本論文的研究目的主要在於時空守恆法的發展，並應用於超音速流場中震波反射與繞射現象之探討。本文主要的研究重點分為三項，茲說明如下：

　　首先，原時空守恆法(CE/SE method)在避免傳統數值方法的限制下，建構在固定及等間格網格上以用來求解非定常守恆定律。此外，雖其架構在數學上極為簡單，但其適用性及精確性極高。本文將對一維原時空守恆法改良成一移動網格時空守恆法，改良後之方法不但保留了原方法之精神外，更在流場變化較大處(如：震波處)細化網格，不僅提升了該處的解析度且提升了整體的計算效率。

　　其次，本文應用時空守恆法來討論穩泰超音速流場下游條件對強馬赫反射結構及馬赫莖高度之影響。當馬赫數為2.84結果中顯示，馬赫莖高度隨流場之入口馬赫數、入射震波波角及楔形塊頂點到對稱軸間之距離之不同而改變。但流場下游條件，如：長度參數(R)及楔形塊頂點外角角度，將不會影響超音速定常流場中之馬赫莖高度或馬赫反射結構。

　　最後，本文以尤拉算子及層流模式之奈維爾-史托克算子分別計算在不同入射震波強度及繞射角度下，震波繞射結構及反射波與渦漩結構間交互作用時流場渦量之變化。在尤拉算子模擬中，震波繞射結構在打到下板發生震波反射之前，流場結構具有自生相似性；而在奈維爾-史托克算子中，流場結構除邊界層成長外，流場結構幾乎呈現自生相似。然而無論採用哪一算子，在滑移線上並無小渦漩結構形成。在尤拉算子模擬之渦量圖中可看出，流場之渦量主要是由滑移線所產生；而尤拉算子與奈維爾-史托克算子在流場之總環流量對時間之變化率不同，主要是由於邊界層與二次渦漩上之渦量貢獻。而未與下板發生震波反射之前，震波繞射結構中之渦量產生將隨入射震波強度及繞射角度改變。

　　The development and application of the Space-Time Conservation Element and Solution Element method in supersonic flows are investigated. This dissertation contains three main parts as organized in the following.

　　First, the original space-time CE/SE method has been developed and designed to avoid the limitations of the traditional numerical methods for stationary grid system in solving hyperbolic type conservation equations. Its foundation is mathematically simple that one can build it with a coherent, robust, and accurate numerical framework. In this research, an adaptively moving mesh scheme for solving one-dimensional hyperbolic conservation law is developed. This approach not only maintains the essential features of original CE/SE method but also clusters the mesh space at the locations where large variation in physical quantities exists. Computation accuracy and efficiency can be improved.

　　Secondly, we employ the CE/SE method to determine the influence of downstream flow conditions on Mach stem height in steady supersonic flows. The results indicate that the Mach stem height depends on the incident shock wave angle and the distance between the trailing edge and the symmetry plane. Furthermore, it is shown that the downstream length ratio and the trailing edge angle do not affect the Mach stem height nor the Mach reflection configuration.

　　Finally, this study employs the Euler and laminar Navier-Stokes solvers, respectively, to investigate the vorticity production in shock diffraction and the interaction of the reflected shock wave with the main vortex for incident shock waves of various strengths diffracting around convex corners with different angles. The numerical results show that in the Euler simulation, the flow structures are self-similar before they impinge on the bottom wall, and that in the laminar Navier-Stokes simulation, the flow structures are virtually self-similar other than when they are influenced by the boundary layer effect. Neither solver yields evidence of the roll-up of small vortices along the slipstream. The major vorticity production is found to be along the slipstream in Euler solution. Different circulation production rates are observed between the Euler and Navier-Stokes solutions as a result of the vorticity contribution of the boundary layer and the secondary vortex in the latter. The degree of vorticity production is found to be dependent on the strength of the incident shock wave and the diffracting angle when the bottom wall effect is not considered.

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