Effect of vibrations on thermal convection in thick rotating annulus

The effect of transverse vibrations on thermal convection in a rotating thick cylindrical fluid layer is investigated experimentally. The layer rotates about the horizontal axis of symmetry. The temperatures of the layer boundaries are different (the outer boundary is cold) and maintained constant. The study is limited to the case of fast rotation. The centrifugal force of inertia plays a stabilizing role, bringing the fluid into a state of a stable mechanical equilibrium. The vibrations affect the equilibrium of the layer in a narrow frequency range close to the rotation frequency. The structure of the convective flows is studied using PIV-method. It is found that when the frequencies of vibrations and rotation definitely coincide, the convective flow has a form of the couple of symmetric two-dimensional vortices, the position of which is stationary in the cavity reference frame. The convection occurs under the action of an induced inertial force field (superposition of vibrational and centrifugal inertial force fields). With a frequency mismatch, the induced force field rotates in the cavity reference frame. The maximum heat transport corresponds to a resonant excitation of the azimuthal two-dimensional inertial oscillations of the non-isothermal fluid layer. The convective heat transport in this case is much higher than in the case of the frequencies coincidence. The dependence of heat transport, both under resonance conditions and with equal frequencies, on vibration parameters is studied. It is shown that the centrifugal and vibrational mechanisms play a key role in the development of the convection.


Introduction
The increasing complexity of production technologies associated with the movement of liquids, mixing in liquid media, as well as with the separation of inhomogeneous mixtures dictates the need to develop effective tools for controlling the processes of mass and heat transfer. The well-studied methods of influencing the behavior of hydrodynamic systems are vibration and rotation [1,2].
Earlier in the works [3] the experimental study of thermal convection in the rotating cylinder under the action of transverse vibrations was carried out for the case of the liquid with internal heat release. It was VII Perm Hydrodynamical Forum (PHD-Forum 2020) Journal of Physics: Conference Series 1809 (2021) 012035 IOP Publishing doi:10.1088/1742-6596/1809/1/012035 2 found that vibrations have a significant effect on the movement of fluid in a narrow frequency range close to the rotation frequency. The maximum heat transfer is achieved when the difference between the rotation frequency and the vibration one is 2 3 % − . The intensive heat transport in this range of parameters is caused by the resonant excitation of inertial circular oscillations of a non-uniformly heated liquid column. Another situation arises when the vibration frequency of the cavity and the rotation frequency strictly coincide: convection is associated with the generation of a static inertial force field which breaks the axial symmetry of the centrifugal force field in the cavity. Despite the detailed experimental studies, the structure of thermal convection in different regimes remained unclear.
In this paper, the structure of thermal convection in a rotating thick cylindrical layer with transverse vibrations is investigated by the PIV-method. The discovered and studied effects can be applied for efficient control over the heat and mass transfer in the rotating systems.

Experimental facility and procedure
The cylindrical layer is formed by an aluminum heat exchanger 1 and an external Plexiglas tube 2 (Fig1) with a wall thickness equal 3 mm. Layer geometrical parameters: thickness 26 h = mm, length 230 L = mm and relative radius . The heating of the layer is carried out from the inside using an electric heater installed on the axis of rotation. The Plexiglas tube 2 is washed outside with a thermostated liquid, which provides the constant cooling of the layer. In the experiment the temperatures of internal heat exchanger 1 T , of the outer layer boundary 2 T and of the coolant 3 T are measured with using integral resistance thermometers. A detailed description of the experimental setup can be found in the paper [4]. The vibrations perpendicular to the axis of the layer rotation are set according to the harmonic law using a mechanical vibrator, which is a crank mechanism mounted on a massive base [4]. The amplitude of vibrations b varies from 1.0 to 25.0 mm.
Before the beginning of the experiment, the layer is filled with a working fluid. In the experiments a distilled water and aqueous-glycerol solutions with a mass fraction 50 C = % are used. The Prandtl number Pr varied in the ranges 5-7 and 29-35 respectively. Further, the heating of the inner boundary of the layer and cooling of the outer one are switched on; the cuvette is brought into the relatively rapid rotation with an angular velocity rot Ω = 12.57, 18.85 or 25.13 rad/s. The action of the centrifugal force of inertia brings the non-isothermal fluid into the state of the stable mechanical equilibrium, the occurrence of which is determined by the readings of temperature sensors. After the establishing a stationary 3 temperature distribution in the layer, vibrations with a frequency exceeding the rotation frequency are switched on. During the experiment, the vibration frequency is reduced stepwise, the rotation frequency remains unchanged. The readings of the temperature sensors at each step of the experiment are averaged over the time interval. The temperature difference between the layer boundaries is 1 2 T T Θ = − ; temperature drop on the wall of a Plexiglas tube is 2 3 T T T ∆ = − The sample includes only the data obtained after the establishment of a stationary convection regime in the layer. A visualization of the fluid flows is carried out using plastic particles with a diameter 50 μm with a density slightly higher than the density of water. The particles are suitable for both photographic recording and PIV-analysis. Observation and shooting on a high-speed video camera are carried out along the axis of rotation through a transparent flange in the plane of the light-sheet created by the laser perpendicular to the axis of rotation. Figure 2a shows the dependence of the temperature difference Θ at the layer boundaries on the frequency difference vib rot ∆Ω = Ω − Ω . The vibrations have an effect in a narrow range of vibration frequencies, close to the rotation one. Decreasing the temperature difference Θ between the layer boundaries indicates the appearance of the convective flow. The graphs show two areas of the convective flow intensification.

The results of the temperature measurements
The paper [1] shows, that the intensification of the heat transport at the small frequency mismatch has a resonant nature. The resonance is observed at both 0 ∆Ω > and 0 ∆Ω < . The position of the maxima is symmetrical about the center of the graph ( 0) ∆Ω = . Temperature difference Θ takes the different values at the peaks: at 0 ∆Ω > the temperature difference is smaller, which indicates more intense convective flows than in resonance at 0 ∆Ω < . It is necessary to note that in the experiments with the heat-generating fluid in a rotating cylinder [3] such a significant asymmetry of the convective flows intensity in the resonance areas was not observed. The convective heat transport is also recorded when the frequencies coincide vib rot Ω = Ω , although it is lower than one in the resonant regions. It should also be noted that the intensity of convection in the  (Fig2a, points 1-3). In the case of the frequencies coincidence the intensity of convection is practically independent on the vibration frequency at constant vibration amplitude. Figure 2b shows the heat transport under resonance conditions depending on the vibration amplitude. The characteristic of the heat transport is the Nusselt number Nu, defined as the ratio of the heat flux at a certain step of the experiment to the heat flux in the absence of convection at a given heat release power. Initial temperature differences Θ are the same in all series of the experiments. The graphs show the results of the experiments with different working fluids. The empty points indicate the heat transport at 0 ∆Ω < , the filled ones at 0 ∆Ω > . With increasing viscosity, the asymmetry of the heat transport in different resonances increases (points 4 and 5).

Study of convective structures
The study of the convective flows structure is carried out using the PIV method. For this, a high-speed camera is installed along the axis of rotation. The camera focuses on a laser knife that dissects the cavity in the cross section in the middle part along the length. The camera is installed at a distance 0.5 m from the transparent flanges. The vibration amplitude is 6.9 mm. The cuvette remains in the frame when shifted to the right and left from the center. This allows fixing any instantaneous phase of the cavity oscillations. Video is recorded at a frequency 200 fps. The cuvette rotates with a frequency rot rot 2 2 f π = Ω = rps. The frames are captured from the video in the same phase of the cavity rotation (every hundredth frame). Due to the fact that the frequency of rotation and vibration do not coincide, the cuvette after one full revolution turns out to be in different phases of vibration. In each frame, the position of the cuvette is determined according the vibration parameters; the frame is cropped along the outer boundary of the layer. Further, the frames are analyzed using the PIVLab program. In one series of frames, at least 150 rotation periods are analyzed. At fast rotation in the absence of vibrations in the middle along the length part of the layer, an azimuthal prograde rotation of the liquid is formed (Fig3a). The cuvette in the laboratory frame rotates counterclockwise. It should be noted that the angular velocity of the prograde rotation of the fluid liq Ω is small. Maximum angular velocity liq Ω is observed near the inner cylinder and is approximately 0.005 rad/s, which corresponds to a full turnover of the liquid in about 20 minutes at the cavity rotation rate equal 2 rps. Let us consider the motion of a fluid in the cavity reference frame under the action of vibrations and start the analysis with the case of coincidence of the vibration frequency and the rotation one. As it shown in [3] a static inertial force field is formed in the cavity. In this case a system of vortices shown in Fig3b is observed in the cavity. The cavity is in the extreme right position; the resulting inertial force field (the superposition of the uniform vibrational and the axisymmetric centrifugal inertial force fields) is directed to the right. The vibration amplitude is 6.9 mm. In the left part of the cavity, an ascending relative to the direction of the induced force field flow is formed, and a descending flow is formed near the outer cylindrical boundary of the layer. This flows form two symmetrical vortices. A stagnant zone forms on the right side of the layer. The convection structure is similar to convective vortices that arise in a horizontal stationary cylindrical layer heated from the inside. In contrast to the stationary case, this vortex system is formed under conditions of the action of the inertial forces: centrifugal force and Coriolis force. This makes significant changes in the flow structure. For example, in the stationary case, the vortices are located mainly near the outer boundary, in the moving cavity -near the inner one.  In the resonance regions, when the maximum heat transfer is recorded, convection develops when the frequency of the inertial force field coincides with one of the natural frequencies of the system inertial oscillations. It is known [5,6], that one of the main eigenmodes is the circular vibration of the system about the axis of rotation. The resonant circular fluctuation of the liquid leads to the fact that one of the vortices is closed around the internal heat exchanger: in the case 0 ∆Ω > the cyclonic vortex (

Conclusion
Thermal convection arising under the action of transverse vibrations in a rotating thick cylindrical liquid layer is investigated experimentally. The case of rapid rotation and a hot inner boundary of the layer is considered The centrifugal inertial force plays a stabilizing role, bringing the liquid into a state of stable mechanical equilibrium. It was found that when the vibration and rotation frequencies coincide, a system of two large-scale vortices similar to ones observed in stationary cylindrical layers when heated from the inside is formed in the layer. The excitation of the convective flows is associated with the generation of an induced inertial field (the superposition of vibrational and centrifugal inertial force fields) stationary in the cavity reference frame. With a slight misalignment of the frequencies (2-4 %) the induced force field rotates in the cavity reference frame. At the same time, a system of convective structures, consisting of two large-scale vortices of opposite vorticity rotates. The direction of the resulting force field rotation depends on the sign of the difference between the frequencies of vibrations and rotation: in the case of a positive difference, the field performs the prograde rotation (coincides with the direction of rotation of the layer in the laboratory reference frame), in the case of a negative difference, a retrograde one. The maximum heat transport is recorded when the field rotation frequency coincides with the natural frequency of inertial oscillations of the system (circular oscillations). A resonant increase in the amplitude of circular oscillations leads to a significant development and closure of one of the vortices around the inner boundary of the layer. The vortices of the opposite vorticity decay. In the case of the prograde resonance a cyclonic vortex closes, in the case of retrograde resonance, an anticyclonic one. It is found that the development of a cyclonic vortex provides a greater heat transport.