1. Introduction
The hypersonic vehicle inlet, leading edge, wing rudder structure, and other key parts form a complex surface design. In the process of a hypersonic flight, the surface heat load distribution gradient is large. At the same time, due to the flight process, hypersonic vehicles often manoeuvre or ultra-manoeuvre orbit changes, have thermal loads with a high impact rate, and have non-linear characteristics. These aspects make the vehicle structure face a serious stress overrun, so a full-scale structural thermal test needs to be carried out. Such extreme working conditions make it very difficult to accurately simulate the aerodynamic thermal environment in ground tests.
Over the years, various kinds of ground test systems have been developed at home and abroad to simulate aerodynamic heat by radiation or convection, such as quartz lamps, electric arcs, high-temperature gas burners, etc., so that the technical parameters of the test, such as the temperature, size, and time, are constantly improved.
The NASA Dryden Flight Research Centre, the NASA Flight Load Laboratory, the Russian National Institute of Aerodynamics, and other institutions have conducted research on quartz lamp heaters to achieve a high temperature of more than 1800 K for a long time [
1,
2]. The thermal test control system developed by Wu Dafang et al. realises the automatic tracking of the preset temperature curve on the surface of the material or structure by adjusting the electric power of the infrared radiant heater. Here, the maximum temperature reaches more than 2000 K, the tracking error between the control curve and the preset curve is not more than 1.5%, and the system control accuracy is high [
3,
4,
5].
NASA Ames conducted the X-33 local large structure investigation using the IHF arc wind tunnel [
6]. The Italian Centre for Aerospace Research (CIRA) carried out a thermal assessment of the HYFLEX vehicle nose cone model using the Scirocco arc wind tunnel. China Aerodynamic Research and Development Centre (CIRA) conducted a large number of evaluation tests on key hypersonic vehicle components, such as the leading-edge structure, wave-transparent structure, wing/rudder rotating parts, etc., using arc wind tunnels and achieved satisfactory simulation results [
7].
The feasibility and method of high-temperature gas flow testing for typical specimens have been studied in the literature [
8]. The research results show that under the premise of existing heating technology, the method of high-temperature gas heating can simulate the stationary heat flow density and stationary temperature on the surface of the specimen under real flight conditions and can simulate the temperature distribution on the surface of the specimen within a certain range of accuracy.
However, the existing testing methods still have certain shortcomings for the thermal testing requirements of complex structures. On the one hand, it is difficult to combine the ability to simulate large-gradient heat flow distribution on the surface with the ability to simulate non-linear, time-varying heat flow with a high impact rate. From the perspective of traditional testing methods, high-temperature wind tunnels are mostly used to achieve non-steady thermal environment simulation in the form of stepped thermal power rise and fall, which makes it difficult to accurately simulate rapid changes in the complex non-linear thermal environment. With quartz lamps and other radiant heating equipment, although the control of the electrical control performance is excellent and they can achieve simulation of high-impact non-linear, time-varying heat flow, it is difficult to accurately simulate the structure of the surface of large-gradient heat flow distribution. On the other hand, due to the installation form of the test system, it is difficult to adapt to the load of heat flow on the surface of large complex structures. Traditional high-temperature wind tunnel specimen size is limited, so it is difficult to achieve the full-size simulation of components. With quartz lamps and other radiant heating equipment, it is difficult to adapt to the complex aerodynamic shape. The high-temperature gas flow test method improves the aerodynamic thermal simulation capability of complex structures to a certain extent, but for specific structures and working conditions, such as inlet structures, there are still problems of insufficient impact rate and poor adaptability [
9].
In this study, an aerodynamic thermal simulation test method based on metal fibre surface combustion is proposed. In this method, metal fibre is used as the attachment surface for combustion, and a pre-mixed homogeneous air–fuel mixture is burned on the surface of the metal fibre fabric to simulate the aerodynamic heating process of the hypersonic vehicle by heating the test specimen by radiation and convection. This method has the ability to simulate high-impact-rate, non-linear, time-varying heat flow; adapt to complex structural profiles; and simulate large-gradient heat flow distributions. For a cylinder specimen, the radial jet heating characteristics are analysed, and structural improvement of the combustion surface is proposed. Furthermore, a kriging surrogate model is established, and a non-dominated sorting genetic algorithm (NSGA-II) with an elite strategy is used to perform bi-objective optimisation, complete the matching design of the thermal test parameters, and obtain reasonable values of the test parameters. This study lays the foundation for carrying out a thermal test of the combustion structure of metal fibre surfaces.
2. Thermal Test Method for the Combustion Structure of Metal Fibre Surfaces
Metal fibre surface combustion is a fully pre-mixed combustion method with metal fibre as the attachment surface. It has high combustion thermal intensity, a large adjustment range, a short response time, and excellent shape adaptability [
10]. This paper proposes a structural thermal test method based on metal fibre surface combustion, which makes use of the above characteristics to provide a new solution to the problem of aerodynamic thermal simulation of complex structures.
The thermal test apparatus of the metal fibre surface combustion structure, as shown in
Figure 1, consists of a fuel supply subsystem, an air supply subsystem, a combustion subsystem, a measurement subsystem, and a control subsystem. The metal fibre can be an iron–chromium–aluminium alloy with a 2 mm thick braid layer.
The fuel supply subsystem is used to provide the specified fuel flow rate to the combustion subsystem and consists of a gaseous fuel source, a manual valve, a pressure-reducing valve, a pressure gauge, a flowmeter, and a check valve. The air supply subsystem is used to provide the specified flow rate of air to the combustion subsystem, which has a similar composition to the fuel supply system. The combustion subsystem is used to mix the gaseous fuel and air and combust on the surface of the metal fibre. It consists of a proportional valve, a mixer, an igniter, and a combustion surface. The proportional valve is used to control the proportion of gaseous fuel and air supplied to the combustion subsystem. The mixer is used to evenly mix the gaseous fuel and air in a confined space. The igniter is used to initiate combustion of the air–fuel mixture. The combustion surface is made of metal fibre and is used to provide support for the flame. The measurement subsystem is used to measure the temperature and surface heat flow of the specimen in real time during the test and to provide measured parameters to the control subsystem, which consists of a sensor and a data collector. The control subsystem is used to control the heat flow in real time during the test and consists of a computer and a flow controller. By controlling the flow controller through the computer, the air–fuel mixture flow is adjusted to control the heat flow, which automatically follows the preset target value of the heat flow and achieves loading with a fast impact rate and large non-linear, time-varying heat flow. The metal fibre burning head can be made into different shapes to meet the testing requirements of specimens with different appearances. In addition, the metal fibre burning surface can be divided into zones so that the heating intensity can be independently controlled according to the different surface heat flow requirements of different areas of the specimen, thus realising simulation of the heat flow gradient.
For a cylindrical specimen, the metal fibre combustion surface can be formed into a cylindrical shape so that the gas flow is injected radially against the inner wall surface of the specimen, as shown in
Figure 2.
4. Surrogate Model and Optimisation Method
The radial jet heating characteristics of the conical combustion surface are complex, and it is difficult to obtain the relationship between the heat flow distribution and the optimisation variables by analytical methods or empirical formulae. An iterative optimisation based solely on CFD numerical calculations requires a large number of calculations, which is not efficient. The use of the surrogate model instead of the original numerical analysis model for iterative optimisation has the characteristics of high accuracy and good robustness [
17,
18], which can greatly improve the computational efficiency when applied to the design of a radial jet heating system for conical combustion surface optimisation.
4.1. Experimental Design Method
Experimental design methods are used to reasonably distribute the design sample points to be simulated in the design space, which is an important statistical method in the optimisation process loop. Among the commonly used experimental design methods, the Latin hypercube design method is characterised by good spatial homogeneity and coverage and can obtain a large amount of information required for constructing a surrogate model with a smaller number of simulations.
The Latin hypercube design method uses a stratified sampling method, and Equation (1) shows the method of generating the design points for its trials. Here,
i denotes the
ith trial,
p denotes the
pth design variable,
n is the number of sample points,
π is the independent random permutation of
π, and
U is the random number in the interval [0, 1].
In this study, the lhsdesign function in Matlab R2014a was used to optimise the sampling method, i.e., for the optimal Latin hypercubic sampling method, the standard parameter of lhsdesign function sampling was set to maximin, and the number of iterations was 100 times.
4.2. Kriging Surrogate Modelling Approach
In the matching design of thermal test conditions and specimen surface heat flow, the surrogate model can be used to obtain the surrogate relationship between thermal test conditions (design variables) and specimen surface heat flow (system response), which can be used as a substitute for direct CFD analysis and improve the optimisation design efficiency of parameter matching under the premise of ensuring credibility of the analysis. Commonly used surrogate modelling methods include the response surface model, the radial basis function model, and the kriging model. Among them, it is easier to obtain ideal fitting results with the kriging model when solving problems with a high degree of non-linearity [
19]. Therefore, this study adopted the kriging method to establish the surrogate model.
The kriging model is a method for finding linearly optimal, unbiased interpolation estimates for data points distributed in space, and its mathematical expression can be expressed as follows:
where
g(
X) is the global approximation model on the matrix
X within the design space, and
z(
X) is a stochastic process with zero mean,
σ2 variance, and non-zero covariance.
The covariance matrix of
z(
X) can be expressed as follows:
where
R is the correlation matrix,
R is the correlation function, and
i = 1, 2, …,
n,
j = 1, 2, …,
n (
n is the number of sample points).
R is a symmetric matrix, and its diagonal element is 1.
R takes the Gaussian correlation function, which can be expressed as follows:
where
m is the number of design variables, and
θ is the vector of unknown correlation parameters.
Introducing the correlation vector,
The kriging model can be expressed as follows:
where
β is the unknown parameter,
σ2 and
R are both functions of
θ, and
y is an
n-dimensional column vector consisting of the response values of the sample points
.
β and σ2 can be obtained by least squares estimation. The relevant parameter θ can be obtained by optimising the great likelihood estimation.
4.3. Bi-Objective Optimisation Method
In this study, the objective was to obtain a uniform heat flow on the inner wall surface of a cylindrical specimen of a certain size, so the heat flow distribution on the surface of the cylindrical specimen predicted by the simulation must be as similar as possible to that of the objective. The cylindrical specimen is divided into two segments along the axial direction on an average basis so that the average relative errors of the heat flow in these two segments are minimised by matching the conical combustion surfaces. Therefore, the objective function can be expressed as follows:
where
denotes the average relative error
of heat flow on the inner wall surface of the specimen with the
x coordinate range of (0, 1/2
L], as shown in
Figure 8, and
denotes the average relative error
of heat flow on the inner wall surface of the specimen with the
x coordinate range of (1/2
L,
L]. The average relative error value can be calculated by Equation (8):
where
is the target heat flow on the inner wall surface of the specimen,
is the predicted heat flow on the surface of the specimen, and
N is the number of nodes on the surface of the specimen in the calculation region.
denotes the optimised design variable matrix, which consists of three adjustable parameters in the thermal test:
where
G is the airflow rate in kg/s,
D11 is the diameter of the top surface of the conical burning surface in mm, and
D12 is the diameter of the bottom surface in mm.
Considering that the diameter of the inner wall surface of the cylindrical specimen was 139 mm and the combustion surface was conical, the range of values of the design optimisation variables are listed in
Table 2.
For the above bi-objective optimisation problem, this study adopted the non-dominated sorting genetic algorithm (NSGA-II) [
20] with an elite strategy to carry out the optimisation design. The computational flow is shown in
Figure 9. First, initial parent populations (
G = 1, where
G is the number of population generations) were randomly generated within the range of values of the optimisation variables. Then, the parent population generated offspring populations of the same size. The parent and offspring populations were merged to form a population of size 2N. The newly generated population was subjected to non-dominated fast sorting, and the degree of crowding was calculated for all individuals in this population of size 2N. According to the degree of crowding relationship between the individuals and the degree of crowding of the individuals, the appropriate individuals were selected to form a new parent population of size N. The new parent population was then selected by the traditional genetic algorithm (NSGA). Then, the crossover, mutation, etc. of the traditional genetic algorithm (NSGA) was used to generate a child population of size N, which was merged with its parent population. The above non-dominated fast sorting process was repeated until the number of population generations was equal to the preset number of generations.
The bi-objective optimisation of the experimental simulation accuracy was performed using NSGA-II by setting the initial population and finally obtaining the constraint-compliant and relatively optimal solution set through selection, crossover, and mutation operations. The relevant parameters during the NSGA-II operations are listed in
Table 3.
4.4. Flow of Optimised Design Based on the Kriging Surrogate Model
Figure 10 shows the flow chart of the design method for matching the thermal test parameters of metal fibre surface combustion structures based on the kriging surrogate model and dual-objective optimisation design. It mainly includes the experimental design, construction of a design scheme for matching the test parameters, construction of the surrogate model for the heat flow on the inner wall surface of the specimen, accuracy analysis of the surrogate model, and bi-objective optimisation design.
First, the Latin hypercubic test design method was applied to sample the design variables in the determined design space to ensure the homogeneity and orthogonality of the sample points in the design space, and the sample point set I(i) was established. According to the values of the sample points, the geometric model of the corresponding test program was established, the grid was divided, and CFD solution batch processing was carried out.
The CFD results were analysed to obtain the heat flow distribution on the inner wall of the cylindrical specimen corresponding to each sample point, and the kriging surrogate model was constructed based on the set of sample points and the corresponding heat flow results. The accuracy of the constructed surrogate model was analysed; if the accuracy did not meet the requirements, the number of sample points was increased according to the point addition criterion, and the surrogate model was reconstructed until it met the accuracy requirements.
Finally, the optimal design was carried out to obtain the optimal solution set. The optimal values of the design variables were selected according to the design requirements, and the optimal design of the parameters based on the surrogate model was completed.
In the above step, the average relative error and correlation coefficient were used to quantitatively evaluate the accuracy of the surrogate model, and for each sample point, the average relative error was calculated by Equation (8), where the CFD calculation was taken as the target heat flow. The correlation coefficient was calculated by Equation (10):
where
is the heat flow on the surface of the specimen calculated by CFD.
i = 1, 2, …,
A, and
A is the number of nodes on the surface of the specimen in the numerical calculation model.