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This work is an analytical investigation of Rayleigh–Bénard convection in the presence of microorganisms in the salt fingering region. The stream function is used to perform a two-dimensional stability study. The amplitudes of the functions are determined using the Lorenz technique, and the stationary curves are drawn with regard to the Rayleigh thermal number and wave number. The stationary curves depicted indicate the stable and unstable zones. The Ginzburg–Landau equation is solved analytically to determine heat and mass transmission. The Nusselt and Sherwood numbers are effectively represented, and a straightforward relationship is discovered using the Lewis number. Based on the results, it is concluded that, in contrast to constant wall temperature, where microorganisms cause Nusselt number attenuation in the majority of the considered cases, microorganisms improve heat transfer in all of the considered cases in linear temperature distribution. These new discoveries are expected to result in significant shifts in the usage of microbes in the heat transfer firms. [ABSTRACT FROM AUTHOR]

Purpose: This paper aims to study the impact of convexity and concavity of the vertical borders on double-diffusive mixed convection. In addition, the study of entropy generation is performed. This numerical study has been carried out for different patterns of wavy edges to reveal their effects on heat and mass transfer phenomena. Design/methodology/approach: Four different flow features are treated by varying the directions of convexity and concavity of the vertical walls. A uniform temperature, as well as concentration distributions, are introduced to the left border while keeping a cold temperature and low concentration for the right border. The horizontal boundaries are in adiabatic condition. The upper border of the chamber is moving in the right direction with an equal speed. The governing Navies–Stokes equations are designed to describe energy and species transport phenomena, and these equations are solved by compact scheme. Findings: The investigated results are analyzed for various parameters, namely, Prandtl number, Richardson number, thermal Grashof number, Lewis number, Buoyancy ratio and amplitude of the wavy walls. It is observed that the thermal and solutal transfer performance becomes effective with lower Richardson numbers. The results reveal that the concavity and convexity of the side borders of the cabinet can control the thermosolutal performance. It is also observed that among all wavy chambers, Case-4 records maximum thermosolutal transfer rate, while Case-3 attains minimum thermosolutal transfer rate. Originality/value: This work is an example of solar thermal power conversion, power collection systems, systems of energy deficiency, etc. [ABSTRACT FROM AUTHOR]

Purpose: The onset of Darcy–Benard convection in a fluid-saturated porous layer within channeled walls can be through either the convectively amplified perturbations or absolutely unstable modes. The convective type route to formation of cellular forms has received much more interest in the literature than the absolute mode. This paper aims to explore the absolute instability mechanism on triggering the Benard convection in the presence of a uniform magnetic field acting perpendicular to the channel walls. Design/methodology/approach: Pursuing a completely theoretical linear stability approach, the locus of wavenumbers and critical Rayleigh numbers with respect to varying Peclet numbers leading to zero complex group velocity, and hence the absolute instability onset, is formulated in closed form. The formulae also incorporate the influence of fluid movement through the porous layer. Findings: Wanenumbers are found to be increasing in magnitude under the effect of the magnetic field. Although the magnetic field expectedly behaves in favor of stabilizing the convection through both convective and absolute instabilities by pushing the threshold value of Rayleigh number to higher values, a peculiarity exists in such a manner that the stabilizing effect of magnetic field can no longer compete against the absolute instability mechanism within the magnetized field after some critical location possessing negative part of the wavenumber. Originality/value: The significant outcome is attributed to the exchange of instabilities as far as the absolute instability mechanism is concerned. Magnetic field is always stabilizing and no such trend is observed regarding the convective type instability. [ABSTRACT FROM AUTHOR]

In this article, a numerical study was conducted on the mixed convection (CNV) of nanofluid (NFD) in a two-dimensional rectangular cavity using the control volume method. The upper and lower walls of the cavity were cold and hot, respectively, and its two vertical walls were insulated. The Upper wall has constant velocity and order to create a forced CNV in the cavity. They positioned three triangle-shaped obstacles on the heated wall that were likewise hot. The velocity and temperature graph values in the cavity, the velocity and streamline contours, and the Nusselt number value were examined independently by varying the height (HIT) of each barrier. Using artificial intelligence, the best values of the parameters to have the strongest flow and the most heat transfer (HTF) were checked. Two-phase method was used to simulate NFD flow. The results of this study showed that the middle obstacle with the highest HIT had more HTF. Increasing the length of the obstacle on the right has reduced the amount of HTF and the highest HTF occurred at the lowest HIT of this obstacle. When the obstacle on the left side has the HIT, the amount of HTF is the highest, while its average value has minimized the HTF. An increase in the HIT of the right fin at different HITs of the other two fins reduces the average Nu value [ABSTRACT FROM AUTHOR]

Energy and mass storage in various single-phase fluid flows is of particular interest, as the world currently faces energy challenges. Double-diffusive natural convection in an n-shaped storage tank is numerically studied which can be a general guideline to maintain a storage tank with higher exergy. Lattice-Boltzmann's approach in an in-house computational code is used to simulate the problem. To display the results, it is considered that the Rayleigh number lies between 103 and 105, and the Lewis number in the range of 0.1 and 10. The average Nusselt and Sherwood number, as well as entropy generation, showing the energy loss, are illustrated. It is observed that the average Nusselt and Sherwood number rises with increasing Rayleigh number and buoyancy ratio. Further, the average Sherwood number boosts by increasing the Lewis number. The most promising parameter in increasing the heat and mass transfer are found to be Rayleigh and Lewis number, respectively, with a maximum 300 percent improvement. The flow friction can be regarded as the main source of entropy generation, with a share of 90 percent. The Rayleigh number increment from 103 to 105 leads to the rise in the total entropy generation by approximately fivefold. [ABSTRACT FROM AUTHOR]

We apply physics-informed neural networks to three-dimensional Rayleigh–Bénard convection in a cubic cell with a Rayleigh number of Ra = 1 0 6 and a Prandtl number of Pr = 0. 7 to assimilate the velocity vector field from given temperature fields and vice versa. With the respective ground truth data provided by a direct numerical simulation, we are able to evaluate the performance of the different activation functions applied (sine, hyperbolic tangent and exponential linear unit) and different numbers of neurons (32, 64, 128, 256) for each of the five hidden layers of the multi-layer perceptron. The main result is that the use of a periodic activation function (sine) typically benefits the assimilation performance in terms of the analyzed metrics, correlation with the ground truth and mean average error. The higher quality of results from sine-activated physics-informed neural networks is also manifested in the probability density function and power spectra of the inferred velocity or temperature fields. Regarding the two assimilation directions, the assimilation of temperature fields based on velocities appears to be more challenging in the sense that it exhibits a sharper limit on the number of neurons below which viable assimilation results cannot be achieved. [Display omitted] • Investigations of 3D turbulent Rayleigh–Benard convection at Ra = 1 0 6 , Pr = 0. 7. • PINNs are used to assimilate temperature fields from velocity fields and vice versa. • Best results are achieved with sine activation in comparison with to tanh and ELU. • Prerequisite of PINN's sufficient expressiveness investigated by varying layer width. • Provision of suitable loss component weights for each assimilation direction. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this study is to explore the phase change (PC) dynamics in a T-shaped ventilated cavity having multiple inlet and outlet ports during nanofluid convection with phase change material (PCM) packed bed-installed system. Design/Methodology/Approach: Finite element method was used to analyze the PC dynamics and phase completion time for encapsulated PCM within a vented cavity during the convection of nanoparticle loaded fluid. The study is performed for different Reynolds number of flow streams (Re1 and Re2 between 300 and 900), temperature difference (ΔT1 and ΔT2 between −5 and 10), aspect ratio of the cavity (between 0.5 and 1.5) and nanoparticle loading (between 0.02% and 0.1%). Findings: It is observed that phase transition can be controlled by assigning different velocities and temperatures at the inlet ports of the T-shaped cavity. The PC becomes fast especially when the Re number and temperature of fluid in the port vary closer to the wall (second port). When the configurations with the lowest and highest Re number of the second port are considered up to 54.7% in reduction of complete phase transition time is obtained, while this amount is 78% when considering the lowest and highest inlet temperatures. The geometric factor which is the aspect ratio has also affected the flow field and PC dynamics. Up to 78% reduction in the phase transition time is obtained at the highest aspect ratio. Further improvements in the performance are achieved by using nanoparticles in the base fluid. The amounts in the phase transition time reduction are 8% and 10.5% at aspect ratio of 0.5 and 1.5 at the highest nanoparticle concentration. Originality/Value: The thermofluid system and offered control mechanism for PC dynamics control can be considered for the design, optimization, further modeling and performance improvements of applications with PCM installed systems. [ABSTRACT FROM AUTHOR]

Miscible density-driven convection in porous media has important implications for the long-term security of geological CO 2 sequestration. In this study, a REV-scale lattice Boltzmann equation method based on the generalized Navier–Stokes equations was used to simulate density-driven convection in porous media, Thus, the effects of the Rayleigh number, the Darcy number, the Schmidt number, and the porosity of porous media can be discussed separately. The results show that density-driven convection only occurs when the Rayleigh–Darcy–Schmidt number Ra D-S exceeds 1 0 2 . The larger the Ra, the more disordered the concentration field, the earlier the convective phenomenon begins, and the more significant the convective mixing; The larger the Da, the finer the generated fingers. These findings provide important insights for the development of geological sequestration technologies. • Using the generalized Navier–Stokes equations to describe the flow of fluids in porous media, allowing for independent investigations of the effects of the Rayleigh number, the Darcy number, and the porosity. • Determining the critical Rayleigh–Darcy–Smith number for the onset of density–driven convection in porous media. • Dividing the process of solute dissolution into two regions: the stable region and the convective region. [ABSTRACT FROM AUTHOR]

Purpose: The purpose of this paper is to develop a subgrid-scale (SGS) model for large eddy simulation (LES) of buoyancy- and thermally driven transitional and turbulent flows and further examine its performance. Design/methodology/approach: Favre-filtered, non-dimensional LES equations are solved using non-dissipative, fully implicit, kinetic energy conserving, finite-volume algorithm which uses an iterative predictor-corrector approach based on pressure correction. Also, to develop a new SGS model which accounts for buoyancy, turbulent generation term in SGS viscosity is properly modified and enhanced by buoyancy production. Findings: The proposed model has been successfully applied to turbulent Rayleigh–Bénard convection. The results show that the model is able to reproduce the complex physics of turbulent thermal convection. In comparison with the original wall-adapting local eddy-viscosity (WALE) and buoyancy-modified (BM) Smagorinsky models, turbulent diagnostics predicted by the new model are in better agreement with direct numerical simulation. Originality/value: A BM variant of the WALE SGS model is newly developed and analyzed. [ABSTRACT FROM AUTHOR]

In thermal multiphase flows, the modulation in the heat transfer rate due to the presence of finite-size solid particles attract growing interest in recent years. This study focuses on developing a robust and efficient simulation method for particle-laden Rayleigh-Bénard convection. The present work utilizes the double distribution function-based thermal lattice-Boltzmann method (TLBM), which enables successful simulations of multiphase fluid-thermal interactions. The no-slip boundary of the moving solid particles is handled by the interpolated bounce-back scheme. Additionally, Galilean invariant momentum exchange and heat exchange approaches are employed for hydrodynamic force and heat transfer calculation at the solid boundaries. The accuracy of the current method is verified with several benchmark cases, including three-dimensional single-phase Rayleigh-Bénard convection, as well as the settling of hot and cold spherical particles in a three-dimensional enclosure. Furthermore, we explore the modulation of Rayleigh-Bénard flows due to the presence of freely moving finite-size particles. The simulations are conducted on a distributed memory cluster with a 3D domain decomposition technique facilitated by the MPI library. A brief discussion on the parallel performance of the simulations is provided for both particle-laden and single-phase flow scenarios. Finally, the effects of finite-size solid particles on the overall heat transfer efficiency and modulation to the flow field in three-dimensional particle-laden turbulent Rayleigh-Bénard convection are discussed. The results show that addition of solid particles results in a moderate increase in the overall Nusselt number, and this enhancement is mainly due to the increased heat flux transported by the particles. • A mesoscopic method for 3D particle-laden thermal flows was developed and validated. • Modulation of 3D particle-laden Rayleigh-Bénard convection flows was studied. • Addition of particles results in a moderate increase in the overall Nusselt number. • This increase is caused by the increased heat flux carried by the moving particles. • The particles damp large-scale fluctuations, but enhance small-scale fluctuations. • The distribution functions are explicitly related to the hydrodynamic fields. [ABSTRACT FROM AUTHOR]

Purpose: Metal-cooled reactors generally use molten metals such as sodium, potassium or a combination of sodium and potassium because of their excellent heat transfer properties so that the reactor can operate at much lower pressures and higher temperatures. The purpose of this paper is to investigate the stability of natural convection in an inclined ring filled with molten potassium under the influence of a radial magnetism. Design/methodology/approach: A numerical simulation of electrically conductive fluid natural convection stability is performed on an inclined cylindrical annulus under the influence of a radial magnetism. The upper and lower walls are adiabatic, while the internal and external cylinders are kept at even temperatures. The equations governing this fluid system are solved numerically using finite volume method. The SIMPLER algorithm is used for pressure-speed coupling in the momentum equation. Findings: Numerical results for various effective parameters that solve the problem in the initial oscillatory state are discussed in terms of isobars, isotherms and flow lines in the annulus for a wide range of Hartmann numbers (0 ≤ Ha ≤ 80), inclination angles (0 ≤ γ ≤ 90°) and radii ratios λ ≤ 6. The dependency stability diagrams between complicated situations with the critical value of the Rayleigh number RaCr and the corresponding frequency FrCr are established on the basis of the numeric data of this investigation. The angle of inclination and the radii ratio of the annulus have a significant effect on the stabilization of the magneto-convective flux and show that the best stabilization of the natural oscillatory convection is obtained by the intensity of the strongest magnetic field, the high radii ratio and inclination of the annulus at γ = 30°. Practical implications: This numerical model is selected for its various applications in technology and industry. Originality/value: To the best of the authors' knowledge, the influence of the inclination of the cylindrical annulus (ring), with various radii ratio, on natural oscillatory convection under a radial magnetism has never been investigated. [ABSTRACT FROM AUTHOR]

• A new high-order kernel function with compact support of R = 3 is constructed. • Two novel high-order gradient and Laplacian operators are proposed in the context of the MPS. • A novel high-order smoothing operator is introduced for treatment of the physical discontinuities in the multi-phase/multi-fluid problems. • A new high-order pressure gradient is proposed based on the tensile instability control (TIC). • Third-order TVD Runge-Kutta scheme is adopted for time discretization. The main objective of the current work is to enhance consistency and capabilities of Moving Particle Semi-implicit (MPS) method for simulating a wide range of free-surface flows and convection heat transfer. For this purpose, two novel high-order gradient and Laplacian operators are derived from the Taylor series expansion and are applied for the discretization of governing equations. Furthermore, the combination of the explicit Third-order TVD Runge-Kutta scheme and two-step projection algorithm is employed to approximate transient terms in the Navier-stokes and energy equations. To further improve the accuracy and performance of the method, a new kernel function is constructed by a combination of the Gaussian and cosine functions and then implemented for modeling the 1D Sod shock tube problem. Validation and verification of the proposed model are conducted through the simulations of several canonical test cases such as: dam break, rotation of a square patch of fluid, two-phase Rayleigh-Taylor instability, oscillating concentric circular drop and good agreement are achieved. The proposed model is then employed to simulate three-phase Rayleigh-Taylor instability and entropy generation due to natural convection heat transfer (Differentially Heated Cavity and Rayleigh-Bénard convection). The obtained results reveal that, the newly constructed kernel function provides more reliable results in comparison with two frequently used kernel functions namely; quartic spline and Wendland. Furthermore, it is found that, the enhanced MPS model is capable of handling multiphase flow problems with low and high density contrast. [ABSTRACT FROM AUTHOR]

We present an experimental facility for the validation of numerical simulations on atmospheric dust devils in a controlled laboratory experiment. Dust devils are atmospheric air vortices with a vertical axis, and are formed by intense solar radiation and the resulting vertical temperature gradient. The structure of a typical dust devil is dominated by a radial inflow near the surface and a vertical upward flow within the vortex. These vortices have been studied in recent years using field observations, in situ measurements, and large-eddy simulation (LES). Field tests suffer from the limited area and their unpredictable behavior, while the LES approach cannot resolve the dust devils well enough. Dust devil-like structures may also occur in direct numerical simulation (DNS) with a Rayleigh number of at least Ra = 107 in Rayleigh-Bénard convection, with the advantage that the structures can be resolved more precisely. In order to validate the DNS approach and provide measurement data, the airflow is measured inside of a large-scale Rayleigh-Bénard cell of similar geometry (i.e. inside the Barrel of Ilmenau) to the DNS set-up for Rayleigh numbers from Ra = 106 to Ra = 1012. For the measurement of the flow in a large volume, an optical measurement method is used to obtain the trajectories of single particles. Since there are no commercial systems that are suitable for such a large measurement volume, we developed our own system. [ABSTRACT FROM AUTHOR]

Purpose This paper aims to conduct numerical simulations to investigate steady-state laminar Rayleigh–Bénard convection of yield stress fluids obeying Bingham model in rectangular cross-sectional cylindrical annular enclosures. In this investigation, axisymmetric simulations have been carried out for nominal Rayleigh number range Ra = 103 to 105, aspect ratio range AR = 0.25 to 4 (i.e. AR = H/L where H is the enclosure height and L is the difference between outer and inner radii) and normalised inner radius range ri/L = 0 to 16 (where ri is internal cylinder radius) for a nominal representative Prandtl number Pr = 500. Both constant wall temperature (CWT) and constant wall heat flux (CWHF) boundary conditions have been considered for differentially heated horizontal walls to analyse the effects of wall boundary condition.Design/methodology/approach The bi-viscosity Bingham model is used to mimic Bingham fluids for Rayleigh–Bénard convection of Bingham fluids in vertical cylindrical annuli. The conservation equations of mass, momentum and energy have been solved in a coupled manner using the finite volume method where a second-order central differencing scheme is used for the diffusive terms and a second-order up-wind scheme is used for the convective terms. The well-known semi-implicit method for pressure-linked equations algorithm is used for the coupling of the pressure and velocity.Findings It is found that the convective transport strengthens (weakens) with an increase in Ra (AR) for both Newtonian (i.e. Bn = 0) and Bingham fluids, regardless of the boundary conditions. Moreover, the strength of convection is stronger in the CWT configuration than that is for CWHF boundary condition due to higher temperature difference between horizontal walls for both Newtonian (i.e. Bn = 0) and Bingham fluids. The mean Nusselt number Nūcy does not show a monotonic increase with increasing Ra for AR = 1 and ri/L = 4 because of the change in flow pattern (i.e. number of convection rolls/cells) in the CWT boundary condition, whereas a monotonic increase of Nūcy with increasing Ra is obtained for the CWHF configuration. In addition, Nūcy increases with increasing ri/L and asymptotically approaches the corresponding value obtained for rectangular enclosures (ri/L → ∞) for both CWT and CWHF boundary conditions for large values of ri/L. It is also found that both the flow pattern and the mean Nusselt number Nūcy are dependent on the initial conditions for Bingham fluid cases, as hysteresis is evident for AR = 1 for both CWT and CWHF boundary conditions.Originality value Finally, the numerical findings have been used to propose a correlation for Nūcy in the range of 0.25 ≤ ri/L ≤ 16, 0.25 ≤ AR ≤ 2 and 5 × 104 ≤ Ra ≤ 105 for the CWHF configuration. [ABSTRACT FROM AUTHOR]

In this work, we consider the multiplicity two dynamic transitions of a broad class of problems. The first main assumption is the existence of two critical eigenmodes of the linear operator which depend on at least two wave indices one of which are consecutive m , m + 1 and the other identical n. The second main assumption is an orthogonality condition on the nonlinear interactions of the basis vectors of the phase space which is typical in many applications. Under this assumption we obtain a reduced system of ODE's which describe the first dynamic transitions. We make a careful analysis of this reduced system to classify all possible transition behavior. We then apply our main theoretical findings to the 2D Rayleigh–Bénard convection with free-slip boundary conditions and show that this problem displays an S 1 attractor bifurcation. • Comprehensive analysis of "multiplicity two dynamic transitions" in a wide range of problems. • Key assumption 1: Existence of two critical eigenmodes with wave indices of the form (m,n) and (m＋1,n). • Key assumption 2: An orthogonality condition on the nonlinear interactions of the basis vectors. • Obtained a universal reduced ODE system describing the dynamics. • Application to the 2D Rayleigh–Bénard convection with free-slip boundary conditions. [ABSTRACT FROM AUTHOR]

The linear and nonlinear stability of axisymmetric convection of a viscous fluid in a cylindrical enclosure heated from below is investigated for various radius to height ratios. A weak vertical throughflow is imposed in a gravity-aligned or a gravity-opposing manner. Symmetric and asymmetric boundaries of free-free, rigid-rigid and rigid-free types are considered for lower and upper boundaries with isothermal temperature boundary condition. The side-walls are assumed to be rigid and adiabatic. A convergent Maclaurin series representation is considered for the finding of axial trial eigenfunctions. In order to corroborate the results of the present study with those of a previous investigation, the critical Rayleigh number and the number of radial rolls manifesting for any given aspect ratio are determined in the case of no throughflow and an exact match is found. Further, the influence of boundaries and the effect of throughflow on chaotic and periodic regimes of motion are studied with the help of a time series solution and the largest Lyapunov exponent as indicators of chaos. The novelty of the present study is the use of a Maclaurin series representation for the eigenfunctions of the linear problem and using the same in determining the solution with the convective mode. • Axisymmetric convection of viscous fluids is considered. • A weak vertical throughflow (gravity-aligned or gravity-opposing) is assumed. • Investigation is made for symmetric and asymmetric boundary conditions. • Results of asymmetric boundaries do not conform with those of symmetric ones. • Features of regular convection are not carried into the chaotic regime. [ABSTRACT FROM AUTHOR]

Entropically Damped Artificial Compressibility (EDAC) equation-based solution method is investigated to simulate the incompressible Navier–Stokes equations for buoyancy-driven flows. The method introduces an evolution equation for pressure, which is used to close the system of equations. The resulting parabolic system removes the need to solve the traditional Poisson's equation at each time step. The energy equation with the Boussinesq approximation and the EDAC system of equations with the low Mach number approximation is solved for the thermal convection problem. This system is discretized using a sixth-order compact difference in space and advanced in time using an explicit fourth-order Runge–Kutta scheme. To investigate the suitability of the EDAC model for buoyancy flows, two widely used benchmark problems, namely: thermal cavity and Rayleigh–Bénard problems, are simulated. The simulation results are compared against the literature data. An excellent agreement is obtained, showing the feasibility and accuracy of the EDAC method in simulating buoyancy-driven flows. The EDAC pressure equation derived from entropy balance, together with the energy equation, are shown in this paper to model thermally dominant flows accurately. • Entropically Damped Artificial Compressibility (EDAC) method for simulating stratified flows. • Unsteady turbulent buoyancy-driven flow using EDAC. • Usage of high-order compact finite difference scheme. [ABSTRACT FROM AUTHOR]

In this article, we study the onset of convection in porous media with through flow based on Darcy formulation. Unlike the impermeable porous medium problems between horizontal two parallel plates frequently accessed in the literature, the thermal instability of convective cells under the influence of a uniform vertical flow is of the prime concern here in an aim to control the thermal instability onset. Three devised thermal boundary constraints representing external heat transfer with isothermal or isoflux lines across the walls are investigated fully taking into account the linear stability equations with small flow/heat wave disturbances with respect to the normal modes. In each case, the neutral stability response of the saturated porous layer separating the stable and unstable perturbation regimes are identified affected by a Peclet number manifesting itself as the through flow strength. The pre-analysis of the analytical solutions of thermal fields proves that the thermal layer thickness is lowered by the opposing through flow, otherwise an assisting through flow enhances the thickness. The critical Darcy–Rayleigh numbers and Benard cell wavenumbers of lateral fluctuating perturbations initiating the convective Darcy–Benard thermal rolls within the porous media for a prescribed Peclet number are ultimately worked out numerically, except an analytical expression is elaborated where the heating is supplied from the walls via varying heat fluxes. The well-documented thresholds of Rayleigh numbers 39.47, 27.09 and 12 with vanishing Peclet number (Barletta, 2019) corresponding to the three studied thermal cases are reproduced successfully. Further results clarified that both opposing and assisting through flows are sources of stabilization when the walls are thermally isothermal. On the other hand, the other thermally conducting porous walls lead to a more unstable convection with the assisting through flow having a thermal layer growth, otherwise stable convection occurs with the opposing through flow. • Control of Darcy-Benard convection by assisting/opposing through flow. • Opposing through flow reduces thermal layer thickness. • Assisting/opposing through flow stabilizes with thermally isothermal walls. • Heat fluxes give rise to unstable or stable convection. • Buoyancy-induced cells appear at lower Rayleigh numbers with assisting flow. [ABSTRACT FROM AUTHOR]

Highlights • We provide a complete analysis of the first and second transition of a low-dimensional dynamical system for the Rayleigh–Bénard convection subject to a vertically applied magnetic field as the Rayleigh number varies. • Our analysis follows the dynamical phase transition theory for dissipative dynamical systems based on the principle of exchange of stability and the center manifold reduction. The framework is applicable to general dissipative dynamical systems. • We find that, as the Rayleigh number increases, the system undergoes two transitions, the first of which is a well-known pitchfork bifurcation, whereas the second one is structurally more complex and can be of different type depending on the system parameters. B • We present numerical results to corroborate the analytic predictions, which provide a more precise description of the transition structure. Abstract In this article, we study the dynamic transitions of a low-dimensional dynamical system for the Rayleigh–Bénard convection subject to a vertically applied magnetic field. Our analysis follows the dynamical phase transition theory for dissipative dynamical systems based on the principle of exchange of stability and the center manifold reduction. We find that, as the Rayleigh number increases, the system undergoes two successive transitions: the first one is a well-known pitchfork bifurcation, whereas the second one is structurally more complex and can be of different type depending on the system parameters. More precisely, for large magnetic field, the second transition is of continuous type and gives to a stable limit cycle; on the other hand, for low magnetic field or small height-to-width aspect ratio, a jump transition occurs where an unstable periodic orbit eventually collides with the stable steady state, leading to the loss of stability at the critical Rayleigh number. Finally, numerical results are presented to corroborate the analytic predictions. [ABSTRACT FROM AUTHOR]

The AFiD code, an open source solver for the incompressible Navier–Stokes equations ( http://www.afid.eu ), has been ported to GPU clusters to tackle large-scale wall-bounded turbulent flow simulations. The GPU porting has been carried out in CUDA Fortran with the extensive use of kernel loop directives (CUF kernels) in order to have a source code as close as possible to the original CPU version; just a few routines have been manually rewritten. A new transpose scheme has been devised to improve the scaling of the Poisson solver, which is the main bottleneck of incompressible solvers. For large meshes the GPU version of the code shows good strong scaling characteristics, and the wall-clock time per step for the GPU version is an order of magnitude smaller than for the CPU version of the code. Due to the increased performance and efficient use of memory, the GPU version of AFiD can perform simulations in parameter ranges that are unprecedented in thermally-driven wall-bounded turbulence. To verify the accuracy of the code, turbulent Rayleigh–Bénard convection and plane Couette flow are simulated and the results are in excellent agreement with the experimental and computational data that have been published in literature. Program summary Program Title: AFiD-GPU Program Files doi: http://dx.doi.org/10.17632/rwjdg7ry66.1 Licensing provisions: MIT Programming language: Fortran 90, CUDA Fortran, MPI External routines: PGI, CUDA Toolkit, FFTW3, HDF5 Nature of problem: Solving the three-dimensional Navier–Stokes equations coupled with a scalar field in a cubic box bounded between two walls and with periodic boundary conditions in the horizontal directions. Solution method: Second order finite difference method for spatial discretization, third order Runge–Kutta scheme in combination with Crank–Nicolson for the implicit terms for time advancement, two dimensional pencil distributed MPI parallelization, GPU accelerated routines. Additional comments including restrictions and unusual features: The code is available and supported on https://github.com/PhysicsofFluids/AFiD_GPU_opensource . [ABSTRACT FROM AUTHOR]

Building physically sound low-order models of atmospheric dynamics is discussed. As an example, the issue of convective rolls vs cells in the atmospheric boundary layer resulting from the interplay of buoyancy and shear was explored using a nonlinear low-order model of 3D Rayleigh–Bénard convection with shear, with results comparable with observations. For moderate values of buoyancy, the shear alone proved capable of controlling the occurrence of rolls or cells, though the whole picture depended on both buoyancy and shear. [ABSTRACT FROM AUTHOR]

Computational codes for direct numerical simulations of Rayleigh–Bénard (RB) convection are compared in terms of computational cost and quality of the solution. As a benchmark case, RB convection at Ra = 10 8 and Pr = 1 in a periodic domain, in cubic and cylindrical containers is considered. A dedicated second-order finite-difference code ( AFID / RBflow ) and a specialized fourth-order finite-volume code ( Goldfish ) are compared with a general purpose finite-volume approach ( OpenFOAM ) and a general purpose spectral-element code ( Nek5000 ). Reassuringly, all codes provide predictions of the average heat transfer that converge to the same values. The computational costs, however, are found to differ considerably. The specialized codes AFID / RBflow and Goldfish are found to excel in efficiency, outperforming the general purpose flow solvers Nek5000 and OpenFOAM by an order of magnitude with an error on the Nusselt number Nu below 5%. However, we find that Nu alone is not sufficient to assess the quality of the numerical results: in fact, instantaneous snapshots of the temperature field from a near wall region obtained for deliberately under-resolved simulations using Nek5000 clearly indicate inadequate flow resolution even when Nu is converged. Overall, dedicated special purpose codes for RB convection are found to be more efficient than general purpose codes. [ABSTRACT FROM AUTHOR]

PaScaL_TCS is an efficient and scalable solver for large-scale direct numerical simulations of natural convective flow that considers temperature-dependent fluid properties. For increased scalability on a massive-scale distributed memory system, the solver decomposes the computational domain into a cubic sub-domain. The numerical procedure is based on the monolithic projection method with staggered time discretization [1] , which is an implicit and non-iterative solver for wall-bounded domains. The PaScaL_TDMA library is used to solve batched tridiagonal systems, which are partitioned according to domain decomposition. For parallel fast Fourier transform in a Fourier Poisson solver for pressure, two transpose schemes with different communicator sizes are proposed and compared according to the number of cores. An explicit intermediate aggregation scheme for MPI-IO is suggested to reduce the number of processes that simultaneously take part in parallel IO. The overall implementation was evaluated using the Nurion cluster system of the Korea Institute of Science and Technology Information with detailed performance profiling. With the efficient communication, the results showed good strong and weak scalability up to 131,072 cores. The file IO performance improved with the suggested MPI-IO scheme using explicit aggregation, especially when many processes are involved in parallel IO. The solver was verified by conducting large-scale differentially heated vertical convection flow simulations under the Oberbeck–Boussinesq (OB) approximation. The capability of the solver to capture and quantitatively describe the non-OB effect was demonstrated via glycerol Rayleigh–Bénard convection flow simulations. Program Title: PaScaL_TCS CPC Library link to program files: https://doi.org/10.17632/3b2bysrcxm.1 Developer's repository link: https://github.com/MPMC-Lab Licensing provisions: MIT Programming language: Fortran90 Nature of problem: Solving the incompressible Navier-Stokes equations for natural convective flow considering the non-Oberbeck–Boussinesq effect, which considers the temperature dependence of fluids properties, in three-dimensional channel domain. Solution method: PaScaL_TCS uses the monolithic projection-based method with staggered time discretization (MPM-STD) [1]. It employs the Crank-Nicolson scheme along with staggered time which evaluates scalar and vector variables at the (n + 1 / 2) and (n + 1) time level, respectively. PaScaL_TDMA [2] is used to solve multiple tridiagonal matrix systems for decoupled energy and momentum equations in a distributed memory system. The Poisson equation solver for the pressure-correction scheme is based on Fourier diagonalization. MPI_Alltoallw is used for parallel FFT. PaScaL_TCS supports a single-file-based parallel IO for efficient data generation in large-scale computing. • PaScaL_TCS is a scalable solver for large-scale direct numerical simulations of turbulent natural convective flows. • PaScaL_TDMA is used for the implicit treatment of convection and diffusion terms in distributed memory systems. • The transpose schemes for the FFT-based pressure solver are used considering the number of cores. • An explicit intermediate aggregation scheme for MPI-IO is implemented to mitigate the IO burden with many cores. • The numerical performance of PaScaL_TCS is investigated for DHVC and RBC flows. [ABSTRACT FROM AUTHOR]

We study the thermodynamic efficiency of the Malkus–Lorenz waterwheel. For this purpose, we derive an exact analytical formula that describes the efficiency of this dissipative structure as a function of the phase space variables and the constant parameters of the dynamical system. We show that, generally, as the machine is progressively driven far from thermodynamic equilibrium by increasing its uptake of matter from the environment, it also tends to increase its efficiency. However, sudden drops in the efficiency are found at critical bifurcation points leading to chaotic dynamics. We relate these discontinuous crises in the efficiency to a reduction of the attractor's average value projected along the phase space dimensions that contribute to the rate of entropy generation in the system. In this manner, we provide a thermodynamic criterion that, presumably, governs the evolution of far-from-equilibrium dissipative systems towards their self-assembly and synchronization into increasingly complex networks and structures. • An exact analytical equation for the thermodynamic efficiency of the Lorenz system is computed, using the Malkus–Lorenz waterwheel. • As the system is posed very far from thermodynamic equilibrium, it transits to chaos through a subcritical Hopf bifurcation. • A discontinuous jump in the thermodynamic efficiency is observed at such critical point. • The entropy generation of the dissipative structure is related to its thermodynamic efficiency. • Numerical simulations are compared to results from Rayleigh–Bénard convection experiments. [ABSTRACT FROM AUTHOR]

Purpose This study aims to carry out the analysis of Rayleigh-Bénard convection within enclosures with curved isothermal walls, with the special implication on the heat flow visualization via the heatline approach.Design/methodology/approach The Galerkin finite element method has been used to obtain the numerical solutions in terms of the streamlines (ψ ), heatlines (Π), isotherms (θ), local and average Nusselt number ( Nut¯) for various Rayleigh numbers (103 ≤ Ra ≥ 105), Prandtl numbers (Pr = 0.015 and 7.2) and wall curvatures (concavity/convexity).Findings The presence of the larger fluid velocity within the curved cavities resulted in the larger heat transfer rates and thermal mixing compared to the square cavity. Case 3 (high concavity) exhibits the largest Nut¯ at the low Ra for all Pr. At the high Ra, Nut¯ is the largest for Case 3 (high concavity) at Pr = 0.015, whereas at Pr = 7.2, Nut¯ is the largest for Case 1 (high concavity and convexity).Practical implications The results may be useful for the material processing applications.Originality/value The study of Rayleigh-Bénard convection in cavities with the curved isothermal walls is not carried out till date. The heatline approach is used for the heat flow visualization during Rayleigh-Benard convection within the curved walled enclosures for the first time. Also, the existence of the enhanced fluid and heat circulation cells within the curved walled cavities during Rayleigh-Benard heating is illustrated for the first time. [ABSTRACT FROM AUTHOR]

The original Oberbeck–Boussinesq (OB) equations which are the coupled two dimensional Navier–Stokes(NS) and heat conduction equations have been investigated by E.N. Lorenz half a century ago with Fourier series and opened the way to the paradigm of chaos. In our former study—Chaos, Solitons and Fractals 78, 249 (2015)—we presented fully analytic solutions for the velocity, pressure and temperature fields with the aim of the self-similar Ansatz and gave a possible explanation of the Rayleigh–Bènard convection cells. Now we generalize the Oberbeck–Boussinesq hydrodynamical system, going beyond the first order Boussinesq approximation and consider a non-linear temperature coupling. We investigate more general, power law dependent fluid viscosity or heat conduction material equations as well. Our analytic results obtained via the self-similar Ansatz may attract the interest of various fields like meteorology, oceanography or climate studies. [ABSTRACT FROM AUTHOR]

This paper describes an improved decoupling algorithm for calculating low Mach number near-critical fluid flows, in which the c p -formulation of the energy equation is used ( c p refers to the specific heat at constant pressure) and the pressure–velocity coupling is treated through the block-coupled method, together with a transformation of the continuity equation. The algorithm is validated by a 2D near-critical CO 2 Rayleigh–Bénard convection problem. After the mesh and time step independence studies, the correctness is presented through the stability behaviors of the hot boundary layer under different heating intensities and the typical evolution of the temperature field. The improved decoupling algorithm owns a better stability and can further reduce the CPU time. The advantages are even more obvious when the flow is convection-dominated. [ABSTRACT FROM AUTHOR]

We present the Fluid Transport Accelerated Solver, FluTAS, a scalable GPU code for multiphase flows with thermal effects. The code solves the incompressible Navier-Stokes equation for two-fluid systems, with a direct FFT-based Poisson solver for the pressure equation. The interface between the two fluids is represented with the Volume of Fluid (VoF) method, which is mass conserving and well suited for complex flows thanks to its capacity of handling topological changes. The energy equation is explicitly solved and coupled with the momentum equation through the Boussinesq approximation. The code is conceived in a modular fashion so that different numerical methods can be used independently, the existing routines can be modified, and new ones can be included in a straightforward and sustainable manner. FluTAS is written in modern Fortran and parallelized using hybrid MPI/OpenMP in the CPU-only version and accelerated with OpenACC directives in the GPU implementation. We present different benchmarks to validate the code, and two large-scale simulations of fundamental interest in turbulent multiphase flows: isothermal emulsions in HIT and two-layer Rayleigh-Bénard convection. FluTAS is distributed through a MIT license and arises from a collaborative effort of several scientists, aiming to become a flexible tool to study complex multiphase flows. Program Title: : Fluid Transport Accelerated Solver, FluTAS. CPC Library link to program files: https://doi.org/10.17632/tp6k8wky8m.1 Developer's repository link: https://github.com/Multiphysics-Flow-Solvers/FluTAS.git. Licensing provisions: MIT License. Programming language: Fortran 90, parallelized using MPI and slab/pencil decomposition, GPU accelerated using OpenACC directives. External libraries/routines: FFTW, cuFFT. Nature of problem: FluTAS is a GPU-accelerated numerical code tailored to perform interface resolved simulations of incompressible multiphase flows, optionally with heat transfer. The code combines a standard pressure correction algorithm with an algebraic volume of fluid method, MTHINC [1]. Solution method: the code employs a second-order-finite difference discretization and solves the two-fluid Navier-Stokes equation using a projection method. It can be run both on CPU-architectures and GPU-architectures. [ABSTRACT FROM AUTHOR]

We investigate instabilities and chaos in Rayleigh-Bénard convection (RBC) of low Prandtl-number ( Pr ) fluids with stress-free boundary conditions. Three dimensional direct numerical simulations (DNS) of the governing equations of RBC are done for investigation. DNS shows the existence of multiple solutions near the onset of convection. To understand the origin of these solutions and their bifurcations, we construct a low dimensional model from the DNS data. The low dimensional model unfolds a rich bifurcation structure in RBC of low Prandtl-number fluids. The bifurcation structure involves various bifurcations including pitchfork, Neimark-Sacker, Homoclinic gluing, Hopf bifrucations, and attractor merging crisis near the onset of convection. The results of the model are consistent with that of the DNS. [ABSTRACT FROM AUTHOR]

A flexible Galerkin method based on proper orthogonal decomposition (POD) is described to construct the bifurcation diagram, as the Rayleigh number R is varied, in the Rayleigh–Bénard convection in a rectangular box for large Prandtl number. The bifurcation diagram is approximated using the POD modes resulting from unconverged snapshots for just one specific value of R , calculated in either Newton iterations or time-dependent runs converging to steady states. Moreover, the selection of the specific value of R is quite flexible. In addition, a horizontal reflection symmetry is taken into account to construct a symmetry-preserving Galerkin system. The resulting un-symmetric and symmetric low-dimensional systems are combined with a basic continuation method, which provide the bifurcation diagram at a quite low computational cost. [ABSTRACT FROM AUTHOR]

A simple lattice Boltzmann thermal model is proposed for simulating the turbulence Rayleigh–Bénard convection. This new model introduces a mesoscopic external force term in the lattice Boltzmann equation to consider the effect of the buoyant body force and introduces a temperature density distribution function to simulate the temperature field. The macroscopic density and velocity fields are simulated using the density distribution function. The results show that this simple model is able to simulate turbulence Rayleigh–Bénard convection in the current simulation range of 10 6 ⩽ Ra ⩽ 10 12 . It is shown that the energy spectra of the velocity and temperature are in good agreement with the Bolgiano–Obukhov-like ( BO59 ) scaling. [ABSTRACT FROM AUTHOR]

In this article we will present pure two-dimensional analytic solutions for the coupled non-compressible Newtonian–Navier–Stokes — with Boussinesq approximation — and the heat conduction equation. The system was investigated from E.N. Lorenz half a century ago with Fourier series and pioneered the way to the paradigm of chaos. We present a novel analysis of the same system where the key idea is the two-dimensional generalization of the well-known self-similar Ansatz of Barenblatt which will be interpreted in a geometrical way. The results, the pressure, temperature and velocity fields are all analytic and can be expressed with the help of the error functions. The temperature field shows a strongly damped single periodic oscillation which can mimic the appearance of Rayleigh–Bénard convection cells. Finally, it is discussed how our result may be related to nonlinear or chaotic dynamical regimes. [ABSTRACT FROM AUTHOR]

Purpose – The purpose of this paper is to focus on the advanced solution of the parametric non-linear model related to the Rayleigh-Benard laminar flow involved in the modeling of natural thermal convection. This flow is fully determined by the dimensionless Prandtl and Rayleigh numbers. Thus, if one could precompute (off-line) the model solution for any possible choice of these two parameters the analysis of many possible scenarios could be performed on-line and in real time. Design/methodology/approach – In this paper both parameters are introduced as model extra-coordinates, and then the resulting multidimensional problem solved thanks to the space-parameters separated representation involved in the proper generalized decomposition (PGD) that allows circumventing the curse of dimensionality. Thus the parametric solution will be available fast and easily. Findings – Such parametric solution could be viewed as a sort of abacus, but despite its inherent interest such calculation is at present unaffordable for nowadays computing availabilities because one must solve too many problems and of course store all the solutions related to each choice of both parameters. Originality/value – Parametric solution of coupled models by using the PGD. Model reduction of complex coupled flow models. Analysis of Rayleigh-Bernard flows involving nanofluids. [ABSTRACT FROM AUTHOR]

We present a numerical scheme geared for high performance computation of wall-bounded turbulent flows. The number of all-to-all communications is decreased to only six instances by using a two-dimensional (pencil) domain decomposition and utilizing the favourable scaling of the CFL time-step constraint as compared to the diffusive time-step constraint. As the CFL condition is more restrictive at high driving, implicit time integration of the viscous terms in the wall-parallel directions is no longer required. This avoids the communication of non-local information to a process for the computation of implicit derivatives in these directions. We explain in detail the numerical scheme used for the integration of the equations, and the underlying parallelization. The code is shown to have very good strong and weak scaling to at least 64 K cores. [ABSTRACT FROM AUTHOR]

Purpose – Natural convection with heat transfer in porous media has been subject of extensive study in engineering due to its numerous applications. A case of particular interest is the Bénard-type flow.The paper aims to discuss this issue. Design/methodology/approach – Based on the network simulation method in order to solve this problem, a numerical model is proposed. Findings – Nusselt-Rayleigh correlation is determined for a broad range of Rayleigh, the dimensionless number that influences the solution, above and below the threshold which separates the conduction and convection pure mechanisms. Originality/value – Based on the network simulation method. [ABSTRACT FROM AUTHOR]

Purpose – The post-critical convective state for Rayleigh-Benard (RB) convection is studied using a nonlinear spectral-amplitude-perturbation approach in a fluid layer heated from below. The paper aims to discuss these issues. Design/methodology/approach – In the spectral method the flow and temperature fields are expanded periodically along the layer and orthonormal shape functions are used in the transverse direction. A combined amplitude-perturbation approach is developed to solve the nonlinear spectral system in the post-critical range, even far from the linear stability threshold. Also, to leading order, the Lorenz model is recovered. Findings – It is found that very small Prandtl numbers (Pr<0.1) can change the Nusselt number, when terms to O(ε5/2) and higher are considered. However, to lower orders the Prandtl number does not affect the results. Variation of the Nusselt number to different orders is found to be highly consistent. Comparison with experimental results is made and a very good qualitative agreement is observed, even far from the linear threshold. Originality/value – Unlike existing nonlinear formulations for RB thermal convection, the present combined spectral-perturbation approach provides a systematic method for mode selection. The number and type of modes to be included are directly related to the post-critical Rayleigh number. The method is not limited to the weakly nonlinear range. [ABSTRACT FROM AUTHOR]

We have recently developed a chaotic gas turbine whose rotational motion might simulate turbulent Rayleigh-Bénard convection. The nondimensionalized equations of motion of our turbine are expressed as a star network of N Lorenz subsystems, referred to as augmented Lorenz equations. Here, we propose an application of the augmented Lorenz equations to chaotic cryptography, as a type of symmetric secret-key cryptographic method, wherein message encryption is performed by superimposing the chaotic signal generated from the equations on a plaintext in much the same way as in one-time pad cryptography. The ciphertext is decrypted by unmasking the chaotic signal precisely reproduced with a secret key consisting of 2^N-1 (e.g., N=101) real numbers that specify the augmented Lorenz equations. The transmitter and receiver are assumed to be connected via both a quantum communication channel on which the secret key is distributed using a quantum key distribution protocol and a classical data communication channel on which the ciphertext is transmitted. We discuss the security and feasibility of our cryptographic method. [ABSTRACT FROM AUTHOR]

Purpose – The purpose of this paper is to optimize the heat transfer rate in square cavity by attaching fin at the bottom wall. Design/methodology/approach – The problem is formulated and solved using finite element method. Accuracy of the method is validated by comparisons with previously published work. Findings – It was found that attaching fin reduces heat transfer rate in the cavity. Originality/value – Although the problem is not very original it is important in that many applications have heating on adjacent walls. [ABSTRACT FROM AUTHOR]

We investigate the effect of radiatively heated solid particles settling in Rayleigh–Bénard turbulent flow. Three-dimensional fluid flow computations were performed using direct numerical simulation, while the evolution of particle temperature, velocities, and positions are obtained by Lagrangian particle tracking. We consider particles whose diameters vary form 20 µm to 80 µm subject to thermal radiation and settle in Rayleigh–Bénard cell at Rayleigh number R a = 2 × 1 0 6 . The results show that the sedimentation behaviors of the particles with different particle sizes are significantly different, and the trajectories of small particles are very chaotic compared to those of large particles. Consequently, small particles can absorb solar energy efficiently and result in gas temperature increase. As the particle size decreases, the thickness of the upper boundary layer becomes thinner, while the thickness of the lower boundary layer becomes thicker. The Nusselt number varies monotonically with a logarithmic function with the diameter of the particle on both the upper and lower walls. • A high-precision parallel multiphase solver is developed. • DNS of radiatively heated particles-laden Rayleigh–Bénard convection is performed. • Particle trajectories and distributions are depicted. • The effects of particles on flow modulation and heat transfer are investigated. [ABSTRACT FROM AUTHOR]

Abstract: The Swift–Hohenberg equation is a central nonlinear model in modern physics. Originally derived to describe the onset and evolution of roll patterns in Rayleigh–Bénard convection, it has also been applied to study a variety of complex fluids and biological materials, including neural tissues. The Swift–Hohenberg equation may be derived from a Lyapunov functional using a variational argument. Here, we introduce a new fully-discrete algorithm for the Swift–Hohenberg equation which inherits the nonlinear stability property of the continuum equation irrespectively of the time step. We present several numerical examples that support our theoretical results and illustrate the efficiency, accuracy and stability of our new algorithm. We also compare our method to other existing schemes, showing that is feasible alternative to the available methods. [Copyright &y& Elsevier]

Abstract: A numerical study of thermorheological effect on Rayleigh–Bénard and Marangoni magnetoconvection in a Newtonian liquid is studied numerically for all possible boundary combinations such as rigid–rigid/free–free/rigid–free and isothermal/adiabatic to know the influence of temperature-dependent viscosity and an externally applied magnetic field on the onset of convection. The higher order Rayleigh–Ritz (HORR) method is used to obtain the eigenvalue of the problem. The Rayleigh–Bénard magnetoconvection is studied in great detail for all the ten possible boundary combinations and the Marangoni magnetoconvection for all the four possible boundary combinations. A general inference is made from the results to show the effects of magnetic field and the variable viscosity on the stability of the system. Also the results obtained for Rayleigh–Bénard magnetoconvection agree quite well with those of for the limiting case. In the case of Marangoni convection, it agrees quite well with those of . The results indicate that the critical Rayleigh/Marangoni number increases with increasing Chandrasekhar number, but decreases with increasing variable viscosity parameter. The results have possible astrophysical and space applications. [Copyright &y& Elsevier]

Abstract: The lattice Boltzmann method (LBM) is the simple numerical simulator for fluids because it consists of linear equations. Excluding the higher differential term, the LBM for a temperature field is also achieved as an easy numerical simulation method. However, the LBM is hardly applied to body fitted coordinates for its formulation. It is then difficult to calculate complex lattices using the LBM. In this paper, the finite element discrete Boltzmann equation (FEDBE) is introduced to deal with this weakness of the LBM. The finite element method is applied to the discrete Boltzmann equation (DBE) of the basic equation of the LBM. For FEDBE, the simulation using complex lattices is achieved, and it will be applicable for the development in engineering fields. The natural convection in a square cavity and the Rayleigh–Bernard convection are chosen as the test problem. Each simulation model is accurate enough for the flow patterns, the temperature distribution and the Nusselt number. This method is now considered good for the flow and temperature field, and is expected to be introduced for complex lattices using the DBE. [ABSTRACT FROM AUTHOR]

Abstract: The use of Karhunen–Loève decomposition (KLD) to explore the complex fluid flows that are common in engineering applications is increasing and has yielded new physical insights. However, for most engineering systems the dimension of the dynamics is expected to be very large yet the flow field data is available only for a finite time. In this context, it is important to establish the amount of data required to compute the asymptotic value of the Karhunen–Loève dimension given a finite amount of data. Using direct numerical simulations of Rayleigh–Bénard convection in a finite cylindrical geometry we compute the asymptotic value of the Karhunen–Loève dimension. The amount of time required for the Karhunen–Loève dimension to reach a steady value is very slow in comparison with the time scale of the convection rolls. We show that the asymptotic value of the Karhunen–Loève dimension can be determined using much less data if one uses the azimuthal symmetry of the governing equations prior to performing a KLD. The Karhunen–Loève dimension is found to be extensive as the system size is increased and for a dimension measurement that captures 90% of the variance in the data the Karhunen–Loève dimension is approximately 20 times larger than the Lyapunov dimension. [ABSTRACT FROM AUTHOR]

Abstract: This paper proposes a new approach to analysis of incompressible 3D fluid motion in Rayleigh–Benard convection in transition from laminar to turbulent regimes. Number of test series were conducted. The analysis indicated that in different test series laminar-turbulent transition follows either the subharmonic bifurcation cascade of two-dimensional tori or the subharmonic bifurcation cascade of limit cycles. Cycles up to the third period were found in the system that indicated the end of the Sharkovskii sequence. All bifurcation cascades agree with the Feigenbaum–Sharkovskii–Magnitskii (FSM) scenario. [Copyright &y& Elsevier]

Abstract: We present a class of a high-resolution Godunov-type algorithms for solving flow problems governed by the incompressible Navier–Stokes equations. The algorithms use high-resolution finite volume methods developed in LeVeque (SIAM J Numer Anal 1996;33:627–665) for the advective terms and finite difference methods for the diffusion and the Poisson pressure equation. The high-resolution algorithm advects the cell-centered velocities using the divergence-free cell-edge velocities. The resulting cell-centered velocity is then updated by the solution of the Poisson equation. The algorithms are proven to be robust for constant-density flows at high Reynolds numbers via an example of lid-driven cavity flow. With a slight modification for the projection operator in the constant-density solvers, the algorithms also solve incompressible flows with finite-amplitude density variation. The strength of such algorithms is illustrated through problems like Rayleigh–Taylor instability and the Boussinesq equations for Rayleigh–Bénard convection. Numerical studies of the convergence and order of accuracy for the velocity field are provided. While simulations for two-dimensional regular-geometry problems are presented in this study, in principle, extension of the algorithms to three dimensions with complex geometry is feasible. [Copyright &y& Elsevier]

Abstract: In our paper we study the localization problem of compact invariant sets of nonlinear systems. Methods of a solution of this problem are discussed and a new method is proposed which is based on using symmetrical prolongations and the first-order extremum condition. Our approach is applied to the system modeling the Rayleigh–Bénard convection for which the symmetrical prolongation with the Lorenz system has been constructed. [Copyright &y& Elsevier]

Abstract: Fluid particle trajectories for the Rayleigh–Bénard problem in a cube with perfectly conducting lateral walls have been investigated. The velocity and temperature fields of the stationary flow solutions have been obtained by means of a parameter continuation procedure based on a Galerkin spectral method. The rich dynamics of the resulting fluid particle paths has been studied for three branches of stationary solutions and different values of the Rayleigh number within the range at a Prandtl number equal to 130. The stability properties and bifurcations of fixed points, which play a key role in the global dynamics, have been analyzed. Main periodic orbits and their stability character have also been determined. Poincaré maps reveal that regions of chaotic motion and regions of regular motion coexist inside the cavity. The boundaries of these three-dimensional regions have been determined. The metric entropy gives an indication of the mixing properties of the large chaotic zone. [Copyright &y& Elsevier]

Generally considered an annoyance on airplane flights, convective turbulence is a ubiquitous and important phenomenon, controlling weather patterns, transport in stellar interiors, and Earth's magnetic field. Moreover, from the perspective of basic physics, Rayleigh-Bénard convection is the paradigm for emergent behavior--the presence of a rich phenomenology at long length scales which bears little, if any, semblance to the microscopic or atomistic description of the same system. In this review, the authors guide the reader through one aspect of convection, namely, heat transfer and flow at large length scales. [ABSTRACT FROM AUTHOR]

Abstract: We present results of a numerical simulation of the thermal convection in the subsurface mushy ice layer of Europa, one of the Jupiter’s moons. Beside fluid dynamics and heat transfer within such a layer, heat conduction in the solid crustal surface and heat exchange between the two phases – mushy ice and solid crust – are included in our model in order to follow also the evolution of the phase front. Since the images of Europa’s crust taken by the spacecrafts Voyager and Galileo got to be known, planetary scientists stimulated this kind of investigations with the aim of studying the origin of such a topographic aspect. Actually the chaotic lineaments and splotches, clearly visible, solicited the conjecture of the existence of an internal ocean of water that is also supported by the most recent Galileo magnetic field data. The presence of water would make life possible on the jovian satellite. However, in the recent literature, just few numerical simulations describing the overall scenario and including either heat transfer and convection flow have been proposed. Here we adopt the stream-function/vorticity formulation of the Navier–Stokes equations for the flow of the mushy ice and a Stefan condition combined with a front-fixing technique for the front evolution. Our numerical discretization is based upon an ENO scheme. Mathematical model and numerical procedure have been thoroughly tested and have the advantage of yielding accurate numerical solutions via relatively coarse space discretization grids. For applications in this field the present one is the first attempt, at our knowledge, to solve a complete Stefan condition with convection flow, obtaining a good match with other numerical solutions in the literature. [Copyright &y& Elsevier]

Abstract: The energetic balance in the Rayleigh–Stokes problem for a Maxwell fluid is studied for several initial and/or boundary conditions. In the case of the first problem of Stokes, in comparison with the Newtonian fluid, the power of the wall shear stress and the dissipation increase while the boundary layer thickness decreases. A similar result is obtained for a series solution. In the case of the decay of the previous steady motion as well as for the Newtonian fluid, the power of the wall shear stress is null and the boundary layer thickness is the same. Finally, the second problem of Stokes is also considered. [Copyright &y& Elsevier]