Problem Of Thermal Convection Research Articles

Qualitative Properties of the Solution of a Conjugate Problem of Thermal Convection

The joint convection of two viscous heat-conducting liquids in a three-dimensional layer bounded by flat solid walls was studied. The upper wall is thermally insulated, and the lower wall has a non-stationary temperature field. The liquids are immiscible and separated by a flat interface with complex conjugation conditions set on it. The evolution of this system in each liquid was described by the Oberbeck–Boussinesq equations. The solution of the problem was sought for velocities that are linear in two coordinates and temperature fields that are quadratic functions of the same coordinates. Thus, the problem was reduced to a system of 10 nonlinear integro-differential equations. Its conjugate and inverse nature is determined by the four functions of time. Integral redefinition conditions were set to find them. The physical meaning of the integral conditions is the closeness of the flow. The inverse initial-boundary value problem describes convection near the temperature extremum point on the lower solid wall in a two-layer system. For small Marangoni numbers, the problem was approximated linearly (the Marangoni number is analogous to the Reynolds number in the Navier–Stokes equations). Using the obtained a priori estimates, sufficient conditions were identified for the non-stationary solution to become a stationary one over time.

Thermal convection with a Cattaneo heat flux model

The problem of thermal convection in a layer of viscous incompressible fluid is analysed. The heat flux law is taken to be one of Cattaneo type. The time derivative of the heat flux is allowed to be a material derivative, or a general objective derivative. It is shown that only one objective derivative leads to results consistent with what one expects in real life. This objective derivative leads to a Cattaneo–Christov theory, and the results for linear instability theory are in agreement with those for a material derivative. It is further shown that none of the theories allow a standard nonlinear, energy stability analysis. A further heat flux due to P.M. Mariano is added and then an analysis is performed for stationary convection, oscillatory convection, and fully nonlinear theory. For the material derivative case, the analysis proceeds and global nonlinear stability is achieved. For Cattaneo–Christov theory, it appears necessary to add a regularization term in the equation for the heat flux, and even then the analysis only works in two space dimensions, and is conditional upon the size of the initial data. For the three-dimensional situation, it is shown how a nonlinear stability analysis may be achieved with a Navier–Stokes–Voigt fluid rather than a Navier–Stokes one.

Sharp Instability Estimates for Bidisperse Convection with Local Thermal Non-equilibrium

We analyse a theory for thermal convection in a Darcy porous material where the skeletal structure is one with macropores, but also cracks or fissures, giving rise to a series of micropores. This is thus thermal convection in a bidisperse, or double porosity, porous body. The theory allows for non-equilibrium thermal conditions in that the temperature of the solid skeleton is allowed to be different from that of the fluid in the macro- or micropores. The model does, however, allow for independent velocities and pressures of the fluid in the macro- and micropores. The threshold for linear instability is shown to be the same as that for global nonlinear stability. This is a key result because it shows that one may employ linearized theory to ensure that the key physics of the thermal convection problem has been captured. It is important to realize that this has not been shown for other theories of bidisperse media where the temperatures in the macro- and micropores may be different. An analytical expression is obtained for the critical Rayleigh number and numerical results are presented employing realistic parameters for the physical values which arise.Article HighlightsA two-temperature regime for a bidisperse Darcy porous medium is proposed to study the thermal convection problem.The optimal result of coincidence between the linear instability and nonlinear stability critical thresholds is proven.Numerical analysis enhances that the scaled heat transfer coefficient between the fluid and solid and the porosity-weighted conductivity ratio stabilize the problem significantly.

Diffuse interface-lattice Boltzmann modeling for heat and mass transfer with Neumann boundary condition in complex and evolving geometries

In this research, a diffuse interface-lattice Boltzmann method is developed to model heat and mass transfer problems with Neumann boundary condition in complex and evolving geometries. In this model, the physical domain with curved and moving boundaries is extended to a larger and regular domain based on the concept of fictitious domain. The Neumann boundary condition is incorporated into the jump conditions of field variable at the inner interface. Then a unified convection-diffusion equation of field variable with two singular source terms is built. Using the smoothing technique, a diffuse interface equation is obtained. Moreover, an energy method is presented to derive the diffuse interface equation in a special case. We utilized a multiple-relaxation-time lattice Boltzmann scheme to numerically solve the diffuse interface equation. A simple example with analytical solutions validates the numerical accuracy of the present model. Finally, several thermal convection and mass transport problems are also simulated. The numerical results denotes the validity of the proposed diffuse interface model.

DNS of buoyancy-driven flows using EDAC formulation solved by high-order method

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.

SIKIŞTIRILAMAZ ISIL TAŞINIM PROBLEMLERİNİN FİZİKLE ÖĞRENEN YAPAY SİNİR AĞLARI İLE ÇÖZÜMÜ

Physics-informed neural networks (PINNs) have drawn attention in recent years in engineering problems due to their effectiveness and ability to tackle problems without generating complex meshes. PINNs use automatic differentiation to evaluate differential operators in conservation laws and hence do not need a discretization scheme. Using this ability, PINNs satisfy governing laws of physics in the loss function without any training data. In this work, we solve various incompressible thermal convection problems, and compare the results with numerical or analytical results. To evaluate the accuracy of the model we solve a channel problem with an analytical solution. The model is highly dependent on the weights of individual loss terms. Increasing the weight of boundary condition loss improves the accuracy if the flow inside the domain is not complicated. To assess the performance of different type of networks and ability to capture the Neumann boundary conditions, we solve a thermal convection problem in a closed enclosure in which the flow occurs due to the temperature gradients on the boundaries. The simple fully connected network performs well in thermal convection problems, and we do not need a Fourier mapping in the network since there is no multiscale behavior. Lastly, we consider steady and unsteady partially blocked channel problems resembling industrial applications to power electronics and show that the method can be applied to transient problems as well.

Study of the Influence of Thermal Convection on Temperature Measurement in Thermal Control Boreholes during Artificial Ground Freezing

This paper considers the problem of thermal convection of a calcium chloride solution in a vertical borehole. A non-uniform temperature distribution with a given vertical gradient is set at the walls of the borehole. The non-stationary temperature distribution along the borehole axis was analyzed, and its deviations from the temperature at the walls were investigated. From a practical point of view, this problem is important for estimating the error in distributed temperature measurements over the depth of thermal control boreholes during artificial ground freezing. In this study, an area near the bottom of the borehole was identified where the fluid temperature at the borehole axis deviates significantly from the temperature at the wall. The maximum deviations of the fluid temperature from the temperature at the walls, as well as the length of the temperature deviation sections, were determined.

Characteristic methods for thermal convection problems with infinite Prandtl number on nonuniform staggered grids

In this paper, two characteristic methods are introduced to solve the thermal convection problems with infinite Prandtl number on nonuniform staggered grids. We obtain spatial second order superconvergence in the discrete L2 norm for the temperature, velocity and pressure. Besides, second order superconvergence in the discrete H1 norm for both the temperature and some terms of the velocity on nonuniform grids is established. First and second order time discretizations of the thermal balance equation are provided. Numerical experiments are presented to show that the convergence rates are in agreement with the theoretical analysis.

Natural convection of a binary liquid in cylindrical porous annuli/rectangular porous enclosures with cross-diffusion effects under local thermal non-equilibrium state

The present article reports an analytical study of the double diffusive natural convection (DDNC) in cylindrical porous annuli (CPA) and rectangular porous enclosures (RPE), which are handled in a unified way using the curvature parameter, saturated by a binary liquid under the assumption of local thermal non-equilibrium (LTNE) state. The buoyancy forces (thermal and solutal) driving the flow are assumed to be induced by the maintenance of constant and uniform heat and mass fluxes applied along the vertical (radial) walls and insulation of both horizontal walls of the annuli/rectangular enclosures. The Darcy-Boussinesq equations with LTNE assumption between the fluid and solid phases are employed to model the problem of DDNC in a binary liquid-saturated porous medium with cross-diffusion effects. The analytical results are obtained by employing the Oseen-linearization transformation technique in the study. The influence of various dimensionless parameters on heat and mass transports of the system are depicted using the Nusselt and Sherwood numbers and isotherms plots, and the obtained results are analysed with the physical explanation. Special attention is given to understand the effect of LTNE parameter and cross-diffusion parameters on heat and mass transports of the system. Different aspect ratio values are chosen to obtain the results of three types of CPA/RPE (shallow, square and tall). Among these CPA/RPE, maximum and minimum heat and mass transports are achieved in the cases of shallow and tall CPA/RPE, respectively. The results of the pure thermal convection problem is obtained at the zero value of buoyancy ratio and solute Rayleigh number. The increasing value of N magnifies the heat and mass transports in the system due to the augmented buoyancy effect resulted from the thermal and solutal gradients. The increase of solid inner cylinder radius, by fixing its volume, makes the annulus slender which yields to decrease the heat and mass transports in the system. The effects of LTNE parameter and cross-diffusion parameters on heat and mass transports of the system are clearly brought out. The results of LTE model are obtained at the infinite value of ratio of porosity modified thermal conductivities, γ, as a particular case of the present model. From the study, we conclude that the shallow porous annulus and tall rectangular enclosure are best suited in the design of heat removal and heat storage systems, respectively.

UPPER MANTLE CONVECTION RELATED TO SUBDUCTION ZONE AND APPLICATION OF THE MODEL TO INVESTIGATE THE CRETACEOUS-CENOZOIC GEODYNAMICS OF CENTRAL EAST ASIA AND THE ARCTIC

A geodynamic model of upper mantle convection related to the Pacific subduction zone is mathematically substantiated and applied to investigate the Cretaceous-Cenozoic evolution of Central East Asia (CEA) and the Arctic. We present a solution for the two-dimensional stationary problem of thermal convection in the upper mantle layer, considering different Rayleigh numbers and taking into account the influence of the subduction process and lithospheric movements along the upper mantle base. We describe the results of 3D modeling of nonstationary upper mantle convection in a subduction zone. Our data give grounds to propose explanations for the entire spectrum of tectonic-magmatic processes developing within CEA in the Cenozoic and the Arctic in the Upper Cretaceous and Cenozoic. We discuss the reasons why the lithosphere in CEA and the Arctic is generally shifting towards the Pacific subduction zone, considering the presence of separate magmatic provinces and rift zones. In our opinion, this is due to the existence of a large horizontally elongated convective cell, which interior is composed of smaller isometric cells. This long cell creates the effect of conveyor dragging of the lithosphere, while its internal cells produce the effect of upper mantle plumes.

A spectral/hp element method for thermal convection

AbstractWe report on a high‐fidelity, spectral/hp element algorithm developed for the direct numerical simulation of thermal convection problems. We consider the incompressible Navier–Stokes (NS) and advection–diffusion equations coupled through a thermal body‐forcing term. The flow is driven by a prescribed flowrate forcing with explicit treatment of the nonlinear advection terms. The explicit treatment of the body‐force term also decouples both the NS and the advection–diffusion equations. The problem is then temporally discretized using an implicit–explicit scheme in conjunction with a velocity‐correction splitting scheme to decouple the velocity and pressure fields in the momentum equation. Although not unique, this type of discretization has not been widely applied to thermal convection problems and we therefore provide a comprehensive overview of the algorithm and implementation which is available through the open‐source package Nektar++. After verifying the algorithm on a number of illustrative problems we then apply the code to investigate flow in a channel with uniform or streamwise sinusoidal lower wall, in addition to a patterned sinusoidal heating. We verify the solver against previously published two‐dimensional results. Finally, for the first time we consider a three‐dimensional problem with a streamwise sinusoidal lower wall and sinusoidal heating which, for the chosen parameter, leads to the unusual dynamics of an initially unsteady two‐dimensional instability leading to a steady three‐dimensional nonlinear saturated state.

Effects of density-affecting scalar on the onset of chaos in a simplified model of thermal convection: a nonlinear dynamical perspective

To explore how density-affecting scalar influences the onset of chaos in a simplified model of thermal convection, we consider three versions of a physically extended Lorenz system obtained from incorporating additional physical ingredients such as rotation and density-affecting scalar. The three versions of the extended Lorenz system correspond to the cases when the density-affecting scalar has positive, neutral, and negative impacts on buoyancy. In general, compared to the case when the density-affecting scalar has a positive (neutral) impact on buoyancy, the case when the density-affecting scalar has a neutral (negative) impact on buoyancy leads to a higher critical Rayleigh parameter, a sign of delayed onset of chaos. For an appropriate choice of parameters that lead to chaotic solutions in all three cases, it is shown that the chaotic attractors for the three cases can exhibit a variety of different characteristics, such as taking on the shape of the classic Lorenz attractor, sharing the phase space with stable point attractors, and taking an unusual shape that is distinguished from the Lorenz attractor. The different characteristics in chaotic attractors lead to qualitative differences in the behaviour exhibited by the corresponding flow and temperature patterns reconstructed from the numerical solutions. Further analysis of this simplified model offering a nonlinear dynamical perspective on the thermal convection problem is expected to serve as an avenue for gaining new insights into this and related problems.

Natural convection in horizontal layers with lateral heating at various Rayleigh and Prandtl numbers

The paper presents the results of numerical solution of the problem of thermal convection of an incompressible liquid in a horizontally elongated layer with lateral heat supply. In this paper the influence of Rayleigh and Prandtl numbers of boundary conditions on the flow structure and on the existence of counter-flows are considered. It is also shown that for laminar convection modes, flows with Prandtl numbers less than 0.01 differ qualitatively from the flow of a liquid with a zero Prandtl number, so the approximation of the zero value of the Prandtl number may not always be correct.

Localised thermal convection in rotating spheres that undergo weak precession

We report a new nonlinear phenomenon discovered in the classical problem of thermal convection in a rapidly rotating, self-gravitating, internally heated fluid sphere that also undergoes weak precession. When the Prandtl number of fluids is sufficiently large, convection-driven columnar rolls – which are nearly geostrophic and marked by small azimuthal scale – cannot have substantial nonlinear interaction with precession-driven flow that is in the form of an equatorially antisymmetric, large-scale inertial wave in the mantle frame of reference. When the Prandtl number of fluids is sufficiently small, convection-driven flow, because of predominant inertial effects in its dynamics, is non-geostrophic and in the form of an equatorially symmetric, large-scale inertial wave, and, hence, is able to interact destructively with the precession-driven large-scale inertial wave via nonlinear effects. We reveal that the destructive interaction between the convection-driven wave and the precession-driven wave leads to a localised convective wave that is nearly equatorially symmetric, progradely travelling in the azimuthal direction, and largely confined within a quarter of the sphere.

Coriolis effect on thermal convection in a rotating bidispersive porous layer

We obtain the linear instability and nonlinear stability thresholds for a problem of thermal convection in a rotating bidispersive porous medium with a single temperature. We show that the linear instability threshold is the same as the nonlinear stability one. This means that the linear theory is capturing completely the physics of the onset of thermal convection.

Reduced-Order Quasilinear Model of Ocean Boundary-Layer Turbulence

AbstractThe combined effectiveness of model reduction and the quasilinear approximation for the reproduction of the low-order statistics of oceanic surface boundary layer turbulence is investigated. Idealized horizontally homogeneous problems of surface-forced thermal convection and Langmuir turbulence are studied in detail. Model reduction is achieved with a Galerkin projection of the governing equations onto a subset of modes determined by proper orthogonal decomposition (POD). When applied to boundary layers that are horizontally homogeneous, POD and a horizontally averaged quasilinear approximation both assume flow features that are horizontally wavelike, making the pairing very efficient. For less than 0.2% of the modes retained, the reduced quasilinear model is able to reproduce vertical profiles of horizontal mean fields as well as certain energetically important second-order turbulent transport statistics and energies to within 30% error. Reduced-basis quasilinear statistics must approach the full-basis statistics as the basis size approaches completion; however, some quasilinear statistics resemble those found in the fully nonlinear simulations at smaller basis truncations. Thus, model reduction could possibly improve upon the accuracy of quasilinear dynamics.

Multiple-relaxation-time lattice Boltzmann model for convection heat transfer in porous media under local thermal non-equilibrium condition

In this work, a thermal multiple-relaxation-time (MRT) lattice Boltzmann (LB) model is proposed for simulating convection heat transfer in porous media under local thermal non-equilibrium (LTNE) condition. In the present model, the thermal non-equilibrium effects are incorporated into the model by adding source terms into the temperature-based MRT-LB equations. The discrete effects of the source terms are eliminated, and the temperatures of the fluid and solid phases are obtained through algebraic operations. Moreover, the error term which exists in the macroscopic temperature equation of the fluid phase is eliminated via a way that is consistent with the philosophy of the LB method. Numerical simulations show that thermal non-equilibrium natural convection problems can be well simulated.

Ultra-weak symmetry of stress for augmented mixed finite element formulations in continuum mechanics

In this paper we propose a novel way to prescribe weakly the symmetry of stress tensors in weak formulations amenable to the construction of mixed finite element schemes. The approach is first motivated in the context of solid mechanics (using, for illustrative purposes, the linear problem of linear elasticity), and then we apply this technique to reduce the computational cost of augmented fully-mixed methods for thermal convection problems in fluid mechanics, in the case where several additional variables are defined. We show that the new approach allows to maintain the same structure of the mathematical analysis as in the original formulations. Therefore we only need to focus on ellipticity of certain bilinear forms, as this property provides feasible ranges for the stabilization parameters that complete the description of augmented methods. In addition, we present some numerical examples to show that these methods perform better than their counterparts that include vorticity, and emphasize that the reduction in degrees of freedom (and therefore, in computational cost) does not affect the quality of numerical solutions.

Analyzing thermal convection in a two-dimensional circular annulus via spatio-temporal Koopman decomposition

This paper deals with the analysis of numerically obtained spatio-temporal data for thermal convection in a two-dimensional circular annulus. These data are post-processed using a recent method called the spatio-temporal Koopman decomposition, which applies to the frequent case in which the underlying dynamics exhibit oscillatory (possibly growing or decaying) behavior in both time and one of the spatial directions (the azimuthal coordinate for the present problem). When this holds, the method decomposes the data into Fourier-like series in both the distinguished spatial direction and time. In the general case, the obtained series account for, not only the involved temporal frequencies and spatial wavenumbers, but also the spatial and temporal growth rates. In the simpler situation of attractors showing spatially periodic behavior, the spatial and temporal growth rates vanish (or are very small). In this case, the analysis of the wavenumber–frequency pairs that are present allows for uncovering the spatio-temporal structure of the flow in the circular annulus. The analysis focuses on periodic and quasi-periodic attractors, which exhibit spatio-temporal symmetries that are identified by the method. In addition, for quasi-periodic attractors, the method gives semi-analytic descriptions for the tori densely covered by particular trajectories. Although the paper concentrates on the thermal convection problem in an annulus, it will become clear that the method applies to other related dynamics as well.

A two-dimensional-three-component model for spanwise rotating plane Poiseuille flow

Spanwise rotating plane Poiseuille flow (RPPF) is one of the canonical flow problems to study the effect of system rotation on wall-bounded shear flows and has been studied a lot in the past. In the present work, a two-dimensional-three-component (2D/3C) model for RPPF is introduced and it is shown that the present model is equivalent to a thermal convection problem with unit Prandtl number. For low Reynolds number cases, the model can be used to study the stability behaviour of the roll cells. It is found that the neutral stability curves, critical eigensolutions and critical streamfunctions of RPPF at different rotation numbers ($Ro$) almost collapse with the help of a rescaling with a newly defined Rayleigh number $Ra$ and channel height $H$. Analytic expressions for the critical Reynolds number and critical wavenumber at different $Ro$ can be obtained. For a turbulent state with high Reynolds number, the 2D/3C model for RPPF is self-sustained even without extra excitations. Simulation results also show that the profiles of mean streamwise velocity and Reynolds shear stress from the 2D/3C model share the same linear laws as the fully three-dimensional cases, although differences on the intercepts can be observed. The contours of streamwise velocity fluctuations behave like plumes in the linear law region. We also provide an explanation to the linear mean velocity profiles observed at high rotation numbers.