Martin Lessen

The stability of a trailing line vortex. Part 1. Inviscid theory

The inviscid stability of swirling flows with mean velocity profiles similar to that obtained by Batchelor (1964) for a trailing vortex from an aircraft is studied with respect to infinitesimal non-axisymmetric disturbances. The flow is characterized by a swirl parameter q involving the ratio of the magnitude of the maximum swirl velocity to that of the maximum axial velocity. It is found that, as the swirl is continuously increased from zero, the disturbances die out quickly for a small value of q if n = 1 ( n is the azimuthal wavenumber of the Fourier disturbance of type exp{ i (α x + n ϕ − α ct )}); but for negative values of n , the amplification rate increases and then decreases, falling to negative values at q slightly greater than 1·5 for n = −1. The maximum amplification rate increases for increasingly negative n up to n = −6 (the highest mode investigated), and corresponds to q ≃ 0·85. The applicability of these results to attempts at destabilizing vortices is briefly discussed.

Experimental Investigation of the Stability of Hagen‐Poiseuille Flow

The stability of pipe Poiseuille flow to azimuthally periodic disturbances is examined experimentally. The disturbance is generated in the fully developed flow region to avoid extraneous effects of possible entrance flow instability. A Reynolds number spectrum to above 3200 was covered with a frequency range from 13 to 1 kc/sec. The peak disturbance velocity amplitude was maintained at a value less than 2% of the maximum steady‐state velocity in the pipe. Some measurements of spacewise changes in velocity and disturbance wavelength were made as well as radial surveys of disturbance amplitude and steady‐state profiles. The results indicate that the stability is both frequency and Reynolds number dependent to the first mode of azimuthal periodicity, with a minimum critical Reynolds number occurring at about 2150.

Stability of Pipe Poiseuille Flow

The stability of pipe Poiseuille flow with respect to linear azimuthally periodic disturbances is investigated. It is found that many radial modes of disturbance exist for each azimuthal periodicity. No linear instability is found for the mode sets with azimuthal periodicity of one.

Effect of small amplitude wall waviness upon the stability of the laminar boundary layer

The effect of small amplitude wall roughness on the minimum critical Reynolds number of a laminar boundary layer is studied under the assumptions normally employed in parallel flow stability problems. The presence of a single Fourier component of roughness enhances the instability of the flow by introducing a point of inflection and increasing the profile curvature at the critical layer, and the effect becomes increasingly important as the wavelength of the roughness component decreases. Numerical calculations show a 10% reduction in minimum critical Reynolds numbers in the presence of small wavelength roughness with an amplitude of only 1% of the boundary layer thickness.

On the inviscid stability of the laminar mixing of two parallel streams of a compressible fluid

The inviscid stability of the laminar mixing of two parallel streams of a compressible fluid is investigated with respect to three-dimensional wavy disturbances. The flow is more unstable as the angle between the disturbance-wave-number vector and the principal-flow direction exist even at very high Mach number, and the flow is still unstable. However, it is also found that increasing the Mach number of the flow tends to stabilize the flow.

Marginal instability in Taylor–Couette flows at a very high Taylor number

We report on a set of turbulent flow experiments of the Taylor type in which the fluid is contained between a rotating inner circular cylinder and a fixed concentric outer cylinder, focusing our attention on very large Taylor number values, i.e. \[ 10^3 \leqslant T/T_c \leqslant 10^5, \] where T c is the critical value of the Taylor number T for onset of Taylor vortices. At such large values of T , the turbulent vortex flow structure is similar to the one observed when T – T c is small and this structure is apparently insensitive to further increases in T . These flows are characterized by two widely separated length scales: the scale of the gap width which characterizes the Taylor vortex flow and a much smaller scale which is made visible by streaks in the form of a ‘herring-bone’-like pattern visible at the walls. These are conjectured to be Gortler vortices which arise as a result of centrifugal instability in the wall boundary layers. Ideas of marginal instability by which we postulate that both the Taylor and Gortler vortex structures are marginally unstable on their own scale seem to provide good quantitative agreement between predicted and observed Gortler vortex spacings.

Poiseuille flow in a pipe with axially symmetric wavy walls

The effect of small amplitude wall waviness on the steady flow in a pipe is studied. Friction factors, Reynolds stresses, and mean velocity profiles are obtained for various Reynolds numbers, wall wave amplitudes, and wavenumbers. The results agree qualitatively with the experiments of Batra, Fulford, and Dullien and contradict the classical ’’Moody’’ diagram which represents the laminar friction factor as independent of wall roughness. The mean velocity profiles obtained suggest that wall roughness may cause pipe flow to be unstable to infinitesimal disturbances at a finite Reynolds number.

Marginal instability of turbulent shearing layers and the break point of a jet

The postulate that fully turbulent free flows are marginally stable is used to predict the break point of the turbulent jet. The theory is consistent with recent experiments, and is used to improve an existing theory of entrainment.

Stability of turbulent jets and wakes

The mean flow of rotationally symmetric turbulent jets and wakes is postulated as marginally stable. The angles of spread of turbulent jets and wakes can be estimated as being those of the laminar flows at their critical Reynolds numbers.

Stability of plane Couette flow with respect to finite two‐dimensional disturbances

The stability of plane Couette flow is studied by applying a finite two‐dimensional disturbance to the basic laminar motion and following the timewise evolution of the quasi‐steady flow and the disturbance. This is accomplished by combining a Meksyn–Stuart type method with a quasi‐steady technique. Results show that plane Couette flow tends to slowly converge to a stable state for the Reynolds numbers and disturbance intensities studied. This is in disagreement with past investigations which were able to obtain nonlinear neutral stability curves.