Motivated by nuclear safety issues, we study the heat transfers in a thin cylindrical fluid layer with imposed fluxes at the bottom and top surfaces (not necessarily equal) and a fixed temperature on the sides. We combine direct numerical simulations and a theoretical approach to derive scaling laws for the mean temperature and for the temperature difference between the top and bottom of the system. We find two asymptotic scaling laws depending on the flux ratio between the upper and lower boundaries. The first one is controlled by heat transfer to the side, for which we recover scaling laws characteristic of natural convection (Batchelor, Q. Appl. Maths, vol. 12, 1954, pp. 209–233). The second one is driven by vertical heat transfers analogous to Rayleigh–Bénard convection (Grossmann & Lohse, J. Fluid Mech., vol. 407, 2000, pp. 27–56). We show that the system is inherently inhomogeneous, and that the heat transfer results from a superposition of both asymptotic regimes. Keeping in mind nuclear safety models, we also derive a one-dimensional model of the radial temperature profile based on a detailed analysis of the flow structure, hence providing a way to relate this profile to the imposed boundary conditions.

Linear and nonlinear stability analyses are performed to determine critical Rayleigh numbers (${Ra}_{cr}$) for a Rayleigh–Bénard convection configuration with an imposed bottom boundary heat flux that varies harmonically in time with zero mean. The ${Ra}_{cr}$ value depends on the non-dimensional frequency $\omega$ of the boundary heat-flux modulation. Floquet theory is used to find ${Ra}_{cr}$ for linear stability, and the energy method is used to find ${Ra}_{cr}$ for two different types of nonlinear stability: strong and asymptotic. The most unstable linear mode alternates between synchronous and subharmonic frequencies at low $\omega$, with only the latter at large $\omega$. For a given frequency, the linear stability ${Ra}_{cr}$ is generally higher than the nonlinear stability ${Ra}_{cr}$, as expected. For large $\omega$, ${Ra}_{cr} \omega ^{-2}$ approaches an $O(10)$ constant for linear stability but zero for nonlinear stability. Hence the domain for subcritical instability becomes increasingly large with increasing $\omega$. The same conclusion is reached for decreasing Prandtl number. Changing temperature and/or velocity boundary conditions at the modulated or non-modulated plate leads to the same conclusions. These stability results are confirmed by selected direct numerical simulations of the initial value problem.

Through laboratory experiments and numerical simulations, we examine the evolution of buoyant plumes as they are influenced by background rotation in a uniform density ambient fluid. The source Rossby number is sufficiently large that rotation does not directly affect the plume at early times. However, on a time scale of the order of half a rotation period, the plume becomes deflected from the vertical axis. For some experiments and simulations, the deflection persists and the flow precesses about the vertical axis. In other cases, shortly after being deflected, the plume laminarizes near the source to form a near-vertical columnar vortex, which we refer to as a ‘tornado’. Tornado formation occurs in some experiments and not in others even if the source and background rotation parameters are identical. However, their formation is more likely if the plumes are ‘lazy’. Simulations reveal that this is a consequence of the competing dynamics that occurs on comparable time scales. As a consequence of entrainment, vertical vorticity builds up within the plume reducing the Rossby number and suppressing vertical motion at distances progressively closer to the source. Meanwhile, the swirl (the ratio of the azimuthal to vertical flow) around the vicinity of the source increases, which tends to suppress three-dimensional turbulence in the near-source flow. Although the former process ultimately acts to deflect the plume off axis, in some instances, the swirl around the source succeeds in laminarizing the flow, resulting in tornado formation.

The dynamics, structure and stability of zonal jets in planetary flows are still poorly understood, especially in terms of coupling with the small-scale turbulent flow. Here, we use an experimental approach to address the questions of zonal jets formation and long-term evolution. A strong and uniform topographic $\beta$-effect is obtained inside a water-filled rotating tank thanks to the paraboloidal fluid free upper surface combined with a specifically designed bottom plate. A small-scale turbulent forcing is performed by circulating water through the base of the tank. Time-resolving particle image velocimetry measurements reveal the self-organization of the flow into multiple zonal jets with a strong instantaneous signature. We identify a subcritical bifurcation between two regimes of jets depending on the forcing intensity. In the first regime, the jets are steady, weak in amplitude, and directly forced by the local Reynolds stresses due to our forcing. In the second one, we observe highly energetic and dynamic jets of width larger than the forcing scale. An analytical modelling based on the quasi-geostrophic approximation reveals that this subcritical bifurcation results from the resonance between the directly forced Rossby waves and the background zonal flow.

This study explores experimentally the flows driven by precession in an oblate spheroid, in the vicinity of the possible resonance with the tilt-over mode. Two main phenomena are reported, combining observations and velocity measurements. First, a hysteretic cycle is quantitatively described between two uniform vorticity solutions, in good agreement with the historical analytical study of Busse (J. Fluid Mech., vol. 33, 1968, pp. 739–752). We then address the destabilization of each branch at low enough Ekman number. We confirm the possible presence of a so-called conical shear instability, recently depicted in the sphere by Lin et al. (Phys. Fluids, vol. 27, 2015, 046601) and in the spheroid by Horimoto et al. (Phys. Rev. Fluids, vol. 5, 2020, 063901). However, available measurements in the accessible parameter range are not sufficient to definitively discard an elliptical or shear origin of the excited instabilities in the spheroid, as first introduced by Kerswell (Geophys. Astrophys. Fluid Dyn., vol. 72, 1993, pp. 107–144).

We explore the near-resonant interaction of inertial waves with geostrophic modes in rotating fluids via numerical and theoretical analysis. When a single inertial wave is imposed, we find that some geostrophic modes are unstable above a threshold value of the Rossby number $kRo$ based on the wavenumber and wave amplitude. We show this instability to be caused by triadic interaction involving two inertial waves and a geostrophic mode such that the sum of their eigenfrequencies is non-zero. We derive theoretical scalings for the growth rate of this near-resonant instability. The growth rate scaled by the global rotation rate is proportional to $(kRo)^2$ at low $kRo$ and transitions to a $kRo$ scaling for larger $kRo$. These scalings are in excellent agreement with direct numerical simulations. This instability could explain recent experimental observations of geostrophic instability driven by waves.

Using water–salt water laboratory experiments, we investigate the mechanism of erosion by a turbulent jet impinging onto a density interface, for moderate Reynolds and Froude numbers. The Froude number is defined by $Fr_{i}=u_{i}/\sqrt{b_{i}g^{\prime }}$, with $u_{i}$ and $b_{i}$, the typical velocity and width of the jet at the interface, and $g^{\prime }$ the reduced gravitational acceleration. The Froude number $Fr_{i}$ characterizes the competition between inertial forces against the restoring buoyancy force. Contrary to previous observations reporting baroclinic destabilization of the interface, we show that the entrainment, in the range of parameters explored here, is driven by interfacial gravity waves. The waves are generated by the barotropic excitation coming from the turbulent fluctuations of the jet; they are then amplified by a mechanism of wave-induced stress; and they finally break and induce entrainment and mixing. Based on those physical observations, we introduce a scaling model for the entrainment rate, which varies continuously from $Fr_{i}^{3}$ to an $Fr_{i}$ power law from small to large Froude numbers, in agreement with the present and some of the previous laboratory data.

We study both experimentally and numerically the steady zonal flow generated by longitudinal librations of a spherical rotating container. This study follows the recent weakly nonlinear analysis of Busse (J. Fluid Mech., vol. 650, 2010, pp. 505–512), developed in the limit of small libration frequency–rotation rate ratio and large libration frequency–spin-up time product. Using particle image velocimetry measurements as well as results from axisymmetric numerical simulations, we confirm quantitatively the main features of Busse's analytical solution: the zonal flow takes the form of a retrograde solid-body rotation in the fluid interior, which does not depend on the libration frequency nor on the Ekman number, and which varies as the square of the amplitude of excitation. We also report the presence of an unpredicted prograde flow at the equator near the outer wall.

In this paper, we analyse the characteristics of the elliptic instability in a finite cylinder in the presence of both background rotation and axial stratification. A general formula for the linear growth rate of the stationary sinuous modes is derived including viscous and detuning effects in the limit of small eccentricity. This formula is discussed and compared to experimental results which are obtained in a cylinder filled with salted water for two different eccentricities by varying the stratification, the background rotation and the cylinder rotation. A good agreement with the theory concerning the domain of instability of the sinuous modes is demonstrated. Other elliptic instability modes, oscillating at the cylinder angular frequency are also evidenced together with a new type of instability mode, which could be connected to a centrifugal instability occurring during the experimental phase of spin-up. The nonlinear regime of the elliptic instability is also documented. In contrast with the homogeneous case, no cycle involving growth, breakdown and re-laminarization is observed in the presence of strong stratification. The elliptic instability in a stratified fluid seems to yield either a persistent turbulent state or a weakly nonlinear regime.

The effects of the Coriolis force on the elliptical instability are studied experimentally in cylindrical and spherical rotating containers placed on a table rotating at a fixed rate . For a given set-up, changing the ratio ΩG of global rotation to flow rotation leads to the selection of various unstable modes due to the presence of resonance bands, in close agreement with the normal-mode theory. No instability occurs when ΩG varies between −3/2 and −1/2 typically. On decreasing ΩG toward −1/2, resonance bands are first discretized for ΩG<0 and progressively overlap for −1/2 ≪ ΩG < 0. Simultaneously, the growth rates and wavenumbers of the prevalent stationary unstable mode significantly increase, in quantitative agreement with the viscous short-wavelength analysis. New complex resonances have been observed for the first time for the sphere, in addition to the standard spin-over. We argue that these results have significant implications in geo- and astrophysical contexts.

The linear stability of a rotating flow in an elliptically deformed cylindrical shell with an imposed radial temperature contrast is studied using local and global approaches. We demonstrate that (i) a stabilizing temperature profile can either increase or decrease the growth rate of the elliptical instability depending on the selected mode and on the strength of the radial buoyancy force; (ii) when the temperature profile is destabilizing, the elliptical instability coexists with two-dimensional convective instabilities at relatively small values of the Rayleigh number, the fastest growing mode depending on the relative values of the Rayleigh number and of the eccentricity; (iii) the elliptical instability totally disappears for larger values of the Rayleigh number. We argue that thermal effects have to be taken into account when looking for the occurrence and influence of inertial instabilities in geophysical and astrophysical systems, especially in planetary cores.

The single-domain Darcy–Brinkman model is applied to some analytically tractable flows through adjacent porous and pure-fluid domains and is compared systematically with the multiple-domain Stokes–Darcy model. In particular, we focus on the interaction between flow and solidification within the mushy layer during binary alloy solidification in a corner flow and on the effects of the chosen mathematical description on the resulting macrosegregation patterns. Large-scale results provided by the multiple-domain formulation depend strongly on the microscopic interfacial conditions. No satisfactory agreement between the single- and multiple-domain approaches is obtained when using previously suggested conditions written directly at the interface between the liquid and the porous medium. Rather, we define a viscous transition zone inside the porous domain, where the Stokes equation still applies, and we impose continuity of pressure and velocities across it. This new condition provides good agreement between the two formulations of solidification problems when there is a continuous variation of porosity across the interface between a partially solidified region (mushy zone) and the melt.