Improved energy normalization function in rocket motor stability calculations

This work details the derivation of the energy normalization function which arises in motor stability calculations. It starts by identifying the flow parameters that appear in Kirchoff's expression defining the acoustic energy density in a given enclosure. Special care is taken to account for the rotational contributions due to unsteady chamber vorticity. Subsequently, these flow parameters are inserted into the energy normalization function and perturbed to the leading order. The resulting expressions are simplified and asymptotically integrated to obtain the average value of the energy normalization function. This procedure is repeated for two basic geometries pertaining to the circular-port and slab rocket motors, respectively. Our results demonstrate that inclusion of unsteady rotational flow components is critically important for the accurate assessment of energy. We also find that the conventional one-dimensional approach is lacking because it omits unsteady rotational contributions. It under-predicts the energy normalization function by 25% in each of the circular and slab motor cases.