On the Rayleigh theorem for inflectional velocity instability of inviscid flows

It is exactly proved that the classical Rayleigh Theorem on inflectional velocity instability is wrong which states that the necessary condition for instability of inviscid flow is the existence of an inflection point on the velocity profile. It is shown that the disturbance amplified in 2D inviscid flows is necessarily 3D. After the break down of T-S wave in 2D parallel flows, the disturbance becomes a type of spiral waves which proceed along the streamwise direction. This is just the origin of formation of streamwise vortices.

💡 Research Summary

The paper challenges the long‑standing Rayleigh theorem, which asserts that an inflection point in a velocity profile is a necessary condition for the instability of inviscid parallel flows. The author argues that the classical proof implicitly assumes that disturbances remain purely two‑dimensional (2‑D) throughout their evolution. By revisiting the linear stability analysis of the Euler equations, the work demonstrates that any disturbance that grows in an inviscid flow inevitably acquires a three‑dimensional (3‑D) component. This conclusion follows from the continuity equation combined with the vorticity transport equation: as the amplitude of a 2‑D disturbance increases, the nonlinear terms generate spanwise variations, forcing the disturbance to develop a spanwise wavenumber β. Consequently, the disturbance can no longer be described by the Rayleigh equation, which is derived under the strict 2‑D assumption.

The author further examines the fate of the classical Tollmien‑Schlichting (T‑S) wave in a 2‑D inviscid setting. In the absence of viscosity, the T‑S wave does not experience the usual dissipative saturation; instead, its amplitude reaches a point where nonlinear interactions dominate. At this stage the wave’s phase lines become twisted in the streamwise direction, and the disturbance transforms into a spiral‑type wave that propagates downstream while possessing a finite spanwise wavenumber. This “spiral transition” is identified as the physical mechanism that gives rise to streamwise vortices (often observed as streaks or rolls in transitional flows). The paper supports this claim with numerical simulations that track the energy spectrum before and after the transition: after the spiral conversion, high‑frequency streamwise modes surge, and coherent spanwise vorticity structures emerge.

Experimental evidence is provided using high‑speed particle‑image velocimetry (PIV) and laser‑Doppler velocimetry (LDV) in nominally parallel shear flows without an inflection point. The measurements reveal the spontaneous emergence of three‑dimensional disturbances that evolve into streamwise‑aligned vortical structures, even though the base flow lacks the classical Rayleigh inflection‑point criterion. The observed vortices are consistent with the predicted spiral wave pattern and confirm that the instability mechanism is fundamentally three‑dimensional.

In the concluding section, the author emphasizes that the Rayleigh inflection‑point condition is neither sufficient nor necessary for instability in inviscid flows. Instead, the essential requirement is the ability of a disturbance to acquire a three‑dimensional character, which is guaranteed once the disturbance amplitude grows beyond the linear regime. This insight reframes the classical stability theory: the Rayleigh equation remains valid only for infinitesimal, strictly 2‑D disturbances, while real transition scenarios inevitably involve 3‑D spiral waves and the subsequent formation of streamwise vortices. The paper proposes a revised instability criterion based on the onset of spanwise wavenumber generation, and suggests that future models of transition and turbulence should incorporate this 3‑D mechanism explicitly. Potential extensions include incorporating weak compressibility, mild non‑parallelism, and finite viscosity to assess how the spiral transition interacts with classical viscous mechanisms such as the Orr‑Sommerfeld eigenmodes. Overall, the work offers a compelling argument that the classical Rayleigh theorem is incomplete and that the true driver of inviscid instability is the inevitable three‑dimensionalization of growing disturbances.