Unstable Flows and Laminar-Turbulent Transition

Abstract

It is common practice to categorize flows as laminar or turbulent, i.e. to employ a special state of the flow to perform a subdivision: into laminar flows, i.e. in such flows in which the momentum, heat and mass transport processes are molecular dependent, and into such flows in which turbulence-dependent transport processes occur in addition. For the considerations presented in this chapter, a further subdivision is appropriate, so that grouping into four sub-groups is made: Stable Laminar Flows. A laminar flow may fulfill all requirements of the basic equations of fluid mechanics. It may also satisfy the initial and boundary conditions characteristic for the flow. Yet it must not represent a solution such as one finds in corresponding experimental investigations. Disturbances of the flow, as always occur in experiments, are often not considered in solutions of the basic equations governing fluid flows. Only such laminar flows that prove stable towards disturbances that act from the outside, i.e. attenuate the imposed disturbances, are defined as stable laminar flows. Unstable Laminar Flows. A laminar flow is considered unstable when disturbances introduced into it are amplified, but a certain “regularity” in the excited disturbance is maintained, i.e. due to the disturbance the investigated flow merges into a new laminar flow state. If this “new laminar flow state” is stable towards newly introduced disturbances, we have a bifurcation of the laminar flow. Here it is important to understand that the flow occurring after the imposed disturbance can be stationary or non-stationary. Transitional Flows. When the disturbances introduced in a laminar flow are amplified and result in flows that appear orderly in parts, but show also temporarily and/or spatially irregular fluctuations of all flow quantities, we speak of a transitional flow state. Intermittent laminar and turbulent flow states occur, i.e. phases occur in the flow in which the flow is laminar and phases in which the flow shows turbulent characteristics. Flows that are in a transitional state still show clear characteristics that depend on the imposed disturbances. Turbulent Flows. It is now easily possible to imagine, on the basis of the considerations presented above, that disturbances are introduced into flows to such an extent that fluid motions result from them that are “out of control.” Such turbulent fluctuations of all flow quantities are superimposed on corresponding mean flow quantities and are characterized by high non-stationarity and by high three-dimensionality. Turbulence-dependent transport processes of momentum, heat and mass are superimposed on the molecular-dependent transport process. A closed treatment of turbulent transport processes is at present only possible for small Reynolds numbers (Re ≤ 40,000) by employing numerical computation procedures. The treatment of turbulent flows at high Reynolds numbers remains a problem of fluid mechanics that has not been solved.