Oberbeck-Boussinesq Equations Research Articles

Computational investigation of magnetohydrodynamic convective flow in a trapezoidal cavity with multiple obstacles via finite element analysis

The present research deals with magnetohydrodynamic (MHD) natural convection within a trapezoidal cavity under the influence of inner solid obstacles of different shapes. The cavity includes two horizontal plates and two heated circular cylinders and is subjected to a uniform horizontal magnetic field. The walls of the domain can either be heated or cooled isothermally. The finite element method is used to solve the Oberbeck-Boussinesq equations that govern the fluid flow and heat transfer process. Three different thermal conditions are considered for the horizontal plates: isothermal heating, isothermal cooling, and adiabatic. The investigation explores the impact of varying Rayleigh numbers (Ra = 104–106), Hartmann numbers (Ha = 0–100), and thermal boundary conditions for inner horizontal plates on the flow patterns and heat transfer efficiency. The results indicate that as the Rayleigh number rises and the magnetic field strength diminishes, a noticeable improvement in heat transfer is observed, particularly in cases where the horizontal plates are subjected to cooling. Conversely, the heated inner horizontal plates exhibit less effective energy transport across the considered three configurations. It should be noted that the heat transfer rate for Ra = 106 can be increased more than 5 % when cooled ones will replace the heated inner plates. Whilst a transition between the adiabatic inner plates to cooled ones allows improving the average Nusselt number at the bottom heated wall at about 3 %. Moreover, the convective flow intensity can be described using the kinetic energy, where a growth of the magnetic field strength from 0 till 100 for Ra = 105 decreases the total kinetic energy at 76 times for heated plates, at 100 times for cooled plates and at 92 times for adiabatic plates.

Exact solutions to the oberbeck–boussinesq equations for convective stokes flows

To study large-scale convective flows (fluid motion in a thin layer), it is possible, for initial studies, to consider the Stokes approximation when integrating the Oberbeck–Boussinesq equation. The convective derivative in the momentum transfer equations and in the heat conduction equation is in this case assumed to be identically equal to zero. The paper discusses several approaches to constructing exact solutions for slow (creeping) flows of a non-uniformly heated fluid. Formulas for three-dimensional flows in the Lin–Sidorov–Aristov class are given for steady flows. The hydrodynamic fields are described by polynomials. Exact solutions are given for the velocity field nonlinearly depending on two spatial coordinates (longitudinal, or horizontal) with coefficients of nonlinear forms, which depend on the third coordinate. The study shows how it is possible to automate the computation of unknown coefficients for the formation of hydrodynamic fields (velocities and temperatures).

Numerical simulation of heat transfer performance in an enclosure filled with a metal foam and nano-enhanced phase change material

Nowadays phase change materials are widely used in different engineering systems to perform effective cooling techniques for heat-generating elements or to accumulate the thermal energy effectively. The disadvantage of the phase change material is a low thermal conductivity, but it can be improved using porous media or metal nanoadditives of high thermal conductivity. The present research is devoted to the mathematical simulation of conjugate thermal convection in a closed chamber partially filled with metal foam saturated with nano-enhanced phase change material. The governing equations have been formulated using the Oberbeck-Boussinesq equations with the Stefan problem approach for the liquid phase and the heat conduction equation for the solid phase. The finite difference technique in combination with non-primitive variables has been used for numerical analysis. The developed in-house numerical program has been verified using the grid independence test as well as the experimental and theoretical outcomes of other authors. The effects of the metal foam properties and nanoadditives characteristics on melt behavior and thermal energy transport parameters have been studied. The main attention is paid to the influence of a filling height ratio of the metal foam on the heat transport process. It has been revealed that the impact of the metal foam on thermal dissipation is essentially higher compared to the nanoadditives influence. Therefore, in the case of full height foam, the effect of nanoparticles is weak, but in the cavity with a partial height of the porous medium, NePCM can be melted faster than pure PCM and has a negative effect on the heater temperature. It has been also found that changing the filling height ratio from 0.167 to 0.5 does not lead to a significant change in the temperature of the heater.

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.

Solving the Hydrodynamical System of Equations of Inhomogeneous Fluid Flows with Thermal Diffusion: A Review

The present review analyzes classes of exact solutions for the convection and thermal diffusion equations in the Boussinesq approximation. The exact integration of the Oberbeck–Boussinesq equations for convection and thermal diffusion is more difficult than for the Navier–Stokes equations. It has been shown that the exact integration of the thermal diffusion equations is carried out in the Lin–Sidorov–Aristov class. This class of exact solutions is a generalization of the Ostroumov–Birikh family of exact solutions. The use of the class of exact solutions by Lin–Sidorov–Aristov makes it possible to take into account not only the inhomogeneity of the pressure field, the temperature field and the concentration field, but also the inhomogeneous velocity field. The present review shows that there is a class of exact solutions for describing the flows of incompressible fluids, taking into account the Soret and Dufour cross effects. Accurate solutions are important for modeling and simulating natural, technical and technological processes. They make it possible to find new physical mechanisms of momentum transfer for the design of new types of equipment.

Exact Solutions of the Oberbeck–Boussinesq Equations for the Description of Shear Thermal Diffusion of Newtonian Fluid Flows

We present a new exact solution of the thermal diffusion equations for steady-state shear flows of a binary fluid. Shear fluid flows are used in modeling and simulating large-scale currents of the world ocean, motions in thin layers of fluid, fluid flows in processes, and apparatuses of chemical technology. To describe the steady shear flows of an incompressible fluid, the system of Navier–Stokes equations in the Boussinesq approximation is redefined, so the construction of exact and numerical solutions to the equations of hydrodynamics is a very difficult and urgent task. A non-trivial exact solution is constructed in the Lin-Sidorov-Aristov class. For this class of exact solutions, the hydrodynamic fields (velocity field, pressure field, temperature field, and solute concentration field) were considered as linear forms in the x and y coordinates. The coefficients of linear forms depend on the third coordinate z. Thus, when considering a shear flow, the two-dimensional velocity field depends on three coordinates. It is worth noting that the solvability condition given in the article imposes a condition (relation) only between the velocity gradients. A theorem on the uniqueness of the exact solution in the Lin–Sidorov–Aristov class is formulated. The remaining coefficients of linear forms for hydrodynamic fields have functional arbitrariness. To illustrate the exact solution of the overdetermined system of Oberbeck–Boussinesq equations, a boundary value problem was solved to describe the complex convection of a vertical swirling fluid without its preliminary rotation. It was shown that the velocity field is highly stratified. Complex countercurrents are recorded in the fluid.

Comparative study of convection characteristics forasystem with deformable interface based on the Navier-Stokes and Oberbeck-Boussinesq equations.

The right choice of the appropriate mathematical model is crucial for evaluating the physical plausibility of modelling results. The issue of the correct application of the classical Boussinesq approximation for studying the heat and mass transfer in fluidic systems with a deformable boundary is a subject of scientific discussions despite the good agreement of numerous theoretical and numerical results obtained within the convection models based on the Oberbeck-Boussinesq equations with the data of physical experiments and observations. A comparative analysis of the results of numerical simulations in the framework of two-sided models based on the Navier-Stokes equations, and their Boussinesq approximation, is performed in the context of a convection problem in a locally heated two-phase system with a deformable interface. It is demonstrated that the application of the standard Boussinesq approximation allows one to give a consistent description of the effect of interface deformations on combined buoyant-thermocapillary driven fluid motions. This article is part of the theme issue 'New trends in pattern formation and nonlinear dynamics of extended systems'.

Convective Heat Transfer of a Pseudoplastic Nanosuspension within a Chamber with Two Heated Wall Sections of Various Heat Fluxes

Cooling of heat-generating elements in different engineering fields is a very important and crucial topic. The present research is devoted to numerical analysis of thermogravitational convection of a pseudoplastic nanosuspension in a chamber with two heated bottom wall sections of various heat fluxes and isothermally cooling vertical walls. A mathematical model formulated employing the time-dependent Oberbeck–Boussinesq equations with non-primitive variables has been worked out by the finite difference technique. It has been revealed that a mixture of 1% carboxymethylcellulose with water can be the most effective medium to cool the heat-generating elements. At the same time, aluminum oxide nano-sized solid particles have a more essential cooling impact on the heated sections.

A playground for compressible natural convection with a nearly uniform density

In the quest to understand the basic universal features of compressible convection, one would like to disentangle genuine consequences of compression from spatial variations of transport properties. For instance, one may choose to consider a fluid with uniform dynamic viscosity, but, then, compressible effects will generate a density gradient and consequently the kinematic viscosity will not be uniform. In the present work, we consider a very peculiar equation of state, whereby entropy is solely dependent on density, so that a nearly isentropic fluid domain is nearly isochoric. Within this class of equations of state, there is a thermal adiabatic gradient and a key property of compressible convection is still present, namely its capacity to viscously dissipate a large fraction of the thermal energy involved, of the order of the well-named dissipation number. In the anelastic approximation, under the assumption of an infinite Prandtl number, the number of governing parameters can be brought down to two, the Rayleigh number and the dissipation number. This framework is proposed as a playground for compressible convection, an opportunity to extend the vast corpus of theoretical analyses on the Oberbeck–Boussinesq equations regarding stability, bifurcations or the determination of upper bounds for the turbulent heat transfer. Here, in a two-dimensional geometry, we concentrate on the structure of numerical solutions. For all Rayleigh numbers, a change in the vertical temperature profile is observed in the range of dissipation number between $0$ and less than $0.4$ , associated with the weakening of ascending plumes. For larger dissipation numbers, the heat flux dependence on this number is found to be well predicted by Malkus's model of critical layers. For dissipation numbers of order unity, and large Rayleigh numbers, dissipation becomes related to the entropy heat flux at each depth, so that the vertical dissipation profile can be predicted, and so does the total ratio of dissipation to convective heat flux.

Temporal Evolution of Cooling by Natural Convection in an Enclosed Magma Chamber

This research numerically studies the transient cooling of partially liquid magma by natural convection in an enclosed magma chamber. The mathematical model is based on the conservation laws for momentum, energy and mass for a non-Newtonian and incompressible fluid that may be modeled by the power law and the Oberbeck–Boussinesq equations (for basaltic magma) and solved with the finite volume method (FVM). The results of the programmed algorithm are compared with those in the literature for a non-Newtonian fluid with high apparent viscosity (10–200 Pa s) and Prandtl (Pr = 4 × 104) and Rayleigh (Ra = 1 × 106) numbers yielding a low relative error of 0.11. The times for cooling the center of the chamber from 1498 to 1448 K are 40 ky (kilo years), 37 and 28 ky for rectangular, hybrid and quasi-elliptical shapes, respectively. Results show that for the cases studied, natural convection moved the magma but had no influence on the isotherms; therefore the main mechanism of cooling is conduction. When a basaltic magma intrudes a chamber with rhyolitic magma in our model, natural convection is not sufficient to effectively mix the two magmas to produce an intermediate SiO2 composition.

Двумерное термокапиллярное движение жидкости в открытом канале

The problem of two-dimensional thermocapillary fluid motion in a flat channel is studied. The temperature in the liquid is distributed according to the quadratic law, which is consistent with the velocity field of the Himentz type. At the bottom of the channel, the temperature depends on the time, which allows you to control the movement inside the layer. The Oberbeck-Boussinesq equations are taken as a mathematical model. The resulting initial - boundary value problem is highly nonlinear and inverse with respect to the pressure gradient along the channel. To solve it, a modified Galerkin method was used, where Legendre polynomials were chosen as the basis functions. The expansion coefficients are functions of time for which a system of nonlinear ODES was obtained. As a result of the application of the Runge-Kutta-Felberg method, a solution was found that, with increasing time, tends to solve a stationary problem if the temperature at the bottom of the channel stabilizes.

Numerical Simulation of Hybrid Nanofluid Mixed Convection in a Lid-Driven Square Cavity with Magnetic Field Using High-Order Compact Scheme.

In this study, the energy transference of a hybrid Al2O3-Cu-H2O nanosuspension within a lid-driven heated square chamber is simulated. The domain is affected by a horizontal magnetic field. The vertical sidewalls are insulated and the horizontal borders of the chamber are held at different fixed temperatures. A fourth-order accuracy compact method is applied to work out the vorticity-stream function view of incompressible Oberbeck–Boussinesq equations. The method used is validated against previous numerical and experimental works and good agreement is shown. The flow patterns, Nusselt numbers, and velocity profiles are studied for different Richardson numbers, Hartmann numbers, and the solid volume fraction of hybrid nanoparticles. Flow field and heat convection are highly affected by the magnetic field and volume fraction of each type of nanoparticles in a hybrid nanofluid. The results show an improvement of heat transfer using nanoparticles. To achieve a higher heat transmission rate by using the hybrid nanofluid, flow parameters like Richardson number and Hartmann number should be considered.

Analytical study of the Ekman angle for the Benard–Marangoni convective flow of viscous incompressible fluid

The paper considers the convective flow of a viscous incompressible fluid over a rotating surface. It studies the angle between the fluid velocity vector in the upper layer and the temperature gradient vector on the free surface. For the study, an analytical solution to the Oberbeck–Boussinesq equations is constructed, which describes the stratified Ekman flow taking into account two components of the Coriolis force. The temperature gradient and the conditions of Marangoni thermocapillary convection are set at the upper (free) boundary, and the condition of fluid adhesion is set on the lower (solid) boundary. The representation of velocities in the form of linear functions of horizontal coordinates is used. It is shown that, when the flow depth tends to infinity, the angle between the upper layer fluid velocity vector and the temperature gradient vector tends to π/2 .

Variable Permeability Effects on Natural Convection in a Vertical Porous Layer with Uniform Heat Flux from the Side

Natural convection heat transfer in a rectangular cavity filled with a saturated porous medium with variable permeability is investigated analytically and numerically. The side walls of the cavity are heated and cooled by constant heat fluxes, while the horizontal walls are adiabatic. The governing parameters for the problem are thermal Rayleigh number, R, the aspect ratio of the cavity, A, and the variable permeability constant, c. On the basis of the Darcy–Oberbeck–Boussinesq equations, a new analytical solution is developed in the limit of a long shallow cavity. A numerical study of the same phenomenon, obtained by solving the complete system of governing equations, is conducted to assess the validity of the analytical results. It is demonstrated that, for a wide range of the governing parameters, the flow structure and the heat transfer were well predicted by the analytical solution.

Non-stationary Unidirectional Motion of Binary Mixture in Long Flat Layer

The properties of solutions of non-stationary Oberbeck–Boussinesq equations are studied. The solutions describe unidirectional motions of binary mixture in a long horizontal layer. New non-stationary solution of inverse problem is constructed. In this problem basic characteristics of solution together with longitudinal pressure gradient are found. The condition of overdefinition for flow rate is used for searching of all unknowns. The convergence of the obtained non-stationary solution to the stationary one is proved with some restrictions on the initial data of the problem.

Numerical modeling of a thin liquid layer flow on an inclined substrate based on classical convection equations and generalized conditions at the interface

The flow of a thin layer of a viscous incompressible liquid along an inclined non-uniformly heated substrate is investigated in the two-dimensional case. Evaporation is taken into account at the thermocapillary interface. The system of Oberbeck-Boussinesq equations is used as a mathematical model. The kinematic, dynamic and energy conditions generalized for the case of a nonzero mass flux are assumed to be fulfilled at the interface. The value of the local mass flux is determined using the Hertz-Knudsen equation. Solutions to problems for the main terms of power decomposition of a small parameter of the problem are obtained. The evolution equation to determine the position of the interface is obtained. A numerical algorithm for solving the problem is constructed. The problem of periodic runoff of an incompressible liquid layer is considered. The effect of additional terms in the energy condition on the dynamics of the thickness of the liquid layer is studied.

On Marangoni shear convective flows of inhomogeneous viscous incompressible fluids in view of the Soret effect

A new exact solution is obtained for the Oberbeck-Boussinesq equations describing the steady-state layered (shear) Marangoni convection of a binary viscous incompressible fluid with the Soret effect. When layered (shear) flows are considered, the Oberbeck-Boussinesq system is overdetermined. For it to be solvable, a class of exact solutions is constructed, which allows one to satisfy identically the “superfluous” equation (the incompressibility equation). The found exact solution allows the Oberbeck-Boussinesq system of equations to be reduced to a system of ordinary differential equations by the generalized method of separation of variables. The resulting system of ordinary differential equations has an analytical solution, which is polynomial. The polynomial velocity field describes counterflows in the case of a convective fluid flow. It is demonstrated that the components of the velocity vector can have one stagnant (zero) point inside the region under study. In this case, the corresponding component of the velocity vector can be stratified into two zones, in which the fluid flows in opposite directions. The exact solution describing the velocity field for the Marangoni convection of a binary fluid has non-zero helicity, the flow itself being almost everywhere vortex.

Invariant states in inclined layer convection. Part 2. Bifurcations and connections between branches of invariant states

Convection in a layer inclined against gravity is a thermally driven non-equilibrium system, in which both buoyancy and shear forces drive spatio-temporally complex flow. As a function of the strength of thermal driving and the angle of inclination, a multitude of convection patterns is observed in experiments and numerical simulations. Several observed patterns have been linked to exact invariant states of the fully nonlinear 3D Oberbeck-Boussinesq equations. These exact equilibria, traveling waves and periodic orbits reside in state space and, depending on their stability properties, are transiently visited by the dynamics or act as attractors. To explain the dependence of observed convection patterns on control parameters, we study the parameter dependence of the state space structure. Specifically, we identify the bifurcations that modify the existence, stability and connectivity of invariant states. We numerically continue exact invariant states underlying spatially periodic convection patterns at Pr=1.07 under changing control parameters for temperature difference between the walls and inclination angle. The resulting state branches cover various inclinations from horizontal layer convection to vertical layer convection and beyond. The collection of all computed branches represents an extensive bifurcation network connecting 16 different invariant states across control parameter values. Individual bifurcation structures are discussed in detail and related to the observed complex dynamics of individual convection patterns. Together, the bifurcations and associated state branches indicate at what control parameter values which invariant states coexist. This provides a nonlinear framework to explain the multitude of complex flow dynamics arising in inclined layer convection.

Invariant states in inclined layer convection. Part 1. Temporal transitions along dynamical connections between invariant states

Thermal convection in an inclined layer between two parallel walls kept at different fixed temperatures is studied for fixed Prandtl number Pr=1.07. Depending on the angle of inclination and the imposed temperature difference, the flow exhibits a large variety of self-organized spatio-temporal convection patterns. Close to onset, these patterns have been explained in terms of linear stability analysis of primary and secondary flow states. At larger temperature difference, far beyond onset, experiments and simulations show complex, dynamically evolving patterns that are not described by stability analysis and remain to be explained. Here we employ a dynamical systems approach. We construct stable and unstable exact invariant states, including equilibria and periodic orbits of the fully nonlinear three-dimensional Oberbeck-Boussinesq equations. These invariant states underlie the observed convection patterns beyond their onset. We identify state-space trajectories that, starting from the unstable laminar flow, follow a sequence of dynamical connections between unstable invariant states until the dynamics approaches a stable attractor. Together, the network of dynamically connected invariant states mediates temporal transitions between coexisting invariant states and thereby supports the observed complex time-dependent dynamics in inclined layer convection.

Thermocapillary Convection of a Vertical Swirling Liquid

A new exact solution of the Oberbeck–Boussinesq equations has been presented which describes the steady-state Marangoni convection of a vertical swirling liquid in the shear flow of a viscous incompressible liquid. The velocity field of the nonuniform vortical liquid flow is studied in detail. It is shown that there are stagnation points in the liquid flow, the number of which can be as large as five. The existence of stagnation points leads to the formation of counterflows with very complex topology resembling cellular flow patterns. The tangential stresses in the liquid flow are not constant. They are also stratified from the zone of tensile and compressive stresses. Because of the structure of the exact solution, the vorticity components coincide to within a dissipative coefficient with the expressions for the tangential stresses, which means that there are several counterrotating vortices in the liquid. Similar studies of polynomial exact solutions for temperature and pressure determined the number of zones of stratification of the corresponding fields.