Uncertainty Quantification of Two-Dimensional Wake Vortices Near the Ground

The study of the interaction of a two-dimensional vortex pair with a wall dates back to Lamb [1], and under the inviscid simplification, the hyperbolic vortex trajectory is independent of the initial circulation. Taking into account the role of the viscosity near the wall, the vortex rebounds due to the interaction with the created secondary vortex formed by the negative pressure gradient produced by the primary vortex, causing the boundary layer to detach (see, e.g., Harvey and Perry [2], Corjon and Poinsot [3], and Zheng and Ash [4]). Wake vortices in ground influence have received particular attention due to their impact in safety and efficiency of airports (see, e.g., the review by Spalart [5]). The physics of wake vortex decay is now well understood (see, e.g., Crow [6], Proctor and Switzer [7], and Gertz et al. [8]), but their application to real scenarios faces the problem of large uncertainties in many of the parameters that affect the motion of lift-generated vortex wakes. Among the uncertainty parameters are the meteorological conditions, wind velocity components (crosswind), turbulence, and thermal stratification, combined with aircraft weight, type, velocity, distance from the ground, etc. These random uncertainties may affect the prediction of the wake-hazard region for single or parallel runways in busy airports (see, e.g., Rossow [9]). Several probabilistic models have been developed (see, e.g., Holzapfel [10]), but it will be relevant to consider also the calculation of the stochastic Navier-Stokes equations to capture the events that are more likely to occur and to establish confidence intervals by providing an ensemble of solutions associated to a certain probability of occurrence. There are several stochastic approaches available nowadays to quantify the uncertainty propagation of the input parameters into the model outputs. Monte Carlo (MC) methods are generic and robust but they are also computationally very expensive and they do not provide direct information regarding the uncertainty propagation through the model. Spectral projection (SP) methods, based on polynomial chaos expansion (see, e.g., [11-13]) have shown to be efficient and suitable for the important issue of the parameters and the accurately calculation of the propagation of the uncertainty through the model. The SP methods may be formulated using an intrusive or nonintrusive approach (see, e.g., Le Maitre et al. [14]). We have selected the intrusive-SP approach, which req