On the behaviour of small disturbances in plane Couette flow

Abstract

The problem considered here is concerned with small disturbances of plane Couette flow. As is usual in such problems it is assumed that the disturbance velocities are sufficiently small to allow the Navier-Stokes equations to be linearized. There results a special case of the well-known Orr-Sommerfeld equation and this is solved by an exact method using a digital computer. The problem has previously been considered by several authors, mostly using approximate methods and their results have been compared where possible with those obtained here. It was possible to proceed to values of αR not in excess of 1000 (α being the wave-number of the disturbance and R the Reynolds number of the basic flow), and the results tend to confirm the belief that Couette flow is stable at all Reynolds numbers.

Pressure distributions on circular cylinders at critical Reynolds numbers

“…This result is not however of primary importance as we shall explain in the next section. Their work was substantiated by Deardorff (1963), who calculated some higher eigenvalues and extended slightly the parameter range covered by Gallagher & Mercer (1962). They carried out an extensive numerical investigation of the basic twodimensional Orr-Sommerfeld problem for moderate values of a and R such that aR was less than about 1000, and indeed found that the flow was stable for this parameter range.…”

“…The first notable numerical attack was made by Gallagher & Mercer (1962). I n a later paper Gallagher & Mercer (1964) also calculated some higher eigenvalues and they found that a mode-crossing phenomenon calculated by Grohne (1954) was erroneous. Their work was substantiated by Deardorff (1963), who calculated some higher eigenvalues and extended slightly the parameter range covered by Gallagher & Mercer (1962).…”

Uniform Asymptotic Approximations to the Solutions of the Orr‐Sommerfeld Equation. Part 1. Plane Couette Flow

“…The roots '0 of equation (5.10). The points labelled z, and z: (5 = 1,2,3) are the zeros of A,(z, I); the points shown on the negative real axis are the zeros of (5.7); and the three circled points for the least stable asymmetric mode are from Gallagher and Mercer (1962).…”

“…2 (say) then). may be found directly from the eigenvalue problem [see, for example, Gallagher and Mercer (1962)] (5.12)…”

Gravitational instability in the thermal entry region with liquid film flow

“…The stability criteria of flowing films subject to a linear temperature profile was studied extensively by Chandra (1938) experimentally and by Gallagher and Mercer (1965) and Deardorff ( 1965) time-dependent problem because of the nonlinearity of the temperature distribution. The spatially-dependent problem is studied in the present paper to predict the stability characteristics.…”