An analytical solution for unconfined, unsteady, inviscid jets; with applications to penetration problem debris cloud formation

Abstract

Here we derive a closed form analytical solution for an unsteady inviscid jet using linearized, acceleration potential theory and classical analytical methods. Use is made of both the Laplace transformation and reduction to a self-similar form to solve the associated governing equations. The convolution theorem provides a closed form mapping from the transformed plane to the real plane. The streamwise diffusion term, e.g., χχ is shown to be second order and is neglected. This analytical model is used to estimate the debris cloud geometry and velocity field as a function of location and time. Though formal solutions of the potential equation yield good results for debris cloud expansion near the initial impact point, χ ≪ 1, the debris cloud expansion front behavior is not recovered. The steady state eigenfunction expansion solution is used to extend the unsteady solution in an approximate manner. The extended solution retains physically correct expansion behavior for χ ≪ 1 but also provides a reasonable model near the debris cloud expansion front. Since debris cloud dynamics and witness plate impact are readily obtained from experimental observations, this model provides a simple, but useful supplement to conventional hydrocode simulation of impact and penetration phenomenon.