Stratified Fluid Layer Research Articles

Enhancing transport processes in stratified fluids: Exploring rotational and stretching effects coupled with magnetic fields, heat, mass transfer and thermal/solutal slip

This research paper presents an in-depth investigation into the complex interactions governing stratified fluid flow in the presence of rotational and stretching phenomena, magnetic field influences, heat and mass transfer effects, and the incorporation of thermal and solutal-slips at fluid interfaces. Stratified fluid systems are commonly encountered in various natural and industrial processes, ranging from geophysical flows to chemical engineering applications, making a thorough understanding of these phenomena paramount.The study commences by formulating a comprehensive set of governing equations that account for the unique interplay of these multifaceted physical phenomena. Integrating thermal and solutal slip conditions at fluid interfaces introduces innovative prospects for precisely controlling and manipulating mass and heat transfer in these systems. This research paper extensively explores layered fluid dynamics, encompassing phenomena such as rotation and stretching, influences from magnetic fields, effects of heat and mass transfer, and the examination of thermal and solutal slip conditions. The insights obtained from this study not only enhance our comprehension of multi-physics fluid dynamics but also offer practical implications for streamlining processes across diverse fields. The principal findings and implications of this research can be summarized as follows: Various factors that affect fluid flow exhibit evident and discernible patterns. The mathematical model includes the Navier-Stokes equations with rotational and stretching terms, the Maxwell equations describing magnetic field effects, and the transport equations for heat and mass transfer. Additionally, thermal and solutal slip conditions at the fluid interfaces are incorporated to capture boundary layer dynamics.Numerical simulations are conducted to explore the intricate dynamics arising from this amalgamation of physical effects. The results reveal complex flow patterns, including stratification, boundary layer development, and vortical structures, offering insights into the behavior of fluids under the joint influence of rotational and stretching phenomena, magnetic fields, and thermal and solutal gradients. Furthermore, the impact of key parameters, such as rotation rate, stretching rate, magnetic field strength, and concentration gradients, is systematically analyzed to elucidate their effects on flow characteristics, heat and mass transfer rates, and concentration profiles within the stratified fluid layers. The Soret effect, or thermophoresis, causes nanoparticles to migrate away from hot regions due to fluid molecule flow and high energy levels. Elevating ξNt results in decreased heat and mass transfer. The Prandtl number shows minimal influence on heat and mass transfer when increased. The disk separation, and fluid viscosity, significantly affect the rotational spectrum. As the rotational parameter increases, heat and mass transfer decrease.

The magnetic non-hydrostatic shallow-water model

We consider the dynamics of a set of reduced equations describing the evolution of a magnetised, rotating stably stratified fluid layer, atop a stagnant dense, perfectly conducting layer. We consider two closely related models. In the first, the layer has, above it, relatively light fluid where the magnetic pressure is much larger than the gas pressure, and the magnetic field is largely force-free. In the second model, the magnetic field is constrained to lie within the dynamical layer by the implementation of a model diffusion operator for the magnetic field. The model derivation proceeds by assuming that the horizontal velocity and the horizontal magnetic field are independent of the vertical coordinate, whilst the vertical components in the layer have a linear dependence on height. The full system comprises evolution equations for the magnetic field, horizontal velocity and height field together with a linear elliptic equation for the vertically integrated non-hydrostatic pressure. In the magneto-hydrostatic limit, these equations simplify to equations of shallow-water type. Numerical solutions for both models are provided for the fiducial case of a Gaussian vortex interacting with a magnetic field. The solutions are shown to differ negligibly. We investigate how the interaction of the vortex changes in response to the magnetic Reynolds number ${Rm}$ , the Rossby deformation radius $L_D$ , and a Coriolis buoyancy frequency ratio $f/N$ measuring the significance of non-hydrostatic effects. The magneto-hydrostatic limit corresponds to $f/N\to 0$ .

Stability of internal gravity wave modes: from triad resonance to broadband instability

A theoretical study is made of the stability of propagating internal gravity wave modes along a horizontal stratified fluid layer bounded by rigid walls. The analysis is based on the Floquet eigenvalue problem for infinitesimal perturbations to a wave mode of small amplitude. The appropriate instability mechanism hinges on how the perturbation spatial scale relative to the basic-state wavelength, controlled by a parameter $\mu$ , compares to the basic-state amplitude parameter, $\epsilon \ll 1$ . For $\mu ={O}(1)$ , the onset of instability arises due to perturbations that form resonant triads with the underlying wave mode. For short-scale perturbations such that $\mu \ll 1$ but $\alpha =\mu /\epsilon \gg 1$ , this triad resonance instability reduces to the familiar parametric subharmonic instability (PSI), where triads comprise fine-scale perturbations with half the basic-wave frequency. However, as $\mu$ is further decreased holding $\epsilon$ fixed, higher-frequency perturbations than these two subharmonics come into play, and when $\alpha ={O}(1)$ Floquet modes feature broadband spectrum. This broadening phenomenon is a manifestation of the advection of small-scale perturbations by the basic-wave velocity field. By working with a set of ‘streamline coordinates’ in the frame of the basic wave, this advection can be ‘factored out’. Importantly, when $\alpha ={O}(1)$ PSI is replaced by a novel, multi-mode resonance mechanism which has a stabilising effect that provides an inviscid short-scale cut-off to PSI. The theoretical predictions are supported by numerical results from solving the Floquet eigenvalue problem for a mode-1 basic state.

Marangoni effect on Rayleigh-Taylor instability in viscous stratified fluid layers

Marangoni effect on Rayleigh-Taylor instability in viscous stratified fluid layers

Buoyancy force and magnetic field effects on laminar vortex breakdown and fluid layers

In this study, the Generalized Integral Transformation Technique (GITT) is used to describe the effect of buoyancy force and magnetic field on the vortex breakdown process generated by the rotation of an electrically conductive fluid. A magnetic field is positioned vertically to stabilize the swirling flow caused by the rotation of the bottom disc of a cylindrical recipient. Three fluids were compared in this study where the range of Richardson number is 0 ≤Ri ≤2.0. When the temperature difference is greater than Ri = 0.1, many layers become visible. These stratified flu id layers act as thermal insulators. In the case of stratification, the increased magnetic field reduces the total number of layers formed in the fluid. The influence of gradient temperature on the distribution of the layers generated is discussed. The limitations between the multilayer structure and the monolayer structure for three fluids are calculated as a function of the flow parameters.

Computational Analysis of Thermal Striping in Primary Sodium System of Liquid Metal Fast Breeder Reactor Using Finite Volume Method

Thermal striping is associated with random fluctuations of temperature that occur at the nonisothermal jet stream interface or across thermally stratified fluid layers due to the high heat transfer coefficient of liquid sodium flow. The temperature fluctuations in the jet mixing or stratified layer regions are transmitted to the adjoining structures after minimal attenuation in a Liquid Metal Fast Breeder Reactor (LMFBR). In turn, the adjoining structure may experience high cycle fatigue and catastrophic failure caused by crack propagation. Investigations have been carried out in detail numerically, and frequency and amplitude of temperature fluctuations in 500-MW(electric) pool-type fast reactor [Prototype Fast Breeder Reactor (PFBR)] structures for practical applications have been observed. The investigations consist of numerical simulations at two levels. First, a published benchmark experiment is analyzed, and then, a suitable computational fluid dynamics (CFD) model is identified for simulating the thermal striping phenomenon numerically. After that, detailed flow and temperature fluctuations are predicted in the reactor structures by analysis carried out based on the CFD model. The values of the temperature fluctuations predicted are found to be within acceptable limits, as required by structural mechanics considerations in the study.

Colloidal nanoparticle-assisted double diffusive gravitational fingering in a vertical Hele-Shaw cell: Theoretical and numerical studies

If one fluid layer contact with another layer having different density or viscosity, the interface can become unstable resulting in motions driven by density or viscous gradient. Here, the onset of colloidal particle assisted fingering motion in a Hele-Shaw cell is analyzed theoretically and numerically. By considering the mobility of colloidal nanoparticles, new governing equation for colloidal particle movement is obtained. Under the linear stability theory, new stability equations are derived and solved analytically and numerically. The linear stability analysis shows that the fingering without an adverse density gradient is possible for the present colloidal nanoparticle assisted fingering system. In addition, through the numerical simulations, we visualize the fingering motion and prove the results of the linear stability analysis. The present linear stability analysis and numerical simulations are in good agreement. Furthermore, the present analyses explain the recent experimental finding: fingering motion can be induced by adding a small amount colloidal particles into the initially stably stratified fluid layer.

MAGNETOTHERMAL CONVECTION ON THE THERMALLY STRATIFIED FLUID LAYER BY PERMANENT MAGNETS

Magnetothermal convection can be induced in paramagnetic fluid by temperature differences and the magnetic field. In this study, convection was studied both experimentally and numerically for the thermally stratified fluid layer attained by heating the top wall and cooling the bottom wall. When a single block magnet was placed above the top hot wall, convection was confirmed near the magnet edges. In the case of multiple magnets with alternating magnetic poles, convection was observed near the magnet junction(s) since the local magnetic force became stronger than that at the magnet edges. This effect depended on the number of magnets. A pair of block magnets with one junction induced a pair of vortex flows, in which the the local Nusselt number reached its maximum. In the case of three magnets, the two maxima of the local Nusselt number near the magnet junctions became smaller than that obtained in the two-magnet case. This was due to the weakened magnetothermal force that occurred by shifting the low-temperature region outward below the magnet junctions.

Convective Rolls in a Compressible Fluid

Rayleigh-Bénard convection occurs between two horizontally infinite plates when the lower plate is heated with respect to the upper one. The temperature gradient between the two plates causes convective rolls to form as the warmer fluid below becomes more buoyant than the cooler fluid above it. The standard (classical) analysis uses the Boussinesq approximation, which neglects the variations of fluid density except in relation to buoyancy forces. This approximation is not accurate for some real-world applications. This project is inspired by the atmosphere, so we will consider the onset of convection in a vertically stratified layer of fluid which we model using the anelastic equations. The standard analysis is presented in many textbooks and is used as a comparison to the analysis for the compressible fluid presented here. Using linear stability analysis, we compute the critical temperature differences required to induce convection. It is not possible to find analytical solutions, and therefore, numerical methods implemented in Python using LAPACK subroutines are used. Results of the critical Rayleigh number at the half-height of the fluid for a range of plate separation distances are computed. For all the cases that are considered, the solution for the compressible problem follows the standard solution for plate separation distances smaller than some viscosity-dependent value that increases with the viscosity of the fluid. For larger d, it is observed that the stratification inhibits the onset of convection. Our solution is far more idealized than any actual convection happening in the atmosphere. However, it does demonstrate, in this context, the limits of the Boussinesq approximation.

Modeling of salt finger convection through a fluid-saturated porous interface: Representative elementary volume scale simulation and effect of initial buoyancy ratio

Simultaneous existence of solute gradients along with temperature gradients in a double diffusive system is favorable to the onset of salt finger convection and this phenomenon has been investigated for several decades due to its efficient mixing mechanism. However, relatively few works were focused on the double diffusive process through a fluid-saturated porous interface (FPI), which could be applied in a variety of scenarios such as the directional solidification of concentrated alloys or mixing zones in coastal freshwater aquifers. In this paper, we consider the evolution of double-diffusive salt finger through FPI with a single-domain approach adopted for the solution of flow in a composite region made up of a fluid layer overlying a porous layer. Comparisons with existing numerical results show great agreement and demonstrate that flows through a fluid–porous system can be predicted with good accuracy by the proposed method. Several cases spanning a range of initial buoyancy ratio N0 from 2 to 7 are conducted to study the structure and behavior of salt finger together with key issues being focused on the effect of initial buoyancy ratio on flux variations. Differing from in stratified fluid layers, salt finger through FPI shows an asymmetric structure in which the growth rate in the fluid layer is much greater and the finger column in the porous layer presents a squarer shape. It is found that under the condition of low initial buoyancy ratios, the potential energy stored in the unstable stratification of salinity converted more into kinetic energy, which enhances the mixture of heat and masses in the vertical direction.

Process of Air Ingress during a Depressurization Accident of GTHTR300

A depressurization accident is the design-basis accidents of a gas turbine high temperature reactor, GTHTR300, which is JAEA’s design and one of the Very-High-Temperature Reactors (VHTR). When a primary pipe rupture accident occurs, air is expected to enter the reactor core from the breach and oxidize in-core graphite structures. Therefore, it is important to know a mixing process of different kinds of gases in the stable and unstable density stratified fluid layer. In order to predict or analyze the air ingress phenomena during the depressurization accident, we have conducted an experiment to obtain the mixing process of two component gases and the characteristics of natural circulation. The experimental apparatus consists of a storage tank and a reverse U-shaped vertical rectangular passage. One side wall of the high temperature side vertical passage is heated and the other side wall is cooled. The other experimental apparatus consists of a cylindrical double coaxial vessel and a horizontal double coaxial pipe. The outside of the double coaxial vessel is cooled and the inside is heated. The results obtained in this study are as follows. When the primary pipe is connected at the bottom of the reactor pressure vessel, onset time of natural circulation of air is affected by not only molecular diffusion but also localized natural convection. When the wall temperature difference is large, onset time of natural circulation of air is strongly affected by natural convection rather than molecular diffusion. When the primary pipe is connected at the side of the reactor pressure vessel, air will enter the bottom space in the reactor pressure vessel by counter-current flow at the coaxial double pipe break part immediately. Afterward, air will enter the reactor core by localized natural convection and molecular diffusion.

Differential Rotation in Solar-like Convective Envelopes: Influence of Overshoot and Magnetism

Abstract We present a set of four global Eulerian/semi-Lagrangian fluid solver (EULAG) hydrodynamical (HD) and magnetohydrodynamical (MHD) simulations of solar convection, two of which are restricted to the nominal convection zone, and the other two include an underlying stably stratified fluid layer. While all four simulations generate reasonably solar-like latitudinal differential rotation profiles where the equatorial region rotates faster than the polar regions, the rotational isocontours vary significantly among them. In particular, the purely HD simulation with a stable layer alone can break the Taylor–Proudman theorem and produce approximately radially oriented rotational isocontours at medium to high latitudes. We trace this effect to the buildup of a significant latitudinal temperature gradient in the stable fluid immediately beneath the convection zone, which imprints itself on the lower convection zone. It develops naturally in our simulations as a consequence of convective overshoot and rotational influence of rotation on convective energy fluxes. This favors the establishment of a thermal wind balance that allows evading the Taylor–Proudman constraint. A much smaller latitudinal temperature gradient develops in the companion MHD simulation that includes a stable fluid layer, reflecting the tapering of deep convective overshoot that occurs at medium to high latitudes, which is caused by the strong magnetic fields that accumulate across the base of the convection zone. The stable fluid layer also has a profound impact on the large-scale magnetic cycles developing in the two MHD simulations. Even though both simulations operate in the same convective parameter regime, the simulation that includes a stable layer eventually loses cyclicity and transits to a non-solar, steady quadrupolar state.

Mixing Dynamics in Double-Diffusive Convective Stratified Fluid Layers

Double-diffusive convection in a linearly stratified fluid in the presence of radiative cooling at the surface has been investigated experimentally and theoretically. The stratification strength, which was varied in the experiments, is characterized by the buoyancy frequency of a stable environment defined as N 2 = (g/ρ 0 )/(dρ/d z ). The surface radiative cooling mimics buoyancy forcing incumbent at the surface of the ocean during the boreal winter months. The significant parameters governing the mixing dynamics for such a system were identified to be the Richardson number (Ri) and flux Rayleigh number (Ra f ). Controlled experiments were performed for Ri = 0–6, while maintaining a constant Ra f = 2.58 x 10 7 . This indicates that the stratification strength N was changed while the cooling flux Q was fixed. The mixing and barrier layers were visualized using a commercial dye solution. The thickness of the mixing layer was quantified from the flow evolution images. It was found that the mixing layer decays exponentially with increase in the stratification strength owing to suppression of downward convective motion due to the buoyancy force. A similar trend was observed for the entrainment velocity. A scaling law was proposed as follows: Δ = CRi –3/4 , where Δ is the mixing layer depth and C is a constant. The experimental results were compared with theoretical analysis and reasonable agreement was found. The results would be useful in parameterizing the mixing and barrier layers in strongly stratified environments such as the Bay of Bengal.

Parametric instability and wave turbulence driven by tidal excitation of internal waves

We investigate the stability of stratified fluid layers undergoing homogeneous and periodic tidal deformation. We first introduce a local model which allows us to study velocity and buoyancy fluctuations in a Lagrangian domain periodically stretched and sheared by the tidal base flow. While keeping the key physical ingredients only, such a model is efficient in simulating planetary regimes where tidal amplitudes and dissipation are small. With this model, we prove that tidal flows are able to drive parametric subharmonic resonances of internal waves, in a way reminiscent of the elliptical instability in rotating fluids. The growth rates computed via direct numerical simulations (DNSs) are in very good agreement with Wentzel–Kramers–Brillouin analysis and Floquet theory. We also investigate the turbulence driven by this instability mechanism. With spatio-temporal analysis, we show that it is weak internal wave turbulence occurring at small Froude and buoyancy Reynolds numbers. When the gap between the excitation and the Brunt–Väisälä frequencies is increased, the frequency spectrum of this wave turbulence displays a $-2$ power law reminiscent of the high-frequency branch of the Garett and Munk spectrum (Geophys. Fluid Dyn., vol. 3 (1), 1972, pp. 225–264) which has been measured in the oceans. In addition, we find that the mixing efficiency is altered compared to what is computed in the context of DNS of stratified turbulence excited at small Froude and large buoyancy Reynolds numbers and is consistent with a superposition of waves.

Effects of variable thermal diffusivity on the structure of convection

The structure of multiscale convection in a thermally stratified plane horizontal fluid layer is investigated by means of numerical simulations. The thermal diffusivity is assumed to produce a thin boundary sublayer convectively much more unstable than the bulk of the layer. The simulated flow is a superposition of cellular structures with three different characteristic scales. In contrast to the largest convection cells, the smaller ones are localised in the upper portion of the layer. The smallest cells are advected by the larger-scale convective flows. The simulated flow pattern qualitatively resembles that observed on the Sun.

Influence of magnetic field configuration on magnetohydrodynamic waves in Earth’s core

We develop a numerical model to study magnetohydrodynamic waves in a thin layer of stratified fluid near the surface of Earth’s core. Past studies have been limited to using simple background magnetic field configurations. However, the choice of field distribution can dramatically affect the structure and frequency of the waves. To permit a more general treatment of background magnetic field and layer stratification, we combine finite volume and Fourier methods to describe the wave motions. We validate our model by comparisons to previous studies and examine the influence of background magnetic field configuration on two types of magnetohydrodynamic waves. We show that the structure of zonal Magnetic-Archimedes-Coriolis (MAC) waves for a dipole background field is unstable to small perturbations of the field strength in the equatorial region. Modifications to the wave structures are computed for a range of field configurations. In addition, we show that non-zonal MAC waves are trapped near the equator for realistic magnetic field distributions, and that their latitudinal extent depends upon the distribution of magnetic field strength at the CMB.

Atmospheric thermal tides and planetary spin

Context. Thermal atmospheric tides can torque telluric planets away from spin-orbit synchronous rotation, as observed in the case of Venus. They thus participate in determining the possible climates and general circulations of the atmospheres of these planets. Aims. The thermal tidal torque exerted on an atmosphere depends on its internal structure and rotation and on the tidal frequency. Particularly, it strongly varies with the convective stability of the entropy stratification. This dependence has to be characterized to constrain and predict the rotational properties of observed telluric exoplanets. Moreover, it is necessary to validate the approximations used in global modelings such as the traditional approximation, which is used to obtain separable solutions for tidal waves. Methods. We wrote the equations governing the dynamics of thermal tides in a local vertically stratified section of a rotating planetary atmosphere by taking into account the effects of the complete Coriolis acceleration on tidal waves. This allowed us to analytically derive the tidal torque and the tidally dissipated energy, which we used to discuss the possible regimes of tidal dissipation and to examine the key role played by stratification. Results. In agreement with early studies, we find that the frequency dependence of the thermal atmospheric tidal torque in the vicinity of synchronization can be approximated by a Maxwell model. This behavior corresponds to weakly stably stratified or convective fluid layers, as observed previously. A strong stable stratification allows gravity waves to propagate, and makes the tidal torque negligible. The transition is continuous between these two regimes. The traditional approximation appears to be valid in thin atmospheres and in regimes where the rotation frequency is dominated by the forcing or the buoyancy frequencies. Conclusions. Depending on the stability of their atmospheres with respect to convection, observed exoplanets can be tidally driven toward synchronous or asynchronous final rotation rates. The domain of applicability of the traditional approximation is rigorously constrained by calculations.

Excitation of Surface Waves Due to Thermocapillary Effects on a Stably Stratified Fluid Layer

In chemical engineering applications, the operation of condensers and evaporators can be made more efficient by exploiting the transport properties of interfacial waves excited on the interface between a hot vapor overlying a colder liquid. Linear theory for the onset of instabilities due to heating a thin layer from above is computed for the Marangoni–Bénard problem. Symbolic computation in the long wave asymptotic limit shows three stationary, non-growing modes. Intersection of two decaying branches occurs at a crossover long wavelength; two other modes co-exist at the crossover point—propagating modes on nascent, shorter wavelength branches. The dispersion relation is then mapped numerically by Newton continuation methods. A neutral stability method is used to map the space of critical stability for a physically meaningful range of capillary, Prandtl, and Galileo numbers. The existence of a cut-off wavenumber for the long wave instability was verified. It was found that the effect of applying a no-slip lower boundary condition was to render all long waves stationary. This has the implication that any propagating modes, if they exist, must occur at finite wavelengths. The computation of 8000 different parameter sets shows that the group velocity always lies within 1 2 to 2 3 of the longwave phase velocity.

Short and long-term sensitivity of lab-scale thermocline based thermal storage to flow disturbances

Molten-salt thermocline-based systems are a low-cost option for single-tank thermal energy storage in concentrated solar power plants. Due to the high variability in solar energy availability, these energy storage devices are subject to transient heat loads during charging that can affect the storage efficiency. Numerical simulations were conducted to analyze the stability characteristics of a lab-scale thermocline tank subject to a flow disturbance during charging under different operating temperatures. The charging process was first simulated at a constant Reynolds number for three different Atwood numbers; a stably stratified fluid layer develops inside the storage tank in all cases. A flow disturbance was then introduced at the inlet of the stratified thermocline tank by inserting colder fluid for a short period of time. The disturbance interacts with the thermocline and causes oscillations and mixing. The thermocline oscillations are under-damped and lead to an increase in thermocline region thickness. The transient behavior of the thermocline and the decay rate in its oscillations were analyzed; the damping time depends on the Atwood number. The persistence of flow disturbance effects during long-term cyclical operation was also investigated. Several charge/discharge cycles were simulated at constant Reynolds number to obtain a time-periodic thermal response for each Atwood number. The characteristic flow disturbance was introduced at the inlet during a single charging process, and the thermocline region was observed during several subsequent charge/discharge cycles to assess the long-term temporal attenuation of the disturbance. The thermocline almost fully recovers to the time-periodic behavior after a single cycle.

鉛直流体層内の2成分気体混合過程に関する研究

The purpose of this study is to investigate the effect of natural convection and molecular diffusion on the mixing process under the stable density stratified fluid layer of two component gas. An experiment has been carried out to investigate a control method of natural circulation of air by injection of helium gas. The experimental apparatus consists of a reverse U-sharped vertical slot and a storage tank. The left side vertical slot consists of the heated wall and cooled wall. The right side vertical slot does not insert anything. Experimental results regarding mixing process of two component gases in vertical fluid layer were as follows. The heavy gas was transported to the slot by the molecular diffusion and natural convection. As time elapses, natural circulation of heavy gas suddenly occurs through the reverse U-shaped slot. As a result of experiments, the onset time of natural circulation is affected by not only molecular diffusion coefficient but also the strength of natural convection. When the helium gas is injected into the channel, it is possible to control the natural circulation of air. The onset time of the reproduction of the natural circulation can be varied by changing the injection rate of the helium gas.