Oberbeck Boussinesq Research Articles

Gyrotactic mixed bioconvection flow of a nanofluid past a circular cylinder with convective boundary condition

Abstract In recent years, Bioconvection phenomenon plays a fundamental role in several applications like biological systems, modern engineering, biotechnology and environmental systems. The term bioconvection indicates to a macroscopic convection movement of fluid produced due to the density gradient generated by collective swimming of microorganisms like bacteria and algae. These self-propelled motile microorganisms enhance the density of the base fluid by swimming in a particular direction, thus causing bioconvection. This research describes the bioconvection of a nanofluid containing motile gyrotactic micro-organisms past a horizontal circular cylinder subject to convective boundary condition. Buongiorno's two-component nanofluid model is deployed with the Oberbeck–Boussinesq approximation. Modeling is based on Brownian movement and thermophoresis aspects. Boundary-layer idea is employed for simplification of governing expressions. Finite-difference algorithm is implemented for computations of nonlinear problems. Besides, velocity, temperature, motile micro-organism density function, local skin friction coefficient, Nusselt number, wall motile density gradient function are analyzed to variation in physically pertinent values of selected control parameters. Comparison amid the previously published results and the current numerical results are made for the limiting cases, which are found to be in an virtuous agreement.

Convection of thermoviscouse fluid in a cell heated from the side

The work is devoted to the peculiarities of free-convective flow of liquids with viscosity temperature anomaly (the presence of extremes on the viscosity curve). Examples of such liquids are polymer solutions, metal melts, well-purified liquid sulfur, and other fluids. The mechanism of the anomalous viscosity behavior of such liquids (in the case of polymers) can be explained by polymerization and depolymerization reactions. At a certain temperature interval, the molecules of a substance interlock, forming long chains and, as a result, increasing the viscosity, then when the upper limit polymerization temperature is reached, the reaction begins that is reverse to the polymerization, which proceeds according to the chain mechanism and results in the sequential cleavage of molecules from the chain and leads to a decrease in viscosity . The features of behavior of such environments are currently not well understood and require increased attention to experimental and theoretical studies, especially now, mainly due to the intensive development of computer technologies and numerical modeling. Based on computational experiments on the process of free convection of a liquid with a Gaussian dependence of viscosity on temperature, the possibility of the existence of isolated regimes of convection of a liquid in a square cell heated from the side is shown. It was assumed that the viscosity function has one extremum and is unambiguously described by two parameters: the ratio of the highest to the lowest viscosity at a given temperature range and the degree of fullness of a given temperature range. As a mathematical model, a system of equations was used in the Oberbeck–Boussinesq approximation. For the numerical solution of the system of equations, the control volume method with the SIMPLE procedure is modified and implemented.

The collisional drift wave instability in steep density gradient regimes

The collisional drift wave instability in a straight magnetic field configuration is studied within a full-F gyro-fluid model, which relaxes the Oberbeck–Boussinesq (OB) approximation. Accordingly, we focus our study on steep background density gradients. In this regime we report on corrections by factors of order one to the eigenvalue analysis of former OB approximated approaches as well as on spatially localised eigenfunctions, that contrast strongly with their OB approximated equivalent. Remarkably, non-modal phenomena arise for large density inhomogeneities and for all collisionalities. As a result, we find initial decay and non-modal growth of the free energy and radially localised and sheared growth patterns. The latter non-modal effect sustains even in the nonlinear regime in the form of radially localised turbulence or zonal flow amplitudes.

The Bénard Problem for Slightly Compressible Materials: Existence and Linear Instability

We study a generalization of the Oberbeck–Boussinesq system, which consists in a buoyancy term where the density depends also on the pressure. A new pressure equation is introduced, which is deduced from the divergence-free condition on the velocity; such an equation cannot be decoupled from the system and is studied under Robin’s boundary conditions. Then, the existence of regular periodic solutions is proved for the full system. In Benard’s problem, the two-dimensional linear instability of the solution depends on a dimensionless parameter that is proportional to the compressibility factor: the related critical Rayleigh number decreases as it increases.

On the conditions for existence of unidirectional motions of binary mixtures in the Oberbeck–Boussinesq model

On the conditions for existence of unidirectional motions of binary mixtures in the Oberbeck–Boussinesq model

Effect of the System Geometry on the Flow Stability of an Evaporating Liquid

In this paper, we study the exact stationary solution of the Oberbeck–Boussinesq equations describing the joint flow of a viscous heat-conducting liquid and the cocurrent flow of a gas-vapor mixture in a flat horizontal layer. The initial formulation of the problem takes into account mass transfer through the interphase interface due to evaporation, vapor diffusion in the gas phase in the presence of diffusion heat conduction, and a longitudinal temperature gradient at the solid impermeable boundaries of the channel. The effect of the layer thickness of working media on the characteristics of the main flow under conditions of equal thermal load on the channel walls was studied. Based on the linear theory, the stability of the exact solution is studied, typical forms of perturbations and their dynamics are determined with a change in the linear dimensions of a system, and neutral curves and maps of the instability regimes are plotted.

Non-Oberbeck-Boussinesq effects due to large temperature differences in a differentially heated square cavity filled with air

We numerically investigate non-Oberbeck-Boussinesq (NOB) effects due to large temperature differences in a two-dimensional differentially heated square cavity using low-Mach-number equations. The working fluid is air and the Prandtl number Pr is 0.71 for the reference state. The considered Rayleigh numbers Ra range from 105 to 109. Various temperature differences ΔT̂ between the hot and cold plates are considered and the maximum value is up to 360K. The critical Rayleigh number for the onset of unsteadiness decrease with increasing temperature differences. The NOB effects on the temperature and velocity fields are investigated. It is found that both the thermal and velocity boundary layers become thicker near the hot plate while they get thinner near the cold plates under NOB conditions. The central temperature is increased compared to Oberbeck-Boussinesq (OB) cases considering NOB effects. The normalized center temperature θc roughly increases linearly with increasing temperature differential ∊. The horizontal velocity near the top plate is normally enhanced by NOB effects while it is normally decreased near the bottom plate under NOB conditions. In spite of these marked qualitative differences in the NOB flow relative to OB flow, the overall integral quantities like the Nusselt number Nu and Reynolds number Re are insensitive to NOB effects and retain their Ra-scaling exponents. The Nu has a minor decrease under NOB conditions. The maximum decrease is only about 3% for temperature difference as large as 360K. Within OB approximation, the Nusselt number Nu is found to scale as ∼Ra0.27, and the inverse thermal boundary layer thickness λθ-1 also scale as ∼Ra0.27. These scaling exponents do not change under NOB conditions. The Reynolds number based on the root mean square (r.m.s) velocity Rerms increases slightly for NOB cases. We also find Rerms∼Ra0.37,Rew∼Ra0.50,Rev∼Ra0.45 for OB cases, where Rew/Rev are Reynolds number based on maximum magnitude of vertical/horizontal velocity over the whole cell. The NOB effects almost have no influence on these scaling exponents.

Impacts of moving wall and heat-generating element on heat transfer and entropy generation of Al2O3/H2O nanofluid

Development of various electronic devices demands to create effective cooling complexes. The present paper deals with computational analysis of mixed convection cooling of heat-conducting and heat-generating element located inside an alumina–water nanofluid enclosure with upper moving wall. Usage of upper moving wall, nanofluid and cooling vertical walls allows to create the effective cooling process. Analysis has been performed numerically using the Oberbeck–Boussinesq equations. The effects of nanoparticles concentration, heat source location and upper wall velocity on flow structures, heat exchange and entropy generation have been investigated. It has been ascertained that effective cooling of the heated element occurs for high Reynolds number and central position of the heat-generating element.

Natural convection of Al2O3/H2O nanofluid in an open inclined cavity with a heat-generating element

Free convection of alumina-water nanofluid in a tilted open cavity with a heat-generating solid element has been studied. The considered topic allows understanding an opportunity of nanofluids for cooling of the heat-generating elements in open cavities. Upper border is supposed to be open where nanofluid penetrates into the cavity. Simulation has been performed using the Oberbeck–Boussinesq equations formulated in non-dimensional stream function, vorticity and temperature. Finite difference method of the second order accuracy has been applied. The effects of cavity inclination angle (0≤γ≤π/2), heater location (0.1≤δ≤0.7, where δ is the dimensionless distance between the heater and left wall) and nanoparticles volume fraction (0≤ϕ≤0.04) have been analyzed. It has been found that the considered parameters allow minimizing average heater temperature. The cavity inclination angle of π/3 characterizes the heat transfer enhancement (average Nusselt number has the maximum value), while the heater average temperature has the minimum value. At the same time, a proximity of the heater to the cavity wall characterizes non-essential cooling of the heat-generating element. The most effective cooling of the heat-generating element occurs for central heater location with cavity inclination angle of π/3. The considered alumina-water nanoparticles do not allow to intensify the cooling process for the heater element.

An extension of Oberbeck–Boussinesq approximation for thermal convection problems

An extension of the Oberbeck–Boussinesq approximation (OB) with expressive thermal effects is investigated to model specific mass with a temperature-dependent approach. An expansion of OB is proposed using a mathematical formulation with null velocity divergence in the continuity equation and variable specific mass in the momentum and energy equations (NOB). The NOB method has been previously tested in the literature, and the present paper aims to carry out a quantitative analysis between NOB and OB. The specific mass is calculated based on the temperature field and a divergence-free velocity is imposed on the NOB due to the small effects of variations of the specific mass in the continuity equation, as previously demonstrated by the OB. The purpose of the NOB is to overcome the OB’s restriction to single-phase flows, as well as to model the effects of specific mass variations more accurately, especially in problems with prominent thermal transfer effects, as that occur in the turbulent regime. Since the effects of a variable specific mass in the momentum and energy equations were taken into account by the NOB, it was expected that the thermal transfer results of the NOB would be improved compared with those of the OB. Then, three-dimensional natural convection simulations for a large range of Rayleigh numbers were performed using OB and NOB in a cubic cavity. Turbulence modeling was performed using Large Eddy Simulation (LES). Numerical results from both approaches were validated and all the results presented good quality. Thermal transfer was evaluated by the calculation of the Nusselt number at the heated wall and compared to experimental data from the literature. The OB and NOB provided adequate thermal transfer rate results; however, the differences between the literature and OB were higher than for NOB. Therefore, NOB is presented as a useful mathematical formulation for modeling incompressible flows with a temperature-dependent specific mass approach in single-phase or multiphase problems instead of solving the full compressible mathematical formulation. In addition, the differences between the literature and OB results increased as the Rayleigh number was raised, especially in the turbulent regime. The results of the NOB were closer to the literature than were those of the OB for the entire range of Rayleigh numbers tested. Therefore, the numerical results confirmed the higher accuracy of NOB compared to OB despite the NOB’s still being an approximate model. Lastly, flow visualization allowed the identification of coherent turbulent structures near the cavity walls in the simulation with Rayleigh number $$10^{10}$$ . The presence of hairpin and Tollmien–Schlichting instabilities revealed the importance of modeling the three-dimensional effects of natural convection in the turbulent regime.

A numerical methodology for efficient simulations of non-Oberbeck-Boussinesq flows

The Oberbeck-Boussinesq (OB) form of the Navier-Stokes equations can provide reliable solutions only for problems where the density differences are relatively small and all other material properties can be considered constant. For heat transfer problems at large temperature differences, the dependence of the material properties on temperature is the main source of non-Oberbeck-Boussinesq (NOB) effects. When a numerical solution is attempted for this category of problems, due to the density variations of the heated medium, a variable coefficient Poisson equation for the pressure emerges which is difficult to solve in a computationally efficient manner.In the present study, an efficient methodology to treat incompressible flows with variable properties outside the range of validity of the OB approximation is proposed. In the context of this methodology, the variable coefficient Poisson equation for the pressure is transformed into a constant coefficient Poisson equation using a pressure-correction scheme. In addition, all thermophysical properties are considered to be temperature dependent for all terms of the conservation equations. The proposed methodology is validated against results provided by previous studies on the natural convection flow of air, water and glycerol. Moreover, the potential of this methodology is demonstrated with the direct numerical simulation (DNS) of the NOB Rayleigh-Bénard convection of water inside a three-dimensional (3D) cavity. The comparison between two-dimensional (2D) and 3D results reveals significant differences, highlighting the need for efficient methodologies capable to accurately simulate NOB problems in 3D.

Ekman Convective Layer Flow of a Viscous Incompressible Fluid

Analytical solutions for generalizing the Ekman stationary flow of a viscous incompressible fluid in an infinite layer are obtained. The solution of an overdetermined system of the Oberbeck–Boussinesq equations is considered. It is suggested to use a class of exact solutions for this problem. It is shown that the structure of the solutions allows one to preserve the advective derivative in the heat-conductivity equation; this makes it possible to model the stratification of the temperature and pressure fields and describe the oceanic countercurrents.

Behavior of the Melt in the Vertical Bridgman Method with a Low Axial Temperature Gradient

The thermal conditions for the growth of Ge crystals with a diameter of 50 mm by the vertical Bridgman method in the case of low thermal-gravitational convection are studied using model experiments. Distilled water being hydrodynamically similar to the Ge melt is used as the model liquid. When modelling by means of the light cut method, it is established that mixture particles move along the heat flow direction from top to bottom. It is shown that an axial temperature gradient of 2 K/cm or more increases the contribution of thermal diffusion to mass transfer at a vertical flow rate of 0.09 mm/s or more. The numerical simulation of thermal convection in the Oberbeck–Boussinesq approximation confirms the absence of convection under the given thermal conditions. However, the deviation of the container axis from the vertical by 0.5° during the process of crystal growth contributes to the increase in the flow rates in the liquid phase up to 0.55 mm/s.

Onset of Convection in a Cylindrical Container with a Free Boundary

The problem of small perturbations of the equilibriumstate of a viscous, heat-conducting fluid in a cylindrical container with a deformable free upper boundary, on which heat exchange with the ambient medium is preassigned, is studied. The mathematical modeling of convection is based on Oberbeck–Boussinesq equations. The spectral problem thus obtained is solved using the tau method. As a result, the dependence of the imaginary part of the complex decrement on the Marangoni number is obtained. In the case of monotonic perturbations the neutral curves are plotted as functions of a geometrical parameter, namely, the cylinder height-to-radius ratio. The dependence of the Marangoni number on the physical parameters of the fluid is also obtained.

Application of a parallel solver to the LES modelling of turbulent buoyant flows with heat transfer

An existing fully implicit, non-dissipative direct numerical simulation (DNS) algorithm is reformulated to utilise the sub-grid scale (SGS) models in large eddy simulation (LES). The Favre-filtered equations with low-Mach number scaling are derived. The wall-adapting local eddy-viscosity (WALE) is used as SGS model. A fully parallel, finite volume solver is developed based on the resulting LES algorithm using PETSc library and applied to buoyancy- and thermally-driven transitional/turbulent flows in Rayleigh-Taylor instability and turbulent Rayleigh-Benard convection. Results verify that the proposed low-Mach number LES approach, which is physically more accurate than pure incompressible methods for flows with variable properties, perfectly captures the evolution and complex physics of turbulent buoyant flows with or without heat transfer by taking the effects of density and viscosity changes into account without the Oberbeck-boussinesq (OB) assumption even at large temperature differences with uniform accuracy and efficiency.

A Priori Estimates of the Solution of the Problem of the Unidirectional Thermogravitational Motion of a Viscous Liquid in the Plane Channel

We consider an initial boundary-value problem describing the unidirectional motion of a liquid in the Oberbeck–Boussinesq model in a plane channel with rigid immovable walls on which the temperature distribution is given (or the upper wall is heat-insulated). For this problem, we obtain a priori estimates, find an exact stationary solution, and determine conditions under which the solution converges to its stationary regime.

Thermal Convection in a Tilted Rectangular Cell with Aspect Ratio 0.5**Supported by the National Natural Science Foundation of China under Grant Nos 11572314, 11232011 and 11621202, and the Fundamental Research Funds for the Central Universities.

Thermal convection in a three-dimensional tilted rectangular cell with aspect ratio 0.5 is studied using direct numerical simulations within both Oberbeck–Boussinesq (OB) approximation and strong non-Oberbeck–Boussinesq (NOB) effects. The considered Rayleigh numbers Ra range from to , the working fluid is air at 300 K, and the corresponding Prandtl number Pr is 0.71. Within the OB approximation, it is found that there exist multiple states for and hysteresis for . For a relatively small tilt angle β, the large-scale circulation can either orient along one of the the vertical diagonal planes (denoted by mode) or orient parallel to the front wall (denoted by mode). Which of the two modes transports heat more efficiently is not definitive, and it depends on the Rayleigh number Ra. For and , the time-averaged flow field contains four rolls in the upper half and lower half of the cell, respectively, and modes only developing in tilted cells. By investigating NOB effects in tilted convection for fixed , it is found that the NOB effects on the Nusselt number Nu, the Reynolds number Re and the central temperature for different β ranges are different. NOB effects can either increase or decrease Nu, Re and when β is varied.

Local Thermal Non-equilibrium Analysis of the Instability in a Vertical Porous Slab with Permeable Sidewalls

Buoyant flow in a fluid-saturated porous vertical slab with isothermal and permeable boundaries is performed. Two reservoirs, maintained at different uniform temperatures, confine the slab. The permeable plane boundaries of the slab are modelled by imposing a condition of hydrostatic pressure. Darcy’s law and the Oberbeck–Boussinesq approximation are employed. The hypothesis of local thermal equilibrium between the fluid and the solid phase is relaxed. A two-temperature model is adopted, so that two local energy balance equations govern the heat transfer in the porous slab. The basic stationary buoyant flow consists of a single convective cell of infinite height. The time evolution of normal mode perturbations superposed onto the basic state is investigated in order to determine the onset conditions for thermal instability. A pressure–temperature formulation is employed. Major asymptotic cases are investigated. It is shown that departure from local thermal equilibrium implies in general a destabilisation of the basic stationary flow.

A numerical study on the impact of thermal alterations in porous media during hot fluid injection process employing a modified Boussinesq model

The Oberbeck-Boussinesq (OB) approximation is widely employed as a simplifying assumption for density-dependent flow problems. It reduces the governing differential equations to simpler forms, which can be handled analytically or numerically. In this study, a modified OB model is formulated to account for the variation of rock permeability and porosity with temperature during the hot fluid injection process in an oil-saturated porous medium under the assumption of local thermal equilibrium (LTE). The mathematical model is solved numerically using a fully implicit control volume finite difference discretization with the successive over relaxation (SOR) method to handle the non-linearity. Subsequently, the numerical model is validated with the analytical solution of the simplified problem successfully. Through detailed sensitivity analyses, the simulation results reveal the hot fluid injection rate as the most important operational parameter to be optimized for a successful thermal flood. The numerical runs show that that for single-phase core-flood simulation, the effect of temperature on the rock absolute permeability and porosity can be neglected without introducing any significant errors in the estimated recovery and temperature profile.

Aiding and opposing of mixed convection Casson nanofluid flow with chemical reactions through a porous Riga plate

In the present study, simultaneous effects of the chemical reaction and electromagnetohydrodynamic on the Casson nanofluid over porous Riga plate has been investigated. The governing flow problem consists of linear momentum, thermal energy, and nanoparticle concentration equation, which is modeled with the help of Oberbeck–Boussinesq approximation. Shooting method is employed to obtain the solution of the resulting nonlinear coupled ordinary differential equation. The behavior of the velocity profile, temperature profile, and nanoparticle concentration profile are discussed against modified Hartman number, porosity parameter, chemical reaction parameter, Prandtl number, Brownian motion parameter, thermophoresis parameter, Schmidt number, nanoparticle flux parameter, and Richardson number, respectively. The governing flow is also discussed for aiding and opposing flow by considering the negative and positive values of the modified Hartman number. The Nusselt number and Sherwood number are also computed numerically. Moreover, it is also found that the porosity parameter also enhances the velocity profile. A numerical comparison is also presented with previously published results to ensure the validity of the current methodology and results.