Orr-Sommerfeld equation

The Orr-Sommerfeld equation, in fluid dynamics, is an eigenvalue equation describing the linear two-dimensional modes of disturbance to a viscous parallel flow.

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The equation takes the form:

${1 \over i \alpha \, Re} \left({d^2 \over d z^2} - \alpha^2\right)^2 \varphi = (U - c)\left({d^2 \over d z^2} - \alpha^2\right) \varphi - U'' \varphi$

with the boundary conditions (for no-slip boundaries at $z = z 1$ and $z = z 2$ )

$\alpha \varphi = {d \varphi \over d z} = 0$ at

z = z 1

and

z = z 2

where the base fluid velocity $\mathbf{u} = (U(z), 0, 0)$ and the disturbance is given by $\mathbf{u'} = \mathbf{\hat{u}}(z) \exp(i \alpha (x - c t))$ (real part understood). $R e$ is the Reynolds number.

The eigenvalue parameter of the problem is $c$ and the eigenvector is $\varphi$ .

Solutions

For all but the simplest of velocity profiles $U$ , numerical or asymptotic methods are required to calculate solutions. For Pouseille flow, it has been shown that the flow is unstable (i.e. one or more eigenvalues $c$ has a positive imaginary part) for some $α$ when $R e > R e c = 5772.22$ and the neutrally stable mode at $R e = R e c$ having $α c = 1.02056$ , $c = 0.264002$ . ^[1]

References

P. G. Drazin, W. H. Reid (1981) 'Hydrodynamic Stability', Cambridge University Press

^ Orszag S. A. (1971) 'Accurate solution of the Orr-Sommerfeld stability equation' J. Fluid. Mech. 50, 689-703

Categories: Fluid dynamics | Equations of fluid dynamics

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Orr-Sommerfeld_equation". A list of authors is available in Wikipedia.

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