A note on disturbances in slightly supercritical plane Poiseuille flow

1967 ◽  
Vol 30 (1) ◽  
pp. 17-20 ◽  
Author(s):  
Christopher K. W. Tam
Keyword(s):  
Asymptotic Solution ◽  
Poiseuille Flow ◽  
Initial Value Problem ◽  
Large Time ◽  
Plane Poiseuille Flow ◽  
Initial Value ◽  
Turbulent Spots ◽  
Supercritical Flow ◽  
Characteristic Features

The evolution of disturbances after a laminar, slightly supercritical flow between parallel planes is disturbed is considered as an initial-value problem. An asymptotic solution of the disturbances for large time possesses the same characteristic features as the turbulent spots observed by Emmons (1951).

scholarly journals The influence of circulation on the stability of vortices to mode-one disturbances

1995 ◽  
Vol 451 (1943) ◽  
pp. 747-755 ◽  
Keyword(s):  
Velocity Field ◽  
Angular Velocity ◽  
Initial Value Problem ◽  
Large Time ◽  
Asymptotic Methods ◽  
Two Dimensional ◽  
Initial Value ◽  
The Stability ◽  
Ultimate Fate ◽  
Solution Behaviour

The initial value problem for the two-dimensional inviscid vorticity equation, linearized about an azimuthal basic velocity field with monotonic angular velocity, is solved exactly for mode-one disturbances. The solution behaviour is investigated for large time using asymptotic methods. The circulation of the basic state is found to govern the ultimate fate of the disturbance: for basic state vorticity distributions with non-zero circulation, the perturbation tends to the steady solution first mentioned in Michalke & Timme (1967), while for zero circulation, the perturbation grows without bound. The latter case has potentially important implications for the stability of isolated eddies in geophysics.

Waves created against a vertical cliff – a uniform asymptotic solution

1978 ◽  
Vol 83 (2) ◽  
pp. 321-328 ◽  
Author(s):  
A. K. Pramanik
Keyword(s):  
Asymptotic Solution ◽  
Asymptotic Analysis ◽  
Surface Pressure ◽  
Initial Value Problem ◽  
Unsteady State ◽  
Initial Value

SummaryThe initial-value problem of waves generated by a moving oscillatory surface pressure against a vertical cliff is solved and a uniform asymptotic analysis of the unsteady state is given. The same problem with no cliff is solved by Kaplan, and by Debnath and Rosenblat but their solutions are not uniform.

Energy growth of three-dimensional disturbances in plane Poiseuille flow

1991 ◽  
Vol 224 ◽  
pp. 241-260 ◽  
Author(s):  
L. Hårkan Gustavsson
Keyword(s):  
Energy Density ◽  
Poiseuille Flow ◽  
Initial Conditions ◽  
Formal Solution ◽  
Three Dimensional ◽  
Propagation Speed ◽  
Inhomogeneous Equation ◽  
Normal Velocity ◽  
Plane Poiseuille Flow ◽  
Initial Value

The development of a small three-dimensional disturbance in plane Poiseuille flow is considered. Its kinetic energy is expressed in terms of the velocity and vorticity components normal to the wall. The normal vorticity develops according to the mechanism of vortex stretching and is described by an inhomogeneous equation, where the spanwise variation of the normal velocity acts as forcing. To study specifically the effect of the forcing, the initial normal vorticity is set to zero and the energy density in the wavenumber plane, induced by the normal velocity, is determined. In particular, the response from individual (and damped) Orr–Sommerfeld modes is calculated, on the basis of a formal solution to the initial-value problem. The relevant timescale for the development of the perturbation is identified as a viscous one. Even so, the induced energy density can greatly exceed that associated with the initial normal velocity, before decay sets in. Initial conditions corresponding to the least-damped Orr–Sommerfeld mode induce the largest energy density and a maximum is obtained for structures infinitely elongated in the streamwise direction. In this limit, the asymptotic solution is derived and it shows that the spanwise wavenumbers at which the largest amplification occurs are 2.60 and 1.98, for symmetric and antisymmetric normal vorticity, respectively. The asymptotic analysis also shows that the propagation speed for induced symmetric vorticity is confined to a narrower range than that for antisymmetric vorticity. From a consideration of the neglected nonlinear terms it is found that the normal velocity component cannot be nonlinearly affected by the normal vorticity growth for structures with no streamwise dependence.

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