## 1 Introduction  
Nowadays, Computational Fluid Dynamics (CFD) has become an indispensable tool in the design and development of engineering products. The target is to reduce the time and cost for the new design of a prototype via computer simulations. Recently, the rapid progress of computation performance and reduction of computational cost have enabled the adoption of the Uncertainty Quantification technique (UQ) [1; 2; 3; 4; 5; 6; 7], which requires many simulation cases. One important part of UQ is uncertainty propagation, which is used to quantify the sensitivity and variation of the simulation results with consideration of single or multiple stochastic inputs such as the freestream velocity, model coefficients, and target geometry. A direct method for UQ is Monte Carlo simulation, but it is expensive to obtain converged solutions. Therefore, the Polynomial Chaos (PC) approach developed by [PERSON] [8; 9] is proposed as an alternative method. Such methods can be divided into two types: intrusive and non-intrusive approaches. The intrusive methods require modification of the existing code into a Galerkin projection-based flow solver [10; 11; 12; 13; 14; 15]. This extra work can be a big burden for CFD engineers. Whereas a non-intrusive method that enables the use of established code without any modification is considered as a promising approach in UQ and has been applied to various CFD fields by many researchers. [PERSON] and [PERSON] [16; 17] applied Point-Collocation Non-Intrusive Polynomial Chaos (NIPC) to selected CFD problems such as aerodynamic flows around the airfoil/wing or in a nozzle. They considered the uncertainty of the freestream velocity, eddy viscosity ratio, and model coefficients of the adopted turbulence models. They demonstrated that NIPC improved the accuracy of the polynomial chaos coefficient in NIPC approaches and reduced the computational expense by achieving the same level of the accuracy with a small number of sample points. [PERSON] [18] introduced a non-intrusive probabilistic collocation approach for efficient propagation of arbitrarily distributed parametric uncertainties. They also showed that the convergences depending on the order of approximation of the probabilistic collocation method and the Galerkin PC method are the same. In the UQ technique for turbulence model coefficients, [PERSON] et al. [19] applied the Probabilistic Collocation method to the model coefficients of the standard k-\(\epsilon\) turbulence model. They found the necessary order for proper quantification of drag coefficients around a flat plate and NACA 0012 airfoil. [PERSON] et al. [20] studied uncertainty quantification of three turbulence models: the Spalart-Allmaras model, Wilcox k-\(\omega\) model, and [PERSON]'s SST model for transonic wall-bounded flows. The significant model coefficients of three turbulence models were investigated through a sensitivity analysis. The mechanism of transitional flow from laminar to turbulent flow is still a difficult problem in engineering and science fields. In particular, many researchers have developed various transition models based on Reynolds-Averaged Navier-Stokes (RANS) equations to apply to engineering problems to reduce the computational cost and time. Recently, [PERSON] and [PERSON] [21; 22; 23; 24] proposed a promising transition model based on local parameters of the intermittency \(\gamma\) and the momentum thickness Reynolds number \(Re_{\theta}\). This transition model has been applied to various engineering problems and the results such as the transition point and skin friction variation are superior to previous transition model results. However, in any transition or turbulence model, many assumptions and simplifications are required to close model coefficients with uncertainties. This factor was the motivation of the present work. We considered the uncertainties of the model coefficient of the \(\gamma-R_{\theta}\) transition model and applied the UQ technique to the parameters.