Uncertainty analysis and robust shape optimisation for laminar flow aerofoils

Uncertainty in the critical amplification factor (Ncr) of the eN transition model is used to approximate the uncertainty in the surface and flow quality of natural laminar flow (NLF) aerofoils. The uncertainty in Ncr is represented by a negative half-normal probability distribution that descends from the largest Ncr achievable with an ideal surface and flow quality. The uncertainty in various aerodynamic coefficients due to the uncertainty in Ncr is quantified using the weighted mean and standard deviation of flow solutions run at different Ncr values. The uncertainty in the aerofoil performance is assessed using this methodology. It is found that the standard deviation of the aerofoil performance due to the uncertainty in Ncr is largest when the transition location is most sensitive to changes in the lift coefficient at the ideal Ncr. Robust shape optimisation is also carried out to improve the mean performance and reduce the standard deviation of the performance with uncertainty in Ncr. This is found to be effective at producing aerofoils with a larger amount of laminar flow that are less sensitivity to uncertainty in Ncr. A trade-off is observed between the mean performance and the standard deviation of the performance. It is also found that reducing the standard deviation of the performance at one Mach number or lift coefficient design point can cause an increase in the standard deviation off-design.