Method of designing the wing airfoil of a transonic transport aircraft based on consequential mathematical modeling and parametric optimization

Context and relevance. The tasks of designing wing airfoils for prospective aircraft require the creation of methods that take into account modern requirements for aerodynamic characteristics. At the same time, the proposed procedures should be based on the accumulated experience of designers and engineers in order to take into account all limitations and features. Objective. The first objective is to formulate the airfoil design problem as a parametric optimization task. The second objective is to form an optimization process based on mathematical modeling of the airfoil flow to calculate the values of the objective function and a group of metaheuristic algorithms for searching for a conditional extremum. Hypothesis. The paper assumes that the airfoil geometry can be approximated using expansions in Bernstein polynomials. Methods and materials. In the article, the description of the upper and lower surface using the expansion in terms of the Bernstein polynomial system is used to parameterize the airfoil geometry. The parameters defining the airfoil are the expansion coefficients. To find their values, the designer specifies a set of airfoil base points, as well as segments of possible values of their coordinates. In this case, the constraints on the airfoil thickness and the curvature of the upper surface should also be taken into account. The problem of finding the profile parameters is formulated as a problem of conditional optimization with interval constraints on the designed parameters. The objective function characterizes the degree of satisfaction of the requirements for the integral aerodynamic characteristics. Its values are calculated based on the information obtained using the solver. To transfer information about the current profile geometry to the solver, a geometric model is formed. The optimization problem is solved using a group of metaheuristic algorithms applied sequentially. These include methods simulating the behavior of a swarm of moths, the method of sequential reduction of the search area, and the path-relinking method. A step-by-step technique for solving the wing airfoil design problem is formed. Results. The RAE 5213 airfoil was adopted as the initial airfoil. The airfoil obtained as a result of solving the optimization problem satisfies the requirements specified by the designer in terms of both geometric and aerodynamic characteristics. Conclusions. A method for designing an aerodynamic airfoil is proposed, which consists in finding the coordinates of its base points used for their approximation by the CST method, in order to obtain the required aerodynamic and geometric characteristics of the airfoil. The scientific novelty of the proposed method lies in the fact that the problem of finding the base points of the airfoil is reduced to the problem of sequential finite-dimensional optimization, which is solved using a set of metaheuristic algorithms for finding a conditional global extremum, where the value of the objective function is found using a specialized solver that implements the procedure for mathematical modeling of the process of flow around the airfoil. The obtained results form the basis for the subsequent stage of designing the theoretical wing contour when solving the problem of flow around the full configuration of the aircraft.