Uncertainty analysis and robust shape optimisation for laminar flow aerofoils

Abstract

AbstractUncertainty in the critical amplification factor ( $N_{cr}$ ) of the $e^N$ transition model is used to approximate the uncertainty in the surface and flow quality of natural laminar flow (NLF) aerofoils. The uncertainty in $N_{cr}$ is represented by a negative half-normal probability distribution that descends from the largest $N_{cr}$ achievable with an ideal surface and flow quality. The uncertainty in various aerodynamic coefficients due to the uncertainty in $N_{cr}$ is quantified using the weighted mean and standard deviation of flow solutions run at different $N_{cr}$ 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 $N_{cr}$ is largest when the transition location is most sensitive to changes in the lift coefficient at the ideal $N_{cr}$ . Robust shape optimisation is also carried out to improve the mean performance and reduce the standard deviation of the performance with uncertainty in $N_{cr}$ . This is found to be effective at producing aerofoils with a larger amount of laminar flow that are less sensitivity to uncertainty in $N_{cr}$ . 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.