F. Moukalled

Natural convection in the annulus between concentric horizontal circular and square cylinders

Numerical solutions are presented for natural convection heat transfer between a heated horizontal cylinder placed concentrically inside a square enclosure. Three different aspect ratios (R/L 0.1, 0.2, and 0.3), and four different Rayleigh numbers (Ra = 10, 10, 10, and 10), are considered. The governing elliptic conservation equations are solved in a boundary-fitted coordinate system using a control volumebased numerical procedure. Results are displayed in the form of streamlines, isotherms, maximum stream function estimates, and local and average normalized Nusselt number values. At constant enclosure aspect ratio, the total heat transfer increases with increasing Rayleigh number. For constant Rayleigh number values, convection contribution to the total heat transfer decreases with increasing values of R/L. For convection-dominated flows, the average Nusselt number correlation is expressed as Nu = 0.92Ra(R/ L)*. Generated results are in good agreement with previously published experimental and numerical data.

TVD schemes for unstructured grids

Abstract A number of approaches have evolved over the last decade for the implementation of total variational diminishing (TVD) schemes within an unstructured grid finite volume method framework. Unfortunately none of these approaches has been comprehensive enough to permit the general implementation of TVD-based schemes in unstructured grids, and/or accurate enough to recover the exact TVD formulation in structured grids. In this paper we propose a simple method that allows the implementation of the full spectrum of TVD schemes in unstructured grids, while recovering their exact formulation on structured grids. Four schemes implemented using this approach, TVD-MINMOD, TVD-MUSCL (monotonic upstream-centered scheme for conservation laws, MUSCL), TVD-SUPERBEE, TVD-OSHER, are tested and compared to Bruner’s TVD formulation [Parallelization of the Euler equations on unstructured grids, AIAA paper 97-1894, 1995], and to the Barth and Jesperson linear reconstruction scheme [The design and application of upwind schemes on unstructured meshes, AIAA paper 89-0366, 1989] by solving four pure advection problems. Results indicate that the Bruner formulation yields, for the same original TVD scheme, overly diffusive results when compared to the current method. The BJ-MUSCL and TVD-MUSCL are shown to be comparable and more accurate than the OSHER scheme. The SUPERBEE performs best though showing tendency for stepping the modeled profile. In all tests the current method is found to retain the behavior of the structured grid TVD formulation.

A UNIFIED FORMULATION OF THE SEGREGATED CLASS OF ALGORITHMS FOR FLUID FLOW AT ALL SPEEDS

In this article, the segregated SIMPLE algorithm and its variants are reformulated, using a collocated variable approach, to predict fluid flow at all speeds. In the formulation, a unified, compact, and easy-to-understand notation is employed. The SIMPLE, SIMPLER, SIMPLEST, SIMPLEM, SIMPLEC, SIMPLEX, PRIME, and PISO algorithms that are scattered in the literature and appear to a non versed computational fluid dynamics (CFD) user as being unrelated, are shown to share the same essence in their derivations and to be equally applicable for the simulation of incompressible and compressible flows. Moreover, the philosophies behind these algorithms in addition to their similarities and differences are explained.

A coupled finite volume solver for the solution of incompressible flows on unstructured grids

This paper reports on a newly developed fully coupled pressure-based algorithm for the solution of laminar incompressible flow problems on collocated unstructured grids. The implicit pressure-velocity coupling is accomplished by deriving a pressure equation in a procedure similar to a segregated SIMPLE algorithm using the Rhie-Chow interpolation technique and assembling the coefficients of the momentum and continuity equations into one diagonally dominant matrix. The extended systems of continuity and momentum equations are solved simultaneously and their convergence is accelerated by using an algebraic multigrid solver. The performance of the coupled approach as compared to the segregated approach, exemplified by SIMPLE, is tested by solving five laminar flow problems using both methodologies and comparing their computational costs. Results indicate that the number of iterations needed by the coupled solver for the solution to converge to a desired level on both structured and unstructured meshes is grid independent. For relatively coarse meshes, the CPU time required by the coupled solver on structured grid is lower than the CPU time required on unstructured grid. On dense meshes however, this is no longer true. For low and moderate values of the grid aspect ratio, the number of iterations required by the coupled solver remains unchanged, while the computational cost slightly increases. For structured and unstructured grid systems, the required number of iterations is almost independent of the grid size at any value of the grid expansion ratio. Recorded CPU time values show that the coupled approach substantially reduces the computational cost as compared to the segregated approach with the reduction rate increasing as the grid size increases.

Convective Schemes for Capturing Interfaces of Free-Surface Flows on Unstructured Grids

ABSTRACT In this article, the general methodology used in constructing interface-capturing schemes is clarified and concisely described. Moreover, a new interface-capturing scheme, denoted STACS and based on a switching strategy, is developed. The accuracy of the new scheme is compared to the well-known CICSAM and HRIC schemes by solving four pure advection test problems. Results, displayed in the form of interface contours for the various schemes, reveal deterioration in the accuracy of the CICSAM and HRIC schemes, with their performance approaching that of the UPWIND scheme as the Courant number increases. On the other hand, predictions obtained with the new STACS scheme are by far more accurate and less diffusive, preserving interface sharpness and boundedness at all Courant number values considered.

Improvements to incompressible flow calculation on a nonstaggered curvilinear grid

The semi-implicit method for pressure-linked equations (SIMPLE) algorithm of Rhie and Chow [6] for flow problems on nonstaggered curvilinear grids is extended, and a SIMPLE-revised (SIMPLER) algorithm is formulated on nonstaggered curvilinear grids. In addition, a new algorithm, SIMPLE-modified (SIMPLEM), is formulated. The performance of these three algorithms is examined through two test problems (driven flow in a cavity and flow in a sudden expansion). The integral form of the continuity equation is shown to be not satisfied in the SIMPLE formulation. In the SIMPLER formulation, the residuals of the momentum equation do not decrease to acceptably low values. The SIMPLEM algorithm shows reasonably good convergence behavior and is superior to both the SIMPLE and SIMPLER algorithms. On the basis of these comparisons, the SIMPLEM algorithm is recommended for use in nonstaggered curvilinear meshes.

The Finite Volume Method

Similar to other numerical methods developed for the simulation of fluid flow, the finite volume method transforms the set of partial differential equations into a system of linear algebraic equations. Nevertheless, the discretization procedure used in the finite volume method is distinctive and involves two basic steps. In the first step, the partial differential equations are integrated and transformed into balance equations over an element. This involves changing the surface and volume integrals into discrete algebraic relations over elements and their surfaces using an integration quadrature of a specified order of accuracy. The result is a set of semi-discretized equations. In the second step, interpolation profiles are chosen to approximate the variation of the variables within the element and relate the surface values of the variables to their cell values and thus transform the algebraic relations into algebraic equations. The current chapter details the first discretization step and presents a broad review of numerical issues pertaining to the finite volume method. This provides a solid foundation on which to expand in the coming chapters where the focus will be on the discretization of the various parts of the general conservation equation. In both steps, the selected approximations affect the accuracy and robustness of the resulting numerics. It is therefore important to define some guiding principles for informing the selection process.

Particle migration in a concentrated suspension flowing between rotating parallel plates: Investigation of diffusion flux coefficients

This paper reports an experimental and numerical study conducted to investigate the behavior of macroscopic monomodal concentrated suspensions (40%) undergoing a creeping torsional flow between two rotating plates. An experimental technique based on the detection of tracers by measurement of light absorption is developed and used to quantify the time evolution of the particles concentration profiles. Contrarily to results reported in the literature, an outward migration of the particles is observed. This shear-induced migration is confirmed by viscometric measurements where an increase in the apparent viscosity of the suspension has been observed for long periods of shear. Moreover, this increase is found to depend solely on the value of the applied strain, which is consistent with a shear-induced migration phenomenon. Experimental results are reproduced using a semi-quantitative model involving the balance of three diffusion fluxes induced, respectively, by the gradient of viscosity (Jη), the gradient of...

NATURAL CONVECTION IN TRAPEZOIDAL CAVITIES WITH BAFFLES MOUNTED ON THE UPPER INCLINED SURFACES

A numerical investigation has been undertaken to study the effects of mounting baffles to the upper inclined planes of trapezoidal cavities (representing a building or attic space). Two thermal boundary conditions are considered: (a) the vertical and upper surfaces are heated while the lower surface is cooled (summerlike conditions); (b) the lower surface is heated while the other surfaces are cooled (winterlike conditions). For each boundary condition, computations are performed for two baffle heights and two baffle locations. Rayleigh number (Ra) values range from 103 to 5 x 107 for summerlike conditions and from 103 to 106 for winterlike conditions. For both boundary conditions, results obtained with air as the working fluid reveal a decrease in heat transfer in the presence of baffles. In winterlike conditions, convection starts to dominate at an Ra much lower than that in summerlike conditions. The decrease in heat transfer becomes increasingly more significant as the baffle gets closer to the heated vertical wall for the bottom-cooled situation and as the baffle gets closer to the symmetry line for the bottom-heated case. In general, this decrease in heat transfer is higher with taller baffles. Average Nusselt number (Nu) correlations for both boundary conditions are presented.

On entropy generation in thermoelectric devices

In this paper, a comparison between the Entropy Generation Minimization method and the Power Maximization technique is presented. The assessment is performed by analyzing, as a typical example of direct conversion heat engines, the thermoelectric generator. The effects of heat-leak and finite-rate heat transfer on the performance of the generator, which is modeled as a Carnot-like engine with internal irreversibilities, are studied. Even though both methods lead to the same conclusions, the entropy generation minimization method, when applied for such an inherently irreversible device, is shown to be less straight forward than the power maximization technique requiring careful accounting of the different sources of irreversibility. Moreover, the entropy generation rate vs. efficiency behavior of the generator reveals that the efficiency at minimum entropy generation rate and the maximum efficiency are distinct.