Global stability of spiral flow. Part 2

Abstract

The stability of spiral flow between rotating and sliding cylinders is considered. In the limit of narrow gap, a’ modified’ energy theory is constructed. This theory exploits the consequences of assuming the existence of a preferred spiral direction along which disturbances do not vary. The flow is also analyzed from the viewpoint of linearized theory. Both problems depend strongly on the sign of Rayleigh's discriminant, – 2Ωζ. Here Ω is the component of angular velocity, and ζ is the component of total vorticity of the basic flow in the direction perpendicular to the spiral ribbons on which the disturbance is constant. When the discriminant is negative, there is evidently no instability to infinitesimal disturbances, and the spiral disturbance whose energy increases at the smallest R is a roll whose axis is perpendicular to the stream. This restores and generalizes Orr's non-linear results for disturbances having a preferred spiral direction. When the discriminant is positive, the critical disturbances of linear theory and the modified energy theory are spiral vortices. The differences between the energy and linear limits can be made smaller in the restricted class of disturbances with coincidence achieved for axisymmetric disturbances in the rotating cylinder problem in the limit of narrow gap. For the sliding-rotating case, the critical disturbance of the linear theory appears as a periodic wave in a co-ordinate system fixed on the outer cylinder. This wave has a dimensionless frequency equal to - ½ a sin (χ-ψ), where a is the wave-number, χ is the angle between the pipe axis and the direction of motion of the inner cylinder relative to the outer one, and ψ is the disturbance spiral angle.Instability limits, frequencies and wave-numbers are computed numerically when the cylinder gap is not narrow. These are in even closer agreement with Ludwieg's experimental results than the approximate results which were given in part 1.

On the competition between centrifugal and shear instability in spiral Couette flow

“…Section 5 deals with the narrow gap case η = 0.8, where the same features are present. The results are compared with the experimental results of Ludwieg (1964) and the linear stability computations of Hung et al (1972), obtaining a very good agreement with both. A detailed analysis of the experimental data shows the presence of hysteresis regions associated with the aforementioned discontinuities.…”

“…The results showed the correctedness of the inviscid Ludwieg (1964) criterion (see figure 12), later improved by Wedemeyer (1967). The general problem was studied by Mott & Joseph (1968) and by Hung, Joseph & Munson (1972) with special emphasis on energy methods; an excellent review can be found in the book Joseph (1976, chap. VI).…”

Connections between centrifugal, stratorotational, and radiative instabilities in viscous Taylor-Couette flow

“…The Euler equations linearised around the basic flow U := rΩ(r)θ (equations (5)(6)(7)(8)(9) with Re → ∞) for an axisymmetric (m = 0) incompressible disturbance can be reduced down to a 2 nd order differential equation for u, the radial perturbation velocity, Here we prove that the only streamwise-independent state that can exist in rotating, stably-stratified plane Couette flow beyond the Rayleigh line is one of simple shear implying that no other axisymmetric state beyond the base state can exist beyond the Rayleigh line in thin-gap stratified Taylor-Couette flow. The proof is a straightforward extension of the unstratified result presented by [23] to include stratification. In a rotating frame Ω = Ωẑ where the shearing boundaries are at y = ±1 and gravity g := −gẑ, there is the simple shear solution U = yx, P = −Ωy 2 + 1 2 Ri z 2 and ρ = −z (stable stratification).…”

“…A Poincare inequality exists for non-slip conditions on the velocity field and either Dirichlet or Neumann conditions on the density field (in the latter case only if no mean flow is allowed in the direction of gravity). (Note that once Re = 177.2 for any Ω, 2D spanwise-invariant disturbances are not assured to decay [23] so that there is no general global stability result for the basic state beyond the Rayleigh line. )…”

Delaying transition in Taylor–Couette flow with axial motion of the inner cylinder

“…In all these studies both cylinders were rotating. In addition, a number of researchers examined the general case of both axial Couette and axial Poiseuille flow (Joseph & Munson 1970;Hung, Joseph & Munson 1972), including a numerical linear stability analysis for a finite flow domain (Ali & Weidman 1993).…”