The stability of pipe flow Part 1. Asymptotic analysis for small wave-numbers

Abstract

The instability of Poiseuille flow in a pipe is considered for small disturbances. An asymptotic analysis is used which is similar to that found successful in plane Poiseuille flow. The disturbance is taken to travel in a spiral fashion, and comparison of the radial velocity component with the transverse component in the plane case shows a high degree of similarity, particularly near the critical point where the disturbance and primary flow travel with the same speed. Instability is found for azimuthal wave-numbers of 2 or greater, although the corresponding minimum Reynolds numbers are too small to compare favourably with either experiments or the initial restrictions on the magnitude of the wave-number.

Stability of the Pipe Poiseuille Flow. With Respect to Axi and Nonaxisymmetric Disturbances

“…In the same way, Salwen and Grosch [10], Garg and Rouleau [11] and Gill [12] were not able to predict instability. Finally we have to mention the contradictory work of Graebel [13] who found instability at an azimuthal wavenumber n > 2. However, the numerical value reported by Graebel, cannot compare favorously with the usual experimental value (the critical value of the Reynolds number given by Graebel is of the order of 27).…”

“…The local potential for the differential equation (13) has been given in details by Mat and Platten [8], and will not be reconstructed here. However we shall outline the construction of the local potential for the non-axisymmetric case.…”

“…(26)-(29), as e. g. v^ = Vf io) + äí^ etc. Using transformations of the type adding the equations side by side and integrating some terms by parts under the boundary conditions(13), one may separate on the right hand side the variation äÖ of a functional such as Ö = / L(v;, v' d , vi, ñ', í; (0) , íß°>, íß°>, p' (0) ) dV . (31) í…”

A pseudospectral approach applicable for time integration of linearized N‐S operator that removes pole singularity and physically spurious eigenmodes

“…Many theoretical investigations on the stability of Poiseuille flow with axisymmetric disturbances are carried out over many decades. Along with some asymptotic study for nonaxisymmetric disturbances, further numerical approaches show no linear instability for Poiseuille flow. An elaborate review of these works can be found in the works of Joseph, Drazin and Reid, and Schmid and Henningson …”

Stability of developing pipe flow subjected to non-axisymmetric disturbances

“…In addition, it has been found that the flow is least stable to the non-axisymmetric disturbance with azimuthal wavenumber n = 1 and is less stable to this disturbance than to an axisymmetric disturbance (n = 0), for both the wall and centre modes. The work of Graebel (1970), on the other hand, has shown that Poiseuille pipe flow is unstable to non-axisymmetric small disturbances at large axial wavenumbers, giving critical Reynolds numbers of the order of 10-100 for azimuthal wavenumbers of 2 and larger. As was pointed out by Salwen & Grosch (1972), this conflicting finding of Graebel is probably due to the breakdown of the approximation in his asymptotic solution, which is valid only for small axial wavenumbers andlarge Reynoldsnumbers.…”