8.2 Solid Wall 固壁[cfd-8-2]

In the case of an inviscid flow, the fluid slips over the surface. Since there is no friction force, the velocity vector must be tangent to the surface. This is equivalent to the condition that there is no flow normal to the surface, i.e.,

where \(\vec{n}\) denotes the unit normal vector at the surface. Hence, the contravariant velocity \(V\) (Eq. (2.22)) is zero at the wall. Consequently, the vector of convective fluxes Eq. (2.21) reduces to the pressure term alone, i.e,

with \(p_w\) being the wall pressure.

Within the cell-centred scheme, the pressure is evaluated at the centroid of the cell. However, \(p_w\) in Eq. (8.2) is required at the face of the boundary cell. We can obtain the wall pressure most easily by extrapolation from the interior of the domain. Considering Fig. 8.2, we could simply set \(p_w=p_2\). Higher accuracy is achieved by using either a two-point

or a three-point extrapolation formula

In order to account for grid stretching, distances to the wall could be employed instead of the constant coefficients [1].

The above extrapolation formulae (8.3), (8.4) do not account for the grid and the surface geometry. An alternative approach - the so-called normal-momentum relation - was developed by Rizzi [2]. It is based on the fact that the wall represents a streamline in inviscid flow. Differentiation of the zero-normal-flow condition in Eq. (8.1) along the surface streamline, and the substitution of the result into the momentum equation yields

Equation (8.5) relates the density, the velocity and the wall geometry to the normal derivative of the pressure. It was demonstrated that the normal-momentum relation gives very accurate results [1]. However, problems can arise with the numerical solution of the normal-momentum relation in the case of complex geometries. Detailed description of the implementation and accuracy comparisons can be found in Ref. [1].

The values of the conservative variables in the dummy cells can be obtained by linear extrapolation from the interior, i.e.,

The indices in Eq. (8.6) correspond to Fig. 8.2. If the dummy cells are to be utilised within the spatial discretisation scheme (e.g., for the evaluation of the dissipation operator), it is important that the calculation of the convective fluxes is compatible to Eq. (8.2).

The implementation of the boundary condition Eq. (8.1) is straightforward for the cell-vertex discretisation scheme with overlapping control volumes (Subsection 4.2.2). The convective fluxes at the wall faces are computed according to Eq. (8.2). The wall pressure \(p_w\) is obtained by averaging the nodal values as indicated in Eq. (4.26) for a 2-D, or in Eq. (4.27) for a 3-D case. The distribution formula (Eq. (4.30) in 2D) accounts now for only two (four in 3D) cells (compare Fig. 4.4 to Fig. 8.3).

Several different ways can be followed in the case of the cell-vertex scheme with dual control volumes (Subsection 4.2.3). One approach is to apply the condition in Eq. (8.2) separately for each face of the control volume which is on the wall. Thus, according to Fig. 8.3, we can write

The pressures \((p_w)_{i-1/4,2}\) and \((p_w)_{i+1/4,2}\) in Eq. (8.7) can be obtained by linear interpolation, e.g.,

The corresponding 3-D formula will be presented further below in the subsection on unstructured grids.

Another possible implementation employs the condition in Eq. (8.2) directly in the respective wall node (i.e., node \((i,2)\) in Fig. 8.3). The wall pressure \(p_w\) is simply set equal to \(p_{i,2}\). The unit normal vector is computed as the average of the normal vectors of all wall facets which share the node \((i,2)\). This approach requires a correction of the velocity vector. After the solution update by the time-stepping scheme, the velocity vector at the wall is projected onto the tangential plane [3], [4], i.e.,

with \(\vec{n}_{av}\) being the averaged unit normal vector. In this way, the flow will become tangential to the wall.

In order to assign values to the dummy points, it is sufficient to extrapolate the conservative variables (Eq. (2.20)) from the interior field by using a relation similar to Eq. (8.6).

The wall boundary condition in Eq. (8.1) can be implemented for a cell-centred unstructured scheme in a way similar to that on structured grids. If the boundary cell is a quadrilateral, hexahedron or a prism (with triangular face on the wall), the pressure can be extrapolated to the wall by using Eq. (8.3). The neighbouring cell (number 3 in Fig. 8.2) is known from the face-based data structure described in Subsection 5.2.1. For the case of a triangular or tetrahedral cell, in Ref. [5] and [6] it was suggested to employ one layer of dummy cells. The velocity components in the dummy cells were obtained by reflecting the velocity vectors in the boundary cells at the wall. For example, in the dummy cell 1 in Fig. 8.2, the velocity would become

where \(V_2=u_2n_x+v_2n_y+w_2n_z\) is the contravariant velocity and \(\vec{n}=[n_x,n_y,n_z]^T\) stands for the wall unit-normal vector. The pressure and density in the dummy cells were set equal to the values in the corresponding boundary cell (this implies \(p_w=p_2\)).

The boundary condition Eq. (8.1) requires more attention in the case of the median-dual unstructured discretisation scheme. The situation is shown in Fig. 8.4 for the 2-D and in Fig. 8.5 for the 3-D case. The convective fluxes in Eq. (8.2) are computed separately at each face of the control volume which is located on the wall. This is identical to the first approach discussed above for the structured cell-vertex scheme with dual control volumes. For quadrilateral elements (like 1-3-4-5 in Fig. 8.4), the pressure is interpolated correspondingly to Eq. (8.8). In the case of hexahedra, prisms or pyramids, where the face of the control volume is a quadrilateral (like face 1-4-5-6 in Fig. 8.5), the interpolation formula reads

If the boundary elements are tetrahedra or prisms (or triangles in 2-D), the pressure at the wall face should be evaluated like in the finite element method

[7]. At the wall segment 1-2 in Fig. 8.4 for example, the pressure at the face 1-2* would be computed as

In the case of a tetrahedra with, e.g., the wall face 1-2-3 in Fig. 8.5, the pressure is evaluated as

It should be stressed that this finite-element type interpolation is important for an accurate flow solution [7].

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

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.

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)

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

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.

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.

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)

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.

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

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.

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

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.,

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 \(p_{i,3}\).