Научный вестник МГТУ ГА (May 2017)
MAXIMUM PRINCIPLE FOR SUBSONIC FLOW WITH VARIABLE ENTROPY
Abstract
Maximum principle for subsonic flow is fair for stationary irrotational subsonic gas flows. According to this principle, if the value of the velocity is not constant everywhere, then its maximum is achieved on the boundary and only on the boundary of the considered domain. This property is used when designing form of an aircraft with a maximum critical value of the Mach number: it is believed that if the local Mach number is less than unit in the incoming flow and on the body surface, then the Mach number is less then unit in all points of flow. The known proof of maximum principle for subsonic flow is based on the assumption that in the whole considered area of the flow the pressure is a function of density. For the ideal and perfect gas (the role of diffusion is negligible, and the Mendeleev-Clapeyron law is fulfilled), the pressure is a function of density if entropy is constant in the entire considered area of the flow. Shows an example of a stationary subsonic irrotational flow, in which the entropy has different values on different stream lines, and the pressure is not a function of density. The application of the maximum principle for subsonic flow with respect to such a flow would be unreasonable. This example shows the relevance of the question about the place of the points of maximum value of the velocity, if the entropy is not a constant. To clarify the regularities of the location of these points, was performed the analysis of the complete Euler equations (without any simplifying assumptions) in 3-D case. The new proof of the maximum principle for subsonic flow was proposed. This proof does not rely on the assumption that the pressure is a function of density. Thus, it is shown that the maximum principle for subsonic flow is true for stationary subsonic irrotational flows of ideal perfect gas with variable entropy.
