Temperature solution of one-dimensional modeling of rapid transient heat transfer around a cylindrical body

Abstract
The aim of the present paper is to provide a numerical solution of the heat transfer equation into and along the wall of a cylindrical profile in a very high-speed flow. A series of diagrams are derived. They allow to calculate the time-dependent temperature at any point on the surface as well as inside the body. The profile, which is initially at a uniform temperature, is suddenly submitted to a convective and/or radiative heat flux around its surface. These diagrams provide useful data on extremely rapid heat transfer phenomena, i.e., at very low Fourier (Fo) numbers. They can be used regardless the dimensions, thermophysical characteristics of the body, and thermal characteristics of the wall, i.e., at any Biot (Bi) number. They provide a reliable shortcut for the understanding of the thermal behavior of the projectile abrasion in a ram accelerator operation. The analytical aspect of the problem is detailed in the case of a long, i.e., one dimensional, cylindrical geometry, but the diagrams can be used for 2D transient phenomena, in relation with the solution of the infinite plate, which has been developed earlier for the same range of Fourier and Biot numbers. The general equation that is presented in the paper is quite complex to solve for some Fo-Bi combinaisons. Therefore, a numerical procedure was developed in order to account fore some important divergence problems that may occur during the calculation in these specific cases. The accuracy of the present diagrams has been strongly improved in comparison with that of the same type of plots, which are available in the literature and turn out to yield a poor accuracy for some specific Fo-Bi combinaisons.