Simple shape model for normal shock trains in straight channels

Normal shock trains are a flow phenomenon of significance to ramjet engines, but it remains unclear what its structure is decided by and how it evolves with the incoming Mach number. To seek a theoretical explanation, the minimum entropy production principle is generalized to the quasi-steady behavior of normal shock trains in two-dimensional straight channels with uniform incoming flow. Numerical simulations are also performed to validate the model together with the data collected from public literature. The analysis suggests that the flow parameters of a normal shock train depend on the inviscid shock-shock interactions rather than the local boundary-layer separations, though the angles of two incident shocks should still be equal as similar to the case that complies with the free-interaction theory. The shock feet’s positions, meanwhile, are allowed to be coincident or not, free from the entropy restriction. This freedom of position explains why both symmetric and partially asymmetric normal shock trains could be found previously. Further theoretical calculations reveal the inclinations of two incident shocks increase first and then decrease with the incoming Mach number, peaking at 48.570 degrees when the Mach number reaches 1.753. It is also indicated that the Mach number range allowing for a normal shock train is 1.652 to 2.254, giving evidence for past observations.