Solid propellant combustion in the low Mach one-dimensional approximation: from an index-one differential-algebraic formulation to high-fidelity simulations through high-order time integration with adaptive time-stepping

Abstract

An unsteady one-dimensional model of solid propellant combustion, based on a low-Mach assumption, is presented and semi-discretised in space via a finite volume scheme. The mathematical nature of this system is shown to be differential-algebraic of index one. A high-fidelity numerical strategy with stiffly accurate singly diagonally implicit Runge-Kutta methods is proposed, and time adaptation is made possible using embedded schemes. High-order is shown to be reached, while handling the constraints properly, both at the interface and for the mass conservation in the gaseous flow field. Three challenging test-cases are thoroughly investigated: ignition transients, growth of combustion instabilities through a Hopf bifurcation leading to a limit cycle periodic solution and the unsteady response of the system when detailed gas-phase kinetics are included in the model. The method exhibits high efficiency for all cases in terms of both computational time and accuracy compared to first-and second-order schemes traditionally used in the combustion literature, where the time step adaptation is CFL-or variation-based.

A new simulation strategy for solid rocket motor ignition: coupling a CFD code with a one-dimensional boundary flame model, verification against a fully resolved approach

“…They must continuously adapt to the evolution of the other variables to ensure the surface balance equations ( 5)-( 8) and the gas phase continuity equation (11) are satisfied. This specific nature has been explicitly identified in our aforementioned work [20] and requires carefully chosen time integrators so that the solution method is consistent, stable, precise and efficient. We have chosen to use high-order singly diagonal implicit Runge-Kutta methods with an explicit first stage (ESDIRK [26]) and we have shown their high efficiency, stability and precision.…”

“…In previous work [19,20], we have conducted extensive analysis of this flame model, demonstrating its well-posedness, its specific mathematical properties, which requires specific numerical methods (see Section III.A.2) and its relevance for solid propellant applications.…”

“…The complete one-dimensional model propellant combustion model (solid, surface, and flame) is implemented and solved in the Vulc1D code. We recall the main aspects of the numerical strategy for the implementation of this model as a standalone tool, which is explained in greater detail in [20]. The previous conservation equations are semi-discretised in space using a finite-volume approach on a staggered grid, with the temperature and mass fractions stored at the cell centers, and the mass flow rate located at the cell faces.…”

“…The second model, inspired by the previous ignition models from ONERA [16,18], constitutes a major step forward, relying on a previous in-depth study of mathematical models and numerical methods for the one-dimensional solid propellant combustion [19,20]. A one-dimensional solid propellant combustion tool, including propellant flame dynamics, has been developed [20] and is introduced as a dynamic boundary model in the CFD solver CHARME from the multiphysics suite CEDRE from ONERA [21].…”

“…The second model, inspired by the previous ignition models from ONERA [16,18], constitutes a major step forward, relying on a previous in-depth study of mathematical models and numerical methods for the one-dimensional solid propellant combustion [19,20]. A one-dimensional solid propellant combustion tool, including propellant flame dynamics, has been developed [20] and is introduced as a dynamic boundary model in the CFD solver CHARME from the multiphysics suite CEDRE from ONERA [21]. At each propellant boundary face of the CFD mesh, this model is fed with the corresponding wall heat flux evaluated by the CFD solver, and computes the unsteady evolution of the temperature profile inside the solid propellant, as well as the gradual development of the one-dimensional propellant flame.…”