Variational principle of a shock stationed in a duct

Abstract

This paper establishes a variational principle of a normal shock stationed in ducts of the various quasi-one-dimensional (quasi-1D) steady non-conservative flow systems. It is found that the locational stability of a shock inside these ducts, namely, the stationary property of the shock, can be judged by analyzing the second-order variation of the potential energy functional of flow impulse with respect to the locational function of the shock. It proves that, for a control volume containing a shock inside a duct, the real stationary location of the shock among all the possible locations satisfying the determined inlet and outlet boundary conditions of the duct is equivalent to that the potential energy of the cross-sectional flow impulse integrated through the entire duct is a minimal. First, the shock location in general duct flows is analyzed by a momentum relaxation method. Then, based on this method, this paper's variational principle is established referring to the principle of minimum potential energy and the principle of virtual displacement. Further, this principle is applied to the flows through a quasi-1D variable-area duct, a one-dimensional (1D) frictional constant-area duct, a 1D heat-exchange constant-area duct, and a 1D mass-additional constant-area duct, which verified the generality of the principle. At last, relevant examples are provided. This variational principle affords a unified and concise theoretical criterion to analyze the stationary property of a normal shock.