This study entails the analysis of the working performance of solid rocket motors (SRM), featuring the essential element of internal ballistic analysis. Therefore, the internal flow field under the condition of burning surface regression needs to be calculated. The boundary of the internal flow field of the SRM moves with the combustion of the propellant; therefore, it is necessary to accurately track the mobile interface to provide boundary conditions for the flow field calculation. The coupling of the level set method and the volume fraction method is utilized to track the burning surface, and the porous media model is used to divide the fluid and solid calculation domains. The interface between the two calculation domains is used to characterize the burning surface, and then, the area of the burning surface is obtained by solving the area of the interface. The calculation and analysis are carried out for SRM with tubular charge and star charge. The results verify that the calculation model can accurately calculate the transient internal flow field of SRM under the condition of burning surface regression.

Solid rocket motors (SRM) are widely used as power plants on missiles and spacecraft because of their superior performance. In the research and development of SRM, a series of experiments need to be carried out to predict performance, which often consume huge funds. With the development of computer technology, numerical calculation methods to simulate the working process of SRM can be used to supplement the experimental means, and they play a highly important role in reducing expenses and shortening the development cycle. Therefore, directions involving such methods have become one of the most active ones in the field of SRM. The main purpose of computer simulation calculation is to obtain the internal ballistic curve of the SRM, which is the law of working pressure in the combustion chamber changing with time and space. Moreover, pressure is a significant parameter in SRM operation, which determines a series of performance parameters such as thrust, working time, and structural integrity of the SRM; hence, it is necessary to accurately calculate the interior ballistic curve. The burning surface area of the grain is a decisive factor for this calculation; thus, it is necessary to use the corresponding algorithm to simulate the burning surface regression, so as to obtain the burning surface area change curve of the grain during the combustion of the propellant [

For the numerical simulation of burning surface regression, accurate mathematical models and robust and effective numerical techniques are required to ensure that the burning surface regression is synchronized with the combustion of the propellant. Currently, there are three main methods for calculating the burning surface regression, including the following: (1) The solid modeling method mainly relies on computer-aided design (CAD) software for secondary development, which can obtain the area of the burning surface at any time; however, it is difficult to realize the coupling calculation with the flow field [

The key problem for the coupled calculation of burning surface regression and transient flow field is to establish mathematical models and numerical techniques. The coupling process is realized through the data transmission between the models, and the algorithm needs to be optimized in the calculation, so that the accuracy, efficiency, and stability of the simulation calculation are realized at a lower calculation cost. Accordingly, the outline of the paper is as follows. In Section

The VOF method tracks the interface by defining the volume fraction

When solving this volume fraction transport equation, in order to ensure the sharpness of its interface, a compression term is artificially added into Equation (

The LS method is a distance function method, where the two-phase interface is represented by the zero point of a high-order LS function

The governing equation of

To ensure the sharpness of the volume fraction

When the LS distance function is expressed by the

Suppose that

Equation (

Schematic diagram of the implementation process of the CLSVOF method.

The main steps of CLSVOF method calculation are as follows:

Calculate the volume fraction

Map the volume fraction

Obtain the final distance function

Solve the interface position by the obtained distance function

After the distance function

The combustion of the propellant causes burning surface regression, which is a complex process. Consequently, it is difficult to calculate the coupling of burning surface regression and transient internal flow field under propellant burning conditions, and thus problematic, it is difficult to realize the coupling calculation of the burning surface regression and the transient flow field of the combustion chamber under propellant burning conditions. In contrast, it is easier to simulate the transient flow field of the combustion chamber under the burning surface regression by using the burning surface regression calculation method instead of considering the burning mechanism. In order to simplify the calculation, the following assumptions are made:

There is uniformity for the physical and chemical composition of the propellant

The entire burning surface of the grains burns at the same time

The burning surface advances toward the inside of the grain along the normal direction at the burning rate of the local propellant

The gas produced by the combustion of propellant is an ideal gas with a single component

Erosive burning is not included in the calculation

The regression velocity of the burning surface is calculated through St. Robert’s Law

The solid properties of the grain are characterized by limiting the flow velocity of the solid domain to zero through the porous media model

Gravity and radiation heat transfer in the combustion chamber, as well as the chemical reaction process of propellant combustion, are ignored in the calculation. When the burning surface regression reaches a certain place, the propellant in that place is burnt and is directly converted into gas; that is, the mass source

Schematic diagram of burning surface in mesh.

The flow in the combustion chamber and nozzle of the SRM is turbulent, which comprises the following aspects: shock wave, boundary layer, shear layer, recirculation zone, and the interaction of all these. Taking into account the complexity of the flow field and the compressibility of the gas, in this paper, we use the SST

The eddy viscosity can be calculated by the following formula:

For the spatial discretization, a structured multiblock finite volume approach with cell-centered data storage is adopted. The diffusion term is discretized by the central difference scheme, and the convection term is discretized by the second-order upwind scheme. For the time discretization, the fourth-order explicit Runge-Kutta method is adopted. The time step is chosen as the minimum value of maximum stable time step in all fluid cells, and the SIMPLE algorithm is applied for the calculation of the flow field.

Porous media are modeled by the addition of a momentum source term to the standard fluid flow equations [

The viscous resistance coefficient

For the solid domain, the values of ^{10}), so that the flow of the solid domain is blocked

For the fluid domain, the values of

The coefficients at the interface can be well transitioned, so that the calculation efficiency is not affected

Resistance coefficient in the computational domain.

The volume fraction

Distribution curve of resistance coefficient.

The combination of CLSVOF and porous media modeling technology is used to perform the coupled calculation of burning surface regression and SRM transient internal flow field. The CLSVOF is used to control the regression of the combustion surface and to provide the area of the burning surface and volume fraction for the porous media model for the calculation of the source term to simulate the generation of gas. Computational fluid dynamics are used to calculate the gas flow in the combustion chamber to obtain the pressure distribution in the flow field, so that the burning rate equation can be used to compute the burning surface regression velocity. The coupled calculation scheme is shown in the flow chart (Figure

Flow chart showing the coupled computing scheme.

Figure

Schematic diagram of the SRM model with tubular charge.

Propellant parameters.

Value | |
---|---|

Propellant density (kg/m^{3}) | 1800 |

Burning rate coefficient | 0.015 |

Burning rate pressure exponent | 0.3 |

The length-diameter ratio of the tubular charge SRM is relatively small, the gas flow rate in the combustion chamber channel is low, and there is no obvious gap along the axis. Therefore, it is appropriate to use the zero-dimensional internal ballistic method to calculate the combustion chamber pressure. The results calculated by the zero-dimensional interior ballistic method and the CLSVOF method are then compared to verify the accuracy of the CLSVOF method. The zero-dimensional interior ballistic calculation process is as follows:

Suppose the thickness of the charge is

When the burning thickness of the charge is

The pressure under each combustion layer is calculated by the formula

The burning rate at step

Figure

Contours of pressure distribution at different time points.

Comparison of combustion chamber pressure.

As presented in Figure

Schematic diagram of burning surface regression.

Comparison of burning surface area.

In summary, the effectiveness and accuracy of the CLSVOF method were verified through the comparative analysis results of the combustion chamber pressure and the combustion surface area. Thus, this method can effectively calculate the transient internal flow field of the tubular charge SRM under the regression of the burning surface.

The calculation and analysis of star charge SRM are carried out to further verify the ability of the CLSVOF method to calculate the complex burning surface. Figure

Cross-section of the star charge.

Propellant parameters.

Value | |
---|---|

Propellant density (kg/m^{3}) | 1800 |

Burning rate coefficient | 0.015 |

Burning rate pressure exponent | 0.3 |

The area of burning surface is extracted during the regression of the burning surface and compared with the result calculated by the CAD method, as seen in Figure

Comparison of burning surface area.

Schematic diagram of burning surface regression.

Figure

Contour of pressure distribution.

Contour of temperature distribution.

Contour of Mach number distribution.

Contour of velocity distribution.

The numerical calculation results and the theoretical calculation results are compared and analyzed to verify the accuracy of the numerical calculation results earlier. The theoretical calculation formula of the parameters at the nozzle outlet is as follows.

The relationship between the nozzle area expansion ratio

The relationship between pressure

The relationship between temperature

The formula for the speed of sound

The theoretical calculation results for parameters at the nozzle outlet is obtained by solving the above formula, and the relative error is obtained by comparison with the numerical calculation result, as shown in Table

Propellant parameters.

Parameter | Theoretical calculation value | Numerical calculation value | Relative error |
---|---|---|---|

Pressure (MPa) | 1.43 | 1.45 | 1.39% |

Temperature (K) | 917.85 | 937.46 | 2.13% |

Velocity (m/s) | 2044.14 | 1999.57 | 2.18% |

Mach number | 3.37 | 3.28 | 2.67% |

In this study, a numerical calculation model for SRM burning surface regression was developed and implemented with the purpose of accurately and efficiently calculating the performance of the SRM. It is necessary for the interface to be clearly tracked during the regression of the burning surface to provide boundary conditions for the flow field calculation. Then, the gas flow in the flow field is calculated numerically according to the obtained boundary conditions, and the pressure, temperature, and velocity of the gas in the flow field can be solved through computational fluid dynamics. The CLSVOF method is used to track the interface, which is a coupled LS and VOF method. This method not only overcomes the shortcomings of the VOF method in that it is difficult to accurately calculate the normal and curvature of the interface but also solves the problem of nonconservation in the calculation procedure of the LS method. The porous media model is used to divide the fluid and solid calculation domains, which avoids the problem of mesh reconstruction in the calculation. The interface between the fluid domain and the solid domain is used to characterize the burning surface, and then, the area of the combustion surface at different moments can be obtained by solving the area of the interface.

The calculation model of this paper is verified by the zero-dimensional interior ballistic method and the CAD method. The results confirm that the proposed calculation model can compute the internal ballistics of SRM with different shapes of charges. However, some remaining issues need to be further studied to achieve more accurate calculations, such as erosion combustion and fluid-structure coupling calculations, of the performance of SRM.

Flow velocity (m/s)

Burning surface regression velocity (m/s)

Time (s)

Level set function (−)

Volume fraction (−)

Interface normal vector (−)

Compression coefficient (Pa)

Initial distance function

Characteristic cell size (−)

Infinitesimal (−)

Density (kg/m^{3})

Gravity (m/s^{2})

Temperature (K)

Dynamic viscosity (Pa·s)

Pressure (Pa)

Thermal conductivity (W/(m·K))

Tangential stress (Pa)

Internal energy (J)

Mass source (kg/(m^{3}·s))

Momentum source (N/m^{3})

Energy source (W/m^{3})

Gas constant (J/(mol·K))

Propellant density (kg/m^{3})

Propellant burning rate (m/s)

Specific heat capacity at constant pressure (J/(kg·K))

Turbulence kinetic energy (m^{2}/s^{2})

Turbulence dissipation rate (m^{2}/s^{3})

Specific dissipation rate (1/s)

Eddy viscosity (Pa·s)

Mach number (−).

The data used to support the findings of this study are included within the article.

The authors declare that they have no conflicts of interest.