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Numerical Methods and Computational Tools
Published in Raj P. Chhabra, CRC Handbook of Thermal Engineering Second Edition, 2017
Atul Sharma, Salil S. Kulkarni, K. Hrisheekesh, Amit Agrawal, Shyamprasad Karagadde, Amitabh Bhattacharya, Rajneesh Bhardwaj
Interface reconstruction is required after every advection sweep in order to compute the geometric fluxes accurately. Here, least-squared VOF interface reconstruction algorithm (LVIRA) of Puckett et al. (1997) is presented. The interface, in each computational grid cell, is represented by a linear segment and can be defined in 2D by the unit normal n^ pointing into the liquid region and a distance, l, of the interface from the center (centroid in 3D) of the computational cell, as shown in Figure 5.7.4b. For e.g., in 2D calculations, a nine-point stencil around a two-phase cell is considered. A line is drawn through the stencil by using a guess n^ (obtained by Youngs’ method (Youngs, 1982)) and l, such that the void fraction at the center cell of the stencil matches the void fraction obtained after advection. The normal is estimated using the following relation.
A fourth-order compact difference algorithm for numerical solution of natural convection in an inclined square enclosure
Published in Numerical Heat Transfer, Part A: Applications, 2021
Xian Liang, Haiyan Zhang, Zhenfu Tian
In this subsection, a fourth-order compact (4OC) method is established to solve the system of Eqs. (1)–(3) governing laminar thermal flows with the novelty of “genuine compactness,” i.e. the compact scheme is strictly with in the nine-point stencil. Note that each of these equations is a special case of the following equation Poisson-type: where Note that each of Eqs. (1), (2) and (3) is a special case of Eq. (6). Notice also that and in Eq. (1); and in Eq. (1); and in Eq. (3).
Heatline Analysis on Heat Transfer and Convective Flow of Nanofluids in an Inclined Enclosure
Published in Heat Transfer Engineering, 2018
To develop a higher-order numerical method for solving the above heat transfer and convective flow of nanofluids in an square enclosure with a heated thin plate inside, we first design a mesh as Xi = ih, Yj = jh, h = 1/M, i, j = 0, 1, …, M, M is a positive integer. We also note that each of Eqs. (19)–(21), (26) can be viewed as the following steady-state convection diffusion equation [38]where Ψ is a transport variable representing Φ, Ω, θ, Q, respectively, and a, b, s denote the corresponding coefficient functions. Thus, a fourth-order compact scheme [39, 40] for solving Eq. (27) can be written as: and the coefficients are given in Appendix A, where δ2XΨi, j, δ2YΨi, j, δXΨi, j, δYΨi, j are the standard second-order central difference operators within the nine-point stencil [41] (see Appendix B). Note that one has to solve four systems based on Eq. (28) for stream function, vorticity, temperature and heatfunction, we introduce the pesudo-time algorithms and alternating direction implicit method [42] to solve Eq. (28) in order to simplify the computation.
Exponential high-order compact finite difference method for convection-dominated diffusion problems on nonuniform grids
Published in Numerical Heat Transfer, Part B: Fundamentals, 2019
The organization of this article is as follows. Section 2 is devoted to construct a three-point EHOC difference scheme on nonuniform grids for the 1D convection–diffusion equation with constant convection coefficient and variable convection coefficients. In Section 3, EHOC difference schemes on nonuniform grids on a nine-point stencil are proposed for the 2D convection–diffusion equation with constant and variable convection coefficients. In Section 4, numerical experiments are performed to demonstrate the feasibility of the proposed EHOC finite difference methods on nonuniform grid. A few remarks and comments are given in Section 5.