Modeling of One-Dimensional Ablation with Porous Flow Using Finite Control Volume Procedure

Abstract

The development and verification of a one-dimensional (planar) material thermal response code with ablation is presented. The mixture energy, gas phase continuity, and solid phase continuity equations are solved with Fourier's law to model heat conduction, Darcy's law to model porous flow, and the ideal gas law to model the state of the pyrolysis gases. Consequently, the temperature, gas density, and solid density profiles are predicted for a decomposing ablator and the resulting gas flux, porosity, and pore pressure can be determined.

The control volume finite element spatial discretization method (CVFEM), the Euler implicit time integrator, and a contracting grid scheme are used for the solution of the mixture energy and gas phase continuity equations. The solid continuity equation is solved through direct integration of decomposition kinetics under the assumption of a constant temperature rise rate within a given time step. The mixture energy and gas phase continuity equations are solved using segregated Newton solvers, which allow for nonlinear iteration on the entire system of nodal equations that are discretized according to a residual formulation. The block Gauss-Seidel segregated solution procedure has been implemented to globally iterate on the system of governing equations resulting in a fully coupled solution. Formal verification studies were performed that show the implemented model exhibits second order spatial accuracy and first order temporal accuracy. In addition, the second order nonlinear convergence of the Newton solvers was verified for temperature dependent material properties, the thermochemical ablation model, the heat of ablation model, decomposing materials, and several nonlinear boundary conditions. While not considered a part of the formal verification process, code-to-code comparisons are also presented.

Timing studies were performed, and when comparable accuracy is considered, the method developed in this study exhibits significant time savings over the property lagging approach typically used in legacy codes. In addition, maximizing the Newton solver's convergence rate by including sensitivities to the surface recession rate for the mixture energy equation reduces the overall computational time when compared to lagging the grid convection terms in the iteration process.