Maximum instability in lamiar flow occurs at hafl pipe radius
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The long-puzzling, unphysical result that linear stability analyses lead to no transition in pipe flow, even at infinite Reynolds number, is ascribed to the use of stick boundary conditions, because they ignore the amplitude variations associated with the roughness of the wall. Once that length scale is introduced (here, crudely, through a corrugated pipe), linear stability analyses lead to stable vortex formation at low Reynolds number above a finite amplitude of the corrugation and unsteady flow at a higher Reynolds number, where indications are that the vortex dislodges. Remarkably, extrapolation to infinite Reynolds number of both of these transitions leads to a finite and nearly identical value of the amplitude, implying that below this amplitude, the vortex cannot form because the wall is too smooth and, hence, stick boundary results prevail.
This work explores the effect of wall roughness on the long known contradiction between the linear stability analysis result of infinitely stable flow in pipes with smooth boundaries and the experimental observation (1) that flows become unstable at a Reynolds number of ≈2,000 for ordinary pipes (2–4). A hint that wall roughness may be important can be gathered from experiments which show that for smoothed pipes, the onset of the instability can greatly exceed 2,000 (5). The present work should serve as a general warning that stick boundary conditions—for example, in narrow biological channels where the surface roughness of the wall can be a significant fraction of the channel width, or, as another example, in the drag reduction problem—may be inappropriate. For the smooth-wall case, there exists a rigorous proof of stability for axisymmetric disturbances (6) and strong evidence (cf. refs. 7 and 8) that all linear perturbations decay for all values of the Reynolds number and axial and azimuthal wavenumbers. Thus, there remains much interest in the cause of this transition and how one may affect the Reynolds number at which transition occurs.
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