On the structure of a class of aerothermodynamic shocks

Abstract

Some recent work on the existence of vibrational de-excitation shocks (δ-shocks) in expanding non-equilibrium nozzle flows is extended to include situations in which an adiabatic shock (δ-shocks) may be embedded within the de-excitation shock. A discussion of some further properties of the shock solution is given and some examples are worked out. Numerical solutions of the full equations are also presented. These solutions confirm the existence of the δ-shocks but bring to light certain anomalies in the simple approximate solution. The modifications necessary to remove these discrepancies are outlined, and the implications of the numerical results are briefly discussed. Finally, some comments on the nature of the asymptotic solution for an arbitrary rate process are made.

Gas dynamics with relaxation effects

“…The precursor to this motion will still be described by the non-linear theory of $4.4.1 in general. All of these results are originally due to Blythe (1969) and illustrate something of the complexities introduced into otherwise elementary flow situations by the presence of relaxation. However, one is not finished yet as the results in this and in the previous subsection imply a limitation on the size of (a2-1).…”

“…The situation covered by these approximate deductions can be recognised as wholly consistent with the exact results of Q3.2 and 4.1, and it actually extends these results in numerous helpful ways; it also constitutes the analytical counterpart to the numerical computations outlined briefly in $4.2. There are some limitations, which it is not vital to make explicit here, especially since there are some more, strongly-related, matters to follow, but the interested reader can consult the original work by Blythe (1969) or any of the reviews and developments by Lick (1970), Becker (1972) and Clarke and McChesney (1976).…”

“…Far-field behaviour, in the sense that (u2-1)x/2a1 o o ~c o is large, has been discussed for the acoustic case in $2.5.2. Blythe (1969) showed that changing to coordinates where the gauge function A ( € ) is still to be found, gives the following equation for V ( 1 ) :…”

“…It transpires that the correct coordinates in the new circumstances, when (a2-~) = O ( E ) , are: (4.17) with u2: €u(y(', Y'). Blythe (1969) and simultaneously, using a somewhat different set of arguments, by Ockendon and Spence (1969). It is a hyperbolic equation, with characteristics x' = constant and , B = constant, where , B is defined so that (cf (4.13) and the remarks which follow about the exact characteristic parameter /3).…”