Time-dependent Natural Convection Research Articles

An artificial compressibility SAV finite element method for the time-dependent natural convection problem

In this article, we present, analyze, and test a first-order implicit-explicit type scheme based on the artificial compressibility (AC) method and the scalar auxiliary variable (SAV) approach for solving the time-dependent natural convection problem. The proposed method is a combination of a mixed finite element approximation for spatial discretization, the first-order backward Euler scheme for temporal discretization, and explicit treatment for nonlinear terms. It effectively decouples the velocity and pressure via artificial compressibility, thereby reducing computational complexity and execution time. Moreover, we use the SAV approach for the convective terms, it leads to the scheme is linear, and only require solving a sequence of linear differential equation with constant coefficients at each time step. It is proved that the solution of this method is bounded and satisfying the law of energy dissipation. The rigorous error estimate of velocity, pressure, and temperature are derived. Finally, some numerical tests are implemented to verify the theoretical analysis and illustrate the efficiency of the presented method.

Numerical analysis of unsteady free convection under the combined influence of inclined magnetohydrodynamic and exothermic chemical reaction in an enclosure filled with nanofluid

Unsteady study of the natural convection of aluminum oxide-water nanofluid within a trapezoidal geometry containing a circular cylinder located at its center. Finite Element method has been considered for the numerical analysis. The proposed investigation handled the impact of Rayleigh number (103–105), chemical reaction parameter (0–4), aluminum oxide nanoparticles volume fraction (0–0.06), magnetic field (0–63) and its inclination angle (0°–90°), and circular obstacle diameter (0.3–0.7) effects on time-dependent natural convection of Al2O3–H2O nanofluid. On the other hand, the value of Prandtl number has kept constant at (Pr = 6.2). Since the nanofluid mobility at φ = 0.02, Ha = 3, and Fk = 1 significantly improved, the heat transfer rate achieved its maximum intensity at Ra = 105. Research also reveals a little effect on heat transfer by increasing the fraction of nanoparticles. Additionally, as Ha intensifies from 0 to 63, a final change in the mean Nusselt number of 28.65 % is displayed. Finally, as the magnetic field angle of rotation is diminished, more enhancement in heat transmission is achieved. This research provides insights into the intricate relationship between natural convection and exothermic reaction under the influences of various conditions. This can illustrate the flow and thermal behaviors of nanofluid in such non-uniform shapes in many engineering applications.

Unconditional stability of first and second orders implicit/explicit schemes for the natural convection equations

In this paper, we consider and analyze the stability of three implicit/explicit (IMEX) schemes for the time-dependent natural convection problem, these considered numerical schemes contain the first order backward Euler scheme, second order Crank-Nicolson IMEX scheme and BDF2-AB2 combination. All numerical schemes deal with the linear terms implicitly and the nonlinear terms explicitly. Then the original nonlinear problem is split into two linearized subproblems with constant coefficient matrixes, which reduces the computational complexity and saves a lot of storage. The biggest feature of this paper is to establish the unconditional stability of three IMEX schemes for incompressible flows, which improves and supplements the published theoretical findings. Finally, some numerical examples are provided to show the performances of the considered numerical schemes.

Superconvergence error analysis of an efficient mixed finite element method for time-dependent natural convection problem

Superconvergence error analysis of an efficient mixed finite element method for time-dependent natural convection problem

Impact of Thermal Stratification on Unsteady Natural Convection Couette Flow Formation in a Vertical Channel Filled with Anisotropic Porous Material

Recently, heat transfer problems where anisotropic porous medium or stably stratified fluid are taken into account have been separately studied. Developing a mathematical model that combines these physical quantities naturally results to complex coupled differential equations. In this paper, a fully developed time dependent natural convection Couette flow of stably stratified fluid between vertical parallel channels filled with anisotropic porous material is investigated. The governing partial differential equations are transformed into ordinary differential equations using Laplace transform techniques and then decoupled using D’Alembert method. Exact solutions in Laplace domain for the velocity and temperature equations are then obtained. A numerical method: Riemann-sum approximation is then used to invert the expressions for the velocity and temperature profiles, as well as the resulting skin friction, rate of heat transfer and volumetric mass flow rate into their corresponding time domain. The research establishes that both the anisotropic and the stratification parameters aid in regulating the fluid temperature and velocity. The research further reveals that the fluid velocity attains its maximum (or minimum) velocity when θ = 900 (or θ = 00) for k*<1 and when k*>1, the fluid velocity is least (or maximum) when θ = 900 (or θ = 00).

A global approach for the time-dependent solution of natural convection in a tilted porous cavity

Natural convection in an inclined cavity filled with saturated porous media has been investigated. A time-dependent solution using a spectral weighted residual method is determined. Preparing the governing equations by the substitution of the stream functions, we obtained a system of nonlinear ordinary differential equations. Integrating in time using the Runge-Kutta method of variable order the final system is determined. The study consisted of the investigation of the problem in the transient and unsteady modes. For each configuration, the solution is developed for different values of Rayleigh numbers and different inclination levels of the cavity. The high accuracy of the results with the reduced computational cost is well proved by the comparison with results determined using a finite element method that has shown an excellent agreement with our developed solution. This work is very efficient to understand from a physical insight the effect of the inclination parameters on the unsteady and transient mode of changes in the mechanisms of natural convection. The results provide a set of high-accurate data that can be used as a benchmark case for the time-dependent natural convection problems. As a result, the inclination effect of the cavity has clearly influenced the heat transfer process and the fluid flow. Therefore, the inclination can be considered as a control parameter for heat and fluid flow effectively.

Effects of Thermal Stratification and Anisotropic Porous Material in a Symmetrically Heated Vertical Channel: EFFECTS OF THERMAL STRATIFICATION AND ANISOTROPIC POROUS MATERIAL

The coexistence in nature of thermally stratified fluids and directionally inclined porous structures abound in geophysical fluid flow. Despite their important applications especially in underground flow, their combined effects on natural convection have not been intensively investigated. This article aims to investigate a fully developed time dependent natural convection of viscous incompressible stably stratified fluid in the presence of anisotropic porous material. The semi analytical solutions of the governing equations for the temperature and velocity fields are obtained using the Laplace Transform Technique, Riemann Sum Approximation and D’Alembert Methods. The choice of the D’Alembert Method is to provide a systematic decoupling procedure for the coupled governing equations while still retaining their original orders. This research establishes that the fluid maximum or minimum velocity can be attained by properly manipulating the anisotropic parameters.

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.

Chemically reactive species of time-dependent natural convection couple stress fluid flow past an isothermal vertical flat plate

The transient two-dimensional natural convective couple stress fluid flow past a semi-infinite vertical flat plate in the presence of first-order homogenous chemical reaction is investigated. The analysis has been carried out by considering the effects of skin-friction coefficient and Nusselt and Sherwood numbers. The unsteady coupled nonlinear governing flow equations have been solved by applying the Crank–Nicolson implicit finite difference scheme. For the different set physical parameters, graphs are shown and examined. A relevant study with existing results is made in a limiting sense. The transient and steady-state velocity profiles decrease as the chemical reaction parameter, Schmidt number, and couple stress parameter increase. The deviations of concentration, temperature, and velocity contours of the couple stress fluid flow are considerably varied in comparison with the Newtonian fluid flow.

Parallel two-grid finite element method for the time-dependent natural convection problem with non-smooth initial data

In this paper, we consider the parallel two-grid finite element method for the transient natural convection problem with non-smooth initial data. Our numerical scheme involves solving a nonlinear natural convection problem on the coarse grid and solving a linear natural convection problem on the fine grid. The linear natural convection problem can be split into two subproblems which can be solved in parallel: a linearized Navier–Stokes problem and a linear parabolic problem. We firstly provide the stability and convergence of standard Galerkin finite element method with non-smooth initial data. Secondly, we develop optimal error estimates of two-grid finite element method for velocity and temperature in H1-norm and for pressure in L2-norm. In order to overcome the difficulty posed by the loss of regularity, some suitable weight functions are introduced in our stability and convergence analysis for the natural convection equations. Finally, some numerical results are presented to verify the established theoretical results.

Numerical Investigation of Transient Natural Convection Heat Transfer of non-Newtonian Nanofluids Between Eccentric Annulus

The two-dimensional time-dependent natural convection heat transfer of non-Newtonian nanofluids is studied numerically between the gap of two horizontal cylinders. The inner and outer cylinders were kept at $$T_{\mathrm{i}}$$ and $$T_{\mathrm{o}}$$ ( $$T_{\mathrm{i}} > T_{\mathrm{o}}$$ ), respectively. The finite volume technique was used in the study. The effects of Ra number ( $$10^{3} \le Ra \le 10^{6}$$ ), aspect ratio ( $$0.4 \le \hbox {AR} \le 1.2$$ ), power law index of the nanofluid ( $$0.6 \le n \le 1.4$$ ), horizontal eccentricity ( $$0.2 \le {\varepsilon }_{\mathrm{H}} \le 0.8$$ ), vertical eccentricity ( $$-0.8 \le {\varepsilon }_{\mathrm{v}} \le +0.8$$ ) and rotation of the inner cylinder ( $$0 \le {\omega } \le 500$$ ) on the flow field and heat transfer were investigated. It was found that a very good agreement exists with the present numerical results and the results of the open literature. The results indicate that with increasing the Ra number from $$10^{3}$$ to $$10^{6}$$ the heat transfer increases by about 80%. The increase in aspect ratio also increases the heat transfer. The results showed that with increasing the power law index from 0.6 to 1.4, the heat transfer decreases by a rate of about 78%. In addition, the time to reach steady-state conditions increases with increasing power law index. Simulations were performed for a power law index of 0.8 to investigate the effect of eccentricity of the inner cylinder, and it was observed that the heat transfer is higher for negative vertical eccentricities, and the time to reach steady conditions shortens with the effect of negative vertical eccentricity and power law index.

A semi-analytical solution for time-dependent natural convection flow with heat generation/absorption in an annulus partially filled with porous material

Purpose The purpose of this paper is to present a semi-analytical solution for time-dependent natural convection flow with heat generation/absorption in an annulus partially filled with porous material. Design/methodology/approach The governing partial differential equations are transformed into the ordinary differential equations using the Laplace transform technique. The exact solution obtained is inverted from the Laplace domain to time domain using the Riemann-sum approximation approach. Justification of the Riemann-sum approximation approach is achieved by comparing the values obtained with those of the implicit finite difference method at both the transient state and the steady state at large time. Findings If is found that the peak axial velocity always occur in the clear fluid region. In addition, there is an indication that heat generating fluid is desirable for optimum mass flux in the annular gap most importantly when the convection current is enhanced by constant heat flux. Originality/value In view of the amount of works done on natural convection with internal heat generation/absorption, it becomes interesting to investigate the influence of this essential activity on natural convection flow in a vertical cylinder partially filled with porous material where the outer surface of the inner cylinder is either heated isothermally or with constant heat flux.

Effect of temperature dependent viscosity on entropy generation in transient viscoelastic polymeric fluid flow from an isothermal vertical plate

A numerical investigation of the viscosity variation effect upon entropy generation in time-dependent viscoelastic polymeric fluid flow and natural convection from a semi-infinite vertical plate is described. The Reiner–Rivlin second order differential model is utilized which can predict normal stress differences in dilute polymers. The conservation equations for heat, momentum and mass are normalized with appropriate transformations and the resulting unsteady nonlinear coupled partial differential equations are elucidated with the well-organized unconditionally stable implicit Crank–Nicolson finite difference method subject to suitable initial and boundary conditions. Average values of wall shear stress and Nusselt number, second-grade fluid flow variables conferred for distinct values of physical parameters. Numerical solutions are presented to examine the entropy generation and Bejan number along with their contours. The outcomes show that entropy generation parameter and Bejan number both increase with increasing values of group parameter and Grashof number. The present study finds applications in geothermal engineering, petroleum recovery, oil extraction and thermal insulation, etc.

Fully decoupled monolithic projection method for natural convection problems

To solve time-dependent natural convection problems, we propose a fully decoupled monolithic projection method. The proposed method applies the Crank–Nicolson scheme in time and the second-order central finite difference in space. To obtain a non-iterative monolithic method from the fully discretized nonlinear system, we first adopt linearizations of the nonlinear convection terms and the general buoyancy term with incurring second-order errors in time. Approximate block lower-upper decompositions, along with an approximate factorization technique, are additionally employed to a global linearly coupled system, which leads to several decoupled subsystems, i.e., a fully decoupled monolithic procedure. We establish global error estimates to verify the second-order temporal accuracy of the proposed method for velocity, pressure, and temperature in terms of a discrete l2-norm. Moreover, according to the energy evolution, the proposed method is proved to be stable if the time step is less than or equal to a constant. In addition, we provide numerical simulations of two-dimensional Rayleigh–Bénard convection and periodic forced flow. The results demonstrate that the proposed method significantly mitigates the time step limitation, reduces the computational cost because only one Poisson equation is required to be solved, and preserves the second-order temporal accuracy for velocity, pressure, and temperature. Finally, the proposed method reasonably predicts a three-dimensional Rayleigh–Bénard convection for different Rayleigh numbers.

Magnetohydrodynamic time-dependent computational natural convection flow, heat and mass transfer in inclined semi-circular enclosures

Purpose The purpose of this paper is to make a numerical analysis on unsteady analysis of natural convection heat and mass transfer to obtain flow field, temperature distribution, and concentration distribution. Design/methodology/approach A finite element method is applied to solve governing equations of natural convection in curvilinear-shaped system for different parameters as thermal Rayleigh numbers (103=RaT=106), inclination angle (0°=φ=60°) and Hartmann numbers (0=Ha=100). Findings Both magnetic field and inclination angle can be used as control parameter on heat and mass transfer. Flow strength decreases almost 100 percent between Ha=0 and Ha=100 on behalf of the higher values of thermal Rayleigh number. Originality/value The originality of this work is to application of magnetic field on time-dependent natural convection flow, heat and mass transfer for curvilinear geometry.

Time-dependent natural convection of micropolar fluid in a wavy triangular cavity

Natural convection of micropolar fluid in a right-angled wavy triangular cavity has been analyzed numerically. Governing equations formulated in dimensionless stream function, vorticity and temperature using the Boussinesq and Eringen approaches with appropriate initial and boundary conditions have been solved by finite difference method of the second-order accuracy. The effects of the dimensionless time, Prandtl number, vortex viscosity parameter, and undulation number on streamlines, isotherms, vorticity isolines as well as average Nusselt number at wavy wall and fluid flow rate inside the cavity have been studied. Obtained results have revealed essential heat transfer reduction and fluid flow attenuation with vortex viscosity parameter.

On error estimates of the projection method for the time-dependent natural convection problem: First order scheme

In this article, a projection method (or fractional step method) is proposed and analyzed for the time-dependent natural convection problem in two dimensions. Based on this method, the considered problem is decoupled into two subproblems: one is for the velocity and temperature, and the other is for the pressure. Firstly, a linear elliptic equation is treated for the temperature; Then, a linear elliptic equation is solved for the velocity; Finally, a Poisson equation is considered for the pressure. Stability is performed and the convergence results of the projection method are derived. Furthermore, a modified projection scheme which leads to some improved error estimates is also proposed. Finally, some numerical examples are presented to verify the established theoretical findings.

Numerical Study of Natural Convection-Surface Radiation Coupling in Air-Filled Square Cavity with Curved Wall Using the Gauss’s Theorem on Triangular Meshes

A numerical code is developed for the solution of the two-dimensional, time-dependent coupling natural convection- surface radiation equations in square cavity with curved wall. The cavity consists of one curved wall and three straight walls while the two vertical walls are kept isothermal, the cold one is curved (right) and the two horizontal walls are considered adiabatic. The computations are performed for an air- filled cavity, whose four walls have the same emissivity. The governing equations are formulated in Helmholtz variables (ψ, ω). The current method was developed for being used on grid made up of triangles. The basic principle of this method is the Gauss’s theorem (integrals over a closed line around an area). The results are presented in the form of main convective and radiative Nusselt numbers distributions for various Rayleigh number, emissivity and amplitude value of curve, and for Prandtl number equal to 0.71 and finally discussed. Streamlines and isothermal lines are also presented. The effect of emissivity and amplitude value of curve on the temperature, net radiative flux and velocity profiles has been analyzed. The influences of emissivity in a square cavity with outwards curve are the same in a square cavity; with the increase of the emissivity, the net radiative flux will increase at the top horizontal wall and will decrease at the bottom, this means that it cools down the top horizontal wall and heats up the bottom. Hence the convective Nusselt numbers are decreased at the active walls and the horizontal velocity is increased near the adiabatic walls. On the contrary with the increase of the inwards curve, the net radiative flux will increase on the curved wall and will decreases on all the other walls; this leads to heat up both the horizontal walls and to increase their temperature respect to the emissivity that results smaller. For consequence, at the hot wall, the convective Nusselt number is decreased with the increasing of emissivity while at the cold wall the Nusselt Number will be increased; further the horizontal velocity is decreased near the top horizontal wall and it is increased near the bottom.

The second order projection method in time for the time-dependent natural convection problem

We consider the second-order projection schemes for the time-dependent natural convection problem. By the projection method, the natural convection problem is decoupled into two linear subproblems, and each subproblem is solved more easily than the original one. The error analysis is accomplished by interpreting the second-order time discretization of a perturbed system which approximates the time-dependent natural convection problem, and the rigorous error analysis of the projection schemes is presented. Our main results of the second order projection schemes for the time-dependent natural convection problem are that the convergence for the velocity and temperature are strongly second order in time while that for the pressure is strongly first order in time.