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The purpose of this paper is to develop a combined method for three-dimensional incompressible flow and heat transfer by the spectral collocation method (SCM) and the artificial compressibility method (ACM), and further to study the performance of the combined method SCM-ACM for three-dimensional incompressible flow and heat transfer.

The partial differentials in space are discretized by the SCM with Chebyshev polynomial and Chebyshev–Gauss–Lobbatto collocation points. The unsteady artificial compressibility equations are solved to obtain the steady results by the ACM. Three-dimensional exact solutions with trigonometric function form and exponential function form are constructed to test the accuracy of the combined method.

The SCM-ACM is developed successfully for three-dimensional incompressible flow and heat transfer with high accuracy that the minimum value of variance can reach. The accuracy increases exponentially along with time marching steps. The accuracy is also improved exponentially with the increasing of nodes before stable accuracy is achieved, while it keeps stably with the increasing of the time step. The central processing unit time increases exponentially with the increasing of nodes and decreasing of the time step.

It is difficult for the implementation of the implicit scheme by the developed SCM-ACM. The SCM-ACM can be used for solving unsteady impressible fluid flow and heat transfer.

The SCM-ACM is applied for two classic cases of lid-driven cavity flow and natural convection in cubic cavities. The present results show good agreement with the published results with much fewer nodes.

The combined method SCM-ACM is developed, firstly, for solving three-dimensional incompressible fluid flow and heat transfer by the SCM and ACM. The performance of SCM-ACM is investigated. This combined method provides a new choice for solving three-dimensional fluid flow and heat transfer with high accuracy.

The purpose of this paper is the development of a new density-based (DB) semi-Lagrangian method to speed up the conventional pressure-based (PB) semi-Lagrangian methods.

The semi-Lagrangian-based solvers are typically PB, i.e. semi-Lagrangian pressure-based (SLPB) solvers, where a Poisson equation is solved for obtaining the pressure field and ensuring a divergence-free flow field. As an elliptic-type equation, the Poisson equation often relies on an iterative solution, so it can create a challenge of parallel computing and a bottleneck of computing speed. This study proposes a new DB semi-Lagrangian method, i.e. the semi-Lagrangian artificial compressibility (SLAC), which replaces the Poisson equation by a hyperbolic continuity equation with an added artificial compressibility (AC) term, so a time-marching solution is possible. Without the Poisson equation, the proposed SLAC solver is faster, particularly for the cases with more computational cells, and better suited for parallel computing.

The study compares the accuracy and the computing speeds of both SLPB and SLAC solvers for the lid-driven cavity flow and the step-flow problems. It shows that the proposed SLAC solver is able to achieve the same results as the SLPB, whereas with a 3.03 times speed up before using the OpenMP parallelization and a 3.35 times speed up for the large grid number case (512 × 512) after the parallelization. The speed up can be improved further for larger cases because of increasing the condition number of the coefficient matrixes of the Poisson equation.

This paper proposes a method of avoiding solving the Poisson equation, a typical computing bottleneck for semi-Lagrangian-based fluid solvers by converting the conventional PB solver (SLPB) to the DB solver (SLAC) through the addition of the AC term. The method simplifies and facilitates the parallelization process of semi-Lagrangian-based fluid solvers for modern HPC infrastructures, such as OpenMP and GPU computing.

The purpose of this paper is to focus on modeling buoyancy driven viscous flow and heat transfer through saturated packed pebble‐beds via a set of homogeneous volume‐averaged conservation equations in which local thermal disequilibrium is accounted for.

The local thermal disequilibrium accounted for refers to the solid and liquid phases differing in temperature in a volume‐averaged sense, which is modeled by describing each phase with its own governing equation. The partial differential equations are discretized and solved via a vertex‐centered edge‐based dual‐mesh finite volume algorithm. A compact stencil is used for viscous terms, as this offers improved accuracy compared to the standard finite volume formulation. A locally preconditioned artificial compressibility solution strategy is employed to deal with pressure incompressibility, whilst stabilisation is achieved via a scalar‐valued artificial dissipation scheme.

The developed technology is demonstrated via the solution of natural convective flow inside a heated porous axisymmetric cavity. Predicted results were in general within 10 per cent of experimental measurements.

This is the first instance in which both artificial compressibility and artificial dissipation is employed to model flow through saturated porous materials.

The purpose of this paper is to present two low diffusive convective-pressure flux split finite volume algorithms for solving incompressible flows in artificial compressibility framework.

The present method follows the framework similar to advection upwind splitting method of Liou and Steffen for compressible flows which is used by Vierendeels et al. to solve incompressible flow equations. Instead of discretizing the total inviscid flux using upwind scheme, the inviscid flux is first split into convective and pressure parts, and then discretized the two parts differently. The convective part is discretized using upwind method and the pressure part using central differencing. Since the Vierendeels type scheme may not be able to capture the divergence free velocity field due to the presence of artificial dissipation term, a strategy to progressively withdraw the dissipation with time step is proposed here that can ascertain the divergence free velocity condition to the level of residual error. This approach helps in reducing the amount of numerical dissipation due to upwind discretization, which is evident from the numerical test examples.

Upwind treatment of only the convective part of the inviscid flux terms, instead of the whole inviscid flux term, leads to more accurate solutions even at relatively coarse grids, which is substantiated by numerical test examples.

The method is presently applicable to Cartesian grid.

Although similar formulation is reported by Vierendeels et al., no detailed study of the accuracy is presented. Discretization and solution reconstructions used in the present approach differ from the approach reported by Vierendeels et al. A modification to Vierendeels type scheme is proposed that can help in achieving divergence free velocity condition. Finally the efficacy of the present approach to produce very accurate solutions even on coarse grids is successfully demonstrated using a few benchmark problems.

A new approach for solving steady incompressible Navier‐Stokes equations is presented in this paper. This method extends the upwind Riemann‐problem‐based techniques to viscous flows, which is obtained by applying modified artificial compressibility Navier‐Stokes equations and fully discrete high‐order numerical schemes for systems of advection‐diffusion equations. In this approach, utilizing the local Riemann solutions the steady incompressible viscous flows can be solved in a similar way to that of inviscid hyperbolic conservation laws. Numerical experiments on the driven cavity problem indicate that this approach can give satisfactory solutions.

This study aims to feature the application of the artificial compressibility method (ACM) for the numerical prediction of two-dimensional (2D) axisymmetric swirling flows.

The respective academic numerical solver, named IGal2D, is based on the axisymmetric Reynolds-averaged Navier–Stokes (RANS) equations, arranged in a pseudo-Cartesian form, enhanced by the addition of the circumferential momentum equation. Discretization of spatial derivative terms within the governing equations is performed via unstructured 2D grid layouts, with a node-centered finite-volume scheme. For the evaluation of inviscid fluxes, the upwind Roe’s approximate Riemann solver is applied, coupled with a higher-order accurate spatial reconstruction, whereas an element-based approach is used for the calculation of gradients required for the viscous ones. Time integration is succeeded through a second-order accurate four-stage Runge-Kutta method, adopting additionally a local time-stepping technique. Further acceleration, in terms of computational time, is achieved by using an agglomeration multigrid scheme, incorporating the full approximation scheme in a V-cycle process, within an efficient edge-based data structure.

A detailed validation of the proposed numerical methodology is performed by encountering both inviscid and viscous (laminar and turbulent) swirling flows with axial symmetry. IGal2D is compared against the commercial software ANSYS fluent – by using appropriate metrics and characteristic flow quantities – but also against experimental measurements, confirming the proposed methodology’s potential to predict such flows in terms of accuracy.

This study provides a robust methodology for the accurate prediction of swirling flows by combining the axisymmetric RANS equations with ACM. In addition, a detailed description of the convective flux Jacobian is provided, filling a respective gap in research literature.

The purpose of this paper is to describe and analyse a class of finite element fractional step methods for solving the incompressible Navier-Stokes equations. The objective is not to reproduce the extensive contributions on the subject, but to report on long-term experience with and provide a unified overview of a particular approach: the characteristic-based split method. Three procedures, the semi-implicit, quasi-implicit and fully explicit, are studied and compared.

This work provides a thorough assessment of the accuracy and efficiency of these schemes, both for a first and second order pressure split.

In transient problems, the quasi-implicit form significantly outperforms the fully explicit approach. The second order (pressure) fractional step method displays significant convergence and accuracy benefits when the quasi-implicit projection method is employed. The fully explicit method, utilising artificial compressibility and a pseudo time stepping procedure, requires no second order fractional split to achieve second order or higher accuracy. While the fully explicit form is efficient for steady state problems, due to its ability to handle local time stepping, the quasi-implicit is the best choice for transient flow calculations with time independent boundary conditions. The semi-implicit form, with its stability restrictions, is the least favoured of all the three forms for incompressible flow calculations.

A comprehensive comparison between three versions of the CBS method is provided for the first time.

The purpose of this paper is to present a numerical method for the simulation of steady and unsteady incompressible laminar flows, including convective heat transfer.

A node centered, finite volume discretization technique is applied on hybrid meshes. The developed solver, is based on the artificial compressibility approach.

A sufficient number of representative test cases have been examined for the validation of this numerical solver. A wide range of the various dimensionless parameters were applied for different working fluids, in order to estimate the general applicability of our solver. The obtained results agree well with those published by other researchers. The strongly coupled solution of the governing equations showed superiority compared to the loosely coupled solution as inviscid effects increase.

Convective heat transfer is dominant in a wide variety of practical engineering problems, such as cooling of electronic chips, design of heat exchangers and fire simulation and suspension in tunnels.

A comparison between the strongly coupled solution and the loosely coupled solution of the Navier-Stokes and energy equations is presented. A robust upwind scheme based on Roe’s approximate Riemann solver is proposed.

The purpose of this paper is to present a novel numerical method to solve incompressible flows with natural and mixed convections using pseudo‐compressibility formulation.

The present method is derived using the framework of Harten Lax and van Leer with contact (HLLC) method of Toro, Spruce and Spears, that was originally developed for compressible gas dynamics equations. This work generalizes the algorithm described in the previous paper to the case where heat transfer is involved. Here, the solution of the Riemann problem is approximated by a three‐wave system.

A few test cases involving incompressible laminar flows inside 2D square cavity for various Rayleigh and Reynolds numbers are considered for validating the present method. The computed results from the present method are found to be quite promising.

Although pseudo‐compressibility formulation has been found to have superior performance and has the potential to have numerical treatments similar to compressible flow equations, only two numerical methods have been applied so far; namely Jameson method and Roes flux difference splitting method. A new sophisticated numerical method, following the framework of HLLC method, is derived and implemented for solving pseudo‐compressibility‐based incompressible flow equations with heat transfer.

This study is concerned with developing laminar flow of an incompressible, Newtonian fluid, having constant viscosity, rotating in circular and rectangular ducts that contain a 180° bend. The Reynolds number ranges from 100 to 400, the rotation number from 0 to 0.4, and the Dean number from 66 to 264. Positive and negative rotation modes are considered. The artificial compressibility method is used for the numerical calculations and new boundary conditions are developed for these flows. It is shown that rotation causes the secondary flow to occur in ducts of any geometry, and that the strength of the secondary flow in the bend due to both rotation and curvature decreases as compared to the no rotation case.