7.3.1 Spatial Filtering 空间滤波[cfd-7-3-1]

LES基于空间滤波（spatial filtering）操作，它把任意流动变量 $$U$$ 分解为滤波后（大尺度、可解析）的部分 $\overline{U}$ 与亚滤波（不可解析）的部分 $$U'$$ ，即en

LES is based on a spatial filtering operation, which decomposes any flow variable $$U$$ into a filtered (large-scale, resolved) part $\overline{U}$ and into a sub-filter (unresolved) part $$U'$$ , i.e.,

U = \overline{U} + U' \tag{7.76}

空间中 $\vec{r}_0$ 处的滤波变量定义为en

The filtered variable at the location $\vec{r}_0$ in space is defined as

\overline{U}(\vec{r}_0, t) = \int_{D} U(\vec{r}, t)\, G(\vec{r}_0, \vec{r}, \Delta)\, d\vec{r}, \tag{7.77}

其中 $\Omega$ 表示整个流动域， $$G$$ 表示滤波函数， $\vec{r}$ 为位置向量。滤波函数决定小尺度的结构与大小，它依赖于差 $\vec{r}_0 - \vec{r}$ 以及滤波宽度 $\Delta = \left(\Delta_1\,\Delta_2\,\Delta_3\right)^{1/3}$ ， $\Delta_i$ 为第 $$i$$ 个空间坐标方向的滤波宽度。最常用的滤波函数有如下几种（见图7.3）：

tophat（盒式）滤波：
锐利傅里叶截断滤波：
Gaussian滤波：

where $\Omega$ denotes the entire flow domain, $$G$$ represents the filter function, and $\vec{r}$ is the position vector, respectively. The filter function determines the structure and size of the small scales. The filter function depends on the difference $\vec{r}_0 - \vec{r}$ and on the filter width $\Delta = \left(\Delta_1\,\Delta_2\,\Delta_3\right)^{1/3}$ , with $\Delta_i$ being the filter width in the $$i$$ -th spatial coordinate. The following filter functions are the mostly used ones (see Fig. 7.3):

the tophat filter:
The sharp Fourier cut-off filter:
The Gaussian filter:

G = \begin{cases} 1/\Delta^3 & \text{if } |(x_0)_i - x_i| \le \Delta_i/2 \\ 0 & \text{otherwise.} \end{cases} \tag{7.78}

G = \prod_{i=1}^{3}\frac{\sin\left(\dfrac{\pi}{\Delta_i}\left[(x_0)_i - x_i\right]\right)}{\pi\left[(x_0)_i - x_i\right]}. \tag{7.79}

G = \left(\frac{6}{\pi\Delta^2}\right)^{3/2}\exp\left(\frac{-6\,\|\vec{r}_0 - \vec{r}\|_2^2}{\Delta^2}\right). \tag{7.80}

tophat滤波和Gaussian滤波会平滑大尺度脉动以及滤波宽度以下的小尺度。截断滤波只影响截止波数以下的尺度。实际中，Gaussian滤波总是与锐利傅里叶截断配合使用。适用于单元尺寸变化的网格的滤波函数见文献[114]、[115]。en

The tophat and the Gaussian filter smooth the large-scale fluctuations as well as the small scales below the filter width. The cut-off filter affects only the scales below the cut-off wave-number. In practice, the Gaussian filter is always employed in conjunction with a sharp Fourier cut-off. Filters suitable for grids with varying cell sizes were proposed in Refs. [114], [115].

图7.3:物理空间中的LES滤波函数

图7.3:物理空间中的LES滤波函数:tophat (a),cut-off (b),Gaussian (c)。图例:G——滤波函数;x——空间坐标; $-\Delta/2$ 、 $+\Delta/2$ ——tophat滤波的支撑区间; $-\Delta$ 、 $+\Delta$ ——滤波宽度标记。