The effect of opening angle on shock patterns in supersonic jets

The initial opening angle of an astrophysical jet is a key indicator of the downstream shock configuration and kinetic efficiency. We analyse here the relationship for hydrodynamic jets of high pressure issuing from a circular nozzle. A jet can eventually settle down, after the initial penetration, into a divergent-convergent shape that forms of one of two well-known shock patterns described as shock diamonds and Mach shock discs. The distance and shape of the standoff shock are investigated here through numerical simulations. As the opening angle increases, the standoff distance to the first shock generally occurs further from the nozzle, reaching a maximum, and then decreasing. A total of six distinct flow patterns are recognized. We establish semi-empirical formula to link the physical parameters to the flow pattern. Large opening angles with Mach shock discs and subsequent plumes provide an interpretation for jet structures such as wide-angle FR I and stunted FR 0 radio galaxies. This can occur even when the jet is pressure matched to the ambient medium. In this case, a half-opening angle of $16^\circ$ is found, beyond which at least the spine of the jet does not remain supersonic. Remarkably, this angle is a tight prediction, independent of Mach number. These results inform discussions of astrophysical jets as well as the release of gas in other contexts. It should be noted that we limit the study to non-relativistic hydrodynamics of adiabatic flow into a uniform medium. Relaxing these constraints requires extensive further studies.