## 3 Flow around a Triangular Column
### Simulation Results  
Isosurfaces of the second invariant of the velocity gradient tensor (Q-criterion) of the fine grid simulations are shown in Figure 2. The left figure shows the result of the Euler simulation and the right figure shows the wall-modeled Navier-Stokes simulation.  
In both figures, flow separation occurs at the downstream-side sharp edges, and a complex vortex structure that contains longitudinal vortices is formed in the wake. The outline of the flow structure by the wall-modelled Navier-Stokes simulation is the same as that of the Euler simulation.  
Instantaneous distribution of pressure and velocity in an \(xy\)-plane of the fine grid simulation is shown in Figure 3.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \(Ma\) & \(Re\) (Wall-Modeled NS) & \(\Delta x/D\) \\ \hline \(0.3\) & \(2.9\times 10^{5}\) & \(0.092,\,0.046\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Simulation conditions for the flow around a triangular column.  
Figure 2: Isosurfaces of the second invariant of the velocity gradient tensor (Q-criterion).  
Free shear layers form in the wake of the sharp edges, and the vortices form by the rollup of the free shear layers. Pressure in the wake of the wall-modelled Navier-Stokes simulation is similar to that of the Euler simulation.  
Figure 3: Pressure contour and velocity vectors at the \(z=\text{const.}\) plane of the fine grid simulation.  
Time-averaged distribution of pressure and velocity in an \(xy\)-plane of the fine grid simulation is shown in Figure 4.  
Figure 4: Time-averaged pressure contour and velocity vectors at the \(z=\) const. plane of the fine grid simulation.  
The length of the wake of the wall-modelled Navier-Stokes simulation is almost equal to that of the Euler simulation. Pressure distribution in the wake of the wall-modelled Navier-Stokes simulation is slightly different from that of the Euler simulation. However, the surface pressure distributions of the triangular column in both cases do not have noticeable differences. This suggests that the flow field hardly depends on the Reynolds number, and the boundary layer on the triangular column does not play an important role.  
The drag coefficient is shown in Table 2.  
The drag coefficient of the fine grid of the Euler simulation is 1.282 and the difference from the experimental value [22] is 3.2%. The difference decreases with grid resolution. The pressure drag coefficients of the fine grid simulation of the wall-modelled Navier-Stokes simulation is 1.284. This value is extremely close to that of the Euler simulation. The difference in the drag coefficient of the wall-modelled Navier-Stokes simulation from the experiment is slightly less than that of the Euler simulation. This improvement is mainly provided by the additional surface friction of the wall-modelled Navier-Stokes simulation. Considering effect of viscosity of the fluid in the wall-modelled Navier-Stokes simulation, this relationship is reasonable.  
The computational cost of the wall-modelled Navier-Stokes simulation is 3/2 or more of that of the Euler simulation. Both the wall-modelled Navier-Stokes simulation and the Euler simulation are valid when the separation lines are fixed to the sharp edges of the object. Consequently, it is possible to select one of these simulation methods according to accuracy and computational cost requirements when the separation lines are fixed to the sharp edges and the flow field hardly depends on the Reynolds number.