The ``Rayleigh line'' $\ensuremath{\mu}={\ensuremath{\eta}}^{2}$, where $\ensuremath{\mu}={\mathrm{\ensuremath{\Omega}}}_{o}/{\mathrm{\ensuremath{\Omega}}}_{i}$ and $\ensuremath{\eta}={r}_{i}/{r}_{o}$ are respectively the rotation and radius ratios between inner (subscript $i$) and outer (subscript $o$) cylinders, is regarded as marking the limit of centrifugal instability (CI) in unstratified inviscid Taylor-Couette flow, for both axisymmetric and nonaxisymmetric modes. Nonaxisymmetric stratorotational instability (SRI) is known to set in for anticyclonic rotation ratios beyond that line, i.e., ${\ensuremath{\eta}}^{2}<\ensuremath{\mu}<1$ for axially stably stratified Taylor-Couette flow, but the competition between CI and SRI in the range $\ensuremath{\mu}<{\ensuremath{\eta}}^{2}$ has not yet been addressed. In this paper, we establish continuous connections between the two instabilities at finite Reynolds number $\text{Re}$, as previously suggested by Le Bars and Le Gal [Phys. Rev. Lett. 99, 064502 (2007)], making them indistinguishable at onset. Both instabilities are also continuously connected to the radiative instability at finite Re. These results demonstrate the complex impact viscosity has on the linear stability properties of this flow. Several other qualitative differences with inviscid theory were found, among which are the instability of a nonaxisymmetric mode localized at the outer cylinder without stratification and the instability of a mode propagating against the inner cylinder rotation with stratification. The combination of viscosity and stratification can also lead to a ``collision'' between (axisymmetric) Taylor vortex branches, causing the axisymmetric oscillatory state already observed in past experiments. Perhaps more surprising is the instability of a centrifugal-like helical mode beyond the Rayleigh line, caused by the joint effects of stratification and viscosity. The threshold $\ensuremath{\mu}={\ensuremath{\eta}}^{2}$ seems to remain, however, an impassable instability limit for axisymmetric modes, regardless of stratification, viscosity, and even disturbance amplitude.
This paper presents advances towards the data-based control of periodic oscillator flows, from their fully developed regime to their equilibrium stabilized in closed loop, with linear time-invariant (LTI) controllers. The proposed approach directly builds upon the iterative method of Leclercq et al. ( J. Fluid Mech. , vol. 868, 2019, pp. 26–65) and provides several improvements for an efficient online implementation, aimed at being applicable in experiments. First, we use input–output data to construct an LTI mean transfer functions of the flow. The model is subsequently used for the design of an LTI controller with linear quadratic Gaussian synthesis, which is practical to automate online. Then, using the controller in a feedback loop, the flow shifts in phase space and oscillations are damped. The procedure is repeated until equilibrium is reached, by stacking controllers and performing balanced truncation to deal with the increasing order of the compound controller. In this article, we illustrate the method for the classic flow past a cylinder at Reynolds number $Re=100$ . Care has been taken such that the method may be fully automated and hopefully used as a valuable tool in a forthcoming experiment.
This paper is the second part of a two-fold study of mixing, i.e. the formation of layers and upwelling of buoyancy, in axially stratified Taylor--Couette flow, with fixed outer cylinder. In a first paper, we showed that the dynamics of the flow was dominated by coherent structures made of a superposition of nonlinear waves. (Mixed)-ribbons and (mixed)-cross-spirals are generated by interactions between a pair of linearly unstable helical modes of opposite `handedness', and appear to be responsible for the formation of well-mixed layers and sharp density interfaces. In this paper, we show that these structures are also fully accountable for the upwards buoyancy flux in the simulations. The mechanism by which this occurs is a positive coupling between the density and vertical velocity components of the most energetic waves. This coupling is primarily caused by diffusion of density at low Schmidt number Sc, but can also be a nonlinear effect at larger Sc. Turbulence was found to contribute negatively to the buoyancy flux at Sc=1,10,16, which lead to the conclusion that mass upwelling is a consequence of chaotic advection, even at large Reynolds number. Artificially isolating the coherent structure therefore leads to excellent estimates of the flux Richardson numbers Ri_f from the DNS. We also used the theoretical framework of Winters et al. (1995) to analyse the energetics of mixing in an open control volume, shedding light on the influence of end effects in the potential energy budget. The potential connection with the buoyancy flux measurements made in the recent experiment of Oglethorpe et al. (2013) is also discussed.
This paper introduces a new operator relevant to input-output analysis of flows in a statistically steady regime far from the steady base flow: the mean resolvent $\mathbf{R}_0$. It is defined as the operator predicting, in the frequency domain, the mean linear response to forcing of the time-varying base flow. As such, it provides the statistically optimal linear time-invariant approximation of the input-output dynamics, which may be useful, for instance, in flow control applications. Theory is developed for the periodic case. The poles of the operator are shown to correspond to the Floquet exponents of the system, including purely imaginary poles at multiples of the fundamental frequency. In general, evaluating mean transfer functions from data requires averaging the response to many realizations of the same input. However, in the specific case of harmonic forcings, we show that the mean transfer functions may be identified without averaging: an observation referred to as `dynamic linearity' in the literature (Dahan et al., 2012). For incompressible flows in the weakly unsteady limit, i.e. when amplification of perturbations by the unsteady part of the periodic Jacobian is small compared to amplification by the mean Jacobian, the mean resolvent $\mathbf{R}_0$ is well-approximated by the well-known resolvent operator about the mean-flow. Although the theory presented in this paper only extends to quasiperiodic flows, the definition of $\mathbf{R}_0$ remains meaningful for flows with continuous or mixed spectra, including turbulent flows. Numerical evidence supports the close connection between the two resolvent operators in quasiperiodic, chaotic and stochastic two-dimensional incompressible flows.
Efforts to model accretion disks in the laboratory using Taylor–Couette flow apparatus are plagued with problems due to the substantial impact the end plates have on the flow. We explore the possibility of mitigating the influence of these end plates by imposing stable stratification in their vicinity. Numerical computations and experiments confirm the effectiveness of this strategy for restoring the axially homogeneous quasi-Keplerian solution in the unstratified equatorial part of the flow for sufficiently strong stratification and moderate layer thickness. If the rotation ratio is too large, however (e.g. ${\it\Omega}_{o}/{\it\Omega}_{i}=(r_{i}/r_{o})^{3/2}$ , where ${\it\Omega}_{o}/{\it\Omega}_{i}$ is the angular velocity at the outer/inner boundary and $r_{i}/r_{o}$ is the inner/outer radius), the presence of stratification can make the quasi-Keplerian flow susceptible to the stratorotational instability. Otherwise (e.g. for ${\it\Omega}_{o}/{\it\Omega}_{i}=(r_{i}/r_{o})^{1/2}$ ), our control strategy is successful in reinstating a linearly stable quasi-Keplerian flow away from the end plates. Experiments probing the nonlinear stability of this flow show only decay of initial finite-amplitude disturbances at a Reynolds number $Re=O(10^{4})$ . This observation is consistent with most recent computational (Ostilla-Mónico, et al. J. Fluid Mech. , vol. 748, 2014, R3) and experimental results (Edlund & Ji, Phys. Rev. E, vol. 89, 2014, 021004) at high $Re$ , and reinforces the growing consensus that turbulence in cold accretion disks must rely on additional physics beyond that of incompressible hydrodynamics.
This work is concerned with the combined effects of eccentricity and pressure-driven axial flow on the linear stability properties of circular Couette flow with a fixed outer cylinder. An example of this flow can be found in oil-well drilling operations. A pseudospectral method is implemented to compute the basic flow, steady and homogeneous in the axial direction, as well as the normal modes of instability. There are four non-dimensional parameters: the radius ratio _ and the eccentricity e for the geometry, the azimuthal and axial Reynolds numbers, Re and Rez, for the dynamics. The first part of the study is devoted to the temporal stability properties. It is found that eccentricity pushes the convective instability threshold towards higher values of Re. The effect of axial advection on the threshold also tends to be stabilising. Eccentricity deforms the modes structure compared to the concentric case. As a result, the mode with the largest temporal growth rate takes the form of 'pseudo-toroidal' Taylor vortices in the absence of axial flow, and 'pseudo-helical' structures with increasing azimuthal order as Rez becomes larger. Results are qualitatively similar for different radius ratios. Agreement with the few available experimental data is good. In a second part, absolute instability is studied by applying the pinch-point criterion to the dispersion relation. Axial flow is found to strongly inhibit absolute instability, the mechanism of which being centrifugal, and the value of Re at the threshold is typically one order of magnitude larger than that of Rez. The effect of eccentricity is more complex: weak stabilisation for low values of e, marked destabilisation for moderate eccentricities and high enough Rez, and finally stabilisation as e is further increased. Unlike temporal instability, the dominant absolutely unstable mode is the 'pseudo-toroidal' Taylor vortex flow over the whole range of parameter space considered.
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