Welcome back. In today’s post, I hope to take a small break from our discussion last week, and present an introduction I wrote to a project on the Orr-Sommerfeld equation. This work examined the effects of non-normality in plane Poiseuille flow, a canonical shear flow between two flat plates, and the overall motivation was to show the limitations of traditional eigenvalue analysis. Next week, we will return with our usual hydrodynamic stability program.

**Introduction to “A Matlab Exploration of the Orr-Sommerfeld Equation”:**

Wall-bounded flows have been studied for more than a century. Notably, Osborne Reynolds derived his famous Reynolds number in 1883 by watching laminar flow become turbulent as it traveled down a pipe. In the 21st century, channel flows have remained relevant case studies for hydrodynamic stability. The specific scenario analyzed in this paper was plane Poiseuille flow. A laminar, incompressible, viscous fluid moved between two flat plates located at y = ± 1. All variables were assumed to be nondimensionalized, and the velocity profile was

U(y) = 1 – y^{2}.

This study uncovered what would happen if this laminar flow were perturbed by a small disturbance **u**’** **= u’**i** + v’**j**. If the disturbance grew with time, then turbulence could ensue. Since turbulence could increase friction, it was desirable to understand how it developed in a channel. To answer this question, the 2D Navier-Stokes equations were linearized about **u**’. The components of **u**’ were then substituted for a stream function 𝜓, which was assumed to have the form

𝜓(x,y,t) = 𝜑(y,t) e^{i𝛼x},

where 𝛼 was a streamwise wavenumber. The derivation in Appendix I showed that the amplitude 𝜓 was governed by

B ∂𝜑/∂t = A𝜑,

where A and B were the following differential operators defined in terms of D = ∂/∂y, the Reynolds number R, and 𝛼:

A = 1/(i𝛼R) (D^{2} – 𝛼^{2})^{2} – (1 – y^{2})(D^{2} – 𝛼^{2}) – 2,

B = 1/𝛼 (D^{2} – 𝛼^{2}).

By further assuming that 𝜑 = 𝜙(y) e^{𝜆t} for some complex number 𝜆, system above turned into the generalized eigenvalue problem

B𝜆𝜙 = A𝜙.

Setting L = B^{-1}A yielded the Orr-Sommerfeld equation

𝜆𝜙 = L𝜙

where L was the Orr-Sommerfeld operator. Because the base flow was shear-dominated, L was non-normal, which meant that its dynamics were not completely determined by its eigenvalues. The specific purpose of this paper was to understand what information the spectrum of L could and could not provide. This led to an investigation of pseudospectra, which gave deeper information about the finite-time behavior of solutions. A secondary goal was to compute and plot the least stable eigenfunctions of L. These were the building blocks of Tollmien-Schlichting waves, which develop in channels as laminar flows become turbulent.

## Leave a Reply