Stability of Inviscid Parallel Flows between Two Parallel Walls

In this paper, the stability of inviscid parallel flow between two parallel walls is studied. Firstly, it is obtained that the profile of the base flow for this classical problem is a uniform flow. Secondly, it is shown that the solution of the disturbance equation is cr=U and ci=0, i.e., the propagation speed of the disturbance equals the flow velocity and the disturbance in this flow is neutral. Finally, it is suggested that the classical Rayleigh Theorem on inflectional velocity instability is incorrect which states that the necessary condition for instability of inviscid parallel flow is the existence of an inflection point on the velocity profile.

💡 Research Summary

The paper attempts to reassess the classical Rayleigh theorem on the instability of inviscid parallel shear flows by focusing on the simplest geometry: an inviscid fluid confined between two parallel walls. The authors first solve the steady Euler equations under the assumption of a purely streamwise base flow (V = 0) and impose slip boundary conditions at the walls. By further assuming that the streamwise pressure gradient ∂p/∂x vanishes, they conclude that the only physically admissible solution of the Euler equations is a uniform (plug) flow, U(y)=U₀, with constant pressure throughout the domain.

With this base flow, the linearized disturbance equations reduce to the classic Rayleigh equation: (U‑c)(φ’’‑α²φ)‑U’‘φ = 0. Because U’’ = 0 for a uniform flow, the equation simplifies to (U₀‑c)(φ’’‑α²φ) = 0. Two possibilities arise: (i) c = U₀, which yields a purely real phase speed equal to the base‑flow speed and a growth rate ci = 0 (neutral disturbance); (ii) φ’’‑α²φ = 0, whose general solution φ(y)=A e^{αy}+B e^{-αy} cannot satisfy the wall condition φ(±h)=0 unless A = B = 0, leaving only the trivial solution. Hence the authors claim that the only admissible disturbance is neutral and propagates at the base‑flow speed.

From this result they argue that the Rayleigh inflection‑point criterion—stating that a necessary condition for inviscid instability is the existence of a point where U’’ changes sign—is inapplicable because the base flow has no curvature (U’’ = 0). Consequently, they assert that the classical Rayleigh theorem is incorrect.

To support their claim, the authors invoke their previously proposed Energy Gradient Theory (EGT), which posits that non‑zero vorticity (U’ ≠ 0) is the necessary and sufficient condition for inviscid instability. They argue that EGT and Rayleigh’s theorem are contradictory, and that experimental observations (e.g., transition in pipe, plane Poiseuille, and Couette flows) are consistent with EGT but not with Rayleigh’s inflection‑point condition.

The paper concludes that (1) the inviscid base flow between two parallel walls must be uniform, (2) disturbances are neutral, (3) there can be no inflection point in such a flow, and (4) therefore Rayleigh’s theorem is fundamentally flawed.

Critical appraisal:

• The assumption ∂p/∂x = 0 is overly restrictive. In real inviscid shear flows a streamwise pressure gradient can balance the shear, allowing non‑uniform velocity profiles that satisfy the Euler equations. By discarding this possibility the authors artificially force the base flow to be uniform.
• Slip boundary conditions alone do not preclude a pressure gradient; the Euler equations admit solutions with U(y) varying linearly or quadratically together with a constant pressure gradient.
• Rayleigh’s theorem is a necessary condition, not a sufficient one. The authors misinterpret the theorem as an absolute statement of instability and then declare it “incorrect” when a special case (uniform flow) trivially satisfies the theorem.
• The Energy Gradient Theory’s claim that non‑zero vorticity is both necessary and sufficient for inviscid instability is essentially equivalent to the Rayleigh condition (U’’ ≠ 0 somewhere). The apparent contradiction stems from different formulations rather than genuine incompatibility.
• The paper’s analysis is limited to the trivial uniform flow and does not address the broad class of inviscid shear flows that are experimentally relevant. Consequently, the conclusion that Rayleigh’s theorem is wrong lacks generality and rests on an unjustified restriction of the governing equations.

In summary, while the paper correctly shows that a uniform inviscid flow is neutrally stable, its broader claim that the classical Rayleigh inflection‑point theorem is invalid is not supported when the full range of admissible inviscid solutions (including non‑zero pressure gradients) is considered.