The excellent agreement between the thrust predicted by the one-dimensional model of the thermally choked propulsive mode and the experimentally determined thrust at projectile velocities below 85% of CJ detonation speed, as noted in prior publications, has been explained by recent theoretical investigations (Knowlen and Bruckner 1992). For accelerations up to 30,000 g, it has been shown that a quasi-steady Hugoniot analysis of a control volume containing an arbitrarily shaped projectile can be used to determine the end state properties of the reacted propellant mixture as a function of the initial conditions and in-tube Mach number. If the flow is assumed to exit the control volume at sonic velocity (with respect to the control volume) then the chemical equilibria and stream thrust (defined here as the sum of the static pressure and momentum flux) of the exhaust gas are uniquely constrained by the gasdynamic conservation equations.
One of the most counter-intuitive aspects of thermally choked operation is the insensitivity of the projectile thrust to the aerodynamic efficiency of the supersonic diffuser. Indeed, the projectile acceleration is to a high degree independent of its geometry, as long as the flow area around the projectile throat (defined here as the point of maximum projectile cross-sectional area) is large enough to accept the incoming gas, yet small enough to keep the normal shock train on the rear half of the projectile. This invariance of thrust with aerodynamic efficiency is a consequence of operating a propulsive cycle that is stabilized with an end state that corresponds to an entropy extremum (i.e., thermal choking). In the quasi-one-dimensional model of the thermally choked mode there is a normal shock that can move on the projectile body to compensate for non-isentropic flow field phenomena, whereas in actuality, there is a complicated pressure-coupling between the combustion region and the shock system on the rear half of the projectile. Regardless of the details of the flow around the projectile, the thermally choked ram accelerator has a unique end state condition that satisfies the conservation equations and, consequently, a unique thrust-velocity profile for every propellant mixture.
The general expression for the quasi-steady thrust, F, of any ram accelerator mode is determined from the one-dimensional gasdynamic conservation equations as a function of projectile in-tube Mach number, M1, and the relative Mach number of the flow leaving the projectile, M2, as follows (Knowlen and Bruckner 1992):
where p1 is the propellant fill pressure, A is the tube cross-sectional area, g1 and g2 are the specific heat ratios before and after combustion, respectively, and Q is a nondimensional heat parameter defined as the ratio of the combustion heat release, Dq, to the product of the constant pressure heat capacity, cp1, and static temperature, T1, of the undisturbed propellant mixture, i.e., Q = Dq/cp1T1.
This thrust coefficient equation applies to all ram accelerator propulsive modes operating in a quasi-steady manner, even though no details of the internal flow have been considered. If one knows how M2 varies with M1 in a given propellant mixture, then the projectile thrust can be readily estimated for any flight velocity. In addition, Eq. 1 indicates that the thrust increases with increasing heat release for a given g1, g2, M1 and M2. The details of the flow field around the projectile must be considered to accurately predict the exit Mach number, M2, for the propulsive cycles operating in the transdetonative and superdetonative velocity regimes. However, thermal choking of the flow behind the projectile (M2 = 1) corresponds to a state of maximum entropy, thus, the details of the process which brings the flow to choking do not affect the end state conditions of the thermally choked ram accelerator mode, and hence, do not have to be known to predict the thrust.