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Initialization Method of Nozzle Flow Field Based on One-Dimensional Isentropic Theory and the Study of Its Application

The two-dimensional axisymmetric nozzle is initialized by two methods which are the hybrid initialization method of the commercial computational fluid dynamics software Fluent and the initialization program based on the one-dimensional isentropic theory. The first-order upwind format is combined with the SA model to solve the two-dimensional axisymmetric nozzle under the same boundary conditions. The convergence process is obtained in the process of numerical calculation, and the influence of two different initialization methods on the convergence process is analysed. The results show that the initialization method based on the one-dimensional isentropic theory effectively improves the convergence of the nozzle numerical model calculation process, accelerates the convergence speed and improves the calculation efficiency.


Introduction
The nozzle is an important part of the aircraft for energy conversion. It can transform the internal energy of high temperature and high pressure gas into kinetic energy, generate reaction force, and provide continuous and stable power for the aircraft. Therefore, it is of great significance to study the flow field of the nozzle. At present, numerical simulation, theoretical analysis, experimental research and other methods are often used in the study of flow field problems. These methods are studied from different perspectives and complement each other. For the nozzle flow field research, compared with the experimental research method, the numerical simulation method is more economical and convenient. However, when we use numerical simulation method to study the nozzle flow field problem, it is possible that the process of convergence speed of the nozzle numerical simulation is too slow and sometimes it can be diverging because of the complexity of the nozzle flow filed and the defects of the numerical calculation method.
Therefore, it is of great significance to solve the problems of slow convergence and easy dispersion in numerical simulation of nozzle flow field. As the most simplified flow model, one-dimensional isentropic flow model can give analytical solution or semi-analytical and semi-numerical results, which is helpful to understand the actual three-dimensional flow law. The initialization program of numerical calculation of nozzle is written by using the one-dimensional isentropic theory, and the numerical simulation calculation of nozzle flow field is initialized by using this initialization problem. The results of one-dimensional isentropic flow are taken as the initial conditions of numerical calculation of nozzle, which is helpful to improve the convergence in numerical calculation of nozzle.

One-dimensional Isentropic theory of Nozzle
For the one-dimensional flow in the nozzle, the sectional area of the nozzle is set as A, which is function of the axial distance X(the inlet center as the origin point) of the nozzle. The parameters inside the nozzle only change with the change of X. The continuity equation, momentum equation and energy equation of one-dimensional isentropic flow are given below, as shown in (1)-(3). continuity equation: energy equation: In order to give an analytical solution, the complete gas equation (4) and the complete gas enthalpy equation (5) need to be supplemented [1]. The analytical solution can be obtained from the equation (1)- (3).Firstly, the Mach number in the nozzle is only related to the area ratio, and the relational expression is (6).
For a given nozzle, its cross-sectional area A is a function of X, so the relation between the Mach number and X can be obtained from equation (6). After the Mach number is solved, the changes of pressure, density, temperature and others are functions of Mach number, and the equation (7)-(9) can be obtained [2]. 2 1 Among them, are the total pressure at the inlet of the nozzle, the total temperature, the density, the specific heat ratio of the gas, and the area of the throat of nozzle. According to the above conditions, the physical parameters in the nozzle can be determined once the nozzle shape, fluid parameters, and inlet conditions are determined.

Initialization Program Based on One-dimensional Isentropic theory
According to the theory of 1.1, when the gas parameter conditions of the nozzle inlet and the shape of the nozzle are determined, the flow parameters inside the entire nozzle can be obtained. The physical parameter values solved in the one-dimensional isentropic theory are used as the initial values of the nozzle simulation calculation, and the physical parameters obtained by the one-dimensional steady flow theory are analyzed as initial conditions to improve the numerical calculation of the nozzle flow field.
First, after meshing the nozzle, the commercial computational fluid dynamics software Fluent is called to generate the initial IP file, and the coordinate information of all the grid points of the nozzle can be obtained; then a file containing the nozzle generatrix coordinate points and the inlet the gas parameters in the program. The calling program calculates the area of each grid node to obtain the area ratio of the position, and then combines the inlet gas parameters into the formula of equation (1)-(9) to obtain the physical parameters of the mesh node. After looping through all the grid points, an initial nozzle flow field can be obtained, and a new IP file is output, which is imported into Fluent as the initial flow field for numerical simulation calculation, The block diagram is shown in figure 1.

Computational model and meshing
In this paper, a two-dimensional axisymmetric nozzle is taken as the research object, and the convergence process under the same boundary conditions but different initial conditions are mainly analyzed. The geometry is shown in figure 2 and the geometric parameters are shown in Table 1.  Expansion ratio 20 The mesh division of some areas of the two-dimensional axisymmetric nozzle is shown in figure  3.In order to reduce the influence of the grid on the numerical simulation calculation, a structured grid is used to encrypt the wall, throat, entrance and exit to ensure that the y + of the first layer is closer to 1 [3][4], the total number of grids is 1000×200.

Numerical calculation parameter setting
In this paper, the SA turbulence model is selected to solve the problem. It is a RANS model with only a small amount of computation for solving a modified eddy viscosity transport equation. In the modified form, the eddy viscosity in the near-wall region is easier to solve and is often used to simulate the inclusion of wall jets [5]. The problem is that the simulation results are better in the boundary layer flow of the inverse pressure gradient. For the discrete format of various parameters, the first-order upwind format is selected. The first-order upwind format has the characteristics of high stability and fast calculation speed, which is beneficial to improve the calculation efficiency. For the selection of the initialization method, one method is to use the hybrid initialization method provided by the commercial computational fluid dynamics software Fluent to initialize, and the other method is to initialize using the initialization program which is written based on the one-dimensional isentropic theory, and compare the convergence process of the two methods. In the numerical calculation, the Coulomb number also has a great influence on the stability and convergence of the calculation, so the Coulomb number in all numerical calculation is set to 0.5.
For this physical model, as shown in figure 2.The nozzle inlet ab is set as a pressure inlet condition, the nozzle outlet ef is set as a pressure outlet condition, the wall surface ace is set as a wall condition, and bdf is set as a symmetry axis condition. In this paper, three pressure inlet conditions are calculated, which are 8MPa,10MPa and 12MPa,and the total temperature is set to 3568.923k.The The density of the gas is calculated using the ideal gas model. The constant pressure heat ratio of the gas is set to 3724.088k, and the thermal conductivity is solved by the kinetic-theory. The gas viscosity coefficient is solved by the Sutherland's law of three coefficients, as shown in equation (10).
In equation (10),the reference temperature ܶ is taken as 273.15k; ܶ ௌ is the Sutherland constant,it is related to the nature of the gas. ߤ is the reference viscosity coefficient when the temperature is ܶ .

Calculation Results and Analysis
First, the flow field pressure and velocity distribution after initialization using two initialization methods are given, when the inlet pressure is 8MPa,as shown in figure 4-figure 5.The physical parameters obtained by the one-dimensional isentropic theory are only related to the area ratio and are a function of the change with X; and the hybrid initialization method is based on the given boundary conditions to pre-iterate the Laplace equation, as shown in equation (11).The initial distribution of velocity and pressure in the calculation region is solved, and other physical quantities( temperature, turbulence ) are automatically obtained from the average interpolation of the flow field.   figure 5 that the pressure fields obtained by the two initial methods are similar, but there is a big difference in the velocity field. It can be seen that the velocity field of the hybrid initialization is a certain area as the "source point". Diffusion to the surrounding, the speed gradually decreases; and the velocity field using the one-dimensional isentropic theory is still only a function related to the area ration. In the contraction section of the nozzle, the gas velocity is subsonic, in the expansion section of the nozzle, the gas velocity is supersonic. The continuous  is reduced by nearly three orders of magnitude compared to the use of hybrid initialization. As the inlet pressure continues to increase, the number of convergence steps of the calculation process using the hybrid initialization method increases from 60000 steps to nearly 80000 steps, while the calculation process using the one-dimensional isentropic theory method has a convergence step of about 30000 steps. There is no significant improvement in the number of steps. It can be seen from figure 4- figure 5 that the initialization method based on the one-dimensional isentropic theory method actually accelerates the convergence speed of the nozzle numerical simulation calculation process, effectively reduces the calculation time and improve the efficiency of the nozzle numerical calculation.
The final calculation results obtained by the two methods are compared below. The inlet pressure condition is 8 MPa, and the calculated pressure, velocity field and streamline diagram are shown in figure 9-figure 10. Comparing the final results obtained by the two initialization methods, we can see that there are some differences in the numerical results of the two final results, but the difference is not large, and the pressure, velocity, and streamline distribution are basically the same. The results obtained are reasonable. Since the iterative calculation is more sensitive to the initial value. Therefore, the initial conditions are well specified and have a great influence on the convergence of the calculation process. Before the iterative calculation, giving an initial value close to the calculated results is beneficial to improve the convergence of the iterative calculation. Conversely, if the initial value is far from the final value, it may cause convergence difficulties. Comparing figure 10 with 5, we can see that the initial velocity field based on the one-dimensional isentropic theory is closer to the final calculated velocity field, that is, the initial field obtained by the one-dimensional isentropic theory is better than the hybrid initialization method. Therefore, the initialization method based on the one-dimensional isentropic theory has a good effect on improving the convergence of the nozzle numerical simulation calculation process.

Conclusion
Comparing the two initialization methods, we can see that the one dimensional isentropic theory is used to initialize the Laval nozzle. Under the same conditions, it is better than the Fluent's own hybrid initialization method, which effectively improves the convergence and stability in the calculation process. It also improves the computational efficiency and has a more stable calculation process. Therefore, the numerical simulation of the problem of nozzles, if combined with the one-dimensional isentropic theory, can greatly improve the calculation efficiency and improve the convergence and stability of the calculation process, which is of great significance for the numerical simulation of the