Airfoil and Wing Aerodynamic Analysis

It's an analysis to predict the lift and drag characteristics of an airfoil, and to evaluate stall behavior and the effects of high-lift devices. It's a fundamental technology underpinning aircraft design.

Starting from the NACA airfoil series, CFD has become indispensable for designing modern supercritical and natural laminar flow airfoils. CFD expands the design space that cannot be fully explored by wind tunnel testing alone.

Wind tunnel testing costs on the order of millions of yen per condition. The standard modern workflow is to narrow down design candidates using CFD before bringing them to the wind tunnel.

The compressible Navier-Stokes equations are fundamental. They are described by a set of three equations: the continuity equation, the momentum equation, and the energy equation.

The lift coefficient and drag coefficient are defined as follows.

Here, $L$ is lift, $D$ is drag, $\rho_\infty$ is freestream density, $V_\infty$ is freestream velocity, and $S$ is wing area.

The chord-based Reynolds number for a passenger aircraft at cruise is around $Re_c \approx 2 \times 10^7$. The Mach number is primarily in the transonic range of $M \approx 0.78$--$0.85$.

In the transonic regime, a local supersonic region forms on the upper surface of the wing, generating a shock wave. The interaction between this shock wave and the boundary layer triggers buffet phenomena and can lead to stall.

They are chosen according to the application. The options change depending on whether transition prediction is needed or if fully turbulent flow is assumed.

That's right. The SA model was originally developed by NASA for airfoil analysis. Both Boeing and Airbus use it extensively. However, for large-scale separation near stall, SST k-omega or DDES becomes necessary.

Let's organize the aerodynamic parameters of typical airfoils.

Exactly. Case 9 ($M=0.73$, $\alpha=2.79°$, $Re=6.5 \times 10^6$) is an industry-standard benchmark. In CFD, the pressure distribution on the upper surface and the shock wave position are compared with wind tunnel data.

The most important is boundary layer resolution. It's necessary to set the $y^+$ value of the first wall layer to 1 or less and ensure sufficient prism layers within the boundary layer.

• $y^+ \approx 1$: Recommended value for SA/SST models. Resolves the viscous sublayer without wall functions.
• Number of prism layers: At least 20 layers or more. Growth rate recommended to be 1.2 or less.
• Trailing edge treatment: Sharp trailing edges become singular points, so round them to a finite thickness (about 0.1% of chord).
• Far-field boundary: Place at a distance 20-50 times the chord length away.

That's right. Drag prediction accuracy is directly linked to boundary layer mesh quality. Sometimes accuracy of $\Delta C_D = 0.0001$ (1 count) is required, so meticulous attention to the mesh is necessary.

The Finite Volume Method (FVM) is mainstream. It uses a cell-centered scheme, discretizing by integrating the governing equations over the volume of each cell.

Discretization of the convection term holds the key to accuracy. Schemes of second-order accuracy or higher are essential, with specific choices like these.

That's right. In Fluent, Roe-FDS is often used; in STAR-CCM+, the AUSM+ scheme is common for transonic airfoils. In OpenFOAM, the rhoCentralFoam solver supports shock wave capture.