5.1 Geometrical Quantities of a Control Volume 控制体的几何量[cfd-5-1]

Generally, we think of the flow in a plane as being a special case of a 3-D problem, where the solution is symmetric with respect to one coordinate direction (e.g., to the \(z\)-direction). Because of the symmetry and in order to obtain correct physical units for volume, pressure, etc., we set the depth of all grid cells and control volumes equal to a constant value \(b\). The volume of a control volume results then in 2D from the product of its area with the depth \(b\). Since the depth \(b\) is arbitrary, we may set \(b = 1\) for convenience. In the following discussion, we restrict ourselves to triangular and quadrilateral elements. Even though the control volume of a median-dual scheme can have a rather complex shape, it can always be decomposed into triangles and/or quadrilaterals.

The area of a general triangle can be most conveniently and exactly calculated by the formula of Gauss. Thus, using a node numbering in accordance with Fig. 5.3a, the volume results from

The nodes have to be numbered in the anti-clockwise direction in order to obtain a positive value for the volume.

The area of a general quadrilateral can be exactly calculated by Gauss' formula, which leads, after some algebra, to the expression

where the nodes are numbered according to Fig. 5.3b in the anti-clockwise direction. In the above, we assumed that the control volume is located in the \(x-y\)-plane and that the \(z\)-coordinate represents the symmetry axis.

The edges of a control volume are given by straight lines in 2D and therefore the unit normal vector is constant along them. When we integrate the fluxes according to the approximation of Eq. (5.2), we have to evaluate the product of the area of a face \(\Delta S\) and the corresponding unit normal vector \(\vec{n}\) which is the face vector \(\vec{S}\). Considering Fig. 5.3, the outward pointing face vector, e.g., at the side 2-3 is given by

Because of the symmetry, the \(z\)-component of the face vectors (and of the unit normal vector) is zero. It is therefore omitted in Eq. (5.6). The unit normal vector can be obtained from Eq. (5.6) with

where \(S_x\), \(S_y\) denote the Cartesian components of the face vector.

The centre of the triangle from Fig. 5.3a is defined as

with \(\vec{r}_{1/2/3}\) representing the Cartesian coordinates of the nodes. The centre of a quadrilateral element can be computed by the formula given in Ref. [34]. For this purpose, the quadrilateral is decomposed into two triangles which share two points. With the node numbering according to Fig. 5.3b, and 1 and 3 being the common nodes, the relation reads

The volumes of the two triangles \(\Omega_{123}\) and \(\Omega_{134}\) are evaluated using Eq. (5.4), and their centroids \(\vec{r}_c\) are obtained from Eq. (5.8).

As opposed to the previous 2-D case, the computation of face vectors and volumes poses in 3D some problems for elements or control volumes with quadrilateral faces. The main reason for this is that, in general, the four vertices of a quadrilateral face of a control volume may not lie in a plane. Then, the normal vector is no longer constant on such face (see Fig. 4.2). In order to overcome this difficulty, we could decompose each quadrilateral face into two or even more triangles. However, the gain in accuracy is hardly noticeable for a second-order scheme on a smooth grid. The additional effort can only be justified - and in fact it becomes necessary - for a third- and higher order spatial discretisations. Therefore, we shall apply a simplified treatment of the quadrilateral faces in the following considerations, which is based on an averaged normal vector.

The face vector \(\vec{S}\) can be exactly computed for a triangular face using Gauss' formula. Defining the nodes according to Fig. 5.4a, we obtain for the edge differences of the triangle 1-2-3

The outward pointing face vector \(\vec{S} = \vec{n}\Delta S\) results then from

The averaged face vector \(\vec{S}\) of a quadrilateral face, like that rendered in Fig. 5.4b, is most conveniently computed using the same Gauss' formula as employed in 2-D for the area of a quadrilateral. Thus, for the face given by the nodes 5, 6, 7 and 8 in Fig. 5.4b, we first define the differences

Then, we obtain the outward pointing face vector \(\vec{S} = \vec{n}\Delta S\) from the relation

The approximation becomes exact when the face approaches a parallelogram, i.e., when the vertices of the face lie all in one plane.

The unit normal vector is obtained in both cases from \(\vec{n} = \vec{S}/\Delta S\) with

where \(S_x\), \(S_y\) and \(S_z\) denote the Cartesian components of the face vector given by Eq. (5.11) or Eq. (5.13), respectively.

As we already stated in the case of 3-D structured finite volume schemes, a very convenient approach for the computation of volumes is based on the divergence theorem [35]. The discussion in Subsection 4.1.2 led finally to the expression

for the volume, where \(N_F\) denotes the number of the faces of the control volume, \((\vec{r}_c)_m\) the centre of the face \(m\) of the control volume, and \(\vec{S}_m\) the face vector (outward directed) of the face \(m\), respectively. The formula (5.15) is directly applicable on unstructured grids. It is exact for a volume with triangular faces, or a volume with planar quadrilateral faces.

The previously mentioned median-dual type of control volume requires the knowledge of the centroid of the grid cell. The centroid of a general volume is defined as

According to the derivation in Ref. [34], the relation in Eq. (5.16) can be discretised as

where the centre of a face \(m\), i.e., \((\vec{r}_c)_m\) is obtained from Eq. (5.8) for a triangular face, or from Eq. (5.9) for a quadrilateral face. As we can note, the denominator in Eq. (5.17) is identical to \(\Omega/3\) from Eq. (5.15).