The Lagrangian and Eulerian acceleration properties of fluid particles in homogeneous turbulence with uniform shear and uniform stable stratification are studied using direct numerical simulations. The Richardson number is varied from $\mathrm{Ri}=0$, corresponding to unstratified shear flow, to $\mathrm{Ri}=1$, corresponding to strongly stratified shear flow. The probability density functions (pdfs) of both Lagrangian and Eulerian accelerations have a stretched-exponential shape and they show a strong and similar influence on the Richardson number. The extreme values of the Eulerian acceleration are stronger than those observed for the Lagrangian acceleration. Geometrical statistics explain that the magnitude of the Eulerian acceleration is larger than its Lagrangian counterpart due to the mutual cancellation of the Eulerian and convective acceleration, as both vectors statistically show an antiparallel preference. A wavelet-based scale-dependent decomposition of the Lagrangian and Eulerian accelerations is performed. The tails of the acceleration pdfs grow heavier for smaller scales of turbulent motion. Hence the flatness increases with decreasing scale, indicating stronger intermittency at smaller scales. The joint pdfs of the Lagrangian and Eulerian accelerations indicate a trend to stronger correlations with increasing Richardson number and at larger scales of the turbulent motion. A consideration of the terms in the Navier-Stokes equation shows that the Lagrangian acceleration is mainly determined by the pressure-gradient term, while the Eulerian acceleration is dominated by the nonlinear convection term. A similar analysis is performed for the Lagrangian and Eulerian time rates of change of both fluctuating density and vorticity. The Eulerian time rates of change are observed to have extreme values substantially larger than those of their Lagrangian counterparts due to the advection terms in the advection-diffusion equation for fluctuating density and in the vorticity equation, respectively. The Lagrangian time rate of change of fluctuating vorticity is mainly determined by the vortex stretching and tilting term in the vorticity equation. Since the advection-diffusion equation for fluctuating density lacks a quadratic term, the Lagrangian time rate of change pdfs of fluctuating density show a more Gaussian shape, in particular, for large Richardson numbers. Hence, the Lagrangian acceleration and time rates of change of fluctuating density and vorticity reflect the dominant physics of the underlying governing equations, while the Eulerian acceleration and time rates of change are mainly determined by advection.

中文翻译：

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湍流分层剪切流中的拉格朗日和欧拉加速度

使用直接数值模拟研究了具有均匀剪切和均匀稳定分层的均匀湍流中流体粒子的拉格朗日和欧拉加速度特性。理查森数从$里=0$, 对应于非分层剪切流, 到 $里=1$，对应于强分层剪切流。拉格朗日和欧拉加速度的概率密度函数 (pdf) 都具有拉伸指数形状，并且它们对理查森数显示出强烈且相似的影响。欧拉加速度的极值强于观察到的拉格朗日加速度。几何统计解释说，由于欧拉加速度和对流加速度的相互抵消，欧拉加速度的幅度大于其拉格朗日加速度，因为这两个向量在统计上显示出反平行偏好。执行拉格朗日和欧拉加速度的基于小波的尺度相关分解。对于较小尺度的湍流运动，加速度 pdf 的尾部变得更重。因此，平坦度随着尺度的减小而增加，表明在较小的尺度上具有更强的间歇性。拉格朗日和欧拉加速度的联合 pdf 表明随着理查森数的增加和湍流运动的更大尺度，相关性更强。考虑 Navier-Stokes 方程中的项，拉格朗日加速度主要由压力梯度项决定，而欧拉加速度主要由非线性对流项决定。对波动密度和涡度的拉格朗日和欧拉时间变化率进行了类似的分析。由于波动密度的平流-扩散方程和涡度方程中的平流项，欧拉时间变化率的极值明显大于拉格朗日对应的极值。脉动涡量的拉格朗日时间变化率主要由涡量方程中的涡旋伸缩项和倾斜项决定。由于波动密度的对流-扩散方程缺少二次项，波动密度的拉格朗日时间变化率 pdf 显示出更多的高斯形状，特别是对于大理查森数。因此，拉格朗日加速度和波动密度和涡度的时间变化率反映了基本控制方程的主要物理特性，