Diagonal Cartesian Method Research Articles

Multidimensional upwind scheme of diagonal Cartesian method for an advection–diffusion problem

The natural calculation region in fluid dynamics involves complex boundaries. When using the Cartesian grid to approximate complex boundaries, two difficulties develop: the boundary zigzag effect and disagreement of direction of grid line and velocity. The multidimensional upwind scheme of the diagonal Cartesian method (DCM), using both Cartesian grid lines and diagonal lines segments, is presented in this paper to simulate the complex boundaries of the multiple-layer quasi 3D model equations. The DCM improves the simulation accuracy for the boundaries and calculation time increases only slightly compared to the Cartesian method. In order to verify the new scheme, a test case is presented which rotates the cavity flow at 45° to compare the numerical calculation results at different Reynolds numbers. The test case shows that the scheme is accurate and efficient in improving the simulation results. Then the three-dimensional advection–diffusion processes in the tidal water of the Hongyanhe Power Plant are simulated using this model. Numerical results show that the scheme is not only efficient on an experiment basis, but also efficient and reliable when applied to a large scale natural water area.

Numerical Simulation of Wind-Induced Motion in Suspended Sediment Transport

In regions with broad water surfaces such as lakes, reservoirs and coastal areas, the wind stress on the flow motion generates a significant impact. The wind stress is an unsteady force which makes numerical simulation difficult. This paper presents a two-dimensional (2-D) mathematical model of the impact of wind-induced motion on suspended sediment transport at Beijing's 13-Ling Reservoir. The model uses the Diagonal Cartesian Method (DCM) with a wetting-drying dynamic boundary to trace variations in the water level. The calculation results have been tested against in situ measurements. The measurements confirm the model's accuracy and agreement with the actual situation at the reservoir. The calculations also indicate that wind stress holds the key to suspended sediment transport at Beijing's 13-Ling Reservoir, especially when westerly winds prevail.

Diagonal Cartesian Method for the Numerical Simulation of Flow and Suspended Sediment Transport over Complex Boundaries

There is increasing demand for simulation tools of flow and suspended sediment transport over complex boundaries in hydraulic engineering. The diagonal Cartesian method, which approximates complex boundaries using both Cartesian grid lines and diagonal lines segments, is presented in the paper to simulate the complex boundaries of two-dimensional shallow-water turbulence equations and nonequilibrium suspended sediment transport equation. The method, which utilizes cell-centered nodes on a nonstaggered grid, uses boundary velocity information at the wall boundary to avoid the specification of water level. An enlarged finite-difference method is introduced for momentum and suspended sediment equations on the complex boundary. This paper describes an application of the diagonal Cartesian method to calculate the tidal current and suspended sediment concentration of Quanzhou Bay in the Fujian province of China. The results show that the method predicts the flow and suspended sediment concentration well, and the calculations agree well with the measurement.

Direct Numerical Simulation on a Cartesian Mesh of the Flow through a Tube Bundle

In this paper, we present a three-dimensional direct numerical simulation (DNS) of a turbulent flow through a staggered tube bundle. The Navier–Stokes equations are discretized on a staggered Cartesian grid with the help of the finite-volume approach. The complex geometry of the tube bundle is implemented by the diagonal cartesian method (DCM). The advantage of this method, in comparison with body-fitted coordinate approaches, is that it leads to less complicated algorithms and, therefore, to more efficient computations. The principle of this method is described in this paper. Simulations of the flow through a tube bundle are carried out at on four different meshes with various grid spacings. Results of the simulations on the finest mesh are compared with the experimental data of Simonin and Barcouda [Measurements of fully developed turbulent flow across tube bundle. In 3ème Conférence Internationale sur les Applications de l'Anéemometrie Laser à la Mécanique des Fluides, 1986] and with the large-eddy simulations (LES) of Rollet-Miet et al. [“LES and RANS of turbulent flow in tube bundles”, Int. J. Heat Fluid Flow, 20 (1999) pp. 241–254]. Based on this comparison we conclude that an accurate simulation of a flow through a tube bundle can be performed with the DCM.

Finite Analytic Method in Flows and Heat Transfer

3R39. Finite Analytic Method in Flows and Heat Transfer. - Ching Jen Chen (Col of Eng, Florida A&M Univ), RA Bernatz (Luther Col), KD Carlson (Univ of Iowa), Wanlai Lin (Tech Dev of Next Generation Prod, Emerson Elec Corp). Taylor & Francis Publ, New York NY. 2000. 332 pp. ISBN 1-56032-898-3. $69.95. Reviewed by G de Vahl Davis (Sch of Mech and Manuf Eng, Univ of New South Wales, Sydney 2052, NSW, Australia).The finite analytic (FA) method is an Eulerian method that solves the differential equations for CFD/HT by representing the solution domain as a series of homogeneous, constant parameter elements. Within each of these elements an algebraic form of the analytic solution of a linearized form of the equations is constructed. The solution at a nodal value in the interior of each element is expressed as the sum of neighboring nodal values weighted by finite analytic coefficients. A system of these finite analytic algebraic equations is then solved to provide a numerical solution for the dependent variable at prescribed discrete locations within the domain.The method was conceived by CJ Chen (one of the authors of this book) and his then student, Peter Li, in 1977. It was first published in 1981 and has since been developed, extended, and implemented by Chen, his students and others. A global web search for “finite analytic method” yielded about 200 hits, while ScienceDirect® (for Elsevier publications) yielded 96 references between 1980 and 2001. The method has been successfully applied to a range of problems in two- and three-dimensional flow and heat transfer, both laminar and turbulent, in regular and irregular domains. The authors claim that, compared to traditional finite difference (FD) methods, the method is stable and accurate over a much broader range of flow and computational parameters such as Reynolds number and grid spacing. Nevertheless, it has not yet been widely adopted, and several well-known texts on CFD/HT published in recent years do not mention it.This book is aimed at graduate students and practitioners of CFD/HT. It is presented in five parts totaling 310 pages, comprising 25 chapters and two appendices. Some knowledge of differential equations and of some analytic methods are assumed. Part I is entitled Introduction to Computational Fluid Dynamics and contains an introduction followed by chapters on Governing Equations (the N-S and energy equations and turbulence modeling); Classification of PDEs; Well-Posed Problems (including existence and uniqueness); Numerical Methods (a brief survey of FD, finite element (FE) and FA methods); and a more extensive chapter on The Finite Difference Method. In the chapter on numerical methods, the authors briefly compare FA with FD and FE, highlighting the advantages and disadvantages of each; they conclude, not surprisingly, that FA wins out. In Part II, The Finite Analytic Method, the method is explained in detail. The seven chapters cover Basic Principles; One-, Two-, and Three-Dimensional Cases; Stability and Convergence; Hyperbolic PDEs; and what is called the Explicit Finite Analytic Method for the 2D transport equation for convection dominated flows, which is less complex than the implicit formulations developed in the preceding four chapters. Part III, Numerical Grid Generation, contains an introduction covering algebraic transformations and a summary of differential methods, followed by chapters on Elliptic Grid Generation; Equations in ξ and η Coordinates; Diagonal Cartesian (DC) Method; and FA Method on DC Coordinates. In Part IV, several Computational Considerations are discussed: Velocity, Pressure and Staggered Grids; Nonstaggered Grid Methods; and Boundary Conditions. Finally, Part V describes some Applications of the FA Method to Turbulent Flows; Turbulent Heat Transfer; Complex Domain Flows; and Conjugate Heat Transfer. It is this section, more than any other, which will allow teachers and practitioners to decide whether to adopt the method. The examples include turbulent flow past disc valves; the sea breeze phenomena (sic), a turbulent atmospheric boundary layer circulation driven by surface temperature gradients; groundwater flow and solute transport; and the design of a compact heat exchanger. The authors say that a 2D laminar Fortran code is available at www.finiteanalytic.com. However, when this reviewer went there, he got the message “This site is currently under construction; please check back at a later time” in several languages. This book should certainly be seriously considered by teachers of CFD/HT, and graduate students should have some exposure to the ideas presented. Time will tell whether the finite analytic method will take a place alongside—or perhaps even surpass—finite difference and finite element methods.

THREE-DIM ENSIONAL DIAGONAL CARTESIAN METHOD FOR INCOMPRESSIBLE FLOWS INVOLVING COMPLEX BOUNDARIES

A diagonal Cartesian method for the three-dimensional simulation of incompressible fluid flows over complex boundaries is presented in this article. The method is derived utilizing the superposition of the finite analytic solutions of a linearized two-dimensional convection-diffusion equation in Cartesian coordinates. The complex boundary is approximated with a structured grid in a series of calculation planes which are perpendicular to the x, y, and z coordinate axis. In the calculation plane, both Cartesian grid lines and diagonal line segments are used. It is observed that this geometry approximation is more accurate than the traditional sawtooth method. Mass conservation on complex boundaries is enforced with an appropriate pressure boundary condition. The method, which utilizes cell-vertex nodes on a staggered grid, uses boundary velocity information to avoid the specification of pressure values on boundaries. An enlarged control-volume method is introduced for the mass conservation and the pressure boundary condition on complex boundaries. The conservation of momentum on complex boundaries is enforced through the use of three-dimensional 19, 15, 11, or 7-point finite analytic elements. The proposed diagonal Cartesian method is verified by the solution of a rotated lid-driven cavity flow. It is shown that this diagonal Cartesian method predicts the fluid flow very well.

Numerical Modeling of Conjugate Heat Transfer on Complex Geometries With Diagonal Cartesian Method, Part II: Applications

Part I of this study discusses the diagonal Cartesian method for temperature analysis. The application of this method to the analysis of flow and conjugate heat transfer in a compact heat exchanger is given in Part II. In addition to a regular (i.e., Cartesian-oriented) fin arrangement, two complex fin arrangements are modeled using the diagonal Cartesian method. The pressure drop and heat transfer characteristics of the different configurations are compared. It is found that enhanced heat transfer and reduced pressure drop can be obtained with the modified fin arrangements for this compact heat exchanger.

Numerical Modeling of Conjugate Heat Transfer on Complex Geometries With Diagonal Cartesian Method, Part I: Methods

In this study, a diagonal Cartesian method for thermal analysis is developed for simulation of conjugate heat transfer over complex boundaries. This method uses diagonal line segments in addition to Cartesian line segments to approximate complex boundaries in Cartesian coordinates. The velocity fields are also modeled using the diagonal Cartesian method. The transport equations are discretized with the finite analytic (FA) method. The current work is validated by simulating a rotated lid-driven cavity flow with conjugate heat transfer, and accurate results are obtained.

Automatic grid generation of complex geometries in Cartesian co-ordinates

A method of automatic grid generation for complex boundaries in Cartesian co-ordinates is proposed in this paper. In addition to the Cartesian grid lines the diagonal segments are used for the approximations of complex geometries in Cartesian co-ordinates. A structured Cartesian grid is employed for the sake of the numerical simplicity and the potential of automatic grid generation. The automatic grid generation is achieved by this diagonal Cartesian method and the accuracy estimations of geometry approximations are given. The approximations of a few complex geometries, such as the multibody system in porous media, lake banks, grooved channels and spheres are shown and analyzed. The proposed method is verified by the numerical solutions of a rotated cavity flow. It is shown that the diagonal Cartesian method improves both the accuracy of geometry approximations and the numerical solution of a rotated cavity flow, comparing with the traditional saw-tooth method in which only Cartesian grid lines are utilized for geometry approximations. The stability and convergence of the proposed method is demonstrated. Finally, the application of the diagonal Cartesian method for the prediction of a grooved channel flow is presented. © 1998 John Wiley & Sons, Ltd.

Modeling of complex flows and heat transfer

This paper presents the numerical modeling of complex flows and heat transfer. The Finite Analytic method is used to discretize the transport equations. The diagonal Cartesian method is proposed to model fluid flows and heat transfer over complex geometries. A three-dimensional channel flow with conjugate heat transfer is simulated. By the diagonal Cartesian method and 5-point Finite Analytic scheme, a grooved channel flow and flow in a casting bank at different Reynolds numbers are modeled. Simulations by both the diagonal Cartesian method and the traditional saw-tooth Cartesian method indicates the diagonal Cartesian method improves the modeling of flows, due to the more accurate approximation of complex boundaries. Heat transfer in two-dimensional finned compact heat exchanger is also studied. An improved heat exchanger is proposed based on the numerical prediction of heat transfer.

DIAGONAL CARTESIAN METHOD FOR NUMERICAL SIMULATION OF INCOMPRESSIBLE FLOWS OVER COMPLEX BOUNDARIES

A diagonal Cartesian method is proposed for the simulation of incompressible fluid flows over complex boundaries in Cartesian coordinates. A structured grid is utilized for the sake of simplicity. The method approximates complex boundaries using both Cartesian grid lines and diagonal line segments. The grid is generated automatically, and the geometry approximation is shown to be more accurate than the traditional sawtooth method. Mass conservation on complex boundaries is enforced with an appropriate pressure boundary condition. The method, which utilizes cell-centered nodes on a nonstaggered grid, uses boundary velocity information to avoid the specification of pressure values on boundaries, An enlarged control-volume method is introduced for mass conservation and pressure boundary conditions on complex boundaries. The conservation of momentum on complex boundaries is enforced through the finite analytic (FA) method, using nine-point and five-point FA elements. Velocity boundary conditions at moving boundaries are analyzed. The proposed diagonal Cartesian method is verified with the solution of a rotated lid-driven cavity flow. It is shown that this diagonal method predicts the fluid flow very well and improves the accuracy of the numerical simulation compared to the traditional sawtooth method. The application of this method to a grooved channel flow is also presented.