Uniqueness and Stability

  • Kolumban HutterEmail author
  • Yongqi Wang
Part of the Advances in Geophysical and Environmental Mechanics and Mathematics book series (AGEM)


This chapter on uniqueness and stability provides a first flavor into the subject of laminar-turbulent transition. Two different theoretical concepts are in use and both assume that the laminar-turbulent transition is a question of loss of stability of the laminar motion. With the use of the energy method one tries to find conditions for the laminar flow to be stable. Energy stability criteria operate with the construction of upper bounds of the rate of the perturbed kinetic energy K(t) of the fluid system, in order to obtain by time integration an inequality of the form \(K(t) < K(0)\mathrm {exp}\,(-t/\tau )\). Here, \(\tau > 0\) guarantees decay and \(\tau < 0\) growth rates of the perturbed energy, \(\tau = 0\) guarantees neutral stability of the perturbation flows. The difficulty of the method is that the condition \(\tau = 0\) generally provides poor, i.e., very safe estimates for stability. More successful for pinpointing the laminar-turbulent transition has been the method of linear instability analysis, in which a lowest bound, is searched for, at which the onset of deviations from the laminar flow is taking place. For plane channel flows the Rayleigh and OrrSommerfeld equations with associated boundary conditions for an ideal and viscous fluid, respectively, are derived and the associated eigenvalue problems are discussed, which leads to the stability chart, separating Reynolds number dependent stable and unstable flow regimes.


Kinetic energy of the difference motion Uniqueness Energy stability of laminar channel flows Rayleigh equation OrrSommerfeld equation 


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© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.c/o Versuchsanstalt für Wasserbau, Hydrologie und GlaziologieETH ZürichZürichSwitzerland
  2. 2.Department of Mechanical EngineeringTechnische Universität DarmstadtDarmstadtGermany

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