Classical Boussinesq Approximation Research Articles

Applying a momentum-based variable formulation in the SIMPLE algorithm to numerically solve thermo-buoyant turbulent flow in enclosures

Natural or buoyant convection flow is an exemplary heat transfer phenomenon, with growing applications in various industries. This article develops a new algorithm, which models and solves the buoyancy-driven turbulent flows in enclosures more accurately than the past similar solvers. A careful literature review shows that the past existing approaches have mostly had serious limitations to apply their algorithms to buoyancy-driven flows with high temperature differences magnitude because of employing the classical Boussinesq approximation. As the novelty of this study, it benefits from a momentum-based variable approach in the context of the semi-implicit method for the pressure linked equations (SIMPLE) algorithm, which lets it accurately solve the strong compressible buoyant flows with high temperature differences. The algorithm is applied to both the Navier-Stokes and the accompanied turbulent flow governing equations using OpenFOAM 4.1 as the platform. To validate the developed algorithm, the current results are compared with experimental data in both square and tall cavities considering low (8.6 × 105), high (1.43 × 106), and very high (1.58 × 109) Rayleigh numbers. As the major contribution of this work, it improves the accuracy of the thermo-buoyant turbulent flow prediction at both low and high Rayleigh numbers. All test cases are carried out employing two different turbulence models of k-ω and k-ε. Furthermore, comparing the results of the present non-Boussinesq algorithm and those of the past developed methods with the experimental data, it is shown that the present algorithm provides a more accurate prediction for the temperature field, that is, <10% differences with the experimental data. Moreover, the present maximum velocity results surpass the solution of the past numerical methods and show <3% differences with the experimental data.

Inhomogeneous Boundary Value Problems for the Generalized Boussinesq Model of Mass Transfer

We consider boundary value problems for a nonlinear mass transfer model, which generalizes the classical Boussinesq approximation, under inhomogeneous Dirichlet boundary conditions for the velocity and the substance’s concentration. It is assumed that the leading coefficients of viscosity and diffusion and the buoyancy force in the model equations depend on concentration. We develop a mathematical apparatus for studying the inhomogeneous boundary value problems under consideration. It is based on using a weak solution of the boundary value problem and on the construction of liftings of the inhomogeneous boundary data. They remove the inhomogeneity of the data and reduce initial problems to equivalent homogeneous boundary value problems. Based on this apparatus we will prove the theorem of the global existence of a weak solution to the boundary value problem under study and establish important properties of the solution. In particular, we will prove the validity of the maximum principle for the substance’s concentration. We will also establish sufficient conditions for the problem data, ensuring the local uniqueness of weak solutions.

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'.

Unsteady nonlinear convective flow of a nanofluid over a vertical plate due to impulsive motion: Optimization and sensitivity analysis

The classical Boussinesq approximation (linear density-temperature variation) is valid for small temperature differences in the system. However, density varies with temperature nonlinearly in various applications, such as heat exchangers, solar collectors, and nuclear reactors. Therefore, unsteady nonlinear mixed convection (quadratic density-temperature variation) in a nanofluid on a vertical plate due to impulsive motion is investigated. Modified Nield and Kuznetsov nanofluid model is incorporated that considers physical realistic boundary conditions. Unsteady flow is governed by the conservation of mass, momentum, energy, and the continuity equation of nanoparticles. Williams-Rhyne transformations are used to parameterize and reduce the infinite time domain to a finite time domain. Subsequent nonlinear differential equations system is solved by using the Finite Difference Method (FDM) based routine. Rayleigh and Hiemenz type equations are obtained respectively in an initial-unsteady and final-steady state. The Response Surface Method (RSM) and Central Composite Design (CCD) are used for the simultaneous optimization of the friction factor and the rate of heat transport. It is found that the quadratic density-temperature variation significantly affects the flow fields. The friction factor has the highest and positive sensitivity toward the thermal buoyancy number. The heat transport rate has the highest and negative sensitivity toward the thermo-migration of nanoparticles.

An efficient and simplified Gay‐Lussac approach in secondary variables form for the non‐Boussinesq simulation of free convection problems

AbstractThe Gay‐Lussac (GL) approach is an incompressible‐based strategy for non‐Boussinesq treatment of the governing equations for free convection problems that is established based on extending the density variations beyond the gravity term. Such a strategy leads to emerging the GL parameter as a non‐Boussinesq prefactor of different terms in the governing equations. In this article, the GL approach is expressed/discussed in terms of the secondary variables, that is, vorticity and stream‐function, for the first time and a simplified version of this approach is proposed by removing density variations from the continuity equation. The difference of results under the simplified and traditional GL approach ranges within a maximum of 1% for different parameters. The lower computational cost of numerical solution of governing equations in the secondary variables formula and the corresponding convergence rate is scrutinized for the simplified GL approach showing around 25% lower computational cost. The performance of this approach is evaluated at high relative temperature differences against the low Mach number scheme and the Boussinesq approximations. In this respect, natural convection in an annulus cavity is numerically simulated using a CVFEM solver under the aforementioned approximations up to Rayleigh number at Prandtl number and high relative temperature differences ( and 0.3). The largest deviations found for either the simplified GL or Boussinesq methods from the low Mach number scheme solution are less than 20% for velocity magnitude, 14% for stream function, 6% for vorticity, and 5% for temperature. Results under the three approximations are also analyzed in terms of the skin friction and local and average Nusselt number, indicating that the Gay‐Lussac approach requires some revisions to act more accurately than the classical Boussinesq approximation at high relative temperature differences in natural convection problems, especially within the convection dominated regime.

Approximation à la Oberbeck-Boussinesq for fluids with pressure-induced stratified density

ABSTRACT We consider a model for convection in compressible fluids in two dimensions. A constitutive limit is studied in which both the mechanical compressibility and thermal expansion affect the buoyancy force. The motion is no longer isochoric as in the classical Boussinesq approximation but has a uniform expansion rate associated to the upward motion: . By using a perturbative approach, we study a Boussinesq-like approximation with pressure-dependent buoyancy force. The existence of weak solutions for the approximated system is proved and their stability is investigated.

A centrifugal buoyancy formulation for Boussinesq‐type natural convection flows applied to the annulus cavity problem

SummaryTraditionally, the Boussinesq approximation is adopted for numerical simulation of natural convection phenomena where density variations are supposed negligible except through the gravity term of the momentum equation. In this study, a recently developed formulation based on a Boussinesq approximation is presented in which the density variations are also considered in the advection terms. Extending density‐variations to the advection terms captures centrifugal effects arising from both bulk enclosure rotation and within individual vortices, and thus more accurate results are expected. In this respect, the results of the proposed formulation are compared against the conventional Boussinesq simulations and weakly compressible approximation in the concentric horizontal annulus cavity. A new relation is established which maps the magnitude of the non‐Boussinesq parameter of incompressible flow to the corresponding relative temperature difference of a compressible flow simulation which is in agreement with the maximum allowed Gay‐Lussac number to avoid unphysical density values. For comparison purposes, variations of different thermo‐fluid parameters including average and local Nusselt number, entropy generation, and skin friction up to Ra = 105 are computed. Results obtained under the proposed approximation agree with the classical Boussinesq approximation up to Ra = 103 for large non‐Boussinesq parameter corresponding to the large relative temperature difference, but at Ra = 105, computed thermo‐fluid parameters via the two approaches are not identical which justifies the inclusion of large Gay‐Lussac number for convection dominated regime in natural convection problems.

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.

On the oscillatory modes of compressible thermal convection in inclined differentially heated cavities

We consider the case of a tall differentially heated cavity (containing air) for which the buoyancy flow is in the so-called boundary-layer regime (displaying thin regions featured by strong temperature gradients adjacent to the vertical walls and thermally stratified fluid in the centre). By using a numerical method able to account for compressibility and variable viscosity effects, we investigate the response of such a system to the application of temperature differences in a range of characteristic numbers for which the resulting flow is expected to be time-periodic or highly time-dependent. More precisely, the problem is examined by allowing both the Rayleigh number and the inclination (θ) of this configuration with respect to the vertical direction to span relatively wide intervals (1.8 × 105 ≤ Ra ≤ 3 × 106, −90° ≤ θ ≤ 90°). Insights into the influence of variable thermodynamic properties are obtained by comparing such numerical results with equivalent simulations based on the classical Boussinesq approximation. The evolution of the considered system towards a fully turbulent state is also discussed to a certain extent.

Effects of compressibility in the mantle convection Equations

The Boussinesq approximation of thermal convection equations results from neglecting the number of the terms which are actually not small in the conditions of the Earth’s mantle. However, the error of calculating the structure of the convective flows is lower than the discarded terms. In this work we analyze the causes of this fact by successively passing from the general equations for a heated viscous compressible fluid to the simpler thermal convection equations by rejecting small quantities with the parameters of the presentday Earth. We consider the anelastic liquid approximation (ALA), truncated anelastic liquid approximation (TALA), extended Boussinesq approximation (EBA), and the simplest classical Boussinesq approximation (BA) which fully disregards the compressibility of a fluid. With the parameters of the mantle, BA is only accurate when describing the flow velocities, while the temperature is predicted with an error of up to a few dozen percent. Therefore, it appears reasonable to consider an intermediate approximation between EBA and BA, in which the effects of compressibility are only taken into account for temperature. This approximation can be referred to as the superadiabatic Boussinesq approximation (SBA) for temperature Tsa. The corresponding equations are structurally similar to the standard Boussinesq approximation but with a superadiabatic temperature Tsa instead of total temperature T. In this simple approximation, the calculated structure of the convective flows and the distribution of total temperature (obtained by adding the known adiabatic Ta to the calculated Tsa) are more accurate than in the classical Boussinesq approximation.

High Gradient Temperature Thermo-Buoyant Flow in a Square Cavity with Magnetoconvection Using a Novel Non-Boussinesq Algorithm

We study numerically the heat transfer of steady laminar flow in a square cavity filled with electrically conducting fluids, in the presence of an external uniform magnetic field. Imposing a large temperature gradient between two opposite vertical walls, there are substantial temperature and density variations in the domain. The fluid is treated as an ideal gas. Indeed, high temperature gradient thermo-buoyant cavity flows result in natural convection flow domains with high Rayleigh number. To implement the temperature variation effect, the fluid properties, including the conductivity and viscosity coefficients, are considered to vary with temperature in accordance to the Sutherland's law. We use a novel non-Boussinesq algorithm to include the density variation in the domain. Of importance, the current results are different from those obtained considering the classical Boussinesq assumption, in which the density variation is replaced with the temperature variation. We use the finite volume method and SIMPLE algorithm to treat the continuity, momentum, and energy equations. The results show that the magnetic field plays a damping role in the flow pattern formations. We show that the increase of Hartmann number would decrease the maximum fluid velocity and Nusselt number magnitudes. Generally, the current study indicates that the computational heat transfer applicants, who need to take into account the effect of MHD in high thermobuoyant flow field treatments, should consider a variable density instead of using the classical Boussinesq approximation if they wish to achieve accurate solutions; specifically in higher Rayleigh number regime applications.

Holmboe instability in non-Boussinesq fluids

We perform the stability analysis for stratified shear flows whose density transition layer is much thinner than, and possibly, displaced with respect to, the velocity shear layer for which Holmboe instability along with the well-known Kelvin-Helmholtz (KH) instability is known to be present. Here, we provide a more complete picture of stability characteristics of stratified shear flows with taking into account the effects of non-negligible density increment for which the classical Boussinesq approximation is no longer valid. It is shown that, in addition to the Kelvin-Helmholtz and Holmboe instabilities for which two unstable modes exist, there is another instability with a single unstable mode so that the unstable waves excited by this instability mechanism propagate only in one direction. Depending on the physical parameters, this unstable mode may not be captured by the stability analysis under the Boussinesq approximation. With a better understanding of the instability mechanisms with including the non-Boussinesq effects, we could validate some of previous experimental results and provide new evidences to observations that have not been fully explained. The results are also expected to be useful in designing laboratory experiments to observe Holmboe waves and estimating their wavelengths and phase speeds.

Modeling of natural convection with Smoothed Particle Hydrodynamics: Non-Boussinesq formulation

A new variant of the Smoothed Particle Hydrodynamics (SPH) simulations of the natural convection phenomena is introduced. In many situations, where the density may vary significantly under the influence of temperature non-uniformities, the classical Boussinesq approximation fails. To characterize such situations, the Gay-Lussac dimensionless number is useful as a measure of density variations in non-isothermal flows. The novel points of the proposed modeling beyond the Boussinesq regime include the proper implementation of the buoyancy force and a smart connection between particle volume and incompressibility constraint. To examine the correctness of the new approach, numerical results are confronted with available reference data.

Numerical Simulation of Thermobuoyant Flow with LargeTemperature Variation

The use of the classical Boussinesq approximation is a straightforward strategy for taking into account the buoyancy effect in incompressible solvers. This strategy is highly effective if density variation is low. However, ignoring the importance of density variation in highly thermobuoyant flow fields can cause considerable deviation from the correct prediction of fluid flow behavior and the accurate estimation of heat transfer rate. In this study, an incompressible algorithm is suitably extended to solve high-density-variation fields caused by strong natural-convection influence. The key point in this research is the way that an ordinary incompressible algorithm is extended to non-Boussinesq-regime applications. The extension results in a unified algorithm capable of solving thermobuoyant flow fields in either a pure incompressible algorithm incorporated with the Boussinesq approximation or an entirely compressible algorithm where the density field is affected by both temperature and pressure fields. The extended algorithm is then verified by solving the benchmark convecting cavity problem at Rayleigh 10 6 and a temperature range of e = 0.01-0.6. The results show that the method can vigorously solve thermobuoyant flow fields with extreme density variation.

Analytical and numerical investigation of double diffusion in thermally anisotropy multilayer porous medium

Double-diffusive natural convection within a multilayer anisotropic porous medium is studied numerically and analytically. The domain composed of two horizontal porous layers is subjected to a uniform horizontal heat flux and a vertical mass flux, where only the lower one is thermally anisotropic. Darcy model with classical Boussinesq approximation is used in formulating the mathematical model. The effect of thermal anisotropy and the relative width of the two layers on the flow and transfers is illustrated with characterising the transitions from the diffusive to the convective solution. Results were well compared with respect to a developed analytical approach, based on a parallel flow approximation for thermally anisotropic multilayer media.

CONJUGATE MIXED CONVECTION IN AIR-COOLED HEAT SINKS USING A NON-BOUSSINESQ APPROACH

This work reports a numerical study of the mixed convection in finned duct flow that occurs in heat sinks devices. The laminar flow is considered fully developed and the convection-conduction coupling is treated by a conjugated approach. The mathematical formulation of this problem is constituted by the mass, momentum and energy equations. The partial differential equations system is solved by the Galerkin finite element method, adopting a pressure Poisson equation to establish the pressure-velocity coupling and to obtain a mass conserving flow. The results using the classical Boussinesq approximation (density varies linearly with the temperature in the buoyancy-term) are compared with the non-Boussinesq approach (density variation in all terms of the governing equations) showing that both the heat transfer and friction factor are affected by the new considerations. The duct aspect ratio and the solid to fluid thermal conductivity ratio influences on the heat transfer rate are also analyzed. This analysis tool was also shown appropriate for the optimization of electronic components air-cooled heat sinks.

CONJUGATE MIXED CONVECTION IN AIR-COOLED HEAT SINKS USING A NON-BOUSSINESQ APPROACH

This work reports a numerical study of the mixed convection in finned duct flow that occurs in heat sinks devices. The laminar flow is considered fully developed and the convection-conduction coupling is treated by a conjugated approach. The mathematical formulation of this problem is constituted by the mass, momentum and energy equations. The partial differential equations system is solved by the Galerkin finite element method, adopting a pressure Poisson equation to establish the pressure-velocity coupling and to obtain a mass conserving flow. The results using the classical Boussinesq approximation (density varies linearly with the temperature in the buoyancy-term) are compared with the non-Boussinesq approach (density variation in all terms of the governing equations) showing that both the heat transfer and friction factor are affected by the new considerations. The duct aspect ratio and the solid to fluid thermal conductivity ratio influences on the heat transfer rate are also analyzed. This analysis tool was also shown appropriate for the optimization of electronic components air-cooled heat sinks.

Transient flow rate behaviour in an external natural convection boundary layer

A theoretical approach is proposed to investigate the transient dynamic behaviour of a free convection boundary layer-type flow. The set of continuity, momentum and energy equations are solved with the classical Boussinesq approximation using the Karman–Pohlhausen integral method. Applying a step variation of the uniform heat flux on a vertical wall, the boundary layer thickness and velocity profiles within the viscous layer, streamline patterns and volumetric flow rate are evaluated as a function of time. In addition, corresponding fully analytical asymptotic solutions are derived to be readily used in engineering applications.

Transient laminar natural convection along rectangular ducts

An analytical approach is proposed to investigate the transient behaviour of a Newtonian, single-phase fluid in natural laminar convection, in a rectangular open-ended duct. The continuity, momentum and energy equations, with the classical Boussinesq approximation, are solved using a twofold sine Fourier transform and the Laplace transform. The unsteady state is due to a step variation of the temperature in the four walls of the duct, which can assume four different, uniform arbitrary values. Considering hydrodynamically developed flow and uniform wall temperatures (UWTs), the velocity and temperature of the fluid are given as series containing two spatial co-ordinates and the time. Some plots show the transient evolution of the velocity and temperature distribution. Then the induced volumetric flow rate, the exchanged power, the mixing cup temperature, and the average Nusselt number are evaluated, as a function of time, emphasising the influence of the duct aspect ratio and the irrelevance of the channel height.

Wavelet spectra and chaos in thermal convection modelling

Thermal convection in planetary bodies represents a chaotic dynamical system, which can be mathematically described by a set of nonlinear equations. We have studied classical Boussinesq approximation of thermal convection with no inertial forces. Even for a low Rayleigh number (105) the convection is chaotic and the wavelet analysis of temperature, kinetic energy and surface heat flow time series demonstrates that wavelet spectra are able to yield an insight into the multiscale temporal dynamics of the convecting systems.