# Numerics: Euler Codes for Steady Flows

Belotserkovsky’s monograph “Thin Wings in Subsonic Flow” published in 1965 [80], became a standard for vortex lattice methods applications. It con­tains a set of results on rectangular wings flying over a solid boundary ob­tained within linear theory. Later on, vortex lattice methods were successfully applied to numerical solutions of nonlinear problems of aerodynamics of lift­ing systems; see Belotserkovsky et al. [81, 82].

The linearized vortex lattice approach was used by Farberov and Plissov [83], Plissov [84], and Konov [85, 86] to determine the characteristics of wings of different planforms and lateral cross section near the interface, and by Plissov and Latypov [87] to compose tables of rotational derivatives for wings of different aspect ratios near the ground. Ermolenko [88] conceived an ap­proximate nonlinear approach to the steady aerodynamics of wings near a solid boundary based on iteration of position of trailing vortices in connec­tion with the induced velocity field in a plane, normal to the wing and passing through the trailing edge.

Treschevskiy and Yushin [89], Pavlovets et al. [90], Yushin [91], and Volkov et al. [92] utilized various modifications of discrete and panel vortex methods to compute the characteristics of wings and wing systems in the ground effect.

Katz [93] used a vortex lattice method incorporating a freely deforming wake to investigate the performance of lifting surfaces close to the ground with application to the aerodynamics of racing cars.

Deese and Agarval [94] employed an Euler solver, based on the finite – volume Runge-Kutta time-stepping scheme to predict the 3-D compressible flow around airfoils and wings in the ground effect. They applied this tech­nique to calculate the aerodynamics of Clark-Y airfoil (infinite aspect ratio) and a low-aspect-ratio wing with a Clark-Y cross section.

Kataoka et al. [95] extended the two-dimensional steady-state approach to treat the aerodynamics of a foil moving in the presence of a water surface. The foil was replaced by a source distribution on its contour and a vortex distribution on its camber line. The free surface effect was taken into account by distributing wave sources on an unperturbed position of the boundary. Mizutani and Suzuki [96] used an iterative approach based on panel meth­ods for the airflow field and the Rankine method for the water flow field to account for the free surface effects and calculated wave patterns generated by a rectangular wing with endplates. In both of these works, the differences in the aerodynamic predictions for a wing near a free surface and near a corresponding solid boundary were found to be insignificant.

By using the method of continuous vortex layers (vortex panel approach ), Volkov [97] carried out some computational investigation into the influence of the geometry of the foil upon its aerodynamic characteristics in proximity to the ground.

Morishita and Tezuka [98] presented some numerical data on the two­dimensional aerodynamics of the airfoil in the compressible ground-effect flow.

Day and Doctors [99] applied the vortex lattice method to calculate the steady aerodynamic characteristics of wings and wing systems in the ground effect with the incorporation of the deformation of the wakes into the numer­ical scheme.

Chun et al. [100] carried out computations for the aerodynamic charac­teristics of both isolated wings and the entire craft of the flying wing configu­ration in ground effect using the potential-based panel method of dipole and source distributions. Standingford and Tuck [101] applied the high-resolution approach developed within the steady lifting surface theory to investigate the influence of endplates upon the characteristics of a thin flat rectangular wing in the ground effect. The authors report this approach yields better accuracy than the standard vortex lattice method near the edges of a junction of a wing-plus-endplates system.

Hsiun and Chen [102] proposed a numerical procedure for the design of two-dimensional airofoils in the ground effect. The corresponding inverse problem was based upon the least square fitting of a prescribed pressure distribution and a vortex panel method to obtain a direct solution of the problem.

Design problems for airfoil sections near the ground have also been con­sidered by Kiihmstedt and Milbradt [103], whose objectives for optimization were stability and maximum lift. These authors used and compared several potential code methods in both 2-D and 3-D formulations.