can be found in reference [10].

Kriging surrogate model
For the m design points , the Kriging model can be described as: [11] ( ) ( ) ( ) Where, ( )  x  represents the global polynomial approximation model, while ( ) z x represents the stochastic process established based on quantified data observations, with a mean value of 0 and a variance denoted as 2   .In the case of an uncertain approximation model, ( ) x    can be taken, and R is a symmetric correlation function.Through the method of least squares, we obtain: T -1 Calculate the correlation coefficient A using the method of maximum likelihood.Let  be the Gaussian correlation function, and the correlation vector at any point x is as follows: Based on this, the mathematical model of Kriging (KRG) can be represented as follows: The Kriging model achieves unbiased and optimal interpolation estimation of spatially distributed data by minimizing the mean squared error as much as possible, thus constructing a surrogate model.

Multi-Island Genetic Algorithm
The Multi-Island Genetic Algorithm (MIGA) is essentially an improvement of the parallel distributed genetic algorithm, and it possesses superior global optimization capability and computational efficiency compared to traditional genetic algorithms.MIGA is built upon the foundation of the traditional genetic algorithm but divides the individuals of each population into several subgroups called "islands."Then, all operations of the traditional genetic algorithm, such as selection, crossover, and mutation, are performed on each island independently.Periodically, selected individuals from each island migrate to other islands to continue the traditional genetic algorithm operations.The migration operation in MIGA preserves the diversity of optimization solutions, increases the chances of including global optimum solutions, and effectively overcomes the premature convergence problem in traditional genetic algorithms.The flowchart of the Multi-Island Genetic Algorithm is shown in Figure 1.

Geometry model
In this paper, numerical simulations of the launch gas flow field were conducted with a simplified model of the launch system by a type of rocket with two nozzles.The simplified model is shown in Figure 2, where the nozzles are distributed on both sides of the symmetry plane of the deflector channel.Due to the symmetry of the entire launch system, a 1/2 model was used when establishing the flow field computational domain to reduce computational cost.Since the focus of this simulation study is on the variation of parameters inside the deflector channel, the rocket body and the regions above it were removed, retaining only the nozzles and the deflector channel for simplification of the calculations.The final computational domain is shown in Figure 3, the evaluation criterion for the impact degree analysis is the injection coefficient γ, which is defined as the ratio of the mass flow rate(QA) at the deflector channel exit surface A to the mass flow rate(QB) at the engine exit surface B.

Grids and boundary conditions
Figure 4 shows the grid division of the computational domain.The schematic diagram of boundary conditions for the rocket deflector channel's launch gas flow field computational domain is provided in Figure 5, including four types of boundary conditions: pressure inlet, pressure outlet, wall, and symmetry plane.Specifically, the top of the engine nozzle is set as the pressure inlet boundary, where gas enters the flow field, and the gas total temperature is set to 3200K, with a pressure of 16MPa in the high-pressure chamber.The pressure outlet boundary is set with the environmental pressure of 101325Pa and environmental temperature of 300K.The wall boundaries consist of the inside of the deflector channel, the ground, and the nozzle walls, all of which are set as adiabatic and no-slip walls.
In this paper, a 1/2 model is used, and the symmetry plane boundary corresponds to the missing part in the figure.

Selection of flow field impact parameters for the deflector channel
Based on the simulation results of the deflector channel flow field and engineering experience, the structural parameters of the deflector channel, including the impingement angle (α), the curvature diameter of the bent section (D), and the depth from the top of the deflector surface to the bottom (H), have significant effects on the deflection efficiency and thermal environment of the deflector channel.
The schematic representations of these three parameters are shown in Figure 6.The impact degree analysis of these three geometric parameters on the flow field of the deflector channel is performed, with the baseline model shown in Figure 3, where the three parameters are 25°for the impingement angle, 3500mm for the curvature radius of the bent section, and 13500mm for the depth from the top of the deflector surface to the bottom.Numerical simulations were conducted for the baseline model, considering a steady-state process with constant high-pressure chamber pressure after ignition of the rocket nozzle.A velocity contour plot was obtained for the symmetric plane where the nozzle is located, as shown in Figure 7.Under the baseline model, the mass flow rate at the deflector channel exit was 1488.2 kg/s, and the mass flow rate at the nozzle exit surface was 359.4 kg/s, resulting in an injection coefficient of 4.141.

Analysis of the Impact of Deflector Channel Flow Field Parameters
The central composite experimental design method was adopted to select 20 sample points for numerical calculations under the same conditions as the baseline case [12].The ranges of each parameter were restricted based on the model's geometric constraints to ensure that the 20 sample points were not redundantly sampled.The sample data and corresponding output responses for the three parameters are shown in Table 1.From Table 1, it can be observed that there exists a complex nonlinear relationship between the deflection efficiency of the deflector channel and the parameters α, D, and H. Based on the input-output data of each group, further polynomial response surface approximation analysis was conducted [13].The results show the impact of the other two parameters on the deflector surface thermal environment evaluation index when H=13000 mm, D=10000 mm, and α=32°, as shown in Figure 8.It can be observed that with the increase in H and D, the injection coefficient γ increases, leading to an improvement in the deflector channel efficiency.As for the impingement angle α, it shows a positive correlation with the deflection efficiency before α approaches 35°, and a negative correlation after that, exhibiting an approximately parabolic relationship.

Deflector channel profile optimization
Based on the numerical simulation results from the previous section, this paper fits the simulation results using a three-parameter Kriging surrogate model.Then, based on the surrogate model, a genetic algorithm is used to predict the maximum injection coefficient of the deflector channel for different combinations of size parameters, aiming to optimize the deflection performance of the deflector channel.
During the fitting of the surrogate model, 10 sets of data were randomly selected from Table 1 and other sample points as the test set.Figure 9 shows the scatter plot of the predicted values of the injection coefficient for the deflector channel against the actual simulation values.The sample points are clustered near the predicted curve, and by calculating the fitting R 2 for the test points as 0.9456, which is very close to 1, it can be demonstrated that the kriging surrogate model used for prediction has sufficient accuracy.Next, the multi-island genetic algorithm model was used for iterative optimization of the results, ensuring that the values of the three design variables remained within the predetermined range during the computation.The iterative process is shown in Figure 10.After 600 iterations, the injection coefficient γ of the deflector channel increased to 5.283 and remained stable, indicating that the model calculation has converged.At this point, the values of the design variables are determined as follows: impingement angle α = 33.45°,curvature diameter of the bent section D = 6733mm, and depth from the top of the deflector surface to the bottom H = 12478mm.The final optimized results are shown in Table 2, where ε represents the error between the optimized numerical values and the surrogate model values.
Based on the Kriging surrogate model and the parameters obtained from the multi-island genetic optimization, the corresponding three-dimensional models of the rocket deflector channel were established for flow field simulation and validation.
Figure 11 shows the velocity contour plot on the symmetric plane of the rocket deflector channel for both the baseline model and the optimized model.It can be observed that the transverse velocity range of the gas above the impingement section and the straight section of the deflector surface has expanded in the optimized model, resulting in a greater amount of gas flowing out from the deflector channel exit.Figure 12 illustrates the streamline distribution of the gas flow on the symmetric plane of the nozzle for both models.Although both models exhibit recirculation in the deflector channel, the optimized model shows denser streamlines and a larger gas outflow from the deflector channel.These simulation results validate that the optimized model has effectively improved the gas flow characteristics and deflection performance of the rocket deflector channel compared to the original model.

Conclusion
This paper investigated the influence of deflector channel configuration on the deflection efficiency during rocket launch.The three design parameters selected for the single-sided deflector channel configuration were the impingement angle α, the curvature diameter of the bent section, and the depth from the top of the deflector surface to the bottom, which served as independent variables in the study.
The injection coefficient was used as the evaluation index to assess the deflection efficiency.The impact of each parameter on the deflection was analyzed, and the optimization results were obtained using the Kriging surrogate model.As the depth H and curvature diameter D increase, the deflection efficiency of the deflector channel improves.Regarding the impingement angle α , it shows a positive correlation with the deflection efficiency until a approaches 35°, but beyond that point, it exhibits a negative correlation.
After validating the flow field calculations, the three-parameter optimization based on the Kriging surrogate model has been shown to improve the deflection efficiency of the deflector channel by approximately 27.58%.The gas deflection environment has significantly improved, achieving the design objectives of the deflector channel.The calculation error is within 1%, indicating the effectiveness and accuracy of the optimization.

Figure 1 .
Figure 1.Flowchart of the Multi-Island Genetic Algorithm.

Figure 9 .
Figure 9. Scatter plot of predicted values against actual values.

Figure 10 .
Figure 10.Iteration process chart of the Multi-Island Genetic Algorithm.

Figure 11 .Figure 12 .
Figure 12.Streamline Distribution of the Gas Flow on the Symmetric Plane.

Table 1 .
Calculation results of the sample points.

Table 2 .
Baseline model and optimization results.