Rayleigh's equation (fluid dynamics)

Example of a parallel shear flow.

In fluid dynamics, Rayleigh's equation is a linear ordinary differential equation to study the hydrodynamic stability of a parallel, incompressible and inviscid shear flow. The equation is:^[1]

${\displaystyle (U-c)(\phi ''-k^{2}\phi )-U''\phi =0,}$

with $U(z)$ the flow velocity of the steady base flow whose stability is to be studied. Further $\phi (z)$ is the complex valued amplitude of the infinitesimal streamfunction perturbations applied to the base flow, $k$ is the wavenumber of the perturbations and $c$ is the phase speed with which the perturbations propagate in the flow direction.

The equation is named after Lord Rayleigh, who introduced it in 1880.^[2]

Background

The parallel shear flow $U(z)$ is in the $x$ direction, and varies only in the cross-flow direction $z.$^[1] The stability of the flow is studied by adding small perturbations to the flow velocity ${\displaystyle u(x,z,t)}$ and ${\displaystyle w(x,z,t)}$ in the $x$ and $z$ directions, respectively. The flow is described using the incompressible Euler equations, which become after linearization – using velocity components ${\displaystyle U(z)+u(x,z,t)}$ and ${\displaystyle w(x,z,t):}$

${\displaystyle {\begin{aligned}&\partial _{t}u+U\,\partial _{x}u+w\,U'=-{\frac {1}{\rho }}\partial _{x}p,\\&\partial _{t}w+U\,\partial _{x}w=-{\frac {1}{\rho }}\partial _{z}p\qquad {\text{and}}\\&\partial _{x}u+\partial _{z}w=0,\end{aligned}}}$

with ${\displaystyle \partial _{t}}$ the partial derivative operator with respect to time, and similarly $\partial _{x}$ and $\partial_z$ with respect to $x$ and $z.$ The pressure fluctuations ${\displaystyle p(x,z,t)}$ ensure that the continuity equation $\partial_x u + \partial_z w = 0$ is fulfilled. The fluid density is denoted as $\rho$ and is a constant. The prime $U'$ denotes differentiation of $U(z)$ with respect to its argument $z.$

The flow oscillations $u$ and $w$ are described using a streamfunction ${\displaystyle \psi (x,z,t),}$ ensuring that the continuity equation is satisfied:

${\displaystyle u=+\partial _{z}\psi }$   and   ${\displaystyle w=-\partial _{x}\psi .}$

Taking the $z$- and $x$-derivatives of the $x$- and$z$-momentum equation, and subtracting, the pressure $p$ can be eliminated:

${\displaystyle \partial _{t}\left(\partial _{x}^{2}\psi +\partial _{z}^{2}\psi \right)+U\,\partial _{x}\left(\partial _{x}^{2}\psi +\partial _{z}^{2}\psi \right)-U''\,\partial _{x}\psi =0,}$

which is essentially the vorticity transport equation, ${\displaystyle \partial _{x}^{2}\psi +\partial _{z}^{2}\psi }$ being (minus) the vorticity.

Next, sinusoidal fluctuations are considered:

${\displaystyle \psi (x,z,t)=\Re \left\{\phi (z)\,\exp \left(i\,k\,\left(x-c\,t\right)\right)\right\},}$

with $\phi (z)$ the complex-valued amplitude of the streamfunction oscillations, while $i$ is the imaginary unit ($i^{2}=-1$) and ${\displaystyle \Re \left\{\cdot \right\}}$ denotes the real part of the expression between the brackets. Using this in the vorticity transport equation, Rayleigh's equation is obtained.

Notes

1. Craik (1988), pp. 21–27
2. Drazin & 2002 (138–154)

References

• Craik, A. D. D. (1988), Wave interactions and fluid flows, Cambridge University Press, ISBN 0-521-36829-4
• Drazin, P.G. (2002), Introduction to hydrodynamic stability, Cambridge University Press, ISBN 0-521-00965-0