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## Über dieses Buch

This book is dedicated to the memory of a distinguished Russian engineer, Rostislav E. Alexeyev, who was the first in the world to develop the largest ground effect machine - Ekranoplan. One of Alexeyev's design concepts with the aerodynamic configuration of a jlying wing can be seen on the front page. The book presents a description of a mathematical model of flow past a lifting system, performing steady and unsteady motions in close proximity to the underlying solid surface (ground). This case is interesting for practical purposes because both the aerodynamic and the economic efficiency of the system near the ground are most pronounced. Use of the method of matched asymptotic expansions enables closed form solutions for the aerodynamic characteristics of the wings-in-ground effect. These can be used for design, identification, and processing of experimental data in the course of developing ground effect vehicles. The term extreme ground effect, widely used through­ out the book, is associated with very small relative ground clearances of the order of 10% or less. The theory of a lifting surface, moving in immediate proximity to the ground, represents one of the few limiting cases that can be treated analytically. The author would like to acknowledge that this work has been influenced by the ideas of Professor Sheila E. Widnall, who was the first to apply the matched asymptotics techniques to treat lifting flows with the ground effect. Saint Petersburg, Russia February 2000 Kirill V. Rozhdestvensky Contents 1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

## Inhaltsverzeichnis

### 1. Introduction

Abstract
This book discusses the aerodynamics of vehicles, that utilize the favorable effect of the proximity to an underlying surface upon their performance. Although this underlying surface may be not only land, but also water, snow, or ice, it will be called ground.
Kirill V. Rozhdestvensky

### 2. Problem Formulation for the Flow Past a Lifting Surface in Proximity to a Solid Boundary

Abstract
Consider a wing of small thickness and curvature, performing an unsteady motion above a solid nonplanar underlying surface in an ideal incompressible fluid1 (see Fig. 2.1). Assume that motion of the wing is the result of superposition of the main translational motion with variable speed U(t) and small vertical motions due to heave, pitch, and possible deformations of the lifting surface. Introduce a moving coordinate system in which the axes x and z are located upon an unperturbed position of the underlying boundary (the ground). Axis x is directed forward in the plane of symmetry of the wing, and axis y is directed upward and passes through the trailing edge of the root chord.
Kirill V. Rozhdestvensky

### 3. The Linear Theory of a Lifting System Moving Close to the Ground

Abstract
To reduce a nonlinear formulation to the linear theory, one has to assume that the deflections of the surfaces of the wing, its vortex wake, and the ground, respectively, from horizontal planes y = h 1 and y = 0 are small compared to the ground clearance h, i.e.,
$$\begin{array}{*{20}{c}} {\left| {{y_{u,l,w}} - h} \right| \ll h,}&{\left| {{y_g}} \right|} \end{array} \ll h.$$
(3.1)
Kirill V. Rozhdestvensky

### 4. Nonlinear Flow Problems for a Lifting System in the Extreme Ground Effect

Abstract
First, we consider an example of a flow problem for a moderately curved thin foil in the ground effect,1 and then present some results for thick foils. Essentially, as discussed at length in section 2, the procedure for the solution uses the assumption that, for h ≪ 1 and ε = O(h), nonlinear effects exhibit themselves mainly in the narrow channel under the foil. The foil and the corresponding coordinate system are shown in Fig. 4.1.
Kirill V. Rozhdestvensky

### 5. Compressible Flow Past a Wing in the Extreme Ground Effect

Abstract
It is practical to extend the analysis of the aerodynamics of a wing in the ground effect to account for the dynamic compressibility of the air. In fact, the cruise speed of ground-effect vehicles can amount to half or more of the speed of sound. At the same time, it is known that the problem of unsteady subsonic flow is one of the most challenging in lifting surface theory; see Belotserkovsky et al. [139]. The complexity of the problem partly stems from the fact that in a compressible fluid the perturbations propagate with finite speeds.
Kirill V. Rozhdestvensky

### 6. The Influence of Endplates, Flaps, and Slots

Abstract
A definite pecularity of the wing-in-ground-effect vehicle compared to the airplane consists of the presence of endplates. Endplates are mounted at the wing’s tips and are intended to decrease leakage of air from under the lifting system. Consequenly, the mounting of endplates results in the augmentation of lift or in a decrease of the induced drag for a given lift. In practice, the configuration of the endplates can vary. Figure 6.1 illustrates schematically some of the possibilities.
Kirill V. Rozhdestvensky

### 7. The Aerodynamics of a Lifting System Near Curved Ground

Abstract
A large seagoing ground-effect vehicle has to combine sufficient seaworthiness with acceptable magnitude of the lift-to-drag ratio, when flying above rough seas. The approach of matched asymptotic expansions furnishes a simplified mathematical model of the unsteady aerodynamics of the wing-in-groundeffect vehicle, based on the idea of domination of the channel flow in the extreme ground effect. This concept seems to be promising for wave perturbations, because the influence of waves upon the aerodynamics of such a vehicle is in fact predominantly due to the corresponding variation of the geometry of the gap between the wing and the sea surface.
Kirill V. Rozhdestvensky

### 8. Schematized Flow Models for a Power-Augmented Lifting System

Abstract
One of the problems that developers of wing-in-ground-effect vehicles have to solve is related to necessity to reduce the power required for detaching the craft from water. An efficient way to facilitate takeoff consists of blowing air under the main wing of the craft from special engines. This mode of vehicle operation is often called power augmentation or, briefly, PAR. Power augmentation provides additional dynamic head to support the vehicle at small speed and alleviates hydrodynamic loads due to the impact of waves upon the structure of the craft. From the viewpoint of aerodynamics and hydrodynamics, the problem of power-augmented takeoff is extremely complicated. It features the interaction of turbulent jets with the vehicle and water surface, the resulting spray effects, and the transient motion of the vehicle. In what follows, only very simplified models of power-augmented flows will be considered for a lifting system moving very close to the underlying surface.
Kirill V. Rozhdestvensky

### 9. The Aerodynamic Efficiency of a Wing in the Extreme Ground Effect

Abstract
The theory of a wing in the extreme ground effect, as developed herein, enables us to formulate a number of extremal problems, which are of both theoretical and practical interest.
Kirill V. Rozhdestvensky

### 10. Integral Formulations for Lifting Surfaces in the Extreme Ground Effect

Abstract
As mentioned in the Introduction, asymptotic analysis can be applied directly to the integral equation of the lifting surface in the ground effect. Essentially, in the extreme ground effect, we study the limiting process, when the distance between two double (vortex) layers, representing the wing and its mirror image, becomes vanishingly small. In the limit, the two double layers merge into a quadruple layer so that the procedure can be characterised as quadruplication.1 The main result of quadruplication is the confluence (for h → 0) of the integral equation of the wing-in-ground effect into a differential equation (ordinary for two-dimensional flow, and in partial derivatives for three-dimensional flow). The resulting differential equation can be shown to be identical to that, obtained in the course of solving the corresponding boundary problem by the method of matched asymptotic expansions. The quadruplication approach in the aerodynamics of wings in the ground effect was first introduced by Panchenkov [64]. In what follows, all derivations will be based on a different scheme of quadruplication proposed in [66].
Kirill V. Rozhdestvensky

### 11. Equations and the Stability of Motion of a Lifting System in the Extreme Ground Effect

Abstract
The analysis of the dynamics of wing-in-ground-effect vehicles provides data for assessing the stability of motion, controllability, and ride comfort of the craft under development.
Kirill V. Rozhdestvensky

### 12. Simple Mathematical Models of Elastic and Flexible Wings in the Extreme Ground Effect

Abstract
The elasticity and flexibility of the lifting surface can play a particular role in ground-effect aerodynamics due to the expected increase of dimensions of wing-in-ground-effect vehicles, use of light materials and fabric, etc. Usually, to account for elastic properties and/or flexibility of the wing, we have to consider simultaneously the equations of aerodynamics and elasticity. In a more profound analysis, the formulations should also cover the equations of the dynamics of the vehicle. As a relationship, linking the deformations of a lifting surface with aerodynamic loading, we normally use equations of unsteady bending of an elastic plate accounting for forces, acting in its camber plane; see Bisplinghoff et al. [151].
Kirill V. Rozhdestvensky

### Backmatter

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