A parallel subgrid stabilized method for the steady natural convection problem

This paper focuses on the parametric study of steady and unsteady forced and natural convection problems by the certified reduced basis method. These problems are characterized by an input-output relationship in which given an input parameter vector — material properties, boundary conditions and sources, and geometry — we would like to compute certain outputs of engineering interest — heat fluxes and average temperatures. The certified reduced basis method provides both (i) a very inexpensive yet accurate output prediction, and (ii) a rigorous bound for the error in the reduced basis prediction relative to an underlying expensive high-fidelity finite element discretization. The feasibility and efficiency of the method is demonstrated for three natural convection model problems: a scalar steady forced convection problem in a rectangular channel is characterized by two parameters — Pe´clet number and the aspect ratio of the channel — and an output — the average temperature over the domain; a steady natural convection problem in a laterally heated cavity is characterized by three parameters — Grashof and Prandtl numbers, and the aspect ratio of the cavity — and an output — the inverse of the Nusselt number; and an unsteady natural convection problem in a laterally heated cavity is characterized by two parameters — Grashof and Prandtl numbers — and a time-dependent output — the average of the horizontal velocity over a specified area of the cavity.

We consider the two-dimensional problem of steady natural convection in a circular cavity filled with viscous fluid and subjected to a cosine temperature variation on the boundary. The solution is expanded for low Grashof number and extended to 35 terms by computer. Analysis shows that convergence is limited by a square root singularity on the negative real axis of the Grashof number. An Euler transformation and extraction of the leading, secondary and tertiary singularities at infinity render the series accurate for all values of the Grashof number. The major conclusion of this investigation is that, while in previous works using the method of series extension-for example the slowly rotating pipe studied by Mansour (1985, J. Fluid Mech., 150, 1–24)- the limiting values of the quantities of interest for high values of the similarity parameters disagreed with other techniques, in the present work partial agreement with other investigations has been achieved.

The problem of steady natural convection from a corrugated plate embedded in an enclosed porous medium is studied numerically. The non-Darcian effects as well as the acceleration terms are taken into consideration in the momentum equation. The governing equations in terms of vorticity, stream function, and temperature are expressed in a body-fitted coordinate system and are solved numerically by the finite difference method. Results are presented in terms of streamlines and isotherms and local and average Nusselt numbers at Prandtl numbers of 0.7 and 7.0, with Darcy-Rayleigh number ranging from 0.1 to 1000 and Darcy number ranging from 10-4to 10-1, for several aspect ratios of the cavity and plate positions. The flow and heat transfer characteristics of a corrugated plate and a flat plate are also compared.

The method of matched asymptotic expansions is applied to the problem of steady natural convection of a Darcian fluid about a semi-infinite inclined heated surface with a power law variation of wall temperature, i.e., Tˆwaxˆλ for xˆ≥0 where 0≤λ<1. The leading edge of the inclined surface intercepts at an angle, Λ0, with another impermeable unheated surface extending upstream. The effects of the inclination angle α0 (0 ≤ α0 < < π/2) of the heated surface as well as the upstream geometry at xˆ<0 (as specified by Λ0) on heat transfer and fluid flow characteristics near the heated surface are investigated. At a given inclination angle α0, it is found that heat transfer from an upward-facing heated inclined surface is larger than that of a downward-facing heated surface, and that decreasing the intercepting angle (Λ0) tends to lower the heat transfer rate. These effects become increasingly pronounced as the Rayleigh number is decreased.

A boundary element investigation of natural convection problems

This paper describes the solution of a steady state natural convection problem in porous media by the dual reciprocity boundary element method (DRBEM). The boundary element method (BEM) for the coupled set of mass, momentum, and energy equations in two dimensions is structured by the fundamental solution of the Laplace equation. The dual reciprocity method is based on augmented scaled thin plate splines. Numerical examples include convergence studies with different mesh size, uniform and non-uniform mesh arrangement, and constant and linear boundary field discretizations for differentially heated rectangular cavity problems at filtration with Rayleigh numbers of Ra*=25, 50, and 100 and aspect ratios of A=1/2, 1, and 2. The solution is assessed by comparison with reference results of the fine mesh finite volume method (FVM). Copyright © 2000 John Wiley & Sons, Ltd.

General nodal expansion method for multi-dimensional steady and transient convection–diffusion equation

Comment on the paper I.A. Badruddin, Z.A. Zainal, Z.A. Khan, Z. Mallick “Effect of viscous dissipation and radiation on natural convection in a porous medium embedded within vertical annulus”, IJTS 46 (3) (2007) 221–227

Influences of Prandtl Number on Transition of Steady Natural Convection Flows in a Square Enclosure with a Circular Cylinder

This paper concerns the investigation of steady thermosolutal natural convection from a vertical plate with the combined effect of streamwise sinusoidal variations of both the surface temperature and the species concentration about their respective constant means. The governing boundary layer equations are reduced to nonsimilar form, and these are solved by the stream-function formulations, and by the subsequent application of the Keller box method. The effects of varying the governing nondimensional physical parameters are obtained in terms of the isolines of temperature, iso-species concentration and the streamlines. An asymptotic solution for large values of X, the distance from the leading edge, has also been obtained. It has been found that the two-term asymptotic analysis and the numerical results obtained from Keller-box method are in excellent agreement.

Shooting methods for some steady diffusion and convection problems

We examine thermal convection in a rotating spherical shell with central gravity and a spatially non-uniformly heated outer surface at two values of the Prandtl number: Pr=7.0, appropriate for water at room temperature, and Pr=0.7, appropriate for air at standard temperature and pressure, by numerical calculation. Four calculations are performed in a sequence: the onset of convection with homogeneous temperature boundary condition, nonlinear boundary-forced steady convection, stability of the forced steady convection to infinitesimal disturbances, and time stepping of subsequent secondary convection. Unlike our previous study of the infinite Prandtl number limit [J. Fluid Mech. 250, 209 (1993)] inertial terms in the equation of motion for moderate Prandtl numbers play a key role in the dynamics. The effects of an inhomogeneous temperature boundary condition on nonlinear convection are illustrated by varying the wavelength and strength of the imposed boundary temperature. It is shown that even a slight inhomogeneity in the thermal boundary condition can lock azimuthally drifting convection and make it stationary, or modify the normal drifting convection rolls to a vacillating structure. In the infinite Prandtl number case, when inertial forces are absent from the equation of motion, resonance occurs when the wavelengths of boundary forcing and natural convection coincide. Fluid inertia destroys this resonance for finite Prandtl number fluids. The same effect reduces in size the stability region where steady convection is locked to the boundary, and steady convection becomes unstable to time-dependent convection. The period of the secondary convection is close to that obtained with uniform temperature boundaries but the spatial structure is dramatically changed, exhibiting vacillations between the wavelength of the boundary temperature and that of the natural convection.

External natural convection is rarely studied by numerical simulation in the literature due to the fact that flow of interest takes place in an unbounded domain and that if a limited computational domain is used the corresponding outer boundary conditions are unknown. In this study, we propose outer boundary conditions for a limited computational domain and make the corresponding numerical implementation in the scope of a projection method combining spectral methods and domain decomposition techniques. Numerical simulations are performed for both steady natural convection about an isothermal cylinder and transient natural convection around a line‐source. An experiment is also realized in water using particle image velocimetry and thermocouples to make a comparison during transients of external natural convection around a platinum wire heated by Joule effect. Good agreement, observed between numerical simulations and experiments, validated the outer boundary conditions proposed and their numerical implementation. It is also shown that, if one tolerates prediction error, numerical results obtained remain at least reasonable in a region near the line‐source during the entire transients. We thus paved the way for numerical simulation of external natural convection although further studies remain to be done for higher heating power (higher Rayleigh number).

The present article studies the effects of suction/injection and variable physical properties on steady natural convection flow through a vertical channel. The investigation was done analytically using Differential Transformation Method (DTM). The variability in viscosity and thermal conductivity are considered linear function of temperature. The governing equations are transformed into a set of coupled nonlinear ordinary differential equations. Results obtained were compared with exact solution when some of the flow conditions were relaxed and results from DTM show an excellent agreement with the exact solution which was obtained analytically. The influence of the flow parameters on fluid temperature, concentration, and velocity are presented graphically and discussed for variations of the governing parameters. From the course of investigation, it was found that an increase in suction (S > 0) through the heated plate causes fluid temperature, velocity as well as concentration to decrease. However, an increase in injection causes heat transfer rate as well as skin friction at the heated plate to decrease.

Steady natural convection in open cavities filled with a porous medium utilizing Buongiorno’s nanofluid model