Eigenvalue Bounds for the Orr-Sommerfeld Equation and Their Relevance to the Existence of Backward Wave Motion

Abstract

Theoretical estimates of the phase velocity cr of an arbitrary unstable, marginally stable or stable wave derived on the basis of the classical Orr–Sommerfeld eigenvalue problem governing the linear instability of plane Poiseuille flow or nearly parallel viscous shear flows in straight channels with velocity U(z) (=1−z2, z∈[−1, +1] for plane Poiseuille flow), leave open the possibility that these phase velocities lie outside the range Umin<cr<Umax but not a single experimental or numerical investigation, concerned with unstable waves in the context of flows with (d2U/dz2)max≤0, has supported such a possibility as yet. Umin, Umax and (d2U/dz2)max are, respectively, the minimum value of U(z), the maximum value of U(z), and the maximum value of (d2U/dz2) for z∈[−1, +1]. This gap between the theory on one hand and experiment and computation on the other has remained unexplained ever since Joseph [3] derived these estimates, first in 1968, and has even led to the speculation of a negative phase velocity in plane Poiseuille flow (i.e., cr<Umin=0) and hence the possibility of a “backward” wave as in Jeffrey-Hamel flow in a diverging channel with backflow [1]. A simple mathematical proof of the nonexistence of such a possibility is given herein by showing that if (d2U/dz2)max≤0 and (d4U/dz4)min≥0 for z∈[−1, +1], then the phase velocity cr of an arbitrary unstable wave must satisfy the inequality Umin<cr<Umax, (d4U/dz4)min is the minimum value of (d4U/dz4) for z∈[−1, +1], and therefore cr cannot be negative when Umin=0. Another result that provides valuable insight into the general modal structure of the problem of instability of the above class of flows with Umin≥0 (e.g., plane Poiseuille flow) is that all standing waves, that is, modes for which cr=0, are stable.