Intermittency in the Transition to Turbulence

It is commonly known that the intermittent transition from laminar to turbulent flow in pipes occurs because, at intermediate values of a prescribed pressure drop, a purely laminar flow offers too little resistance, but a fully turbulent one offers too much. We propose a phenomenological model of the flow, which is able to explain this in a quantitative way through a hysteretic transition between laminar and turbulent "states," characterized by a disturbance amplitude variable that satisfies a natural type of evolution equation. The form of this equation is motivated by physical observations and derived by an averaging procedure, and we show that it naturally predicts disturbances having the characteristics of slugs and puffs. The model predicts oscillations similar to those which occur in intermittency in pipe flow, but it also predicts that stationary "biphasic" states can occur in sufficiently short pipes.

[1]  N. Itoh.  Nonlinear stability of parallel flows with subcritical Reynolds numbers. Part 2. Stability of pipe Poiseuille flow to finite axisymmetric disturbances , 1977, Journal of Fluid Mechanics.

[2]  J. Burgers.  A mathematical model illustrating the theory of turbulence , 1948 .

[3]  A. Davey,et al.  Finite-amplitude stability of pipe flow , 1971, Journal of Fluid Mechanics.

[4]  F. T. Smith,et al.  Amplitude-dependent neutral modes in the Hagen-Poiseuille flow through a circular pipe , 1982, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[5]  B. Launder,et al.  Progress in the development of a Reynolds-stress turbulence closure , 1975, Journal of Fluid Mechanics.

[6]  J. Spurk.  Boundary Layer Theory , 2019, Fluid Mechanics.

[7]  O. Reynolds.  III. An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous, and of the law of resistance in parallel channels , 1883, Proceedings of the Royal Society of London.

[8]  T. Bohr,et al.  A mechanism for localised turbulence , 1991 .

[9]  O. Tietjens,et al.  Applied hydro- and aeromechanics , 1934 .

[10]  R. Deissler.  Spatially growing waves, intermittency, and convective chaos in an open-flow system , 1987 .

[11]  Julian F. Scott,et al.  An Introduction to Turbulent Flow , 2000 .

[12]  L. P.,et al.  Modern Developments in Fluid Dynamics , 1939, Nature.

[13]  N. Rashevsky,et al.  Mathematical biology , 1961, Connecticut medicine.

[14]  H. Sakaguchi,et al.  Stable localized solutions of arbitrary length for the quintic Swift-Hohenberg equation , 1996 .

[15]  S. Orszag,et al.  Subcritical transition to turbulence in plane channel flows , 1980 .

[16]  E. R. Lindgren,et al.  THE TRANSITION PROCESS AND OTHER PHENOMENA IN VISCOUS FLOW , 1957 .

[17]  I. Wygnanski,et al.  On transition in a pipe. Part 1. The origin of puffs and slugs and the flow in a turbulent slug , 1973, Journal of Fluid Mechanics.

[18]  J. Rotta,et al.  Experimenteller Beitrag zur Entstehung turbulenter Strömung im Rohr , 1956 .

[19]  S. Rosenblat,et al.  Bifurcation from Infinity , 1979 .

[20]  Y.-N. Huang,et al.  On the transition to turbulence in pipe flow , 1989 .

[21]  T. Mullin,et al.  Transition to turbulence in constant-mass-flux pipe flow , 1995, Journal of Fluid Mechanics.

[22]  R. Deissler.  Turbulent bursts, spots and slugs in a generalized Ginzburg-Landau equation , 1987 .

[23]  Anthony T. Patera,et al.  Secondary instability of wall-bounded shear flows , 1983, Journal of Fluid Mechanics.