Laminar viscous compressible airfoil flow (FEM)

Figure: Pressure coefficient for laminar viscous flow about a naca012 airfoil
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Figure: Friction coefficient for laminar viscous flow about a naca012 airfoil
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A further example is the laminar viscous compressible flow about a naca012 airfoil. Results for this problem were reported by [57]. 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, an entrance unity velocity and a entrance unity 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 [*] and [*], 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. From Figure [*] it is clear that a reduction of the shock smoothing coefficient improves the results. For a zero shock smoothing coefficient, however, the results oscillate and do not make sense any more. Taking into account that the reference results do not totally agree either, a shock smoothing coefficient of 0.025, which is the smallest smoothing coefficient yielding non-oscillating values, leads to the best results. The friction coefficient is defined by \tau_w/(0.5 \rho_\infty v_\infty^2), where \tau_w is the local shear stress. Here too, a too large shock smoothing coefficient clearly leads to wrong results. A value of 0.05 best agrees with the results by Mittal, however, in the light of the c_p-results from the literature a value of 0.025 might be good as well. The c_f-peak at the front of the airfoil is not very well hit: the literature result is 0.17, the CalculiX peak reaches only up to 0.15. While decreasing the shock smoothing coefficient increases the peak, a too coarse mesh density at that location may also play a role. The general advice is to use as little shock smoothing as possible.