發展一具最佳數值色散關係的方法以求解多維不可壓縮流

本論文是開發一色散關係式得以保持的格式，以求解不可壓縮之 Navier-Stokes 方程，利用此一格式可使數值求解的過程與計算結果能夠更佳的符合物理意義。求解過程中利用 Clebsch 速度分解理論，可將方程式分解為三個步驟，並由其分別的求解對流、擴散與壓力項次。此論文中，針對對流項次的求解發展了一具色散關係式得以保持的格式，並使時間離散格式能滿足具辛結構之漢米爾頓性質與喀什米爾性質，另將計算域引入頻域當中，使此一格式在頻域當中之誤差能夠保有最小值。擴散項次部分使用六階準確緊緻格式處理。對於壓力項次部分則使用了無散度定理求解。希望對於數值求解 Navier-Stokes 方程能夠得到更佳與真實物理意義匹配的解。

關鍵字

Clebsch 速度分解理論；具辛結構；色散關係式得以保持之格式；漢米爾頓與喀什米爾性質

並列摘要

This thesis is aimed to develop a flow solver for properly simulating the incompressible fluid flow at high-Reynolds numbers. The strategy of getting a higher simulation quality is to retain some rich geometrical properties embedded in its limiting Euler flow equations. Following the theory of Clebsch velocity decomposition, I can rigorously decompose the velocity vector as the sum of the velocity components that account, respectively, for the flow potential, rotation, and dissipation. Simulation of the Navier-Stokes equations cast in velocity-pressure primitive variables can be therefore fractionally split into three corresponding steps. The equations are coupled to each other. In the pure advection solution step, we employ the mid-point symplectic time integrator to approximate the temporal derivative term so as to retain the discrete Hamiltonian and Casimir properties embedded in the lossless Euler equations. For the sake of reducing numerical dispersion error, the upwinding spatial scheme with the smallest difference between the numerical and exact dispersion relation equations for the time-dependent pure advection equation is developed in wavenumber space. In the diffusion step, I approximate the time derivative term shown in the time-dependent parabolic equation using the time-stepping scheme that can be different from the one used in the first solution step for the calculation of Euler solutions. I then update the velocity vector in projection step that is solved subject to the divergence-free constraint condition. The proposed method has been validated through some chosen benchmark tests. The predicted results for the incompressible Navier-Stokes equations are also justified by solving two problems at high-Reynolds numbers.

並列關鍵字

Clebsch velocity decomposition ； symplectic ； dispersion relation equations ； Hamiltonian and Casimir properties

參考文獻

[1] A. Clebsch,

Uber die Integration der hydrodynamischen Gleichungen,

J.Reine. Angew. Math., vol. 56, pp. 110, 1895

[2] Christopher K. W. Tam, Jay C. Webb, Dispersion-relation-preserving finite difference schemes for computational acoustics, J. Comput. Phys.,

Space-time discretizing optimal DRP schemes for flow and wave propagation problems,

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發展一具最佳數值色散關係的方法以求解多維不可壓縮流

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