8.3 Farfield 远场[cfd-8-3]

Depending on the sign of the eigenvalues of the convective flux Jacobians (Appendix A.11, Eq. (A.84) or (A.88)), the information is transported out of or into the computational domain along the characteristics. For example, in the case of subsonic inflow there are four incoming characteristics (in 3D) and one outgoing (\(\lambda_5\) in Eq. (A.88)). The situation reverses for subsonic outflow. According to the one-dimensional theory of Kreiss [17], the number of conditions to be imposed from outside at the boundary should be equal to the number of incoming characteristics. The remaining conditions should be determined from the solution inside the domain.

The approach of Whitfield and Janus [12] is based on the characteristic form of the Euler equations (2.45) normal to the boundary. The methodology was found to perform very well on structured and unstructured grids in a large variety of flow cases. It can be applied not only to farfield boundaries but also to inviscid solid walls (Subsection 8.2.1).

The two basic flow situations at the farfield boundary are sketched in Fig. 8.6. The flow can either enter or it can leave the domain. Therefore, depending on the local Mach number, four different types of farfield boundary conditions have to be treated:

• supersonic inflow,
• supersonic outflow,
• subsonic inflow, and
• subsonic outflow.

For supersonic inflow, all eigenvalues have the same sign. Since the flow is entering the physical domain, the conservative variables on the boundary (point \(b\) in Fig. 8.6) are determined by freestream values only. Thus,

The values \(\vec{W}_a\) are specified based on the given Mach number \(M_\infty\) and on two flow angles (angle of attack, side-slip angle).

In this case, all eigenvalues have also the same sign. However, the flow leaves now the physical domain and all conservative variables at the boundary must be determined from the solution inside the domain. This can be accomplished simply by setting

Here, four characteristics enter and one leaves the physical domain. Therefore, four characteristic variables are prescribed based on the freestream values. One characteristic variable is extrapolated from the interior of the physical domain. This leads to the following set of boundary conditions [12]

where \(\rho_0\) and \(c_0\) represent a reference state. The reference state is normally set equal to the state at the interior point (point \(d\) in Fig. 8.6). The values in point \(a\) are determined from the freestream state.

In the case of subsonic outflow, four flow variables (density and the three velocity components) have to be extrapolated from the interior of the physical domain. The remaining fifth variable (pressure) must be specified externally. The primitive variables at the farfield boundary are obtained from [12]

with \(p_a\) being the prescribed static pressure.

Physical properties in the dummy cells can be obtained by linear extrapolation from the states \(b\) and \(d\).

The above characteristic farfield boundary conditions assume zero circulation, which is not correct for a lifting body in sub- or transonic flow. For this reason, the farfield boundary has to be located very far away from the body. Otherwise, the flow solution will be inaccurate. The distance to the farfield can be significantly shortened (by one order of magnitude), if the freestream flow includes the effect of a single vortex (horse-shoe vortex in 3D). The vortex is assumed to be centred at the lifting body. The strength of the vortex is proportional to the lift produced by the body. In the following, we shall present implementations of the vortex correction in 2D and in 3D.

The approach, which we want to describe here, was suggested by Usab and Murman [18]. The components of the corrected freestream velocity are given by the expressions (compressible flow assumed)

with \(\Gamma\) being the circulation, \((d,\theta)\) the polar coordinates of the farfield point, \(\alpha\) the angle of attack, and \(M_\infty\) denoting the freestream Mach number, respectively. The circulation is obtained from

by using the theorem of Kutta-Joukowsky. In Equation (8.25), \(a\) represents the chord length of the airfoil and \(C_L\) is the lift coefficient evaluated by the integration of the surface pressure. The polar coordinates in Eq. (8.24) are computed as

where \(x_{ref}\) and \(y_{ref}\) are the coordinates of the reference point (location of the vortex - e.g., at 1/4 chord).

The modified freestream pressure \(p_\infty^{*}\) is given by

with \(\|\vec{v}_\infty^{*}\|_2^2=(u_\infty^{*})^2+(v_\infty^{*})^2\). The corrected freestream density is obtained from the equation of the state

The corrected quantities \(u_\infty^{*}\), \(v_\infty^{*}\), \(p_\infty^{*}\), and \(\rho_\infty^{*}\) are inserted into Eq. (8.22) or Eq. (8.23) instead of \(u_a\), \(v_a\), \(p_a\), and \(\rho_a\).

The above vortex correction Eqs. (8.24)-(8.28) is strictly valid for subsonic flow only. However, the modification of the freestream conditions proved to be helpful in transonic flow as well. This is demonstrated in Fig. 8.7, where the dependence of the lift coefficient on the distance to the farfield boundary was investigated. The farfield radius was set to 5, 20, 50, and 99 chords. As we can see, simulations without the vortex correction experiences a strong dependence on the farfield distance. On the contrary, simulations with the vortex remain sufficiently accurate up to a distance of about 20 chords. This leads to a significant reduction of the number of grid cells/points. It was demonstrated in Ref. [19] that by using higher-order terms in the vortex correction, the farfield boundary can be placed only about 5 chords away without loss of accuracy.

The effect of a wing on the farfield boundary can be approximated by a horse-shoe vortex. In the case of compressible flow, the modified freestream velocity components can be obtained from [20], [21]

where \(\Gamma\) denotes the circulation, \((x,y,z)\) the Cartesian coordinates of the farfield point, and \(l\) stands for the half span, respectively. Furthermore, in Eq. (8.29) it was assumed that the flow is in the positive \(x\)-direction with the wing being oriented along the \(z\)-axis. The terms \(\mathcal{A}\), \(\mathcal{B}\) and \(\mathcal{C}\) in Eq. (8.29) read

The abbreviations are given by

with \(M_\infty\) being the freestream Mach number. The circulation \(\Gamma\) is calculated using Eq. (8.25), where \(a\) represents the mean chord. The corrected values of pressure (\(p_\infty^{*}\)) and of density (\(\rho_\infty^{*}\)) are obtained from the formulae (8.27) and (8.28), respectively. The quantities \(u_a\), \(v_a\), \(w_a\), \(p_a\), and \(\rho_a\) in Eq. (8.22) or Eq. (8.23) are replaced by their corrected values \(u_\infty^{*}\), \(v_\infty^{*}\), \(w_\infty^{*}\), \(p_\infty^{*}\), and \(\rho_\infty^{*}\).

The expressions for the corrected velocity components \(v_\infty^{*}\) and \(w_\infty^{*}\) in Eq. (8.29) becomes infinite at locations, where the vortex lines cross the outflow boundary. These are the points

and \(x=x_{farf}\). In order to avoid the numerical singularity, in Ref. [21] it was suggested to constrain the values of

in Eq. (8.29) to the 1/4 wingspan, i.e., \(l/2\). This measure reduces the corrections to the velocities \(v_\infty\) and \(w_\infty\) within the distance \(l/2\) around the vortex lines \(z=l\) and \(z=-l\).

The numerical results presented in [21] indicate a reduced sensitivity of the lift and drag coefficient with respect to the farfield distance, if the vortex correction in Eq. (8.29) is applied. It was found that a distance of \(7\cdot l\) to the farfield boundary is sufficient for accurate results.