A new finite difference approach for MHD flow

Date

2023-7-25

Author

Öğütcü, Gamze

Metadata

Show full item record

This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Item Usage Stats

138
views

72
downloads

Cite This

The magnetohydrodynamic (MHD) flow equations with mixed-type boundary conditions, which are presented in this thesis as modified Helmholtz equations, are numerically solved using the finite difference methods (FDM), precisely the standard and new (novel) finite difference approaches. A new finite difference (NFD) method is derived from the standard finite difference (SFD) approach by eliminating the error due to truncation in the finite difference formulas that approximate the derivatives. In light of the fact that the NFD method involves no truncation error, it is regarded as a numerically exact method. First, the NFD formulas are established for approximating first-order derivatives in mixed-type boundary conditions and second-order derivatives in one- and two-dimensional modified Helmholtz equations. Second, the reduced modified Helmholtz form of MHD flow equations and the related mixed boundary conditions are discretized using these formulas. In this regard, we focus primarily on the magnetohydrodynamic flow problems in parallel infinite plates and infinite channels with rectangular cross-sections. The one-dimensional MHD flow between parallel infinite plates, for which the analytical solution exists, is initially solved to build and validate the FDM codes and perform a quantitative comparison of the standard and new finite difference approaches. The two-dimensional MHD flow problem in a rectangular duct is the next application to which we extend the usage of these finite difference approaches. In order to examine the combined impacts associated with the conductivity parameter and slip length on the velocity $V$ and induced magnetic field $B$, numerical simulations with SFD and NFD methods are carried out for several cases in both problems using a variety of Hartmann numbers. The progressive numerical outcomes are conceptually presented as distributions of velocity and induced magnetic field across the entire computational interval in one dimension and along straight lines close to slippery or variably conducting duct walls in two dimensions. It has been observed that, even with a coarse mesh, the NFD method produces numerical results that are noticeably more precise when compared to the findings of the SFD methodology in each case. Additionally, a finer mesh is required in both NFD and SFD approaches to accurately capture the characteristics of the flow velocity and the induced magnetic field at high Hartmann numbers, and with slippery and variably conducting boundaries of the computational domain.

Subject Keywords

new FDM, standard FDM, modified Helmholtz equation, Hartmann flow, MHD duct flow, slip velocity condition, variable conductivity

URI

https://hdl.handle.net/11511/104859

Collections

Graduate School of Natural and Applied Sciences, Thesis