# Numerics: Euler Codes for Unsteady Flows

Efremov [104] used a collocation method to solve numerically the integral equation describing oscillations of a flat plate of infinite aspect ratio in motion near the interface of fluids of different densities.

Gur-Milner [105] used a continuous representation of loading with a Birnbaum-Prantdtl double series to calculate the steady and unsteady characteristics of wings of arbitrary planform near the ground within the framework of linear theory.

Vasil’eva et al. [106] formulated a three-dimensional linearized unsteady theory of a lifting surface moving in the presence of a boundary of two media with different densities. The authors were able to calculate the aerodynamic derivatives for unsteady motions of the lifting surface of a given planform. In particular, they performed calculations for an air-water interface, concluding that, within the assumptions of the mathematical model adopted, the surface of water under a wing behaves as if it were a solid wall.

Avvakumov [107] performed calculations of the aerodynamic derivatives of a wing of finite aspect ratio in motion above the a wavy wall surface, using the vortex lattice method to model the unsteady aerodynamics of the lifting system and the distribution of three-dimensional sources on the underlying surface.

A 2-D estimate of what the authors designate “dynamic ground effect” was made by Chen and Schweikhard [108] by using of the method of discrete vortices of such transitional maneuvers of a flat plate airfoil as descent or climb. In this work, a simplifying assumption was made that the foil unsteady vortex wake is directed along the flight path.

A 3-D numerical simulation of the aerodynamics of wings of finite thickness in the ground effect was carried out by Nuhait and Mook [109, 110], Mook and Nuhait [111], Elzebda et al.[112], and Nuhait and Zedan [113]. In these works, the authors employed the vortex lattice method and modelled the shedding of the vorticity into the wake by imposing the Kutta-Zhukovsky condition along the trailing edges and wing tips. The position and distribution of the vorticity in the wake were determined by requiring the wake to be force free. The model is not restricted by planform, camber, angle of pitch, roll, or yaw as long as stall and vortex bursting do not occur. The method is sufficiently general, and the authors gave calculated examples of the unsteady ground effect related to the descent of the wing and performed computations of the steady-state aerodynamics of wings and wing-tail combinations.

Ando and Ishikawa [114] investigated the aerodynamic response of a thin airfoil at zero pitch angle moving in proximity to a wavy wall which advances in the same direction but with different speed. The authors showed that the wavelength and speed of advancement have considerable influence on what they called “the second-order ground effect.”

Kornev and Treshkov [115] developed an approximate method for calculating the aerodynamic derivatives of a complex lifting configuration based on the vortex lattice approach and the linearization of the unsteady flow components in the vicinity of a nonlinear steady state of the system. They made some numerical estimates of the contribution of different nonlinear factors to the aerodynamics of wing-in-ground-effect craft.

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