臺灣博碩士論文加值系統

在論文中，我們主要探討等向(isotropic)、均勻(homogeneous)且穩態(stationary)之不可壓縮紊流場。該研究是發展一套遞迴重整群分析(recursive renormalization group (RG) analysis)的方法來研究黏性流、熱輸送和磁化流的紊流場特性。
在第一章中，首先介紹紊流場之數值模擬的近況，並且勾勒重整群分析的概略發展史，同時說明重整群分析之於紊流場的觀點和須努力的地方。在第二章中，簡單回顧幾個基本方程式，以便後面章節的引用，其中包括Navier-Stokes 方程式在波數空間中的解析形式，以及相關符號算子的定義。該論文的主要內容放在第三至第五章。
在第三章中，我們假設大尺度渦漩(large-scale eddies)與小尺度渦漩為統計獨立，就牛頓流(Newtonian fluids)作重整群分析並且有相當成果。在遞迴重整群程序中，為了簡化不必要的複雜性，出現在Navier-Stokes方程式中非線性的三重項(triple term)可視為小擾動的微小量，並且被忽略。藉由假設一個具有self-consistent之換尺律 (scaling law) 的能量光譜(energy spectrum)，在重整群程序中，可以完全決定能量光譜中所有被引用的指數值。透過重整群分析的極限運算可以把RG轉換轉變成一階常微分方程式，根據此微分方程的解析解，我們可以推得用來作紊流場大尺度渦漩模擬的Smagorinsky 模型，並且證明該模型常數與流場中幾個特定大尺度渦漩的幾何尺寸和數值運算格點尺度之函數關係。
在第四章中，延續對不可壓縮紊流場之重整群分析，去探討熱輸送的物理性質，其中，溫度場$T$在流場中可視為被動純量(passive scale)。藉由微分計算的運用，我們得到一個紊流Prandtl數 $Pr_t$ 與紊流Peclet數 $Pe_t$ 的關係式。 $Pr_t$和$Pe_t$的函數關係式與Yakhot等人的結果類似，並且與直接數值模擬法(direct-numerical simulation)的結果和實驗數據接近。透過極限運算所得的微分方程式可解出恆定熱渦漩擴散係數(invariant thermal eddy diffusivity) 的解析形式。若結合該解析式與Batchelor 能量光譜，則可計算出Batchelor常數與幾個特定大渦漩的關係式，以及適用於熱輸送的Smagorinsky模型和模型常數。
在第五章中，主要運用我們所發展的遞迴重整群分析來探討三維磁化流的流場特性。其中，重整群分析的固定點(fixed point)是在以下兩個條件下計算得到。(i)平均磁感應場(magnetic induction)相對較弱於平均速度場。(ii)Alfven 效應成立，也就是速度場的擾動與磁場的擾動幾乎平行且近似相等。結果發現，重整群程序不會對磁阻率(magnetic resistivity)產生增量，並且偶對效應(coupling effect)可以化簡恆定渦漩黏滯係數(invariant eddy viscosity)，並且證明速度場與磁感應場的能量光譜在慣性子區都遵循Kolmogorov光譜定律$k^{-5/3}$，此結果與實驗室量測和天文物理上的觀測結果一致。透過假設速度場與磁感應場的能量光譜具有相似解析形式，我們可以決定Kolmogorov常數和Smagorinsky常數的關係以及兩者分別跟幾個特定大尺度渦漩的函數關係。
最後，第六章是整體的結論，以及一些未來研究方向的展望。

In this thesis, we investigate incompressible turbulence which is further assumed to be isotropic, homogeneous and stationary. The methodology developed for the present investigation is a recursive renormalization group (RG) analysis, and the subjects treated include fluid flow, thermal transport and magneto-hydrodynamics.
In Chapter 1, we start with a brief account of the current status of numerical simulation of turbulence, give a short sketch of the history of the RG analysis and explain the points of interest related to the present RG analysis. In Chapter 2, we review briefly the basic equations and formulas used for later chapters; this consists of formulating the Navier-Stokes equation in the wavenumber domain and developing symbolic calculations of the relevant operators. Chapters 3-5 contain the major results of the thesis.
In Chapter 3, the renormalization group analysis is carried out for ordinary Newtonian fluids. It is found fruitful to take the simple hypothesis that large-scale eddies are statistically independent of those of smaller scales. A recursive renormalization procedure is then proposed for turbulence governed by the Navier-Stokes equation in an exact manner that a nonlinear triple term appearing in early treatment can be dispensed with in the present formulation. By assuming a self-consistent scaling law for the energy spectrum, the relevant exponents appearing in the energy spectrum are completely determined. Proceeding with the limiting operation of the renormalization group analysis yields an ordinary differential equation for the RG transformation, to which a closed-form solution is readily obtained. The closed-form solution of the equation facilitates derivation of the Smagorinsky model for large-eddy simulation of turbulent flow, which reveals the explicit dependence of the model constant on the cutoff size and other characteristic wavenumbers.
In Chapter 4, we continue with the RG analysis of incompressible turbulence, aiming at determination of various thermal transport properties. In particular, the temperature field $T$ is considered a passive scalar. A differential argument leads to derivation of the turbulent Prandtl number $Pr_t$ as a function of the turbulent Peclet $Pe_t$ number, which in turn depends on the turbulent eddy viscosity $\nu_t$. The functional relationship between $Pr_t$ and $Pe_t$ is comparable to that of Yakhot et al., and is in close consistency with direct-numerical-simulation results as well as measured data from experiments. Proceeding further with limiting operation of renormalization group analysis yields an ordinary differential equation for an invariant thermal eddy diffusivity $\sigma$. Simplicity of the equation renders itself a closed-form solution of $\sigma$ as a function of the wavenumber $k$. This solution, when combined with a modified Batchelor''s energy spectrum for the passive temperature $T$, facilitates determination of the Batchelor constant $C_B$ and a parallel Smagorinsky model and the model constant $C_P$ for thermal turbulent energy transport.
In Chapter 5, we continue with our previous re-normalization group (RG) analysis of incompressible turbulence,aiming at determination of various properties of three-dimensional magneto-hydrodynamics (MHD). In particular, we are able to locate the fixed points of RG transformation under the following conditions. (i) The mean magnetic induction is relatively weak compared to the mean flow velocity. (ii) The Alfv$\acute{e}$n effect holds, that is, the fluctuating velocity and magnetic induction are nearly parallel and are approximately equal in magnitude. It is found under these conditions that re-normalization does not incur an increment of the magnetic resistivity, while the coupling effect tends to reduce the invariant effective eddy viscosity. Both the velocity and energy spectra are shown to follow the Kolmogorov $k^{-5/3}$ in the inertial subrange; this is consistent with some laboratory measurements and observations in astronomical physics. By assuming that the velocity and magnetic induction share the same form of energy spectrum, we are able to determine the dependence of the Kolmogorov constant $C_K$ and the proportional constant $C_S$ of the Smagorisnky model for large-eddy simulation.
Finally, Chapter 6 is devoted to conclusion in overview and a proposal for further research.

1 Introduction 5
2 Basic Equations and Formulas 9
2.1 introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 The Navier-Stokes equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Fourier analysis of the turbulent velocity eld . . . . . . . . . . . . . . . . . 11
3 Renormalization Group Analysis of Flow Turbulence 17
3.1 introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Renormalization group analysis . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3 Determination of the energy spectrum . . . . . . . . . . . . . . . . . . . . . 25
3.4 Equation of the invariant eective eddy viscosity . . . . . . . . . . . . . . . . 26
3.5 Evaluation of the Kolmogorov constant . . . . . . . . . . . . . . . . . . . . . 30
3.6 The Smagorinsky model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4 Renormalization Group Analysis for Thermal Turbulent Transport 37
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Renormalization of the passive-scalar equation . . . . . . . . . . . . . . . . . 38
4.3 The Turbulent Prandtl number Prt . . . . . . . . . . . . . . . . . . . . . . . 43
4.4 The invariant eective thermal eddy diusivity . . . . . . . . . . . . . . . . . 47
4.5 Evaluation of the Batchelor constant . . . . . . . . . . . . . . . . . . . . . . 50
4.6 The Smagorinsky model for passive scalar . . . . . . . . . . . . . . . . . . . 54
4.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5 Renormalization Group of MHD Turbulence with The Alfv en Eect 59
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.2 Magnetohydrodynamic equations . . . . . . . . . . . . . . . . . . . . . . . . 60
5.3 Renormalization group analysis for MHD uid . . . . . . . . . . . . . . . . . 63
5.4 Determination of energy spectrum . . . . . . . . . . . . . . . . . . . . . . . . 69
5.5 Equation of the invariant eective eddy viscosity . . . . . . . . . . . . . . . . 75
5.6 Evaluation of the Kolmogorov constant and Smagorinsky model . . . . . . . 77
5.7 Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6 Conclusion 81
6.1 Summary of the main results . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.2 A proposal for further research . . . . . . . . . . . . . . . . . . . . . . . . . . 82

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