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. The equation is:
( U − c ) ( φ ″ − k 2 φ ) − U ″ φ = 0 , {\displaystyle (U-c)(\varphi ''-k^{2}\varphi )-U''\varphi =0,}
with U ( z ) {\displaystyle U(z)} the flow velocity of the steady base flow whose stability is to be studied and z {\displaystyle z} is the cross-stream direction (i.e. perpendicular to the flow direction). Further φ ( z ) {\displaystyle \varphi (z)} is the complex valued amplitude of the infinitesimal streamfunction perturbations applied to the base flow, k {\displaystyle k} is the wavenumber of the perturbations and c {\displaystyle c} is the phase speed with which the perturbations propagate in the flow direction. The prime denotes differentiation with respect to z . {\displaystyle z.}
Background The equation is named after Lord Rayleigh, who introduced it in 1880. 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. Rayleigh's equation, together with appropriate boundary conditions, most often poses an eigenvalue problem. For given (real-valued) wavenumber k {\displaystyle k} and mean flow velocity U ( z ) , {\displaystyle U(z),} the eigenvalues are the phase speeds c , {\displaystyle c,} and the eigenfunctions are the associated streamfunction amplitudes φ ( z ) . {\displaystyle \varphi (z).} In general, the eigenvalues form a continuous spectrum. In certain cases there may further be a discrete spectrum of complex conjugate pairs of c . {\displaystyle c.} Since the wavenumber k {\displaystyle k} occurs only as a square k 2 {\displaystyle k^{2}} in Rayleigh's equation, a solution (i.e. φ ( z ) {\displaystyle \varphi (z)} and c {\displaystyle c} ) for wavenumber + k {\displaystyle +k} is also a solution for the wavenumber − k . {\displaystyle -k.}
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.
If a real-valued phase speed c {\displaystyle c} is in between the minimum and maximum of U ( z ) , {\displaystyle U(z),} the problem has so-called critical layers near z = z c r i t {\displaystyle z=z_{\mathrm {crit} }} where U ( z c r i t ) = c . {\displaystyle U(z_{\mathrm {crit} })=c.} At the critical layers Rayleigh's equation becomes singular. These were first being studied by Lord Kelvin, also in 1880. 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 c . {\displaystyle c.}
Derivation Consider a parallel shear flow U ( z ) {\displaystyle U(z)} in the x {\displaystyle x} direction, which varies only in the cross-flow direction z . {\displaystyle z.} The stability of the flow is studied by adding small perturbations to the flow velocity u ( x , z , t ) {\displaystyle u(x,z,t)} and w ( x , z , t ) {\displaystyle w(x,z,t)} in the x {\displaystyle x} and z {\displaystyle z} directions, respectively. The flow is described using the incompressible Euler equations, which become after linearization – using velocity components U ( z ) + u ( x , z , t ) {\displaystyle U(z)+u(x,z,t)} and w ( x , z , t ) : {\displaystyle w(x,z,t):}
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