Cross-flow transition over a delta wing is systematically studied in a Mach 6.5 hypersonic wind tunnel, employing the Rayleigh scattering flow visualisation, high-speed schlieren and fast-response pressure sensors. Direct numerical simulations and analysis based on linear stability theory under the same flow conditions are applied to analyse the transition mechanism. Three unstable modes are identified: the travelling cross-flow instabilities, the second mode and the low-frequency waves. It is shown that the travelling cross-flow vortices first appear in the cross-flow region near the leading edge of the model. These vortices can modulate the mean profile of the flow, which benefits the growth of second mode. A phase-locked interaction mechanism transfers energy from the cross-flow instabilities to the high-frequency second mode, leading to amplification at the expense of the cross-flow instability. As the second mode grows to a critical amplitude, it triggers a $z$-type secondary instability within a similar frequency range, which introduces secondary finger-like structures connecting to the cross-flow vortex. It is further found that the generation of these finger-like structures is related to the expansion and compression of the second mode. These finger vortices further evolve along the streamwise direction into low-frequency waves and corresponding hairpin-like structures that finally trigger turbulence. An interaction mechanism likely exists between the secondary instability and the low-frequency waves, since their phase speeds are approaching each other. These observations of the interaction mechanism are consistent with those of previous studies on hypersonic boundary layers (Zhang et al., Phys. Fluids, vol. 32 (7), 2020, 071702; Li et al., Phys. Fluids, vol. 32 (5), 2020, 051701).

In this paper we discuss the dynamics of vorticity at partial-slip boundaries. We consider the total vector circulation, which includes both the total vorticity of the fluid and the slip velocity at the boundary (the interface vortex sheet). The generation of vector circulation is an inviscid process, which does not depend on either viscosity or the slip length at the boundary. Vector circulation is generated by the inviscid relative acceleration between the fluid and the solid, due to either tangential pressure gradients or tangential acceleration of the partial-slip wall. While the slip length does not affect the creation of vector circulation, it governs how vector circulation is distributed between the total vorticity of the fluid and the interface vortex sheet. Specifically, the partial-slip boundary condition prescribes the ratio between boundary vorticity and the strength of the interface vortex sheet, and the viscous boundary flux transfers vector circulation between the interface vortex sheet and the fluid interior to maintain this condition. The interaction between a vortex ring and a partial-slip wall is examined to highlight various aspects of this formulation. For the head-on collision, the quantity of vector circulation diffused into the fluid as secondary vorticity increases as the slip length is decreased, resulting in a stronger secondary vortex and increased rebound of the vortex ring. For the oblique interaction, the extent to which the vortex ring connects to the boundary increases as the slip length is increased.

This study extends our previous work (McCloughan & Suslov, J. Fluid Mech., vol. 887, 2020, A23), where the existence of a saddle-node bifurcation of steady axisymmetric electrolyte flows driven by the Lorentz force in a shallow annular domain was first reported. Here we perform further weakly nonlinear analysis over a wider range of the governing parameters to demonstrate that the previously reported saddle-node bifurcation is a local feature of a global fold catastrophe, which, in turn, is a part of cusp catastrophe occurring as the thickness of the fluid layer increases. The amplitude equation characterising multiple flow solutions in the finite vicinity of catastrophe points is derived. The sensitivity of its coefficients and solutions to the distance from the catastrophe points is assessed demonstrating the robustness of the used analytical procedure. The asymptotic flow solution past the catastrophe point is subsequently obtained and its topology is explored confirming the existence of the secondary circulation in the bulk of flow (two-tori background flow structure). The latter is argued to lead to the appearance of experimentally observable vortices on the fluid surface. The rigorous justification of this conjecture is to be given in Part 2 of the study.