4.2.3 Cell-Vertex Scheme: Dual Control Volumes 单元顶点格式:对偶控制体[cfd-4-2-3]

在这一格式中，控制体围绕各个网格节点(顶点)构造，所有流动变量都存储在节点上[18]、[19]。如图4.5所示，在二维情形中，对偶控制体通过连接共享该节点的四个单元的中点而构成。在三维中，则须把八个单元的形心连接起来，以构成控制体的各个面。另一种可能的做法是(在二维中)把某个单元形心连接到边中点，再连接回相邻单元的形心[20]。这样一来，控制体的面将由法向量不同的两部分构成，这在非结构网格中十分常见(参见下一章)。然而，对于结构网格，这种控制体定义只有在边界处才有理由采用，因为在边界处若不如此定义，表面的离散方式将会改变。图4.6演示了这一情形。在相当光滑的网格上，不能指望第二种做法在内部流场精度方面带来显著优势。因此，下文将采用较简单的对偶控制体定义。边界处理将在第8章中讨论。en

In this scheme, the control volumes are centred around the particular grid node (vertex), where all flow variables are stored [18], [19]. As depicted in Fig. 4.5, in the 2-D case the dual control volumes are constructed by joining the midpoints of the four cells which share the node. In three dimensions, the centroids of eight cells have to be connected in order to form the faces of the control volume. Another possibility would be to join one cell centroid to the edge midpoint (in two dimensions) and then to the neighbouring cell centroid again [20]. Thus, the face of the control volume would consist of two parts with different normal vectors, as it is common for unstructured grids (see next Chapter). However, in the case of structured grids, such a definition of the control volume is justified only at boundaries, where the surface discretisation would be otherwise changed. This is demonstrated in Fig. 4.6. Significant advantages with respect to accuracy in the interior field cannot be expected from the second approach on reasonably smooth grids. Therefore, in what follows, the simpler definition of the dual control volume will be employed. The boundary treatment is discussed in Chapter 8.

在计算离散化流动方程(4.2)时，需要计算控制体各面上的对流通量和黏性通量。这可以按以下三种做法之一来完成:

  • 通量平均(average of fluxes)——由控制体面左右两侧节点处的值分别计算通量，再取平均，但使用同一面向量(一般只用于对流通量);
  • 变量平均(average of variables)——对存储在控制体面左右两侧节点处的变量取平均;
  • 由分别插值到控制体面左右两侧的流动量计算通量(只用于对流通量)。
en

When we evaluate the discretised flow Equations (4.2), we have to compute the convective and the viscous fluxes at the faces of the control volume. This can be done according to one of the following three approaches:

  • by the average of fluxes computed from values at the nodes to the left and to the right of the face of the control volume, but using the same face vector (generally applied only to the convective fluxes);
  • by using an average of variables stored at the nodes to the left and to the right of the face;
  • by computing the fluxes from flow quantities interpolated separately to the left and to the right side of the face (employed only for the convective fluxes).

于是，例如对图4.5中的单元面\(\vec{n}_{i+1/2,j}\)，第一种做法——通量平均——在二维中写作en

Thus, for example at the cell face \(\vec{n}_{i+1/2,j}\) in Fig. 4.5 the first approach - average of fluxes - reads in two dimensions

\[\left(\vec{F}_c\,\Delta S\right)_{i+1/2,j} \approx \frac{1}{2}\left[\vec{F}_c(\vec{W}_{i,j}) + \vec{F}_c(\vec{W}_{i+1,j})\right]\Delta S_{i+1/2,j} \tag{4.35}\]

其中\(\Delta S_{i+1/2,j}\)分别按方程(4.6)与(4.7)，由单元形心的已知坐标计算。en

with \(\Delta S_{i+1/2,j}\) being computed according to Eq. (4.6) and (4.7), respectively, from the known coordinates of the cell centroids.

第二种可能的做法——变量平均——可以表述为en

The second possible approach - average of variables - can be formulated as follows

\[\left(\vec{F}\,\Delta S\right)_{i+1/2,j} \approx \vec{F}(\vec{W}_{i+1/2,j})\,\Delta S_{i+1/2,j}\,, \tag{4.36}\]

其中，控制体面\(\vec{n}_{i+1/2,j}\)上的守恒变量(或因变量)由两个相邻节点处数值的算术平均得到。于是en

where the conservative (or the dependent) variables at the face \(\vec{n}_{i+1/2,j}\) of the control volume result from arithmetic averaging of values at the two neighbouring nodes. Hence,

\[\vec{W}_{i+1/2,j} = \frac{1}{2}\left(\vec{W}_{i,j} + \vec{W}_{i+1,j}\right)\,. \tag{4.37}\]

图4.5:二维中单元顶点格式的对偶控制体

图4.5:二维中单元顶点格式的对偶控制体。图例:阴影四边形\(\Omega_{I,J}\)为围绕网格节点\(i,j\)的对偶控制体，由共享该节点的四个单元的中点连接而成;节点\(i,j\)以及相邻节点\(i-1,j\)、\(i+1,j\)、\(i,j+1\)、\(i,j-1\)均以实心圆点标记;\(\vec{n}_{i+1/2,j}\)、\(\vec{n}_{i-1/2,j}\)、\(\vec{n}_{i,j+1/2}\)、\(\vec{n}_{i,j-1/2}\)为对偶控制体各面的单位法向量(箭头所示)。

图4.6:边界处对偶控制体的定义(二维);上排:连接边中点构成;下排:连接边中点及边界上的中心节点构成

图4.6:边界处对偶控制体的定义(二维)。上排:通过连接边中点构成;下排:通过连接边中点以及边界上的中心节点构成。图例:阴影四边形为边界网格点(实心圆点)处的对偶控制体;阴影斜线区域为物面(壁面);左列为平直边界上的情形，右列为折角(斜坡)边界上的情形。

方程(4.36)中的通量向量\(\vec{F}\)既可以指对流通量，也可以指黏性通量。en

The flux vector \(\vec{F}\) in Eq. (4.36) represents either the convective or the viscous fluxes.

第三种方法把流动量(大多为速度分量、压力、密度和总焓)分别插值到面的两侧。插值得到的量——称为左(left)状态与右(right)状态(见4.3节开头)——在两侧一般并不相同。通过控制体面的通量随后利用某个非线性函数由左、右状态之差求出。于是en

The third methodology utilises an interpolation of flow quantities (being mostly velocity components, pressure, density and total enthalpy) separately to both sides of the face. The interpolated quantities - termed the left and the right state (see the begin of Section 4.3) - differ in general between both sides. The fluxes through the face of the control volume are then evaluated from the difference of the left and right state using some non-linear function. Thus,

\[\left(\vec{F}_c\,\Delta S\right)_{i+1/2,j} \approx f_{Flux}\left(\vec{U}_L,\,\vec{U}_R,\,\Delta S_{i+1/2,j}\right), \tag{4.38}\]

其中en

where

\[\begin{aligned} \vec{U}_L &= f_{Interp}\left(\cdots,\,\vec{U}_{i-1,j},\,\vec{U}_{i,j},\,\cdots\right) \\ \vec{U}_R &= f_{Interp}\left(\cdots,\,\vec{U}_{i,j},\,\vec{U}_{i+1,j},\,\cdots\right) \end{aligned} \tag{4.39}\]

它们表示插值得到的状态。当然，与(4.35)–(4.39)类似的关系对控制体的其他面同样成立。en

stand for the interpolated states. Of course, similar relations like (4.35)–(4.39) apply in the same way to other faces of the control volume.

同样的近似也用于三维。例如，在单元面\(\vec{n}_{i+1/2,j,k}\)(例如与图4.1b中的\(\vec{n}_2\)相同)处，方程(4.35)的通量平均变为en

The same approximations are employed in three dimensions. For example, at the cell face \(\vec{n}_{i+1/2,j,k}\) (e.g., identical to \(\vec{n}_2\) in Fig. 4.1b) the average of fluxes in Eq. (4.35) becomes

\[\left(\vec{F}_c\,\Delta S\right)_{i+1/2,j,k} \approx \frac{1}{2}\left[\vec{F}_c(\vec{W}_{i,j,k}) + \vec{F}_c(\vec{W}_{i+1,j,k})\right]\Delta S_{i+1/2,j,k}, \tag{4.40}\]

其中\(\Delta S_{i+1/2,j,k}\)由方程(4.8)与(4.9)得到。与方程(4.36)类似，流动变量的平均给出为en

where \(\Delta S_{i+1/2,j,k}\) is obtained from the Equations (4.8) and (4.9). The average of the flow variables results similarly to Eq. (4.36) in

\[\left(\vec{F}\,\Delta S\right)_{i+1/2,j,k} \approx \vec{F}(\vec{W}_{i+1/2,j,k})\,\Delta S_{i+1/2,j,k} \tag{4.41}\]

其中en

with

\[\vec{W}_{i+1/2,j,k} = \frac{1}{2}\left(\vec{W}_{i,j,k} + \vec{W}_{i+1,j,k}\right)\,. \tag{4.42}\]

最后，流动变量的插值对应于方程(4.38)，给出en

Finally, the interpolation of the flow variables leads correspondingly to Eq. (4.38) to

\[\left(\vec{F}_c\,\Delta S\right)_{i+1/2,j,k} \approx f_{Flux}\left(\vec{U}_L,\,\vec{U}_R,\,\Delta S_{i+1/2,j,k}\right), \tag{4.43}\]

其中\(\vec{U}_L\)与\(\vec{U}_R\)表示在面上插值得到的值。关于几种可能做法的详细描述，将在4.3节中针对对流通量、在4.4节中针对黏性通量分别介绍。en

where \(\vec{U}_L\) and \(\vec{U}_R\) denote the interpolated values at the face. A detailed description of several possible approaches will be presented in Section 4.3 for the convective and in Section 4.4 for the viscous fluxes.

离散化流动方程(4.2)中最后一项需要计算的是源项\(\vec{Q}\)。如引言中所述，源项大多假定在控制体内部为常数。因此，它用相应网格点处的流动变量计算。于是，我们可以定义en

The last term in the discretised flow Equations (4.2) to be evaluated is the source term \(\vec{Q}\). As already stated in the introduction, the source term is mostly supposed to be constant inside the control volume. For this reason, it is computed using the flow variables from the corresponding grid point. Hence, we may define

\[\left(\vec{Q}\,\Omega\right)_{i,j,k} = \vec{Q}(\vec{W}_{i,j,k})\;\Omega_{i,j,k}\,. \tag{4.44}\]

利用上述关系，可以算出通过各面的通量，并可按方程(4.2)沿\(\Omega_{i,j,k}\)的边界完成数值积分。这样，我们便得到包含源项的完整残差\(\vec{R}_{i,j,k}\)。守恒变量随时间的变化随后对每个网格点由下式给出en

Using the above relations, the fluxes through the faces can be computed and the numerical integration over the boundary of \(\Omega_{i,j,k}\) can be carried out according to Eq. (4.2). In this way, we obtain the complete residual \(\vec{R}_{i,j,k}\) including the source term. The change in time of the conservative variables follows then for each grid point from

\[\frac{d\vec{W}_{i,j,k}}{dt} = -\frac{1}{\Omega_{i,j,k}}\,\vec{R}_{i,j,k}\,. \tag{4.45}\]

适当的求解方法将在第6章中介绍。en

Suitable solution methods will be presented later in Chapter 6.