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.

The solution to the Navier–Stokes equations for a parallel, laminar flow can become unstable if certain conditions on the flow are satisfied, and the Orr–Sommerfeld equation determines precisely what the conditions for hydrodynamic stability are.

The equation is named after William McFadden Orr and Arnold Sommerfeld, who derived it at the beginning of the 20th century.

Using this knowledge, and the streamfunction representation for the flow, the following dimensional form of the Orr–Sommerfeld equation is obtained: where

Then the equation takes the form where is the Reynolds number of the base flow.

is positive, then the base flow is unstable, and the small perturbation introduced to the system is amplified in time.

Any solution to the three-dimensional equation can be mapped back to a more unstable (lower Reynolds number) solution of the two-dimensional equation above due to Squire's theorem.

It is therefore sufficient to study only two-dimensional disturbances when dealing with the linear stability of a parallel flow.

, numerical or asymptotic methods are required to calculate solutions.

The first figure shows the spectrum of the Orr–Sommerfeld equation at the critical values listed above.

At the critical values of Reynolds number and wavenumber, the rightmost eigenvalue is exactly zero.

For higher (lower) values of Reynolds number, the rightmost eigenvalue shifts into the positive (negative) half of the complex plane.

Then, a fuller picture of the stability properties is given by a plot exhibiting the functional dependence of this eigenvalue; this is shown in the second figure.

The third figure shows the neutral stability curve which divides the

On the other hand, the spectrum of eigenvalues for Couette flow indicates stability, at all Reynolds numbers.

[3] However, in experiments, Couette flow is found to be unstable to small, but finite, perturbations for which the linear theory, and the Orr–Sommerfeld equation do not apply.

It has been argued that the non-normality of the eigenvalue problem associated with Couette (and indeed, Poiseuille) flow might explain that observed instability.

Even if the energy associated with each eigenvalue considered separately is decaying exponentially in time (as predicted by the Orr–Sommerfeld analysis for the Couette flow), the cross terms arising from the non-orthogonality of the eigenvalues can increase transiently.

Thus, the total energy increases transiently (before tending asymptotically to zero).

Although that theory does include linear transient growth, the focus is on 3D nonlinear processes that are strongly suspected to underlie transition to turbulence in shear flows.

The theory has led to the construction of so-called complete 3D steady states, traveling waves and time-periodic solutions of the Navier-Stokes equations that capture many of the key features of transition and coherent structures observed in the near wall region of turbulent shear flows.

It is postulated that transition to turbulence involves the dynamic state of the fluid evolving from one solution to the next.

The theory is thus predicated upon the actual existence of such solutions (many of which have yet to be observed in a physical experimental setup).

Thus, even though not as rigorous as previous approaches to transition, it has gained immense popularity.

An extension of the Orr–Sommerfeld equation to the flow in porous media has been recently suggested.

[14] For Couette flow, it is possible to make mathematical progress in the solution of the Orr–Sommerfeld equation.

In this section, a demonstration of this method is given for the case of free-surface flow, that is, when the upper lid of the channel is replaced by a free surface.

Note first of all that it is necessary to modify upper boundary conditions to take account of the free surface.

This is a single equation in the unknown c, which can be solved numerically or by asymptotic methods.

and for sufficiently large Reynolds numbers, the growth rate