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Instability of flow through pipes of general cross‐section, Part 1
Author(s) -
Smith F. T.
Publication year - 1979
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300009761
Subject(s) - instability , reynolds number , aspect ratio (aeronautics) , mathematics , cross section (physics) , flow (mathematics) , mechanics , geometry , laminar flow , physics , turbulence , optoelectronics , quantum mechanics
Summary The temporal and spatial linear instability of Poiseuille flow through pipes of arbitrary cross‐section is discussed for large Reynolds numbers ( R ). For a pipe whose aspect ratio is finite, neutral stability (lower branch) is found to be governed by disturbance modes of large axial wavelength (of order hR , where h is a characteristic cross‐sectional dimension). By contrast, spatial instability for finite aspect ratios is governed by length scales between O ( h ) and O( hR ). When the aspect ratio is increased to O ( R 1/7 ), however, these two characteristic length scales both become O ( R 1/7 h ) and a match with plane channel flow instability is achieved. Thus the general cross‐section produces temporal and spatial instability if the aspect ratio is O ( R 1/7). Further, in the flow in a rectangular pipe neutral stability (lower branch) exists for some finite aspect ratios, while for the flow in any non‐circular elliptical pipe spatial instability is possible. It is suggested that both temporal and spatial instability occur for a wide range of pipe cross‐sections of finite aspect ratio. Part 2 (Smith 1979a), which studies the upper branch neutral stability, confirms the importance of the O ( hR ) scale modes in neutral stability for finite aspect ratios.