Problem Of Natural Convection Research Articles

Augmentation and diminution of non-Boussinesq effects due to non-Newtonian power-law behavior in natural convection

This study investigates the extent and significance of augmentation and diminution of non-Oberbeck–Boussinesq (NOB) effects due to power-law rheology in natural convection. The relative significance of each temperature dependent thermophysical property of a non-Newtonian power-law fluid was first evaluated using an order of magnitude analysis. The limiting criteria were derived for a class of fluids having an Arrhenius-type thermodependent consistency index, and a constant power-law index. Significant fluid property dependencies were identified, incorporated into the conservation equations, and solved numerically. The benchmark problem of natural convection around an isothermal vertical flat plate, immersed in a quiescent power-law fluid, was re-investigated from the viewpoint of NOB effects. It was found that power-law rheology significantly augmented or diminished the acceleration of flow caused by NOB effects. Compared to the Newtonian case, strong shear thinning behavior more than doubled the NOB acceleration of the flow field, and shear thickening inhibited the acceleration by nearly half. Numerical solutions facilitated the visualization of velocity and temperature distributions, and thus provided insights to the underlying physics. It is demonstrated that the cumulative effect of the power-law behavior and temperature dependence of properties is more than a mere superposition of the individual effects. Important contributions from the present work include modifications to the OB approximation’s limiting criteria applicable to power-law fluids, insights into the effect of temperature dependent properties on the flow field and consequent heat transfer, and their correspondence to the results of the order of magnitude analysis. The present investigation of the underlying physics of augmentation and diminution facilitate a better understanding for future studies of NOB effects combined with non-Newtonian behavior.

CFD-based analysis of entropy generation in turbulent double diffusive natural convection flow in square cavity

The present study concerns the problem of natural and double diffusive natural convection inside differentially heated cavity filled with a binary mixture composed of air and carbon dioxide (CO2). Temperature and CO2 concentration gradients are imposed on both perpendicular left and right walls. Simulations have been performed using the CFD commercial code ANSYS Fluent by solving continuity, momentum, energy and species diffusion equations. Numerical results obtained have been compared to data from the literature for both natural convection thermosolutal cases under laminar and turbulent regimes. For turbulent runs the RNG k-ε model has been selected. A good agreement has been noted between the different types of data for both cases for Rayleigh number ranging between 103 and 1010 and buoyancy ratio between -5 and +5. Entropy generation rates due to thermal, viscous and diffusive effects have been calculated in post processing for all cases.

Natural Convection in a Partially Heated and Cooled Square Enclosure Containing a Diamond Shaped Heated Block

Finite element method is used to solve the two dimensional governing mass, momentum and energy equations for steady state, natural convection problem inside a square enclosure. The enclosure consists of adiabatic vertical walls, heated middle part of bottom wall and the cold (top wall and the rest part of bottom wall) walls and a uniformly heated diamond shaped solid body located somewhere inside the enclosure. The aim of this study is to describe the effect of different sizes and positions of diamond shaped heated block on natural convection. The investigations are conducted for different values of Rayleigh number (Ra), block length (l) and location of block center (Cx, Cy) inside the enclosure by using COMSOL multiphysics. Various results such as streamlines, isotherms, heat transfer rate in terms of the average Nusselt number and average fluid temperature inside the enclosure are presented for different parameters. The results indicate that the average Nusselt number at the heated surface and average temperature of the fluid inside the enclosure are strongly dependent on the configuration of the system under different geometrical and physical conditions. The average Nusselt number decreases with the increasing value of block size and increases in the free convection dominated region, it is maximum for Ra=106 and minimum for Ra=103. Block size also has significant effect on thermal fields. Average temperature increases with the increasing value of heated block.

Novel fractional time-stepping algorithms for natural convection problems with variable density

This work presents novel fractional time-stepping finite element approaches for solving incompressible natural convection problems with variable density. Compared with other projection methods, the main merit of these methods is that we only need to solve one Poisson equation per time step for the pressure, which is computationally more efficient. First-order stability analyses and error estimates are proved and stability analysis of second-order version is established. Numerical tests which confirm the theoretical analyses are provided.

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

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

Efficient high order accurate staggered semi-implicit discontinuous Galerkin methods for natural convection problems

In this article we propose a new family of high order staggered semi-implicit discontinuous Galerkin (DG) methods for the simulation of natural convection problems. Assuming small temperature fluctuations, the Boussinesq approximation is valid and in this case the flow can simply be modeled by the incompressible Navier-Stokes equations coupled with a transport equation for the temperature and a buoyancy source term in the momentum equation. Our numerical scheme is developed starting from the work presented in [1, 2, 3], in which the spatial domain is discretized using a face-based staggered unstructured mesh. The pressure and temperature variables are defined on the primal simplex elements, while the velocity is assigned to the dual grid. For the computation of the advection and diffusion terms, two different algorithms are presented: i) a purely Eulerian upwind-type scheme and ii) an Eulerian-Lagrangian approach. The first methodology leads to a conservative scheme whose major drawback is the time step restriction imposed by the CFL stability condition due to the explicit discretization of the convective terms. On the contrary, computational efficiency can be notably improved relying on an Eulerian-Lagrangian approach in which the Lagrangian trajectories of the flow are tracked back. This method leads to an unconditionally stable scheme if the diffusive terms are discretized implicitly. Once the advection and diffusion contributions have been computed, the pressure Poisson equation is solved and the velocity is updated. As a second model for the computation of buoyancy-driven flows, in this paper we also consider the full compressible Navier-Stokes equations. The staggered semi-implicit DG method first proposed in [4] for all Mach number flows is properly extended to account for the gravity source terms arising in the momentum and energy conservation laws. In order to assess the validity and the robustness of our novel class of staggered semi-implicit DG schemes, several classical benchmark problems are considered, showing in all cases a good agreement with available numerical reference data. Furthermore, a detailed comparison between the incompressible and the compressible solver is presented. Finally, advantages and disadvantages of the Eulerian and the Eulerian-Lagrangian methods for the discretization of the nonlinear convective terms are carefully studied.

The interfacial nanolayer role on magnetohydrodynamic natural convection of an Al2O3-water nanofluid

The present analytical study deals with the laminar steady-state two-dimensional magnetohydrodynamic natural convection within a shallow enclosure. The latter is filled with an Al2O3-water nanofluid, which is determined by volumetrically internal heat sources subject to an externally applied uniform vertical magnetic field. The analytical model incorporates the role of a nanolayer, which is adjacent to the solid particle, in order to comprehend the anomalous heat conducting properties of nanofluids. Besides, the effects of the nanoparticle volume fraction and its size are examined on the cooling process. All the aforementioned factors seem to play a key role on the heat transfer. In brief, via shrinking the nanoparticles, increasing their volume fraction as well as enlarging the nanolayer thickness results (especially for comparable nanoparticle radii and nanolayer thicknesses) in deceleration of the nanofluid motion and, thus, in deterioration of the heat transfer. Finally, the analytical results are expected to be of particular importance regarding the comprehension of the versatile problem of magnetohydrodynamic natural convection of nanofluids given the fast-growing interest in this scientific field.

Natural convection of a non-Newtonian ferrofluid in a porous elliptical enclosure in the presence of a non-uniform magnetic field

In the present study, laminar natural convection of a non-Newtonian ferrofluid inside an elliptical porous cavity was numerically simulated in the presence of a non-uniform external magnetic field. This natural convection problem was relevant to the cooling of micro-sized electronic devices. The well-known finite volume method was employed to discretize the governing equations for ferrofluid flow under the effect of an external magnetic field. The effects of pertinent non-dimensional numbers including the Rayleigh number, the magnetic number, the power-law index, and the Darcy number were studied on the flow pattern and the heat transfer rate of the non-Newtonian ferrofluid. The results showed that by applying the magnetic field by a wire, the overall heat transfer rate increased significantly. Moreover, to achieve the maximum heat transfer enhancement, the wire should have been placed at the center of the elliptical walls of the enclosure. It was also shown that the impact of the power-law index on the heat transfer rate was considerable, and using a shear-thinning liquid increased the average Nusselt number in the porous elliptical enclosure.

A “poor man’s” approach for high-resolution three-dimensional topology design for natural convection problems

This paper treats topology optimization of natural convection problems. A simplified model is suggested to describe the flow of a steady-state incompressible fluid, similar to Darcy’s law for porous media. By neglecting inertia and viscous boundary layers, the flow model is significantly simplified. The fluid flow is coupled to the thermal convection-diffusion equation through the Boussinesq approximation. The coupled non-linear system of equations is discretized with stabilized finite elements and solved in a parallel framework that allows for the optimization of high-resolution three-dimensional problems. A density-based topology optimization approach is used, where the permeability and conductivity of the distributed material is interpolated. Due to the simplified model, the proposed methodology significantly reduces the computational effort required in the optimization. At the same time, it is notably more accurate than even simpler models that rely on Newton’s law of cooling. The methodology discussed herein is applied to the optimization-based design of three-dimensional heat sinks. The final designs are compared with previous work obtained from solving the full set of Navier–Stokes equations, both in terms of design performance and computational cost. The computational time is shown to be decreased to around 5−20% in terms of core-hours, allowing for the possibility of optimizing designs during the workday on a small computational cluster and overnight on a high-end desktop. However, due to the use of a simplified model, the performance of the final designs are evaluated using the full Navier–Stokes equations. This ensures verification of performance, as well as systematic comparison with reference results.

Assessment of Smoothed Particle Hydrodynamics (SPH) models for predicting wall heat transfer rate at complex boundary

Nowadays, the use of Smoothed Particle Hydrodynamics (SPH) approach in thermo-fluid application has been starting to gain popularity. Depending on the SPH boundary condition treatment, different methods can be devised to compute the total wall heat transfer rate. In this paper, for the first time, the accuracies of using the popular dummy particle methods, i.e. (a) the Adami Approach (AA) and (b) the higher-order mirror + Moving Least Square (MMLS) method in predicting the total wall heat transfer rate are comprehensively assessed. The modified equation of the 1D wall heat transfer rate is formulated using Taylor's series. For uniform particle layout, MMLS is first-order accurate. Nevertheless, for an irregular particle layout, its order of accuracy drops to ~O(1), the order similar to that of the computationally simpler AA. The AA method is then used to simulate several steady and unsteady natural convection problems involving convex and concave wall geometries. The estimated wall heat transfer rate and the flow results agree considerably well with the available experimental data and benchmark numerical solutions. In general, the current work shows that AA can offer a practical means of estimating wall heat transfer rate at reasonable accuracy for problems involving complex geometry.

A localized meshless approach using radial basis functions for conjugate heat transfer problems in a heat exchanger

An understanding of conjugate heat transfer is important for enhancing the thermal performance of heat transfer devices. This study discusses the numerical solutions of the conjugate natural convective heat transfer problem using a local radial basis function meshless method. The temperature difference is assumed to be small enough to ensure the validity of the Boussinesq approximation. An artificial compressibility method is used to couple pressure to the continuity equation. The meshless method requires only one set of nodes, which are distributed in the spatial domain. The semi-algebraic equations are integrated using the second-order Runge–Kutta method. Three benchmark problems of conjugate natural convection with one vertical wall, conjugate natural convection with centrally vertical partition, and natural convection with a conducting solid in the middle of the square cavity are analyzed. The numerical results are compared with three benchmark solutions and experimental results; it was observed that the experimental and predicted results showed good agreement with each other. This numerical study is to contribute to understanding the knowledge of heat and fluid flow mechanisms for conjugate heat transfer in a heat exchanger acquired through experiments.

A discrete Hopf interpolant and stability of the finite element method for natural convection

The temperature in natural convection problems is, under mild data assumptions, uniformly bounded in time. This property has not yet been proven for the standard finite element method (FEM) approximation of natural convection problems with nonhomogeneous partitioned Dirichlet boundary conditions, e.g., the differentially heated vertical wall and Rayleigh-Benard problems. For these problems, only stability in time, allowing for possible exponential growth of $\| T^{n}_{h} \| $, has been proven using Gronwall's inequality. Herein, we prove that the temperature approximation can grow at most linearly in time provided that the first mesh line in the finite element mesh is within $\mathcal{O} (Ra^{-1})$ of the nonhomogeneous Dirichlet boundary.

Effect of an uniform magnetic field on unsteady natural convection of nanofluid

In this work, the problem of the unsteady natural convection in an -water filled nanofluids influenced by an uniform magnetic field is analysed numerically. Governing equations are given in terms of the streamlines, the vorticity and the temperature of the fluid. Dual Reciprocity Boundary Element Method (DRBEM) is performed as a solution technique. Both the fundamental solutions of Laplace equation and modified Helmholtz equation are used in the derivation of the method. The numerical analysis are done for physical parameters Rayleigh number (), Hartmann number () and particle volume fraction (). From the results, it is observed that the behaviours of all the variables are influenced by the changing values of parameters. The solution procedure of the all variables needs considerably small number of iterations and large time increments with suitable values of relaxation parameters which occur in the argument of Bessel function .

Exponential Solution for the Natural Convection of a Darcian Fluid About a Full Cone in a Porous Medium

In the current work, a new approach based on exponential functions is presented to reach the exact solutions to the problem of natural convection about a full cone which is set in a porous medium for the both states of prescribed surface heat flux and wall temperature. The solutions are described in the form of the exponential functions series with unknown coefficients while the operational matrices of derivative and product of these functions are utilized to find these coefficients iteratively without solving the system of nonlinear equations. The solutions demonstrate the validity and applicability of the proposed technique by means of the residual function error norm.

Magnetohydrodynamic Natural Convection Heat Transfer of Hybrid Nanofluid in a Square Enclosure in the Presence of a Wavy Circular Conductive Cylinder

Abstract In this paper, steady natural convective heat transfer and flow characteristics of Al2O3-Cu/water hybrid nanofluid filled square enclosure in the presence of magnetic field has been investigated numerically. The enclosure is equipped with a wavy circular conductive cylinder. The natural convection in the cavity is induced by a temperature difference between the vertical left hot wall and the other right cold wall. The steady 2-D equations of laminar natural convection problem for Newtonian and incompressible mixture are discretized using the finite volume method. The effective thermal conductivity and viscosity of the hybrid nanofluid are calculated using Corcione correlations taking into consideration the Brownian motion of the nanoparticles. A numerical parametric investigation is carried out for different values of the nanoparticles volumic concentration, Hartmann number, Rayleigh number, and the ratio of fluid to solid thermal conductivities. According to the results, the corrugated conductive block plays an important role in controlling the convective flow characteristic and the heat transfer rate within the system.

Recovery-based error estimator for the natural-convection problem based on penalized finite element method

Purpose This paper aims to proposes and analyzes a novel recovery-based posteriori error estimator for the stationary natural-convection problem based on penalized finite element method. Design/methodology/approach The optimal error estimates of the penalty FEM are established by using the lower-order finite element pair P1-P0-P1 which does not satisfy the discrete inf-sup condition. Besides, a new recovery type posteriori estimator in view of the gradient recovery and superconvergent theory to deal with the discontinuity of the gradient of numerical solution. Findings The stability, accuracy and efficiency of the proposed method are confirmed by several numerical investigations. Originality/value The provided reliability and efficiency analysis is shown that the true error can be effectively bounded by the recovery-based error estimator.

Efficient monolithic projection method with staggered time discretization for natural convection problems

For a more efficient algorithm, we introduce staggered time discretization to improve the previous method (Pan et al., 2017), fully decoupled monolithic projection method with one Poisson equation (FDMPM-1P), to solve time-dependent natural convection problems. The momentum and energy equations are discretized using the Crank–Nicolson scheme at the staggered time grids, in which temperature and pressure fields are evaluated at half-integer time levels (n+12), while the velocity fields are evaluated at integer time levels (n+1). Numerical simulations of two-dimensional (2D) Rayleigh–Bénard convection show that the proposed method is more computationally efficient and stable than FDMPM-1P, while preserving the second-order spatial and temporal accuracy. Further, the proposed method provides accurate predictions of 2D Rayleigh–Bénard convection under different thermal boundary conditions for a Rayleigh number up to 1010, three-dimensional turbulent Rayleigh–Bénard convection in the range of 1×105–2×107 in horizontal periodic domain, and three-dimensional turbulent Rayleigh–Bénard convection in the range of 1×106–1×107 in bounded domain. Finally, we theoretically confirmed that the proposed method is second-order in time and it is more stable than FDMPM-1P for small Ra.

Parallel two-step finite element algorithm based on fully overlapping domain decomposition for the time-dependent natural convection problem

Purpose This paper aims to propose two parallel two-step finite element algorithms based on fully overlapping domain decomposition for solving the 2D/3D time-dependent natural convection problem. Design/methodology/approach The first-order implicit Euler formula and second-order Crank–Nicolson formula are used to time discretization respectively. Each processor of the algorithms computes a stabilized solution in its own global composite mesh in parallel. These algorithms compute a nonlinear system for the velocity, pressure and temperature based on a lower-order element pair (P1b-P1-P1) and solve a linear approximation based on a higher-order element pair (P2-P1-P2) on the same mesh, which shows that the new algorithms have the same convergence rate as the two-step finite element methods. What is more, the stability analysis of the proposed algorithms is derived. Finally, numerical experiments are presented to demonstrate the efficacy and accuracy of the proposed algorithms. Findings Finally, numerical experiments are presented to demonstrate the efficacy and accuracy of the proposed algorithms. Originality/value The novel parallel two-step algorithms for incompressible natural convection problem are proposed. The rigorous analysis of the stability is given for the proposed parallel two-step algorithms. Extensive 2D/3D numerical tests demonstrate that the parallel two-step algorithms can deal with the incompressible natural convection problem for high Rayleigh number well.

Lattice Boltzmann method for conjugate natural convection with heat generation on non-uniform meshes

In this paper, the Taylor series expansion and least square-based lattice Boltzmann method (TLLBM) is combined with a local lattice Boltzmann (LB) conjugate heat transfer scheme to simulate different conjugate natural convection problems in a differentially heated enclosure on non-uniform meshes. Four case studies are designed not only to verify the developed basic TLLBM FORTRAN code but also to validate the results of the TLLBM for different conjugate natural convection problems. The convergence study of the presented TLLBM is also carried out and the order of accuracy of the method is reported in different mesh regions. It is concluded that the results of streamlines, isotherms, temperature distributions along a test line, and average Nusselt numbers calculated on the cold and hot walls of the enclosure are compared well with those available in the literature from conventional numerical methods as well as the standard LBM. The computational efficiency of the TLLBM is also examined through comparing the results of simulations on uniform (standard LBM) and non-uniform (TLLBM) meshes in terms of the accuracy and CPU time needed for solutions to converge. It is shown that the TLLBM is able to give accurate results with lower CPU time and mesh sizes than the standard LBM.

Numerical modeling of natural convection in horizontal and inclined square cavities filled with nanofluid in the presence of magnetic field

In this paper, we have studied the problem of natural convection in an inclined square cavity filled with a Cu/water nanofluid; this is done under a differentially heated configuration in the presence of a magnetic field which tracks the cavity in its inclination. The dimensionless governing equations formulated using stream function, vorticity and temperature have been solved by the finite difference method of second-order accuracy, where the upwind scheme is employed to discretize the convective terms. It turned out that the developed code requires a numerical validation. To do this, a comparison with previously published numerical and experiments results, as well as with the result of simulations under the COMSOL Multiphysics software was performed. The results thus found express very good agreement with the results of the developed code. Heat transfer and fluid flow are examined for parameters of nanoparticles volume fraction ($0\% \leq \phi \leq 20\%$), Grashof number ($ 10^{3}\leq Gr\leq 10^{5}$, Hartmann number ($ 0\leq Ha^{2}\leq 100$), and the cavity inclination angle ($ 0^{\circ}\leq\gamma \leq\pi/2$). Thus, it is found that a good enhancement in the heat transfer rate can be obtained by adding copper nanoparticles to the base fluid, as well as the increasing of Grashof number. However, the presence of the magnetic field in the fluid region causes a significant reduction in the fluid flow and heat transfer characteristics.