Rayleigh's equation (fluid dynamics)

In fluid dynamics, Rayleigh's equation or Rayleigh stability equation is a linear ordinary differential equation to study the hydrodynamic stability of a parallel, incompressible and inviscid shear flow.

is the complex valued amplitude of the infinitesimal streamfunction perturbations applied to the base flow,

is the phase speed with which the perturbations propagate in the flow direction.

The prime denotes differentiation with respect to

The equation is named after Lord Rayleigh, who introduced it in 1880.

[2] The Orr–Sommerfeld equation – introduced later, for the study of stability of parallel viscous flow – reduces to Rayleigh's equation when the viscosity is zero.

[3] Rayleigh's equation, together with appropriate boundary conditions, most often poses an eigenvalue problem.

In general, the eigenvalues form a continuous spectrum.

In certain cases there may further be a discrete spectrum of complex conjugate pairs of

[3] Rayleigh's equation only concerns two-dimensional perturbations to the flow.

From Squire's theorem it follows that the two-dimensional perturbations are less stable than three-dimensional perturbations.

the problem has so-called critical layers near

At the critical layers Rayleigh's equation becomes singular.

These were first being studied by Lord Kelvin, also in 1880.

[4] His solution gives rise to a so-called cat's eye pattern of streamlines near the critical layer, when observed in a frame of reference moving with the phase speed

[3] Consider a parallel shear flow

[1] The stability of the flow is studied by adding small perturbations to the flow velocity

The flow is described using the incompressible Euler equations, which become after linearization – using velocity components

the partial derivative operator with respect to time, and similarly

ensure that the continuity equation

The fluid density is denoted as

and is a constant in the present analysis.

ensuring that the continuity equation is satisfied: Taking the

can be eliminated: which is essentially the vorticity transport equation,

Next, sinusoidal fluctuations are considered: with

the complex-valued amplitude of the streamfunction oscillations, while

denotes the real part of the expression between the brackets.

The boundary conditions for flat impermeable walls follow from the fact that the streamfunction is a constant at them.

So at impermeable walls the streamfunction oscillations are zero, i.e.

For unbounded flows the common boundary conditions are that