Abstract
Corresponding to Oswatitsch’s Mach number independence principle the Mach number of hypersonic inviscid flows, , does not affect components of various non-dimensional formulations such as velocity and density, pressure coefficients and Mach number behind a strong shock. On this account, the principle is significant in the development process for hypersonic vehicles. Oswatitsch deduced a system of partial differential equations which describes hypersonic flow. These equations are the basic gasdynamic equations as well as Crocco’s theorem which are reduced for the case of very high Mach numbers, . Their numerical solution can not only result in simplified algorithms prospectively utilized to describe the flow around bodies flying mainly in the lower stratosphere with very high Mach numbers. It also offers a deeper understanding of similarity effects for hypersonic flows. In this paper, a solution method for Oswatisch’s equations for perfect gas, based on a 4-step Runge-Kutta-algorithm, is presented including a fast shock-fitting procedure. An analysis of numerical stability is followed by a detailed comparison of results for different Mach numbers and ratios of the specific heats.
Highlights
Depending on the type of flight vehicle configuration and its mission, the atmospheric re-entry into the earth’s atmosphere takes place at Mach numbers between 20 and 30 [8]
This effect is typical for flows at very high Mach numbers and is besides this amplified by the low ratio of specific heats which leads to a higher density increase
We present the governing equations, characterize the system and show that a time-dependent solution method is applied in order to take into account the character of subsonic flows at blunt bodies
Summary
Depending on the type of flight vehicle configuration and its mission, the atmospheric re-entry into the earth’s atmosphere takes place at Mach numbers between 20 and 30 [8]. Oswatitsch’s Mach number independence principle explains why above a sufficiently high Mach number, for blunt bodies and behind a strong shock with Ma∞ ≈> 5 , some aerodynamic properties (such as CL , CD , Cp , CM , relative velocities u u∞ , v u∞ , the bow-shock stand-off distance ∆0 , the patterns of streamlines and Mach lines) become independent of Ma∞ [15] Having applied this approach in the 70s’s of the last century to the US space-shuttle, essential differences in the pitching moment between the wind tunnel test results and the data from the first flight occured. The Mach number principle itself is very common in hypersonics, no detailled numercial solution of Oswatitsch’s system of equations has been presented, so far Based on these equations’ solutions, general similarities between the body and the bow shock could be determined enabling the provision of more accurate initial solutions for advanced algorithms, e.g. a coupled Euler/second-order boundary-layer method [10]. A brief explanation of the implemented code and some numerical results of selected cases are presented in Section 4 and 5
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