Laminar viscous compressible airfoil flow

Figure 35: Pressure coefficient for laminar viscous flow about a naca012 airfoil

Figure 36: Friction coefficient for laminar viscous flow about a naca012 airfoil

A further example is the laminar viscous compressible flow about a naca012 airfoil. Results for this problem were reported by [66]. The entrance Mach number is 0.85, the Reynolds number is 2000. Of interest is the steady state solution. In CalculiX this is obtained by performing a transient CFD-calculation up to steady state. The input deck for this example is called naca012_visc_mach0.85.inp and can be found amoung the CFD test examples. Basing the Reynolds number on the unity chord length of the airfoil, a unit entrance velocity and a unit entrance density leads to a dynamic viscosity of $\mu=5 \times 10^{-4}$. Taking $c_p=1$ and $\kappa=1.4$ leads to a specific gas constant $r=0.2857$ (all in consistent units). Use of the entrance Mach number determines the entrance static temperature to be $T_s=3.46$. Finally, the ideal gas law leads to a entrance static pressure of $p_s=0.989$. Taking the Prandl number to be 1 determines the heat conductivity $\lambda=5 \time 10^{-4}$. The surface of the airfoil is assumed to be adiabatic.

The results for the pressure and the friction coefficient at the surface of the airfoil are shown in Figures 35 and 36, respectively, as a function of the shock smoothing coefficient. The pressure coefficient is defined by $c_p=(p-p_\infty)/(0.5 \rho_\infty v_\infty^2)$, where p is the local static pressure, $p_\infty$, $\rho_\infty$ and $v_\infty$ are the static pressure, density and velocity at the entrance, respectively. Figure 35 shows that the result for a shock smoothing coefficient of 0.004, which is the smallest value not leading to divergence is in between the results reported by Cambier and Mittal. The friction coefficient is defined by $\tau_w/(0.5 \rho_\infty v_\infty^2)$, where $\tau_w$ is the local shear stress. The CalculiX results with a shock smoothing coefficient of 0.004 are smaller than the ones reported by Mittal. The $c_f$-peak at the front of the airfoil is also somewhat too small: the literature result is 0.17, the CalculiX peak reaches only up to 0.15. The shock coefficient is already very small and it is the smallest feasible value for this mesh anyway, so decreasing the shock coefficient, which would further increase the peak, is not an option. A too coarse mesh density at that location may also play a role.