## 1 Research Background  
The increase in Reynolds number leads to the forward movement of the transition point on airfoils. According to experience formula, the friction drag coefficient of turbulent flow is 3.5 times than that of laminar flow at the Reynolds number \(1.0\times 10^{6}\). Hence, the friction drag is reduced by delaying the laminar flow transition. The National Advisory Committee for Aeronautics (NACA) first proposed the concept of laminar flow airfoils, and developed the NACA-6 series airfoils, then began the development of NLF airfoils in the 1970s. After that, there were many studies on the design and optimization of NLF airfoils. For example, [PERSON] et al. [1] used multi-island, parallel, multi-objective evolutionary algorithms combined with the \(e^{N}\) transition prediction method to optimize the NLF airfoil shape. [PERSON] et al. [2] developed a discrete, adjoint-based optimization framework, and successfully minimized lift-constrained drag of NLF airfoils for a single-point design and a multi-point design. In addition, [PERSON] et al. [3] proposed an efficient global optimization method for NLF wings based on surrogate models, which used dual N factors for Tollmien-Schlichting and crossflow instabilities prediction. However, various uncertainties in the use of NLF airfoils directly affects the laminar flow transition on its surface, so it is difficult to achieve the expected performance in the actual use of NLF airfoils designed according to only a certain working condition. None of the above studies take the uncertainties into account. To solve the problem, robust optimization must be introduced. Robust optimization is an optimization method considering the impact of uncertainty compared with the traditional optimization method. It mainly includes sample extraction method, sensitivity-based method, analytical method, and agent-model-based method,among which the sample extraction method has the best optimization effect, but is also computationally intensive in dealing with complex problems. There are many works in the field of robust aerodynamic optimization. [PERSON] et al. [4] introduced the principal component analysis (PCA) deep belief net-work (DBN) based surrogate model for the robust optimization of the whole aircraft, [PERSON] et al. [5] used the genetic algorithm (GA) algorithm to optimize DU93-W-210 airfoil with the assumption of Reynolds number and Mach number following normal distribution. [PERSON] et al. [6] used the polynomial chaos method to improve both the ice tolerance and the cruise performance of an airfoil, in recognition of the icing parameters' uncertainty. Most of the robust optimizations for NLF airfoils only consider the uncertainty of the flight Mach number. For low-speed aircrafts whose Mach number is less than 0.3 and whose sensitivity to Mach number changes is not high, so optimizations only based on Mach number cannot play an important role. Instead, the roughness of the airfoil surface becomes an important factor in laminar flow transition. However, all the above studies do not take random surface contamination into consideration. Surface contamination is usually referred to as the destruction of the laminar flow characteristics due to the lack of smoothness of airfoils. As shown in Figure 1, the surface characteristics of the unmanned aerial vehicle (UAV) wing demonstrate the obvious scratches and pits, which usually shift the laminar flow transition point forward and affect the aerodynamic efficiency of airfoils. There are several studies on this, for example, [PERSON] [7] experimentally studied the effect of different types of leading edge roughness on a NACA-633-418 airfoil, [PERSON] [8] studied the effect of leading edge roughness on the aerodynamic characteristics of diamond-type wings, while [PERSON] [9] investigated the effects of roughness bands with different heights.