2. The Couette-Taylor experiment
The CT experimental apparatus consists of two coaxial rotating cylinders of radius and separated by a gap , which is filled with a fluid of viscosity . The cylinders can rotate with different angular velocities and ; their height is in general much larger than their radius, to minimize the effect of the boundaries. The Reynolds number is usually defined in terms of the differential rotation and by taking the gap width as characteristic length:
where R is the mean radius .
2.1. Critical parameters and transition to turbulence
When the inner cylinder is rotating and the outer one is at rest, angular momentum decreases outward and the flow is linearly unstable for Reynolds numbers higher than (for narrow gap, cf. Taylor 1923). This well known instability takes first the shape of steady toroidal, axisymmetric cells (the Taylor vortices); it is very efficient in transporting angular momentum, whose gradient is strongly reduced. At higher Re a wavy pattern appears and after a series of bifurcations the flow becomes fully turbulent. This case has been studied by many experimental teams, and it is extremely well documented (see Andereck et al. 1986). It has also been modeled successfully in three-dimensional numerical simulations (Marcus 1984; Coughlin & Marcus 1996).
In the opposite case, when the outer cylinder is rotating and the inner one is at rest, the angular momentum increases outward and the flow is linearly stable. The only theoretical prediction concerning the non-linear behavior is that by Serrin (1959), later refined by Joseph and Munson (1970), who established that the flow is stable against finite amplitude perturbations below (in the narrow gap limit).
To this date, no numerical simulation has been able to demonstrate the finite amplitude instability. But this instability does occur in the laboratory, and it has been described already by Couette (1890). It was studied in detail by Wendt (1933) and Taylor (1936), who showed that for Reynolds numbers exceeding , the flow becomes unstable and immediately displays turbulent motions. The critical Reynolds number depends on whether the angular speed is increased or decreased in the experiment, a typical property of finite amplitude instabilites. Moreover, it is sensitive to gap width, as demonstrated by Taylor. Fig. 1 displays results from Wendt and Taylor: is roughly constant below , but above it increases as , as was already noticed by Zeldovich (1981), a behavior for which an explanation was proposed by Dubrulle (1993). In the latter regime one can define another critical Reynolds number involving, instead of gap width, the gradient of angular velocity; since
the instability condition becomes
We see that two conditions must be satisfied for the finite amplitude instability to occur: the first is the classical criterion of shear instability, valid also for plane parallel flows, whereas the second , involving what we shall call the gradient Reynolds number, is genuine to differential rotation. In addition, to trigger the instability the strength of the perturbation must exceed a certain threshold, which presumably also depends on Re or .
2.2. Transport of angular momentum
In the turbulent regime, the torque measured by Taylor scales approximately as for a given gap width, where the exponent n tends to 2 for large . The measurements made by Wendt confirm that scaling with . It suggests that the transport of angular momentum may be considered as a diffusive process, and that the mean turbulent viscosity increases linearly with , or . It is then natural to examine whether this viscosity may be expressed as
with being a constant of order unity, since the largest turbulent eddies would have a size and a peripheral velocity . The parameter is easily derived from the torque measurements, and the surprising result is that it decreases with gap width (Fig. 2). For the smallest gaps, scales as the inverse of , but the slope steepens farther as if the scaling would tend asymptotically to
(Comparing Wendt's experimental data of his two largest gaps, one deduces an exponent )
We may therefore conclude that, in the limit of large gap, the mean turbulent viscosity actually scales as
r being the radial coordinate.
In principle, one should be able to verify this prescription for by examining the rotation profiles measured by Taylor and Wendt. According to (7), the conservation of angular momentum requires that its flux, given by
varies as between the cylinders. Therefore should be constant in the turbulent part of the profile (as it is in the laminar flow).
But this constancy can be expected only if the transport of angular momentum is achieved by the viscous and turbulent stresses alone. That is not the case in Taylor's experiment: as acknowledged by him, an Ekman circulation is induced by the ends of his apparatus, although he tries to minimize the boundary effects by chosing a large aspect ratio (height/radius). Moreover his rotation profiles are deduced from pressure measurements made with a Pitot tube located at half height of the cylinders, where the radial return flow has its maximum intensity. Consequently, the torque inferred from these profiles is about half of that measured directly at the inner cylinder.
The aspect ratio is less favorable in Wendt's experiment, but there the top boundary is a free surface, and most of his measurements have been made with the bottom boundary split in two annuli, attached respectively to the inner and the outer cylinders, which reduces drastically the circulation and renders his results more reliable. We examined his rotation profiles obtained for the largest gap width () and with 4 different speeds of the inner cylinder (); we found that in the bulk of the fluid, these profiles are compatible with the constancy of the gradient Reynolds number, as predicted by (8). However we cannot rule out a mild variation of within the profile. Unfortunately, Wendt gives the torque only for the case where the inner cylinder is at rest; we draw from it the following value of the coefficient :
The estimated uncertainty reflects that of the velocity measurements: results from second derivation of the shape of the fluid surface.
© European Southern Observatory (ESO) 1999
Online publication: June 30, 1999