Stability analysis of unbounded uniform dense granular shear flows

本篇論文是利用線性穩定理論去分析稠密顆粒材料在均勻剪力下之無邊界的流況行為。此分析是根據修正後的運動理論組成方程式去進行稠密狀態下之研究，這邊採用乾顆粒，每一個顆粒都為等大小之光滑非彈性球體。在本篇主要進行的研究，乃利用漸進與暫態穩定理論方法去分析完全運動模型在稠密狀態下的情形。暫態現象可以實用性地提供對於有限振幅之發展是否藉由瞬間的觸發而產生的。此線性擾動系統可由凱文模式去得解，凱文模式的幾何意義為擾動波向量會因均勻剪力流況而發生旋轉現象。在空間獨立模式下，整個系統將隨著時間演進而呈現不穩定的現象。系統在沒有流向方向的擾動波情況下，會呈現漸進穩定的現象，另外證明了邊界穩定曲線不可能存在此組成律所定義的範圍內。發現當固體體積比率與恢復係數增加時，初始暫態成長率與最大的暫態成長將會隨之提高。但是在初始縱向擾動波增加時，只有初始暫態成長率會跟著提升，而最大的暫態成長卻反之下降。當擾動波沒有限定只能縱向傳遞時，暫態函數隨著時間的演進將會出現多重的高峰且最大的暫態成長值將明顯地隨著初始縱向擾動波減少而跟著提高。這是由於本研究的系統矩陣乃為非正則矩陣，此將造成初始暫態成長率在極短的瞬間有了正的值，因此可向上發展，且非正則矩陣也是造成暫態成長函數有出現極大值與震盪發生的可能。另外，藉由初始條件下各分量隨時間之發展，任何一個分量在不同時間下之主宰情形亦可被觀察出來。我們在最後將討論加入半線性應力知結果。由於此結果呈現非常混亂的情形，我們依據此結果進而提出對於此修正後的組成方程式之相關建議。This thesis presents a linear stability analysis of unbounded uniform dense granular shear flows. The analysis is based on the revised constitutive equations of the kinetic theory (Savage 2008) for dry, identical spherical, smooth and inelastic particles in the dense state. In the present work, the purely kinetic model in dense state has been studied by asymptotic and transient stability analyses. Transient phenomena can provide a viable way to trigger finite amplitude effects. The solution of the linear perturbed system is obtained by the Kelvin-mode which means that the wavenumber vector of the disturbances is turned by the mean shear flow. The result of spatially uniform mode is unstable when time proceeds. Disturbances of zero streamwise wavenumber are always stable by the asymptotic results and the marginal stability curve has proved that it does not exist. As the solids volume fraction and coefficient of restitution are increased, the initial transient growth rate and the maximum transient growth are both enhanced. As the initial transversal wavenumber is increased, the initial transient growth rate is also enhanced but the maximum transient growth is reduced. Disturbances of nonzero streamwise wavenumber can produce multiple and significant peaks as the initial transversal wavenumber are small. Since the matrix of our linear system is the non-normality matrix, the initial growth rate can be a positive value at initial period. The transient growth function may cause a significant transient growth value and the oscillatory peaks. The temporal evolution of each component from the initial condition is presented in this thesis, the dominated component can be observed. The case of adding the quasi-static stresses is discussed in the end. Because the results are disorderly, we propose a suggestion to the model (Savage 2008).誌謝 III要 IVbstract Vigure list VIable list IXymbols Xhapter 1 Introduction 1.1 Motivation 1.2 Literature survey 2.3 Organization of thesis 7hapter 2 Mathematical model 11.1 Governing and constitutive equations 11.2 Basic-state derivation and non-dimensional process 14.2.1 Basic-state derivation 14.2.2 Non-dimensional process 15.3 Linear perturbation process 17.4 Fourier mode process 21hapter 3 Method of stability analysis 26.1 Criterion of asymptotic stability analysis 26.2 Criterion of transient stability analysis 27.3 Criterion of transient stability analysis for non-layering modes 29hapter 4 Results and discussions 31.1 The spatially uniform mode for purely kinetic model 31.2 Layering modes for purely kinetic model 33.2.1 Results for asymptotic stability analysis 34.2.2 Results for transient stability analysis 36.3 Non-layering modes for purely kinetic model 42.3.1 Results for asymptotic stability analysis 42.3.2 Results for transient stability analysis 43.4 Results for frictional-kinetic model 47hapter 5 Conclusions and future works 74.1 Conclusions 74.2 Future works 76 eferences 77ppendix A Elements of M(t) 79ppendix B Theoretical proof of non-existence of the neutral stability curve for layering modes 81ersonal information of author 8