A comprehensive review of zonal flow phenomena in plasmas is presented. While the emphasis is on zonal flows in laboratory plasmas, planetary zonal flows are discussed as well. The review presents the status of theory, numerical simulation and experiments relevant to zonal flows. The emphasis is on developing an integrated understanding of the dynamics of drift wave–zonal flow turbulence by combining detailed studies of the generation of zonal flows by drift waves, the back-interaction of zonal flows on the drift waves, and the various feedback loops by which the system regulates and organizes itself. The implications of zonal flow phenomena for confinement in, and the phenomena of fusion devices are discussed. Special attention is given to the comparison of experiment with theory and to identifying directions for progress in future research.

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Zonal flows in plasma—a review

An experimental study of Rayleigh-Benard convection in helium gas at roughly 5 K is performed in a cell with aspect ratio 1. Data are analysed in a ‘hard turbulence’ region (4 × 107 < Ra < 6 × 1012) in which the Prandtl number remains between 0.65 and 1.5. The main observation is a simple scaling behaviour over this entire range of Ra. However the results are not the same as in previous theories. For example, a classical result gives the dimensionless heat flux, Nu, proportional to . A new scaling theory is described. This new approach suggests scaling indices very close to the observed ones. The new approach is based upon the assumption that the boundary layer remains in existence even though its Rayleigh number is considerably greater than unity and is, in fact, diverging. A stability analysis of the boundary layer is performed which indicates that the boundary layer may be stabilized by the interaction of buoyancy driven effects and a fluctuating wind.

Scaling of hard thermal turbulence in Rayleigh-Bénard convection

Within the past decade, satellite pictures have shown persistent cloud patterns which indicate that the flow in the atmospheric planetary boundary layer is often organized into helical secondary circulations aligned parallel to the mean flow. Theory and observation agree that both convection in the presence of shear and the dynamic inflection point instabilities of the Ekman layer lead to these flow patterns. The observations and theory are reviewed with emphasis on the dynamics-dominated flow.

Longitudinal instabilities and secondary flows in the planetary boundary layer: A review

A historical review of work at TU Delft on the e N method for transition prediction is presented. The method is shown to be applicable to boundary layers with pressure gradient, suction and separation. Some applications to airfoil design will be discussed. An explanation will be given of the reason for the success of this linear theory to “predict” the position of transition which itself is a highly non-linear phenomenon. It will also be explained why the velocity profile shape factor H is a suitable parameter to characterize stability diagrams for flows with and without suction. Finally a new database method will be presented that is based on the observation that many stability diagrams show a remarkable similarity when properly scaled and shifted. The discussion will be restricted to two-dimensional incompressible flow.

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The eN method for transition prediction: Historical review of work at TU Delft

The development of turbulent spots in plane Couette flow was studied by means of direct numerical simulation. The Reynolds number was varied between 300 and 1500 (based on half the velocity difference between the two surfaces and half the gap width) in order to determine the lowest possible Reynolds number for which localised turbulent regions can persist, i.e. a critical Reynolds number, and to determine basic characteristics of the spot in plane Couette flow. It was found that spots can be sustained for Reynolds number above approximately 375 and that the shape is elliptical with a streamwise elongation that is more accentuated for high Reynolds numbers. At large times though there appears to be a slow approach towards a circular spot shape. Various other features of this spot suggest that it may be classified as an interesting intermediate case between the Pouseuille and boundary-layer spots. In the absence of experiments for this case the present results represent a true prediction of the physical situation.

Direct simulation of turbulent spots in plane couette flow

In studying the stability of a thermally stratified fluid in the presence of a viscous shear flow, we have a situation in which there is an important interaction between the mechanism of instability due to the stratification and the Tollmien-Schlichting mechanism due to the shear. A complete analysis has been carried out for the Benard problem in the presence of a plane Poiseuille flow and it is shown that, although Squire's transformation can be used to reduce the three-dimensional problem to an equivalent two-dimensional one, a theorem of Squire's type does not follow unless the Richardson number exceeds a certain small negative value. This conclusion follows from the fact that, when the stratification is unstable and the Prandtl number is unity, the equivalent two-dimensional problem becomes identical mathematically to the stability problem for spiral flow between rotating cylinders and, from the known results for the spiral flow problem, Squire's transformation can then be used to obtain the complete three-dimensional stability boundary. For the case of stable stratification, however, Squire's theorem is valid and the instability is of the usual Tollmien—Schlichting type. Additional calculations have been made for this case which show that the flow is completely stabilized when the Richardson number exceeds a certain positive value.

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The stability of thermally stratified plane Poiseuille flow

This paper presents a discussion of some aspects of the linear stability problem for the asymptotic suction profile. An exact solution of the inviscid equation is first obtained in terms of the usual hypergeometric function and its analytical continuation. This exact solution provides both a corrected version of an earlier treatment by Freeman and an independent check on the more general method suggested for solving the inviscid equation numerically. Various approximations to the characteristic equation, and hence to the curve of neutral stability, are then considered. In particular, it is found that, in a consistent asymptotic treatment of the related adjoint problem, at least one viscous correction to the singular inviscid solution must be considered. Based on the present results for the adjoint problem, it is suggested that Tollmien's original treatment of the viscous corrections must be slightly modified.

On the stability of the asymptotic suction boundary-layer profile

The effect of an adverse pressure gradient on the stability of a laminar boundary layer is considered in the limiting case when the skin friction at the wall vanishes, i.e. when U′(0) = 0. Such flows are not absolutely unstable as might have been expected but have a minimum critical Reynolds number of the order of 25. General results are given for the asymptotic behaviour of both the upper and lower branches of the neutral curve and a complete neutral curve is obtained for Pohlhausen's simple fourth-degree polynomial profile at separation.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S0022112065001659

W. H. Reid

Papers

The stability of thermally stratified plane Poiseuille flow

On the stability of the asymptotic suction boundary-layer profile

The stability of laminar boundary layers at separation