Oberbeck Boussinesq Research Articles

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.

Bénard Problem for Slightly Compressible Fluids: Existence and Nonlinear Stability in 3D

This paper shows the existence, uniqueness, and asymptotic behavior in time of regular solutions (a la Ladyzhenskaya) to the Bénard problem for a heat-conducting fluid model generalizing the classical Oberbeck–Boussinesq one. The novelty of this model, introduced by Corli and Passerini, 2019, and Passerini and Ruggeri, 2014, consists in allowing the density of the fluid to also depend on the pressure field, which, as shown by Passerini and Ruggeri, 2014, is a necessary request from a thermodynamic viewpoint when dealing with convective problems. This property adds to the problem a rather interesting mathematical challenge that is not encountered in the classical model, thus requiring a new approach for its resolution.

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.

A fully-mixed finite element method for the steady state Oberbeck–Boussinesq system

A fully-mixed finite element method for the steady state Oberbeck–Boussinesq system

Thermal fluctuations and boundary layer properties of turbulent natural convection inside open cavities of different dimensions heated from below

We report the experimental measurement of the temperature fluctuation in the vicinity of different zones of the thermal boundary layer in water-filled open cubic cavities heated from below and open at the top. The experiments are performed on the cubic cavity of aspect ratio 1 and lateral dimension 30 mm; the results of our previously reported open cubic cavities of aspect ratio 1 and lateral dimensions (120 mm and 240 mm) are also considered here. The transient nature of the temperature has been measured from the temperature–time series recorded across the central axis of the cavity at different vertical positions z from the heated bottom plate. The Prandtl number and Rayleigh number ranges reported in this paper are 4 ≤ Pr ≤ 6 and 105 ≤ Ra ≤ 109, respectively. The different basic statistical properties, of temperature fluctuation such as mean temperature, root mean square, and probability density function, are studied and discussed. The power-law of power spectral density of the temperature fluctuations at different regions of the thermal boundary layer is studied, and the different roles of rate are compared with the previously established theories and models. The validity criteria for the Oberbeck–Boussinesq approximation are fulfilled. The trend of the dimensionless Nusselt number (Nu) representing the global convective heat transfer is obtained and discussed. We also study the variation in Raδth−1 3 for the heat transfer representation in the range of 0.04–0.24, where δth is the boundary layer thickness.

A Study of Rayleigh-Bénard-Taylor Convection in Very-Shallow, Shallow, Square and Tall Enclosures

The paper deals with an analytical-study of Rayleigh-Benard-convection (RBC) within four types of enclosures in a non-inertial frame of reference(NIFR) due to uniform rotation of the system. Using a double Fourier trigonometric series with least number of modes, an extended Lorenz model which has in it the effect of an NIFR is derived under the assumption of the Oberbeck-Boussinesq (OB) approximation. The method of multi-scales is employed to arrive at the first-order, autonomous Stuart-Landau equation. Its solution is analysed and it is shown that though the Stuart-Landau model is lower in order than the Lorenz model it retains some of the features of the Lorenz model and, further, facilitates an analytical solution. Using this solution, the expression of the Nusselt number is obtained in closed form. The study reveals that the NIFR-effect leads to a stabilized-system and a decrease in the amount of heat-transport. The rate of heat-transport-diminishment is discussed using an Ozoe coefficient. Besides these findings, the study reveals that from the results of the classical problem of RBC in an NIFR the results of the other three enclosures may be obtained by suitable redefinition of the parameters.

Effects of pore scale on the macroscopic properties of natural convection in porous media

Natural convection in porous media is a fundamental process for the long-term storage of CO2 in deep saline aquifers. Typically, details of mass transfer in porous media are inferred from the numerical solution of the volume-averaged Darcy–Oberbeck–Boussinesq (DOB) equations, even though these equations do not account for the microscopic properties of a porous medium. According to the DOB equations, natural convection in a porous medium is uniquely determined by the Rayleigh number. However, in contrast with experiments, DOB simulations yield a linear scaling of the Sherwood number with the Rayleigh number ( $Ra$ ) for high values of $Ra$ ( $Ra\gg 1300$ ). Here, we perform direct numerical simulations (DNS), fully resolving the flow field within the pores. We show that the boundary layer thickness is determined by the pore size instead of the Rayleigh number, as previously assumed. The mega- and proto-plume sizes increase with the pore size. Our DNS results exhibit a nonlinear scaling of the Sherwood number at high porosity, and for the same Rayleigh number, higher Sherwood numbers are predicted by DNS at lower porosities. It can be concluded that the scaling of the Sherwood number depends on the porosity and the pore-scale parameters, which is consistent with experimental studies.

Computation of electroconductive gyrotactic bioconvection under nonuniform magnetic field: Simulation of smart bio-nanopolymer coatings for solar energy

The water-based bioconvection of a nanofluid containing motile gyrotactic micro-organisms (moves under the effects of gravity) over a nonlinear inclined stretching sheet in the presence of a nonuniform magnetic field has been investigated. This regime is encountered in the bio-nanomaterial electroconductive polymeric processing systems currently being considered for third-generation organic solar coatings, anti-fouling marine coatings, etc. Oberbeck–Boussinesq approximation along with ohmic dissipation (Joule heating) is considered in the problem. The governing equations of the flow are nonlinear partial differential equations and are converted into ordinary differential equations via similarity transformations. These equations are then solved by the Finite Element Method. The effect of various important parameters on nondimensional velocity, temperature distribution, nanoparticle concentration, the density of motile micro-organisms is analyzed graphically in detail. It is observed from the obtained results that the flow velocity decreases with rising angle of inclination [Formula: see text] while temperature, nanoparticle’s concentration and density of motile micro-organisms increase. The local skin friction coefficient, Nusselt number, Sherwood number, motile micro-organism’s density number are calculated. It is noticed that increasing the Brownian motion and thermophoresis parameter leads to an increase in temperature of fluid which results in a reduction in Nusselt number. On the contrary, the Sherwood number rises with an increase in Brownian motion and thermophoresis parameter. Also, interesting features of the flow dynamics are elaborated and new future pathways for extension of the study identified in bio-magneto-nano polymers (BMNPs) for solar coatings.

Self-similar analysis of a viscous heated Oberbeck–Boussinesq flow system

One of the simplest model to couple viscous flow to heat conduction is the Oberbeck–Boussinesq (OB) system which were also investigated by E N Lorenz. In our former studies—2015 Chaos Solitons Fractals 78 249, 2017 Chaos Solitons Fractals 103 336—we derived analytic solutions for the velocity, pressure and temperature fields. Additionally, we gave a possible explanation of the Rayleigh–Bénard convection cells with the help of the self-similar Ansatz. Now we generalize the OB hydrodynamical system, including a viscous source term in the heat conduction equation. Our analysis show that the viscous heating term smooths out any kind of Bénard oscillations and stabilizes the flow. All the velocity, pressure and temperature distributions are free of oscillations. These results may attract interest in micro or nanofluidics.

Variable property DNS of differentially heated cavities filled with air

Direct numerical simulations (DNS) of the thermally-driven flow of air inside differentially heated cavities (DHC) are presented. The variation of air properties is taken into account using the low-Mach approximation, to obtain solutions outside the limits of the Oberbeck–Boussinesq (OB) approximation. The effects of the cavity’s aspect-ratio A and large temperature differences ΔT were investigated over the range 1 ≤ A ≤ 8 and ΔT ≤ 720 K. The Rayleigh and Prandtl numbers were set at Ra=109 and Pr=0.7, considering a reference temperature of T0=600 K. Time-averaged and r.m.s. statistics of the temperature field along with other derived quantities are presented and discussed. It is found that the time-averaged temperature fields vary significantly between non-Oberbeck–Boussinesq (NOB) and OB cases for larger aspect-ratios. This is also true for the instantaneous temperature and temperature r.m.s. fields which demonstrate significant qualitative differences. In addition, as the temperature difference increases and the property variations become more intense, the boundary layers at the vertical walls become turbulent at lower positions. The relocation of the laminar-turbulent transition point is found to be strongly dependent on the cavity’s aspect-ratio. An explanation of this phenomenon is given, based on the disruption of the vertical boundary layers by horizontal streams originating from the downstream part of the vertical walls.

Mixed Magnetoconvection of Nanofluids in a Long Vertical Porous Channel

Abstract This paper aimed to study the flow of a nanofluid in a long vertical porous channel when an external uniform magnetic field is impressed. The Buongiorno two-phase model of nanofluid is supposed to be slightly compressible in order to assume the Oberbeck–Boussinesq approximation. The velocity, the induced magnetic field, the temperature, and the nanoparticle volume fraction are analytically obtained. Detailed considerations are drawn for the occurrence of the reverse flow phenomenon. Moreover, a selected set of plots illustrating the influence of various parameters involved in the problem is presented and discussed.

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.

Numerical simulations of turbulent thermal convection with a free-slip upper boundary

In this paper, we report on direct numerical and large-eddy simulations of turbulent thermal convection without invoking the Oberbeck–Boussinesq approximation. The working medium is liquid water and we employ a free-slip upper boundary condition. This flow is a simplified model of thermal convection of water in a cavity heated from below with heat loss at its free surface. Analysis of the flow statistics suggests similarities in spatial structures to classical turbulent Rayleigh–Bénard convection but with turbulent fluctuations near the free-slip boundary. One important observation is the asymmetry in the thermal boundary layer heights at the lower and upper boundaries. Similarly, the budget of the turbulent kinetic energy shows different behaviour at the free-slip and at the lower wall. Interestingly, the work of the mean pressure is dominant due to the hydrostatic component of the mean-pressure gradient but also depends on the density fluctuations which are small but, critically, non-zero. As expected the boundary-layer heights decrease with the Rayleigh number, due to increased turbulence intensity. However, independent of the Rayleigh number, the height of the thermal boundary layer at the upper boundary is always smaller than that on the lower wall.

A Lorenz Model for Almost Compressible Fluids

Starting from the almost compressible 2-D Oberbeck–Boussi nesq system, characterized by an extra buoyancy term which is pressure-dependent, we write the corresponding Lorenz system and observe a larger instability with respect to the classical model in which the buoyancy is proportional to the temperature deviation only.

Analysis of Characteristics of Two-Layer Convective Flows with Diffusive Type Evaporation Based on Exact Solutions

The theoretical approaches for mathematical modelling of the convective flows with mass transfer through the liquid–gas interface are discussed. The special attention is payed to modelling with use of the classical Boussinesq approximation of the Navier–Stokes equations. The diffusion equation and the effects of thermodiffusion and thermal diffusivity (the Soret and Dufour effects) are taken into account additionally to describe vapor and heat transfer processes in the gas-vapor phase. The use of the Oberbeck–Boussinesq equations allows one to apply the group-analytical methods in the theory of the evaporative convection and to construct the exact solutions of special type of the governing equations. Joint flows of the evaporating liquid and gas-vapor mixture are studied with the help of a partially invariant solution for the convection equations. The 2D and 3D solutions are demonstrated to simulate two-phase flows in the infinite channels with interface being under action of a longitudinal temperature gradient and perpendicularly directed gravity field. In the present paper the fluid flows with diffusive evaporation/condensation in the terrestrial and microgravity conditions are studied in the steady case. The new results obtained for combined thermal regime on the external rigid boundaries are presented.

Direct numerical simulations of Rayleigh–Bénard convection in water with non-Oberbeck–Boussinesq effects

Rayleigh–Bénard convection in water is studied by means of direct numerical simulations, taking into account the variation of properties. The simulations considered a three-dimensional (3-D) cavity with a square cross-section and its two-dimensional (2-D) equivalent, covering a Rayleigh number range of $10^{6}\leqslant Ra\leqslant 10^{9}$ and using temperature differences up to 60 K. The main objectives of this study are (i) to investigate and report differences obtained by 2-D and 3-D simulations and (ii) to provide a first appreciation of the non-Oberbeck–Boussinesq (NOB) effects on the near-wall time-averaged and root-mean-squared (r.m.s.) temperature fields. The Nusselt number and the thermal boundary layer thickness exhibit the most pronounced differences when calculated in two dimensions and three dimensions, even though the $Ra$ scaling exponents are similar. These differences are closely related to the modification of the large-scale circulation pattern and become less pronounced when the NOB values are normalised with the respective Oberbeck–Boussinesq (OB) values. It is also demonstrated that NOB effects modify the near-wall temperature statistics, promoting the breaking of the top–bottom symmetry which characterises the OB approximation. The most prominent NOB effect in the near-wall region is the modification of the maximum r.m.s. values of temperature, which are found to increase at the top and decrease at the bottom of the cavity.

Boundary value and extremum problems for generalized Oberbeck–Boussinesq model

Boundary value and extremum problems for generalized Oberbeck–Boussinesq model

On low-Prandtl-number convection in an inclined layer of liquid mercury

This paper reports on a theoretical analysis of convection in an inclined layer of mercury, a common low-Prandtl-number fluid ($Pr=0.025$). The investigation is based on the standard Oberbeck–Boussinesq equations, which are explored as a function of the inclination angle $\unicode[STIX]{x1D6FE}$ and for Rayleigh numbers $R$ in the vicinity of the convection onset. Along with the conventional Galerkin methods to study convection rolls and their secondary instabilities, we employ direct numerical simulations for fluid layers with quite large aspect ratios. It turns out that, even for small inclination angles $\unicode[STIX]{x1D6FE}\lesssim 6^{\circ }$, the secondary instabilities of the basic rolls lead either to oscillatory three-dimensional patterns or to stationary ones, which appear alternately with increasing $\unicode[STIX]{x1D6FE}$. Due to the competition of these instabilities the patterns may show a complex dynamics.

Stability of the Horizontal Throughflow in a Power-Law Fluid Saturated Porous Layer

Double-diffusive convective instability of horizontal throughflow in a power-law fluid saturated porous layer is investigated. The boundaries of this horizontal porous layer are impermeable, isothermal and isosolutal. The generalized Darcy’s equation for power-law fluid is employed in the momentum balance, along with Oberbeck–Boussinesq approximation for buoyancy effect in the system. It is considered that the viscous dissipation is non-negligible. The basic temperature and concentration profiles for the considered horizontal throughflow are determined analytically, and the linear stability analysis is performed on this basic solution. The eigenvalue problem is solved numerically using bvp4c in MATLAB. The conditions for the onset of instability of base flow in terms of critical Rayleigh number and the corresponding wave number are obtained. It is found that critical values are affected by the flow governing parameters such as the Peclet number (Pe), power-law index (n), viscous dissipation (Ge), angle of inclination $$(\alpha )$$ , buoyancy ratio (N) and Lewis number (Le). For the considered range of these parameter values, the instability of the basic flow is discussed for the convective rolls. It is observed that instability of any general convective roll lies between that of the longitudinal and transverse rolls. Also, the effect of viscous dissipation on the onset of instability for both the longitudinal and transverse rolls is discussed in detail. This source of heating inside the porous layer increases the thermal instability of base flow. Further, the asymptotic analysis for vanishingly small Peclet number is presented as $$Pe = 0$$ is a singular point. The boundary between the stationary and the oscillatory instabilities is given using the buoyancy ratio parameter and the Lewis number in this limiting case of $$Pe\rightarrow 0$$ .

On the Conditions for Existence of Unidirectional Motions of Binary Mixtures in the Oberbeck—Boussinesq Model

Compatibility conditions are obtained for the nonstationary Oberbeck—Boussinesq equations describing the unidirectional motions of a liquid binary mixture in a horizontal strip. We examine the case of polynomial dependence of temperature on the longitudinal coordinate is considered, and the influence of the dependence of the kind on the remaining unknown functions from the original system. It is shown that a nonstationary unidirectional motion between two solid walls can be described by the Oberbeck—Boussinesq model only for the quadratic or linear law of temperature distribution along the horizontal coordinate. Some initial-boundary value problems are posed.