8.2.2 Viscous Flow 黏性流动[cfd-8-2-2]

对于流经固壁的黏性流体，假设表面与紧贴表面的流体之间的相对速度为零。因此，我们称之为无滑移（noslip）边界条件。对于固定的壁面，笛卡儿速度分量变为en

For a viscous fluid which passes a solid wall, the relative velocity between the surface and the fluid directly at the surface is assumed to be zero. Therefore, we speak of noslip boundary condition. In the case of a stationary wall surface, the Cartesian velocity components become

u=v=w=0\quad\text{at the surface}. \tag{8.14}

无滑移条件带来两个基本推论。第一，我们不需要在壁面上求解动量方程——单元顶点格式正是利用了这一点。第二，通过无滑移壁面的对流通量仍由式(8.2)给出，而式(2.24)中的各项简化为 $\vec{\Theta}=k\vec{\nabla}T$ 。因此，对流通量中的壁面压力仍按上文无黏流动中所述的方式求得。不过，虚单元（点）的处理方式有所不同。en

There are two basic consequences of the noslip condition. First, we do not need to solve the momentum equations on the wall. This fact is utilised in the cell-vertex scheme. Second, the convective fluxes through the noslip wall are given again by Eq. (8.2), and the terms in Eq. (2.24) simplify to $\vec{\Theta}=k\vec{\nabla}T$ . Hence, the wall pressure in the convective fluxes is obtained in the same way as described above for the inviscid flow. However, the dummy cells (points) are treated in a different way.

Cell-Centred Scheme 单元中心格式

利用虚单元可以简化式(8.14)无滑移边界条件的实现。对于绝热壁（无热流通过壁面），可以令（见图8.2）en

The implementation of the noslip boundary condition in Eq. (8.14) can be simplified by the utilisation of dummy cells. In the case of an adiabatic wall (no heat flux through the wall), we can set (see Fig. 8.2)

\begin{aligned} \rho_1&=\rho_2\,,\quad E_1=E_2\\ u_1&=-u_2\,,\quad v_1=-v_2\,,\quad w_1=-w_2 \end{aligned} \tag{8.15}

对单元0和3同理。这一做法既适用于结构格式，也适用于非结构格式（参见文献[6]）。en

and likewise for the cells 0 and 3. The approach is applicable to both, structured and unstructured schemes (cf. Ref. [6]).

如果给定壁温，速度分量仍按式(8.15)那样反号。虚单元中的温度利用给定的壁温从内场线性外推。由于垂直于壁面的压力梯度为零，边界元素中的压力也规定用于虚单元（即 $$p_0=p_1=p_2$$ ）。虚单元中的密度和总能由插值得到的值算出。en

If the wall temperature is given, the velocity components are still reversed as in Eq. (8.15). The temperature in the dummy cells is linearly extrapolated from the interior field by using the specified wall temperature. Since the pressure gradient normal to the wall is zero, the pressure in the boundary element is prescribed also in the dummy cells (i.e., $$p_0=p_1=p_2$$ ). The density and the total energy in the dummy cells are evaluated from the interpolated values.

Cell-Vertex Scheme 单元顶点格式

由于不必求解动量方程，壁面上没有来自对流通量（式(8.2)）的贡献。式(2.23)的黏性通量对能量方程只贡献垂直于壁面的温度梯度。对于绝热壁， $\vec{\nabla}T_w\cdot\vec{n}$ 为零。因此，我们完全不必计算通过壁面的对流通量或黏性通量。为了防止在壁面节点上产生非零速度分量，应把动量方程的残差置零。en

Since the momentum equations need not to be solved, there is no contribution from the convective fluxes (Eq. (8.2)) at the wall. The viscous fluxes in Eq. (2.23) contribute only the temperature gradient normal to the wall to the energy equation. For an adiabatic wall, $\vec{\nabla}T_w\cdot\vec{n}$ is zero. Hence, we do not have to compute any convective or viscous fluxes through the wall. The residuals of the momentum equations should be set to zero, in order to prevent the generation of nonzero velocity components at the wall nodes.

如果给定壁温，在完全气体假设下可以直接设置壁面上的总能（例如图8.3中的节点 $$(i,2)$$ ）en

In the case of a prescribed wall temperature, we can directly set the total energy at the wall (e.g., node $$(i,2)$$ in Fig. 8.3) using (perfect gas assumed)

(\rho E)_{i,2}=\frac{c_p}{\gamma}\,\rho_{i,2}\,T_w\,, \tag{8.16}

其中 $$T_w$$ 表示给定的壁温。动量方程和能量方程的残差都必须置零。同一策略也适用于非结构格式。en

where $$T_w$$ denotes the given wall temperature. The residuals of the momentum and of the energy equation have to be zeroed out. The same strategy is applicable also to unstructured schemes.

对于非绝热壁，另一种做法在某些应用中似乎更为稳健：它完全不在壁面上求解控制方程，而是直接给定密度和能量en

Another approach for non-adiabatic walls, which seems to be more robust for some applications, does not solve the governing equations at the wall at all. Both, the density and the energy are directly specified

\rho_{i,2}=\frac{p_{i,3}}{T_w\,R}\quad\text{and}\quad(\rho E)_{i,2}=\frac{p_{i,3}}{\gamma}\,. \tag{8.17}

式(8.17)中的关系假设垂直于壁面没有压力梯度（因此 $p_{i,2}=p_{i,3}$ ）。由于所有守恒变量都已给定， $$(i,2)$$ 处所有方程的残差都应置零。该技术同样可用于非结构网格。不过，在三角形或四面体网格上，压力的外推需要额外的运算。en

The relations in Eq. (8.17) assume that there is no pressure gradient normal to the wall (therefore $p_{i,2}=p_{i,3}$ ). Since all conservative variables are prescribed, the residuals of all equations should be set to zero at $$(i,2)$$ . This technique can be utilised on unstructured grids as well. However, the extrapolation of the pressure requires additional operations on triangular or tetrahedral grids.

如果壁面绝热，虚点中的值按如下方式得到en

If the wall is adiabatic, the values in the dummy points are obtained as follows

\begin{aligned} \rho_{i,1}&=\rho_{i,3}\,,\quad E_{i,1}=E_{i,3}\\ u_{i,1}&=-u_{i,3}\,,\quad v_{i,1}=-v_{i,3}\,,\quad w_{i,1}=-w_{i,3}\,. \end{aligned} \tag{8.18}

对节点0和4同样如此。如果给定壁温，虚点中的温度由内场外推，即en

The same applies to the nodes 0 and 4. If the wall temperature is given, the temperature in the dummy points is extrapolated from the interior, i.e.,

T_{i,1}=2T_w-T_{i,3}\quad\text{and}\quad T_{i,0}=3T_w-2T_{i,3} \tag{8.19}

下标按图8.3。速度分量仍按式(8.18)那样反号。密度和能量用插值得到的温度值以及压力 $p_{i,3}$ 计算。en

with the indices according to Fig. 8.3. The velocity components are again reversed as in Eq. (8.18). The density and energy are computed with the interpolated temperature value and with the pressure $p_{i,3}$ .