Etudiant : EL GHAZOUANI EL HASSAN

Filière : LF Sciences Mathématiques et Applications

Encadrant : Pr. AZROUL ELHOUSSINE

Annèe : 2024

Résumé : Summary: Since the early 1950's, when Schwartz published his theory of distributions, generalized functions have found many applications in various fields of science and engineering. One of the most useful aspects of this theory in applications is that discontinuous functions can be handled as easily as continuous or differentiable functions. This provides a powerful tool in formulating and solving many problems of aerodynamics and acoustics. Furthermore, generalized function theory elucidates and unifies many ad hoc mathematical approaches used by engineers and scientists in these two fields. In this paper, we define generalized functions as continuous linear functionals on the space of infinitely differentiable functions with compact support, then introduce the concept of generalized differentiation. Generalized differentiation is the most important concept in generalized function theory and the applications we present utilize mainly this concept. First, some results of classical analysis, such as Leibniz rule of differentiation under the integral sign and the divergence theorem, are derived with the generalized function theory. The divergence theorem is shown to remain valid for discontinuous vector fields provided that all the derivatives are viewed as generalized derivatives. An implication of this is that all conservation laws of fluid mechanics are valid, as they stand for discontinuous fields with all derivatives treated as generalized derivatives. When the derivatives are written as the sum of ordinary derivatives and the jump in the field parameters across discontinuities times a delta function, the jump conditions can be easily found. For example, the unsteady shock jump conditions can be derived from mass and momentum conservation laws. Generalized function theory makes this derivation very easy. Other applications of the generalized function theory in aerodynamics discussed here are the derivations of general transport theorems for deriving governing equations of fluid mechanics, the interpretation of the finite part of divergent integrals, the derivation of the Oswatitsch integral equation of transonic ow, and the analysis of velocity field discontinuities as sources of vorticity. Applications in aeroacoustics presented here include the derivation of the Kirchho formula for moving surfaces, the noise from moving surfaces, and shock noise source strength based on the Ffowcs Williams{Hawkings equation.