Studies of thermal convection in rotating cavities, both experimental and numerical, are an important task for the development of fundamental and applied aspects of convective fluid motion [1,2].

Various phenomena associated with the effect of vibrations on hydrodynamic systems [3], led to the formation of a separate section—vibrational thermal convection [3,4]. Vibrational thermal convection employs the averaging method for the equations of thermal convection in the Boussinesq approximation [4]. This method is founded upon the partitioning of all variables in the equations of motion and heat and mass transfer into two categories [4]: “slow”, representing smoothly changing variables that are averaged over the period of oscillations, and “fast”, denoting oscillating variables with a frequency equal to that of the external influence.

The convective processes occurring under vibration conditions are determined by the interaction of temperature fields and pulsation velocity fields. The equations of averaged (vibrational) thermal convection obtained by the averaging method along with the Rayleigh Ra=gβΘh3/νχ and Prandtl Pr=ν/χ numbers contain an additional term with a lifting force characterized by a vibrational analogue of the Rayleigh number Rav=(bΩoscβΘh)2/2νχ. Here h is characteristic cavity size, b and Ωosc are amplitude and cyclic frequency of the cavity’s translational vibrations. The remaining designations are generally accepted.

The case of the influence of linear translational vibrations on a non-isothermal liquid was studied in detail in [3]. The equations of vibrational thermal convection in oscillating force fields of elliptical polarization were derived in [5]. In the case of circular polarization vibration, in rotating inertial force fields, the control parameter Rav has the same form. For a flat layer, the fundamental difference lies in the degeneration of the solution in the direction. In particular, vibro-convective structures have the form of cells arranged in a hexagonal order, motionless relative to the cavity.

The description developed in [5] proved to be effective in describing thermal vibrational convection in rotating systems [6]. We apply the term “Thermo-vibrational convection” to the averaged convection caused by oscillations under the action of gravity in a non-isothermal fluid inside a cavity rotating around a horizontal axis. Note that there are no vibrations, but the mechanisms are similar, at this, the averaged convection is characterized by a modified vibrational parameter Rv. One particular instance is that of averaged thermal convection excited in a cavity uniformly rotating in an external static force field. In the context of a liquid rotating around a horizontal axis within a gravitational field, gravitational acceleration g can be considered to play the role of vibration acceleration bΩosc2. In this instance, the oscillations of the non-isothermal liquid are initiated by a stationary field, thereby exhibiting the characteristics of tidal oscillations. It is important that despite the absence of vibrations, the equations of averaged convection are similar to the equations of vibrational convection [5]. This formulation corresponds to a special case when the rotation velocity of the force field in a non-inertial system coincides in magnitude with the rotation velocity of the cavity and is opposite in direction. The dimensionless oscillation frequency takes the value ω=Ωosc/Ωrot=−1, and a modified dimensionless control vibration parameter is introduced Rv=(gβΘh)2/2νχΩrot2 [6]. Another defining parameter is the dimensionless rotation velocity ωrot=Ωroth2/ν, which determines the influence of the Coriolis force on convective (averaged) structures. This parameter is analogous to the well-known Taylor number Ta≡4Ωrot2h4/ν2 [7].

The phenomenon of “vibrational” thermal convection (averaged convection) has been the subject of extensive experimental and theoretical investigation in a range of cavity geometries. The experimental study was started with the case of a vertical flat layer with boundaries of different temperatures, performing uniform rotation around a horizontal axis oriented perpendicular to the plane of the layer [8]. This was subsequently developed in a series of works, as referenced [9]. The experiments [9] conducted on relatively thin layers with a circular boundary and a large aspect ratio a=R/h, where R is a radius of the outer boundary of the layer with a thickness h, showed the emergence of vibrational thermal convection in the form of a system of convective cells arranged in a hexagonal order (Fig. 1a).

The observations [8,9] showed that up to the excitation threshold of cellular vibrational thermal convection, toroidal convective vortex flows were evident in the vicinity of the outer lateral boundary (Fig. 1b). Presumably, these flows are caused by inertial waves [10]. It is known that in a rotating liquid, the development of specific wave mode with inertial effects [10], associated with the action of the Coriolis force—is possible. The region of such waves existence is limited by the value of the dimensionless frequency of cavity oscillations ω<2.

Studies of such wave processes in an isothermal setting are carried out quite widely [11–16] since it is a relevant task in connection with geophysical applications. Inertial waves and specific “inertial modes”—resonant conditions for mode with inertial and wave characteristics—have been studied in cavities of varying geometries: spherical [17–20], cylindrical [21–24], in axisymmetric cylindrical layers with parallel boundaries [25], and a case where one of the cylindrical layer boundaries has a conical shape [26,27]. A detailed overview of these studies can be found in [28]. It should be noted that inertial modes have been primarily investigated from a theoretical perspective.

With regard to rotating non-isothermal systems, the occurrence of inertial waves has been documented in experimental studies, particularly in liquids exhibiting internal heat generation [29]. In a rotating cylindrical layer, in the case of heating the layer both from the inside and from the outside, the existence of convective flows in the form of toroidal vortices located in the subthreshold region of averaged convection was discovered. It was shown that these flows arise in a non-threshold manner, have a relatively low intensity and are generated by inertial waves propagating in the layer.

In previous studies [8,9], thermovibrational convection in rotating systems was investigated. Earlier, only hypotheses were proposed regarding the existence of a region for inertial wave propagation. When thermovibrational convection is excited by oscillations of an external force field at a frequency ω=1, inertial modes can exist in a rotating cavity. This study presents, for the first time, a theoretical model focusing on the development of inertial modes in a layer. The studies are carried out in a vertical flat layer with a circular lateral boundary, rotating around a horizontal axis of symmetry (Fig. 2). The boundaries of the layer have different temperatures, the rotation velocity and temperature drop vary. The paper provides a velocity field in a cross-section (Fig. 3), demonstrating that the development of toroidal structures in the layer is associated with the excitation of standing inertial waves (inertial modes).

In practical applications, the problem under consideration involves studying the dynamics of non-isothermal systems rotating around a horizontal axis. In rotating containers (in particular, a flat layer), fluid oscillations can lead to the generation of inertial waves. In our geometry, the flow takes the form of toroidal structures. In the case of a different geometry, the flow will also adopt a toroidal shape, but its location relative to the rotation axis will differ [29]. The transformation of toroidal structures into thermovibrational cells in industry can be utilized to optimize mixing, heating, and cooling processes, resulting in cost reduction and improved product quality.

The working layer (Fig. 2) is a cavity formed between two heat exchangers. A ring gasket made of heat-insulating material limits the working cavity along the perimeter. The heat exchanger (Tcold), designed for the observation of convective structures is made of organic glass with glass flat boundaries. Another heat exchanger (Thot) is made of aluminum. The working fluid is distilled water. The annular gasket and heat exchangers with flat boundaries create a closed cylindrical cavity with specified geometric parameters: layer thickness h and radius R. The flat boundaries of the heat exchangers act as the layer boundaries. The temperature of the heat exchangers is maintained constant by pumping thermostatic liquid, which is supplied from the thermostats. The hydraulic collector ensures the uninterrupted circulation of liquid from the jet thermostats in the heat exchangers of the cuvette, fixed on a rotating platform. During the experiment, the temperature is measured with precision 0.1∘C. Temperature control at the cavity boundaries is carried out using temperature sensors (resistance thermometers). The signal from the sensors is transmitted to the computer via the measuring module “Termodat-13IK” using an electric collector. The rotation of the platform is controlled by a stepper motor, enabling precise adjustment of the angular rotation velocity with an accuracy of 0.06 rad/s.

Experimental methodology. The cuvette is filled with distillate water and positioned on the platform of the laboratory stand. Then the stepper motor sets the platform with the cuvette fixed to it in relatively fast motion. The rotation speed was varied discretely with a sequential decrease, with a step size of 0.05 rps. The step size did not affect the process of inertial wave generation in the fluid layer. At each stage of the experiment, a waiting period of approximately 10 min was maintained to ensure the stabilization of the system’s thermal regime. The focus of the study was on the rotation frequency range of 0.5–1.5 rps. At lower rotation frequencies, the amplitude of fluid oscillations over the period significantly increases, leading to a weakening of the inertial wave regime. The temperature sensors transmit the real-time temperature data of the flat boundaries of heat exchangers (Fig. 2) to the PC. Based on the temperature difference between the layer boundaries Θ and the temperature drop ΔT on the heat flux sensor, the Nusselt number Nu is calculated to characterize the heat transfer. It is the ratio of the temperature difference across the heat flux sensor to the temperature difference under molecular heat transfer conditions ΔT0 at the same value of Θ measured in the absence of rotation with the horizontal layer heated from above. Proceeding to the subsequent phase of the experiment entails a reduction in the rotation velocity of the cavity. This reduction is continued stepwise until the cuvette reaches a complete stop. The excitation threshold of the averaged convection is determined by plotting the Nusselt number in relation to the rotation velocity (Fig. 4).

Methodology of the velocity fields measuring is based on the utilisation of the tracer particles. Light-scattering Resin Amberlite particles with a size of 50 microns are introduced into the working fluid to visualize its motion. The particle density of approximately 1.05 g/cm³ is close to that of the working fluid, allowing them to move with the fluid. During the experiment, particle motion is recorded by a high-speed camera, specifically a Sony RX0 II, in the area illuminated by a laser sheet. A green line laser module with a wavelength of 520 nm and a Powell lens type is used as the laser sheet. The tracer particles upon entering the plane of the laser knife scatter the light, and the high-speed camera record allows for the analysis of the particles movement. The resulting image of particle movement provides a comprehensive representation of the flow velocities within the plane of the laser knife. In the experiment, the velocity field is analyzed in two sections: the axial (Fig. 3a) and the transverse (Fig. 3b) ones.

The digital correlation of successive images enables the construction of two-dimensional maps of velocity fields. The PIV method provides detailed spatio-temporal data on the flow in a rotating cavity.

Inertial waves are defined as wave oscillations that occur in rotating fluids under the influence of the Coriolis force, which leads to stable oscillatory processes. The theoretical description of inertial waves, given in [10], allows one to calculate the parameters of the inertial mode arising in a closed rotating cavity: m—radial, n—axial, and k—azimuthal wave numbers.

In the problem of a rotating cylindrical flat layer of a given geometry a=R/h, it is pertinent to consider the structure and natural frequencies of inertial oscillations of liquid–inertial modes. In [27], the solutions were found for the inertial modes neglecting viscous forces (small Ekman numbers) and in the approximation of small oscillation amplitude (small Rossby numbers). The solution obtained in [27] for pressure during the excitation of liquid inertial oscillations with a dimensionless frequency ω=Ωosc/Ωrot in a rotating cylindrical layer of liquid is presented here. If we consider the pressure disturbance in a liquid in a rotating cylindrical layer in the form P=F(r)eikθcos⁡nπz, then for the function F(r) the next equation is obtained in [27]. (1)r2∂2F∂r2+r∂F∂r+[(4/ω2−1)n2π2r2−k2]F=0.

In this context, the unit of distance measurement is the length of the cavity h. Introducing a radial variable x=σr, where σ≡nπ(4/ω2)−1, the Eq. (1) is reduced to the Bessel equation, the solution of which has the form (2)F(x)=A(Jk(x)+BYk(x)).

Here, Jk(x) and Yk(x) are the Bessel functions of the first and second kinds. The resulting solution must satisfy the boundary conditions(3)x∂F/∂x+k(σ/πn)2+1)F=0on the cylindrical boundary of the liquid layer x=σr. Note that r—dimensionless radius of the outer boundary of the liquid layer, the unit of measurement is the thickness of the cavity h. In the solution of the equation, the Bessel function of the second kind is absent. These boundary conditions permit the calculation of the m-th eigenvalue of the mode with azimuthal wave number k and axial wave number n denoted by ωmnk. Here ωmnk is the dimensionless frequency of the wave, with a unit of measurement equal to the frequency of the cavity rotation.

In the case under consideration, a wave in a rotating system with a non-isothermal liquid is excited by a gravity field directed perpendicular to the rotation axis, rotating in the cavity reference frame with the rotation velocity of the cavity itself, but in the opposite direction. It should be expected that the resonance condition is met by a wave with a frequency |ωmnk|=1 and azimuthal wave number k=1. Determination of the conditions for resonant excitation of inertial modes for a cavity of a given relative length a=R/h is reduced to solving Eq. (1) taking into account Eq. (3). It follows from the solution that for the relative length of the liquid layer a=3.33, the resonance conditions are met when m=6 at k=1 and n=1.

In the experiment R=7.0 cm, h=2.0 cm, the ratio of the layer radius to the layer thickness is a=R/h=7.0/2.0=3.5. As it can be seen from Fig. 7c, the radial wave number value m=6, obtained in the experiment, corresponds to the theoretical condition of resonance at the value of the wave number m=6 eigenwave mode with azimuthal wave number k=1 and axial wave number n=1. The spatial period of the structures discovered in the experiment is consistent with the calculated value.

A comparison of the theoretical data on the number of vortex structures in the radial direction with experimental results on the plane of parameters m,a (Fig. 10) reveals a high degree of correlation. The solid line is the theoretical curve of the radial value number m depending on the geometry of the cavity a, for the case k=1, n=1. The colored symbols show the experimental results obtained in this work at different layer thicknesses. The experimental data and theoretical curve are in good agreement.

It is assumed that certain values of m may correspond to the aspect ratio a values within a certain interval (Fig. 10, colored lines). Changing the aspect ratio of the cavity leads to a change in the number of vortex structures in the layer; under resonance conditions, the number m takes the integer values.

An analysis of the instantaneous velocity field (Fig. 7) and the theoretical calculations of the inertial mode (Fig. 10) shows that for a frequency |ωmnk|=1 when n=1, k=1 the selected geometry a=3.5 meets the condition of the resonant excitation of the mode m=6. In this case, an oscillatory mode is excited in space, which is a system of toroidal rolls localized between pressure nodes and performing the rotation coordinated with each other. An important feature is that the movement in each toroidal roll changes with an angle according to the harmonic law, that is, in the laboratory coordinate system on different sides of the axis of rotation, the toroidal rolls rotate in opposite directions. From the condition |ωmnk|=1 it follows that the desired wave and the phase of its oscillations remain unchanged in the laboratory reference frame.

The averaged thermal convection in a flat layer rotating around a horizontal axis was experimentally investigated. The attention was paid to the inertial-wave regime of thermal convection. The study was conducted using the PIV method in transverse and axial sections. The analysis of the instantaneous velocity field shows that the emergence of toroidal structures in the subthreshold region of averaged convection occurs during the excitation of inertial modes of fluid oscillations (standing inertial waves). In this case, the entire wave system remains stationary in the laboratory system since the wave propagates in the rotating cavity with an azimuthal velocity equal to the rotation velocity of the cavity (in the opposite direction).

It has been established that the wavelength of the toroidal structure is determined by the radial component of the wave number of the inertial mode for the liquid layer of a given geometry. The spatial period of the radially located vortex structures has been theoretically determined depending on the aspect ratio. The results of experiments with the cavities of different geometries are consistent with the results of theoretical analysis performed in the approximation of low viscosity of the liquid and high rotation velocity (in the limit of small values of the Ekman and Rossby numbers).

The performed studies shed light on the inertial-wave nature of specific averaged convective flows excited in rotating cavities with a non-isothermal liquid. The experiments performed are limited by the cavity shape (flat layer) and a certain value of the Prandtl number. Taking into account that the issues of heat transfer in rotating systems (under the action of various aggravating factors) are the object of attention [31–33], a detailed study of a new, relatively simple to implement mechanism—averaged thermal convection in a cavity rotating uniformly around a horizontal axis—is an urgent task. The results can inform the design of rotating systems [34] where thermal management is critical, such as in turbines, reactors, or space applications, by leveraging the predictable behavior of inertial-wave-driven convection.

The goal of further research is to clarify the mechanisms of generation of averaged flows by inertial waves, as well as to study the effect of the Prandtl number and the cavity shape (aspect number) on the structure and intensity of averaged thermo-convective flows under conditions of inertial-wave oscillations of the liquid.