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The purpose of this paper is to present outcomes of efforts made over the last 20 years to extend the applicability of the Richardson extrapolation procedure to numerical predictions of multidimensional, steady and unsteady, fluid flow and heat transfer phenomena in regular and irregular calculation domains.

Pattern-preserving grid-refinement strategies are proposed for mathematically rigorous generalizations of the Richardson extrapolation procedure for numerical predictions of steady fluid flow and heat transfer, using finite volume methods and structured multidimensional Cartesian grids; and control-volume finite element methods and unstructured two-dimensional planar grids, consisting of three-node triangular elements. Mathematically sound extrapolation procedures are also proposed for numerical solutions of unsteady and boundary-layer-type problems. The applicability of such procedures to numerical solutions of problems with curved boundaries and internal interfaces, and also those based on unstructured grids of general quadrilateral, tetrahedral, or hexahedral elements, is discussed.

Applications to three demonstration problems, with discretizations in the asymptotic regime, showed the following: the apparent orders of accuracy were the same as those of the numerical methods used; and the extrapolated results, measures of error, and a grid convergence index, could be obtained in a smooth and non-oscillatory manner.

Strict or approximate pattern-preserving grid-refinement strategies are used to propose generalized Richardson extrapolation procedures for estimating grid-independent numerical solutions. Such extrapolation procedures play an indispensable role in the verification and validation techniques that are employed to assess the accuracy of numerical predictions which are used for designing, optimizing, virtual prototyping, and certification of thermofluid systems.

The purpose of this paper is to numerically investigate steady, laminar natural and mixed convection heat transfer in a two-dimensional cavity by using a finite volume method with a fourth-order approximation of convective terms, with and without the presence of nanoparticles. Highly accurate benchmark results are also provided.

A finite volume method on a non-uniform staggered grid is used for the solution of two-dimensional momentum and energy conservation equations. Diffusion terms, in the momentum and energy equations, are approximated using second-order central differences, whereas a non-uniform four-point fourth-order interpolation (FPFOI) scheme is developed for the convective terms. Coupled mass and momentum conservation equations are solved iteratively using a semi-implicit method for pressure-linked equation method.

For the case of natural convection problem at high-Rayleigh numbers, grid density must be sufficiently high in order to obtain grid-independent results and capture reality of the physics. Heat transfer enhancement for natural convection is observed up to a certain value of the nanoparticle volume fraction. After that value, heat transfer deterioration is found with increasing nanoparticle volume fraction.

Developed a non-uniform FPFOI scheme. Highly accurate benchmark results for the heat transfer of Al₂O₃-water nanofluid in a cavity are provided.

The purpose of this paper is to present a pressure‐correction procedure for incompressible flows using unstructured meshes. A method of implementing high‐order spatial schemes on unstructured grids was introduced.

The procedure used a collocated cell‐centered unstructured grid arrangement. In order to improve the accuracy of calculation, the widely used high‐order schemes for convection, developed for structured grids and in the form of either the normalized variable and space formulation (NVSF) or the total variation diminishing (TVD) flux limiters (FL), were introduced and implemented onto the unstructured grids. This implementation was carried out by constructing a local coordinate and introducing a virtual upstream node.

The procedure was validated by calculating the lid‐driven cavity flows which had benchmark numerical solutions. For comparison, these flows were also computed by a commercial package, the FLUENT. The results obtained by the present procedure agreed well with the benchmark solution although very coarse grids were used. For the FLUENT, however, worse agreements with the benchmark solutions were obtained although the grids used for computation were the same. These demonstrated the robustness of the presented numerical procedure.

With the present method, high‐order schemes in either NVSF or TVD FL forms for structured grids can be easily implemented onto unstructured grids. This provides more choices of high‐order schemes for calculating complex flows.

The purpose of this paper is to numerically investigate the heat transfer enhancement in a tube filled partially with porous media under non-uniform porosity distribution and thermal dispersion effects. The optimum porous thickness ratio [R_(r,Nu)] for the heat transfer enhancement under these conditions with and without considering required pumping power is evaluated.

The local thermal non-equilibrium and Darcy–Brinkman–Forchheimer models are used to simulated thermal and flow fields in porous region. The tube wall and flow regime are assumed to be isothermal and laminar, respectively. The impacts of Darcy number (Da = 10^-6 - 10^-1) and inertia parameter (F = 0 − 2) on the Nusselt number and friction factor are studied for non-uniform porosity distribution.

First, the effect of Nusselt number indicates that there are two different behaviors with respect to uniform and non-uniform porosity for partially and fully filled porous pipe. Second, variable porosity in porous region has significant influence on the optimum thickness ratio with considering required pumping power. For negligible inertia term, it depends on the Darcy number, whereas it is 0.9 at F > 1. Third, the plug flow assumption cannot be valid even at lower Darcy number under non-uniform porosity, while this assumption is applicable at Da < 10^-3 for constant porosity distribution in porous region.

According to the best knowledge of authors, the optimum porous thickness ratio for the heat transfer enhancement considering the pressure loss effects under variable porosity has not reported up to now. Also the plug flow assumption in such physics is not discussed.

One major challenge in turbulent flow applications is to control the recirculation zone behind the backward‐facing step (BFS). One simple idea to do so is to modify the original BFS geometry, of course, without causing adverse or undesirable impacts on the original characteristics of the primary stream. The main objective of this work is to examine the solidity of the recirculation zone behind several different geometries which are slightly to moderately different from the original BFS geometry.

The implemented modifications cause complicated irregularities at the boundaries of the domain. The experience shows that the mesh distribution around these irregularities plays a critical role in the accuracy of the numerical solutions. To achieve the most accurate solutions with the least computational efforts, we use a robust hybrid strategy to distribute the computational grids in the domain. Additionally, a suitable numerical algorithm capable of handling hybrid grid topologies is properly extended to analyze the flow field. The current fully implicit method utilizes a physical pressure‐based upwinding scheme capable of working on hybrid mesh.

The extended algorithm is very robust and obtains very accurate solutions for the complex flow fields despite utilizing very coarse grid resolutions. Additionally, different proposed geometries revealed very similar separated regions behind the step and performed minor differences in the location of the reattachment points.

The current study is fulfilled two‐dimensionally. However, the measurements in testing regular BFS problems have shown that the separated shear layer behind the step is not affected by 3D influences provided that the width of channel is sufficiently wide. A similar conclusion is anticipated here.

The problem occurs in the pipe and channel expansions, combustion chambers, flow over flying objects with abrupt contraction on their external surfaces, etc.

A novel pressure‐based upwinding strategy is properly employed to solve flow on multiblocked hybrid grid topologies. This strategy takes into account the physics associated with all the transports in the flow field. To study the impact of shape improvement, several modified BFS configurations were suggested and examined. These configurations need only little additional manufacturing cost to be fabricated.

Aiming at finding the velocity distribution profile and other flow characteristic parameters such as the Poiseuille (Po) number, this study aims to focus on the three-dimensional forced convective flow inside rectangular ducts filled with porous media commonly used in air-based solar thermal collectors to enhance the thermal performance. The most general model for the fluid flow (i.e. the non-linear Darcy–Brinkman–Forchheimer partial differential equation subjected to slip and no-slip boundary conditions) is considered.

The general governing equations are solved analytically based on the perturbation technique and the results are validated against numerical simulation study based on a finite-difference solution over a non-uniform but structured grid.

The analytical velocity distribution profile based on exponential functions for the above-mentioned general case is obtained, and accordingly, expressions for the Po are introduced. It is found that the velocity distribution tends to be uniform by increasing the aspect ratio of the duct. Moreover, a criterion for considering/neglecting the nonlinear drag term in the momentum equation (i.e. the Forchheimer term) is proposed. According to the sensitivity analysis, results show that the nonlinear drag term effects on the Nusselt number are important only in porous media with high Darcy numbers.

A general analytic solution for three-dimensional forced convection flows through rectangular ducts filled with porous media for the general model of Darcy–Brinkman–Forchheimer and the general boundary condition including both no-slip and slip-flow regimes is obtained. An analytic expression to calculate Po number is obtained which can be practical for engineering estimations and a basis for validation of numerical simulations. A criterion for considering/neglecting the nonlinear drag term in the momentum equation is also introduced.

The present numerical investigation examines the magnetohydrodynamic (MHD) double diffusion natural convection of power-law non-Newtonian nano-encapsulated phase change materials (NEPCMs) in a trapezoidal cavity.

The governing Navier-Stokes, energy and concentration equations based on the Cartesian curvilinear coordinates are solved using the collocated grid arrangement’s finite volume method. The in-house FORTRAN code is validated with the different benchmark problems. The NEPCM nanoparticles consist of a core-shell structure with Phase Change Material (PCM) at the core. The enclosure, shaped as a trapezoidal hollow, features a warmed (T_h) left wall and a cold (T_c) right wall. Various parameters are considered, including the power law index (0.6 ≤ n ≤ 1.4), Hartmann number (0 ≤ Ha ≤ 30), Rayleigh number (10⁴ ≤ Ra ≤ 10⁵) and fixed variables such as buoyancy ratio (Br = 0.8), Prandtl number (Pr = 6.2), Lewis number (Le = 5), fusion temperature (Θ_f = 0.5) and volume fraction (ϕ = 0.04).

The findings indicate a decrease in local Nusselt (Nu) and Sherwood (Sh) numbers with increasing Hartmann numbers (Ha). Additionally, for a shear-thinning fluid (n = 0.6) results in the maximum local Nu and Sh values. As the Rayleigh number (Ra) increases from 10⁴ to 10⁵, the structured vortex in the streamline pattern is disturbed. Furthermore, for different Ra values, an increase in n from 0.6 to 1.4 leads to a 67.43% to 76.88% decrease in average Nu and a 70% to 77% decrease in average Sh.

This research is for two-dimensioal laminar flow only.

PCMs represent a class of practical substances that behave as a function of temperature and have the innate ability to absorb, release and store heated energy in the form of hidden fusion enthalpy, or heat. They are valuable in these systems as they can store significant energy at a relatively constant temperature through their latent heat phase change.

As per the literature review and the authors’ understanding, an examination has never been conducted on MHD double diffusion natural convection of power-law non-Newtonian NEPCMs within a trapezoidal enclosure. The current work is innovative since it combines NEPCMs with the effect of magnetic field Double diffusion Natural Convection of power-law non-Newtonian NEPCMs in a Trapezoidal enclosure. This outcome can be used to improve thermal management in energy storage systems, increasing safety and effectiveness.

The purpose of this paper is to develop a simplified but accurate finite element procedure for the analysis of the conjugate conduction-convection heat transfer in cross-flow micro heat exchangers.

The velocity fields in single microchannels are calculated by solving the parabolised form of the momentum equations and later mapped onto the three-dimensional grid, corresponding to an appropriate portion of the micro heat exchanger, which is used for the solution of the energy equation in its elliptic form. To allow the use of finite elements elongated in the flow direction, layers of perpendicular microchannels can be meshed independently with grids that do not match at the common interface (domain decomposition).

An original and easy-to-implement method has been developed to deal with non-matching grids. Computed results show that increasing the number of microchannels per layer yields relative pressure drop increments that are larger than those displayed by the relative heat flow rates.

The simplified procedure requires the assumption of constant thermophysical properties. The adopted domain decomposition technique yields non-symmetric system matrices.

The procedure can be very useful in the design of cross-flow micro heat exchangers.

The finite element procedure described in the paper requires only limited computational resources for the analysis of the conjugate conduction-convection heat transfer in cross-flow micro heat exchangers with a large number of microchannels per layer.

The aims of this study is to numerically investigate the thermal phenomena during magnetohydrodynamic (MHD) free convection in an oblique enclosure filled with porous media saturated with Cu–Al₂O₃/water hybrid nanofluid and heated at the left wavy wall. The thermophysical phenomena are explored thoroughly by varying the amplitude (λ) and undulation (n) of the wavy wall and the inclination of the enclosure (γ) along with other pertinent physical parameters. Darcy–Rayleigh number (Ra_m), Darcy number (Da), Hartmann number (Ha) and nanoparticle volumetric fraction (ϕ). The effect of all parameters has been analyzed and represented by using heatlines, isotherms, streamlines, average Nusselt number and local Nusselt number.

The finite volume method is used to work out the transport equations coupled with velocity, pressure and temperature subjected to non-uniform staggered grid structure after grid-sensitivity analysis by an indigenous computing code and the semi-implicit method for pressure linked equations (SIMPLE) algorithm. The solution process is initiated following an iterative approach through the alternate direction implicit sweep technique and the tridiagonal matrix algorithm (TDMA) algorithm. The iterative process is continued until successive minimization of the residuals (<1e-8) for the governing equations.

This study reveals that the increase in the heating surface area does not always favor heat transfer. An increase in the undulation amplitude enhances the heat transfer; however, there is an optimum value of undulation of the wavy wall for this. The heat transfer enhancement because of the wall curvature is revealed at higher Ra_m, lower Da and Ha and lower volume fraction of nanoparticles. In general, this augmentation is optimum for four undulations of the wavy wall with an amplitude of λ = 0.3. The heat transfer enhancement can be more at the cavity inclination   γ = 45°.

The technique of this investigation could be used in other multiphysical areas involving partial porous layers, conducting objects, different heating conditions, wall motion, etc.

This study is to address MHD thermo-fluid phenomena of Cu–Al₂O₃/water-based hybrid nanofluid flow through a non-Darcian porous wavy cavity at different inclinations. The amplitude and number of undulations of the wavy wall, permeability of the porous medium, magnetic field intensity, nanoparticle volumetric fraction and inclinations of the enclosure play a significant role in the heat transfer process. This analysis and the findings of this work can be useful for the design and control of similar thermal systems/devices.

Many researchers have examined the problem of buoyancy-induced free convection in a wavy-porous cavity packed with regular fluids or nanofluids. However, the effect of magnetic fields along with the amplitude (λ) at different undulations (n) of the heated wavy wall of an inclined enclosure is not attended so far to understand the transport mechanisms. Most often, the evolutions of the thermo-fluid phenomena in such complex geometries invoking different multiphysics are very intricate. Numerical implementations for simulations and subsequent post-processing of the results are also challenging.

Laminar natural convection of nanofluids in a square cooled cavity enclosing a heated horizontal cylinder is studied numerically. This paper aims to investigate in what measure the nanoparticle size and average volume fraction, the cavity width, the cylinder diameter and position, the average temperature of the nanofluid and the temperature difference imposed between the cylinder and the cavity walls, affects the basic heat and fluid flow features, as well as the thermal performance of the nanofluid relative to that of the base liquid.

The four-equation system of the mass, momentum and energy transfer governing equations has been solved using a computational code incorporating three empirical correlations for the evaluation of the effective thermal conductivity, the effective dynamic viscosity and the coefficient of thermophoretic diffusion, all based on a high number of experimental data available in the literature. The SIMPLE-C algorithm has been used to handle the pressure-velocity coupling. Simulations have been performed using Al₂O₃ + H₂O, for different values of the average volume fraction of the suspended solid phase in the range 0-0.04, the diameter of the nanoparticles in the range 25-75 nm, the temperature difference imposed between the cylinder and the cavity walls in the range 5-20 K, the average nanofluid temperature in the range 300-330 K, the ratio between the cylinder diameter and the cavity width in the range 0.1-0.5 m, the ratio between the distance of the cylinder axis from the bottom wall and the cavity width in the range 0.2-0.8 and the ratio between the distance of the cylinder axis from the left sidewall and the cavity width in the range 0.2-0.5.

The main results obtained may be summarized as follows: the overall solid phase migration from hot to cold results in a cooperating solutal buoyancy force which tends to compensate the friction increase consequent to the viscosity growth due to the dispersion of the nanoparticles into the base fluid; the effect of the increased thermal conductivity consequent to the nanoparticle dispersion into the base fluid plays the major role in determining the heat transfer enhancement of the nanofluid, at least in the upper range of the investigated average temperatures; at high temperatures, the nanofluid heat transfer performance relative to that of the pure base liquid increases with increasing the average volume fraction of the suspended solid phase, whereas at low temperatures, it has a peak at an optimal particle loading; the relative heat transfer performance of the nanofluid increases notably with increasing the average temperature, and just moderately as the imposed temperature difference, the width of the cavity and the distance of the cylinder from the bottom of the cavity, are increased; the relative heat transfer performance of the nanofluid increases as the nanoparticle size, the cylinder diameter and the distance of the cylinder from the sidewall, are decreased; as a consequence of the local competition between the thermal and the solutal buoyancy forces, a periodic flow arises when the cylinder is located in the vicinity of one of the cooled walls of the enclosure.

Framed in this general background, a comprehensive numerical study on buoyancy-driven convection of alumina-water nanofluids inside a cooled square cavity containing a heated circular cylinder is executed by the way of a two-phase model based on the double-diffusive approach accounting for the effects of the Brownian diffusion and thermophoresis.