7R42. Metode Matematice in Aerodinamica. Mathematical Methods in Aerodynamics. (Romanian). - L Dragos (Fac of Math, Univ of Bucharest, Str Academiei 14, Bucharest, 70109, Romania). Editura Acad Romane, Bucharest. 2000. 560 pp. Soft-cover. ISBN 973-27-0781-X.

Reviewed by L Librescu (Col of Eng, ESM Dept, VPI and State Univ, Blacksburg VA 24061-0129).

Professor Lazar Dragos belongs to the younger generation of Romanian scientists in applied mathematics and mechanics and represents the best product of the brilliant school of aerodynamics illustrated by its founders, Victor Valcovici, Elie Carafoli, and Caius Jacob. His book contains the results generated by him during a period of three decades or so, in which he has interacted with his former students, now established names in the area of aerodynamics.

The present book presents a thorough approach of the aerodynamics of aircraft wings in various flight speed regimes, including the subsonic, transonic, and the supersonic ones. One of the distinguished features of this work consists of an unifying approach for all these cases, approach based on the fundamental solution methodology. The book is divided into 11 chapters arranged in the order of the complexity of the exposed, material and contains six appendices.

Chapter 1 provides a brief review of the equations of the ideal compressible fluids, and of their linearized and irrotational counterparts. Within the same chapter, a review of the theory of shock waves is also provided.

Chapter 2 is devoted to the equations of the linearized aerodynamics and their fundamental solutions. Within this chapter, the small disturbance concept is used to derive the governing system of equations and the boundary conditions necessary for determining the disturbance quantities related to the pressure, velocity, and air density. In this chapter, the fundamental solutions of the equation for the velocity potential are obtained for each speed regime, and for both the stationary and nonstationary cases. In this context, a wide use of the generalized functions (distributions) was made. In addition, alternative methods, among these, that one based on the concept of the fundamental matrices was provided.

The third chapter treats the case of wings of infinite aspect ratios in a subsonic gas flow. In this context, the method of the singular integral equations considered in conjunction with that of the fundamental solution methodology is used to determine the pressure jump, and implicitly, the aerodynamic lift and twist moment that appear in the aeroelastic governing equations. A variety of situations are considered for which the pressure jump is determined. Such situations concern, eg, the wing in the presence of the ground effect; the case of the symmetric profile; the wing in a wind tunnel; the case of a biplane, and of wings in tandem, etc. Notice that in all these cases, the concept of fundamental solutions developed by the author was used.

In Chapter 4, the boundary element method (BEM) is used to address the problem of aircraft wings of infinite aspect ratio in a subsonic flow field. As is shown via the application of this method, it is also possible to consider the case of thick profiles. The results obtained by this method are validated by comparing the obtained predictions with the exact ones obtained for the case of circular obstacle and an incompressible flow.

Chapter 5 deals with the theory of aircraft wings of finite span in a subsonic flow. Herein the author develops an original mathematical model of lifting surfaces that is based on the method of the fundamental solutions.

The power of these methodologies appears again from the fact that the resulting equations contain, implicitly, the famous assumptions used by Prandtl in his lifting surface model, assumptions that, in contrast to Prandtl lifting surface model, have not been a priori postulated. Powerful analytical and numerical methods are supplied for the solution of the equation of lifting surfaces.

The case of the wings of small aspect ratio is also addressed in this chapter.

Chapter 6 is devoted to lifting line (straight and curved), theory aerodynamics in which framework the author makes use of the method of the fundamental solutions.

The application of this method yields the famous Praudtl’s equation. The solution of this singular integro-differential equation, and its extensions done by Weissinger and Reissner, as well as its exact solution done by Vekua are contained in this chapter.

In Chapter 7, the problems addressed in Chapter 4 for the case of the 2D wing, based on the boundary integral method, are extended to the case of the 3D wing.

In Chapter 8, the case of the wing in a stationary, supersonic flow is considered. A wealth of results that concern the 3D wing, obtained via the application of the fundamental function methodology, are supplied.

In Chapter 9, the problem of the stationary transonic aerodynamics is considered. As a first step, the general transonic small-perturbation equation that contains the nonlinear term due to von-Karman is derived. The fundamental solutions for the 2D and 3D flows, when the motion is accompanied by a shock-wave or is free of it, are developed.

Chapter 10 is devoted to the unsteady motion of a gas in various flight speed regimes, and in the context, the disturbance pressure on an harmonically oscillating wing is determined. For the subsonic compressible flows, Possio’s integral equation is derived, and as special case, its counterpart for the incompressible and transonic flows is obtained. A thorough analysis related with the various representations of the aerodynamic kernels and the solution of the integral equations for these cases is presented.

A similar analysis is done for the case of 2D and 3D harmonically, oscillating wings in a supersonic, and in the limit, in a transonic flow.

Chapter 11 is devoted to the aerodynamics of slender bodies in both subsonic in supersonic flow fields. Finally, the six appendices provide details about the mathematical tools that have been used in the treatment of the various problems.

Appendix A supplies a concise treatment of Fourier transform and of the distribution theory. Appendix B includes basic items related to the determination of some intricate integrals; Chapter 6 deals with the solution of singular integral equations of the Cauchy type.

Appendix D discusses the concept of the finite part of improper integrals, with special emphasis on such integrals appearing in aerodynamics. Appendix E presents some basic results on multiple singular integrals, while Appendix F surveys some results connected with the quadrature formulas of the Gauss type used to evaluate the principal value of integrals.

The book ends with an extensive list of references related to each chapter. In the reviewer’s opinion, Metode Matematice in Aerodinamica represents a valuable contribution to the treasure of the literature in the area of the aerodynamics of aircraft wings and aeroelasticity. Research workers and graduate students of the faculties of applied mathematics or from the departments of aeronautics of polytechnic universities, as well as engineers with a need for results of modern aerodynamics and a willingness to accept modern analytical tools will certainly find, each one in his own way, this book an exceptional value.

A single barrier, that of the language remains, but this difficulty can be removed through a translation of the book into English; a fact that is highly desirable.