Modeling and numerical simulation of solid rocket motors internal ballistics

In the design and development of solid propellant rocket motors, the use of numerical tools able to simulate, predict and reconstruct the behavior of a given motor in all its operative conditions is particularly important in order to decrease all the planning times and costs. This work is devoted to present an approach to the numerical simulation of SRM internal ballistic during the entire combustion time (ignition transient, quasi steady state and tail-off) by means of a Q1D unsteady numerical simulation model, named SPINBALL (Solid Propellant rocket motor INternal BALListic). SPINBALL comes out from the updating and further development of the numerical, mathematical and physical models of the SPIT model (Solid Propellant rocket motor Ignition Transient), that allows to extend the numerical simulation of the SRM internal ballistic, from the ignition transient, to quasi steady state and tail-off. SPINBALL core model is a quasi-1D unsteady gasdynamics model of the internal ballistic, with source terms that take into account the contribution to the bore flowfield conditions due to the igniter, the grain propellant and thermal protections. The flow is assumed as a non-reacting mixture of perfect gases with space and time varying thermophysical properties (standard thermodynamics approach). The governing equations are discretized by a Godunov-type scheme, first or second order in space and time. The use of a such approach allows to consider the addition into the chamber due to both ablation phenomena from thermal protections and combustion reactions from the grain propellant, but even to take into account the equilibrium point of the grain propellant exothermic reactions, as function of the local pressure and the variation from that nominal condition, through the combustion efficiency. This main model is completed by several sub-models, in order to describe all the driving phenomena that lead the internal ballistic for the entire combustion time: an igniter model, an heat transfer model for convection and radiation, propellant ignition criterion, a cavity model to account submergence and slot regions, a grain combustion model with the pressure term (APN model) and the erosive term (Lenoir-Robillard model with the modifications due to Lawrence and Beddini). Some of them coming from the SPIT model. Focusing on the driving phenomena that characterize the internal ballistic over the ignition transient, it is known that, during quasi steady state and tail-off, the motor bore flowfield conditions are led mainly by the grain burning surface evolution in time and the possible nozzle throat area ablation phenomena. The grain burning surface evolution model is a 3D numerical grain regression model (named GREG) based on a full matrix level set approach, on both rectangular and cylindrical structured grids, that gives to the gasdynamical model the evolution in time of port area, wet perimeter and burn perimeter along the motor axis and in the submergence zone. The numerical scheme for the numerical integration of the Level Set equation is built from the strong link between Hamilton-Jacobi equations and conservation laws and it is a first or second order (minmod flux limiter and Heun's method) in time and space time marching scheme based of an exact Riemann solver. The use of a 3D model is mandatory to carry out the grain burnback analysis in the case of general and complex 3D grain shapes, as finocyl grains, whatever bore flowfield dimensional model (0D or Q1D) is adopted. GREG module can handle 3D complex geometries directly from CAD tools, building up its initial condition as a narrow signed distance function from STL (stereolithography) files of the grain propellant and the thermal protections and insulation shapes by a completely automatic procedure. The grain burning rate can be variable, both in space and time. The use of grain propellant shape symmetries is exploited with the setting up of mirroring, or periodic boundary conditions, reducing the computational costs. The evaluation of the grain geometrical parameter, as areas, volumes and perimeters, is made with a robust second order regularization of the Dirac Delta and Heaviside functions to avoid the typical problems related with the use of the standard regularization techniques. While potentially GREG module can be completely coupled with the Q1D unsteady flowfield model, in this work, we will consider a decoupling between the grain burnback model and the flowfield model, in order to reduce the computational cost required. GREG model is, hence, used as a pre-processor that generates tables of pre-evaluated grain geometrical properties for a constant burning rate in time and space. During the numerical simulation, these tables of port area, wet perimeter and burning perimeter are then interpolated using the local increment of the web variable, defined by the local grain burning rate, coming from APN and LR models. The final objective is, hence, to develop an analysis/simulation capability of SRM internal ballistic for the entire combustion time, with simplified physical models, in order to reduce the computational cost required, but ensuring, in the meanwhile, an accuracy of the simulation greater than the one usually given by 0D quasi steady models. In this framework, the comparison between the results obtained with a 0D quasi steady chamber model and SPINBALL will be made and the effects of the increased detail level of the internal ballistic simulation on the overall prediction capability will be discussed for three SRM: Z23 second stage of the new European launcher VEGA and two military motors, NAWC n. 6 and 13, on which there are different work in literature, and all motor data are available. Comparisons with the experimental data and with other codes results will be also made.