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Direct numerical simulations based on high-resolution pseudospectral methods are carried out for detailed investigation into the instabilities arising in a differentially heated, rotating annulus, the baroclinic cavity. Following previous works using air (Randriamampianina et al., J. Fluid Mech., vol. 561, 2006, pp. 359–389), a liquid defined by Prandtl number $Pr=16$ is considered in order to better understand, via the Prandtl number, the effects of fluid properties on the onset of gravity waves. The computations are particularly aimed at identifying and characterizing the spontaneously emitted small-scale fluctuations occurring simultaneously with the baroclinic waves. These features have been observed as soon as the baroclinic instability sets in. A three-term decomposition is introduced to isolate the fluctuation field from the large-scale baroclinic waves and the time-averaged mean flow. Even though these fluctuations are found to propagate as packets, they remain attached to the background baroclinic waves, locally triggering spatio-temporal chaos, a behaviour not observed with the air-filled cavity. The properties of these features are analysed and discussed in the context of linear theory. Based on the Richardson number criterion, the characteristics of the generation mechanism are consistent with a localized instability of the shear zonal flow, invoking resonant over-reflection.

Three-dimensional direct numerical simulations (DNS) of the nonlinear dynamics and a route to chaos in a rotating fluid subjected to lateral heating are presented here and discussed in the context of laboratory experiments in the baroclinic annulus. Following two previous preliminary studies, the fluid used is air rather than a liquid as used in all other previous work. This study investigates a bifurcation sequence from the axisymmetric flow to a number of complex flows. The transition sequence, on increase of the rotation rate, from the axisymmetric solution via a steady fully developed baroclinic wave to chaotic flow, followed a variant of the classical quasi-periodic bifurcation route, starting with a subcritical Hopf and associated saddle-node bifurcation. This was followed by a sequence of two supercritical Hopf-type bifurcations, first to an amplitude vacillation, then to a three-frequency quasi-periodic modulated amplitude vacillation (MAV), and finally to a chaotic (MAV). In the context of the baroclinic annulus this sequence is unusual as the vacillation is usually found on decrease of the rotation rate from the steady wave flow. Further transitions of a steady wave with a higher wavenumber pointed to the possibility that a barotropic instability of the sidewall boundary layers and the subsequent breakdown of these barotropic vortices may play a role in the transition to structural vacillation and, ultimately, geostrophic turbulence.

We report on small-scale instabilities in a thermally driven rotating annulus filled with a liquid with moderate Prandtl number. The study is based on direct numerical simulations and an accompanying laboratory experiment. The computations are performed independently with two different flow solvers, that is, first, the non-oscillatory forward-in-time differencing flow solver EULAG and, second, a higher-order finite-difference compact scheme (HOC). Both branches consistently show the occurrence of small-scale patterns at both vertical sidewalls in the Stewartson layers of the annulus. Small-scale flow structures are known to exist at the inner sidewall. In contrast, short-period waves at the outer sidewall have not yet been reported. The physical mechanisms that possibly trigger these patterns are discussed. We also debate whether these small-scale structures are a gravity wave signal.

We present a numerical investigation of the flow between corotating disks with a stationary outer casing – the enclosed corotating disk pair configuration. It is known that in such a geometry, axisymmetric and three-dimensional flow regimes develop depending on the value of the rotation rate. The three-dimensional flow is always unsteady flowing to its wavy structure in the radial-tangential plane. Axisymmetric regimes exhibit first a pitchfork bifurcation, characterized by a symmetry breaking with respect to the inter-disk midplane, before a Hopf bifurcation is established. The regime diagrams for these bifurcations are given in the (Re, G)-plane, where Re(= Ωb2/ν) is the rotational Reynolds number and G(= s/(b−a)) is the gap ratio. For values of G smaller than a critical limit Gc ∼ 0.26, there exists a range of rotation rates where the motion becomes time-dependent before bifurcating to a steady symmetry breaking regime. It is shown that for G [ges ] Gc the transition to unsteady three-dimensional flow occurs after the pitchfork bifurcation, and the flow structure is characterized by a shift-and-reflect symmetry. The transition to three-dimensional flow is consistent with experimental observations made by Abrahamson et al. (1989) where multiple solutions develop (known as the intransitivity phenomenon) with the presence of quasi-periodic behaviour resulting from successive vortex pairings. On the other hand, for smaller values of gap ratio, the three-dimensional flow shows a symmetry breaking. Finally, it is found that the variation of torque coefficient as a function of the rotation rate is the same for both the axisymmetric and three-dimensional solutions.

In many engineering and industrial applications, the investigation of rotating turbulent flow is of great interest. In rotor-stator cavities, the centrifugal and Coriolis forces have a strong influence on the turbulence by producing a secondary flow in the meridian plane composed of two thin boundary layers along the disks separated by a non-viscous geostrophic core. Most numerical simulations have been performed using RANS and URANS modelling, and very few investigations have been performed using LES. This paper reports on quantitative comparisons of two high-order LES methods to predict a turbulent rotor-stator flow at the rotational Reynolds number Re=400000. The classical dynamic Smagorinsky model for the subgrid-scale stress (Germano et al., Phys Fluids A 3(7):1760-1765, 1991) is compared to a spectral vanishing viscosity technique (Séverac & Serre, J Comp Phys 226(2):1234-1255, 2007). Numerical results include both instantaneous data and postprocessed statistics. The results show that both LES methods are able to accurately describe the unsteady flow structures and to satisfactorily predict mean velocities as well as Reynolds stress tensor components. A slight advantage is given to the spectral SVV approach in terms of accuracy and CPU cost. The strong improvements obtained in the present results with respect to RANS results confirm that LES is the appropriate level of modelling for flows in which fully turbulent and transition regimes are involved.

Baroclinic instability is recognized to be one of the dominant energetic processes in the large-scale atmospheres of terrestrial planets, such as Earth and Mars, e.g., Pierrehumbert and Swanson, and in the oceans. With the exponential increase in computing power these last decades, direct numerical simulation has become an indispensable tool for investigating the complex spatiotemporal behaviors of baroclinic instability in the laboratory, complementarily with experiments. It is useful to explore the different nonlinear flow regimes in the parameter space in order to accurately delineate a bifurcation diagram. Moreover, direct numerical simulation provides relevant information about the small-scale fluctuations that progressively destroy the regularity of the flow during the transition toward geostrophic turbulence.

Baroclinic instability is recognized to be one of the dominant energetic processes in the large‐scale atmospheres of terrestrial planets, such as Earth and Mars, e.g., Pierrehumbert and Swanson, and in the oceans. With the exponential increase in computing power these last decades, direct numerical simulation has become an indispensable tool for investigating the complex spatiotemporal behaviors of baroclinic instability in the laboratory, complementarily with experiments. It is useful to explore the different nonlinear flow regimes in the parameter space in order to accurately delineate a bifurcation diagram. Moreover, direct numerical simulation provides relevant information about the small‐scale fluctuations that progressively destroy the regularity of the flow during the transition toward geostrophic turbulence.

A three-dimensional direct numerical simulation (3D DNS) is performed to describe the turbulent flow in an enclosed rotor-stator cavity characterized by a large aspect ratio \\(G=(b-a)/h=18.32\\) and a small radius ratio \\(a/b=0.15\\) (\\(a\\) and \\(b\\) the inner and outer radii of the rotating disk and \\(h\\) the interdisk spacing). Recent comparisons with velocity measurements have shown that, for the rotational Reynolds number \\(Re=\\Omega b^2/\\nu=95000\\) (\\(\\Omega\\) the rate of rotation of the rotating disk and \\(\\nu\\) the kinematic viscosity of water) under consideration, the stator boundary layer is 3D turbulent and the rotor one is still laminar. Budgets for the turbulence kinetic energy are here presented and highlight some characteristic features of the effect of rotation on turbulence. A quadrant analysis of conditionally averaged velocities is also performed to identify the contributions of different events (ejections and sweeps) on the Reynolds shear stress.

A three-dimensional (3D) direct numerical simulation is combined with a laboratory study to describe the turbulent flow in an enclosed annular rotor-stator cavity characterized by a large aspect ratio G=(b-a)/h=18.32 and a small radius ratio a/b=0.152, where a and b are the inner and outer radii of the rotating disk and h is the interdisk spacing. The rotation rate Omega under consideration is equivalent to the rotational Reynolds number Re=Omegab2/nu=9.5 x 104, where nu is the kinematic viscosity of the fluid. This corresponds to a value at which an experiment carried out at the laboratory has shown that the stator boundary layer is turbulent, whereas the rotor boundary layer is still laminar. Comparisons of the 3D computed solution with velocity measurements have given good agreement for the mean and turbulent fields. The results enhance evidence of weak turbulence at this Reynolds number, by comparing the turbulence properties with available data in the literature. An approximately self-similar boundary layer behavior is observed along the stator side. The reduction of the structural parameter a1 under the typical value 0.15 and the variation in the wall-normal direction of the different characteristic angles show that this boundary layer is three-dimensional. A quadrant analysis of conditionally averaged velocities is performed to identify the contributions of different events (ejections and sweeps) on the Reynolds shear stress producing vortical structures. The asymmetries observed in the conditionally averaged quadrant analysis are dominated by Reynolds stress-producing events in this B\"{o}dewadt layer. Moreover, case 1 vortices (with a positive wall induced velocity) are found to be the major source of generation of special strong events, in agreement with the conclusions of Lygren and Andersson.