Modeling the Gravitational Field by Using CFD Techniques

Abstract

The Laplace equation represents harmonic (i.e., both source-free and curl-free) fields. Despite the good performance of spherical harmonic series on modeling the gravitational field generated by spheroidal bodies (e.g., the Earth), the series may diverge inside the Brillouin sphere enclosing all field-generating mass. Divergence may realistically occur when determining the gravitational fields of asteroids or comets that have complex shapes, known as the Complex-boundary Value Problem (CBVP). To overcome this weakness, we propose a new spatial-domain numerical method based on the equivalence transformation which is well known in the fluid dynamics community: a potential-flow velocity field and a gravitational force vector field are equivalent in a mathematical sense, both referring to a harmonic vector field. The new method abandons the perturbation theory based on the Laplace equation, and, instead, derives the governing equation and the boundary condition of the potential flow from the conservation laws of mass, momentum and energy. Correspondingly, computational fluid dynamics (CFD) techniques are introduced as a numerical solving scheme. We apply this novel approach to the gravitational field of comet 67P/Churyumov-Gerasimenko with a complex shape. The method is validated in a closed-loop simulation by comparing the result with a direct integration of Newton’s formula. It shows a good consistency between them, with a relative magnitude discrepancy at percentage level and with a maximum directional difference of 5°. Moreover, the numerical scheme adopted in our method is able to overcome the divergence problem and hence has a good potential for solving the CBVPs.