在自然界和工业生产中携带颗粒的两相湍流无处不在，湍流对颗粒的相对弥散起重要作用。在1950年代Batchelor对颗粒对的相对弥散做出了开创性的研究，他通过对拉格朗日速度关联的积分建立了颗粒对的相对分离距离的标度律。在这几十年间，研究者们建立了不同的拉格朗日速度关联模型，包括本文重点研究的尺度相似模型(He et al., Physics Review E. 80(6): 066313, 2009)。该尺度相似模型是对各向同性湍流中流体质点两点两时间拉格朗日速度关联的二阶近似，它通过弥散速度这一参数建立了与欧拉速度空间关联的联系，这种相似性表现为具有不同初始分离距离的颗粒对在弥散过程中速度关联函数可重合为一条普适的曲线。

本文利用直接数值模拟方法对湍流场通过伪谱方法求解，将颗粒简化为点颗粒模型，在单向耦合的假设下求解颗粒运动方程。通过分析数值计算结果，我们研究了流体质点的拉格朗日速度关联尺度相似模型的雷诺数效应以及惯性颗粒拉格朗日弥散的尺度相似性。本文的创新性工作包括：

1.     在高雷诺数下验证了拉格朗日速度关联尺度相似模型的有效性，并验证了理论推导得出的弥散速度所满足的标度律。在高雷诺数下空间尺度会出现清晰的惯性子区，前人的工作只在低雷诺数下证明了尺度相似模型在耗散区的有效性，本文在高雷诺数下验证了尺度相似模型在惯性区内也具有有效性。

2.     本文通过四组雷诺数下得到的拉格朗日速度关联统计量，探究尺度相似模型对雷诺数的依赖性。通过对重新标度的拉格朗日速度关联曲线以及弥散速度的标度律的分析，我们发现尺度相似模型对雷诺数并不敏感，但是随着雷诺数的增加，弥散速度逐渐变大，在耗散段满足雷诺数的1.39次标度律。

3. 探究不同斯托克斯数St下惯性颗粒弥散的尺度相似性。本文选取St = 0.2, 1,1.5, 3, 5, 10六组惯性不同的颗粒，在两组雷诺数下进行数值分析。我们发现在St小于等于1.5时惯性颗粒的尺度相似性和流体质点高度一致；当St = 3, 5, 10时，高雷诺数下颗粒的尺度相似性更好，但与小St数的惯性颗粒相比，其尺度相似性较弱。

4.为了分析尺度相似性的惯性效应，在小St数假设下，我们推导出惯性颗粒的拉格朗日速度关联函数的近似表达式，其中代表惯性效应的关联值单独出现在表达式中。通过数值计算，发现代表惯性效应的关联值与颗粒本身的速度关联相比非常小，但随着St数的增大这部分关联值会有所增大，使得拉格朗日速度时空关联的尺度相似性变弱。

It is well known that many industrial and environmental turbulent flows are laden with particles, and turbulence plays a significant role in the mixing and relative dispersion of particles. Early in 1950s, Batchelor developed the scaling laws of the relative dispersion between two particles by integrating the Lagrangian velocity correlations. For many decades, researchers have developed models for Lagrangian velocity correlations including the scale-similarity model discussed in this thesis (He et al. Physics Review E. 80(6): 066313,2009). This scale-similarity model is a second-order approximation of Taylor series expansion of the two-point two-time Lagrangian velocity correlation. It builds the relationship of Lagrangian space-time velocity correlation and Eulerian space velocity correlation with the only parameter defined as dispersion velocity. The scale-similarity model indicates that the Lagrangian velocity correlations with different initial space separations in dispersion progress follow a universal form.

In this thesis, we use direct numerical simulation method to calculate turbulent flow fields through the pseudo-spectral method. Lagrangian method is used to track the motion of particles. From our simulation data, we first analyse the Reynolds number effect on the scale-similarity model and then extend to explore the scale-similarity of Lagrangian dispersion for inertial particles. Based on these two issues, the main innovative contributions include:

1.     We verify the scale-similarity model and the corresponding scaling laws of dispersion velocity at high Reynolds number turbulent flows. A well-defined inertial subrange is observed in the turbulent flows at higher Reynolds numbers, thus we are able to verify that scale-similarity exists in inertial subrange, while the previous work only verified the model in dissipation subrange.

2.     In this thesis, we compare the simulation results at four Reynolds numbers to explore the Reynolds number dependence of the scale-similarity model. Through analyzing the rescaled Lagrangian velocity correlation functions and the scaling laws of the dispersion velocity, we find that the scale-similarity model is insensitive to Reynolds number effect. However, the value of dispersion velocity becomes larger with the increasing Reynolds number, particularly, it follows 1.39 power law of Reynolds number in the dissipation range, where is the dispersion velocity, is the Kolmogorov velocity scale and is the Taylor microscale Reynolds number.

3.     We then explore if the scale-similarity exists in the dispersion of inertial particles at our considered Stokes numbers St=0.2, 1, 1.5, 3, 5,10. We perform the simulation at two Reynolds numbers . It shows that the scale-similarity for inertial particles with St no larger than 1.5 is highly consistent with the scale-similarity of fluid particles. For particles with St=3, 5,10, the scale-similarity performs better at higher Reynolds number. By contrast, the scale-similarity of larger St is not as strong as small-St particles.

4.     In order to analyze the inertial effect of scale-similarity separately, we develop an approximation formula of Lagrangian velocity correlation for inertial particles at small Stokes numbers. According to the new formula, we calculate the velocity-acceleration cross-correlation involving inertial effect. The results show that the cross-correlation is a small quantity compared with Lagrangian velocity correlation of inertial particles. It increases as the Stokes number increases.