L. Morino

Singularities in BIEs for the Laplace Equation; Joukowski Trailing–Edge Conjecture Revisited

The paper deals with trailing-edge issues connected with the analysis of three-dimensional incompressible quasi-potential flows (i.e. flows that are potential everywhere, except for a zero-thickness vortex layer, called the wake). Specifically, following the Joukowski conjecture of smooth flow at the trailing edge, all the trailing-edge conditions that are required to avoid singularities in the boundary integral representation for the velocity, in a quasi-potential incompressible flow around a wing, are identified. In particular, these include the Kondratev and Oleinik singularity as well as the vortex-line and the edge-jet singularities of Epton. Also, following Mangler and Smith, the behavior of the wake geometry at the trailing edge is determined, using the Kutta condition of no pressure discontinuity at the trailing edge. Specific theoretical issues are addressed which include (1) the relationship between Joukowski conjecture and Kutta condition, and (2) identification of those trailing-edge conditions that are necessary to assure the uniqueness of the solution (as opposite to relationships that are automatically satisfied by the solution). Regarding the first issue, in the main body of the paper, the Joukowski conjecture and the Kutta condition are used as if they were independent assumptions; then, in Appendix A, it is shown that the Kutta condition need not be invoked as a separate assumption since it may be obtained as a consequence of the governing equations and of the Joukowski conjecture. In order to clarify the second issue, the theoretical analysis is coupled with a numerical one. In particular, the conditions necessary to insure uniqueness are inferred (not proven) through numerical experimentation: only the no-vortex-line condition appears to be necessary to insure uniqueness. This is accomplished by using a piecewise-cubic boundary-element method for quasi-potential flows that is an extension of a high-order formulation introduced by the authors and their collaborators (the order of the formulation is adequate to address all the theoretical trailing-edge conditions uncovered). The emphasis is on steady flows in simply connected regions; however, some issues related to unsteady flows in multiply connected regions are also examined. Finally, several open problems that require additional work are identified.

A Velocity Decomposition for Viscous Flows: Lighthill Equivalent-Source Method Revisited

This paper deals with a new decomposition of the velocity field for the analysis of incompressible viscous flows around bodies in arbitrary motion. Classical decompositions used in fluid dynamics include those by Helmholtz and Clebsch (see, e.g., Serrin, 1959). In the decomposition presented here the velocity field is expressed as the sum of a rotational contribution that is zero in the irrotational region (i.e., the region where the vorticity is negligible) and of a potential contribution (see Eq. 2). The resulting formulation for the analysis of incompressible viscous flows has considerable advantages for flows in which the vortical region has small thickness (e.g., attached high Reynolds-number flows). For this type of flows, it is common to use a potential-flow solution with a viscous-flow correction based on the equivalent-sources approach, as discussed by (1958). The aim of this paper is to introduce the decomposition and to analyze the relationship between this formulation and the Lighthill’s approach.

Recent Developments on a Boundary Element Method in Aerodynamics

A high-order boundary-element method for quasi-potential flows, i.e., flows that are potential everywhere except for the wake surface (vortex layer), is presented. This formulation is an extention of the high-order formulation introduced by the authors and their collaborators. The unknown is expressed through a bicubic Hermite interpolation, with derivatives at the nodes expressed in terms of suitable finite-difference approximations. The innovative aspects are related to the trailing-edge conditions; in particular, the issues related to the problem of three-dimensional multiply-connected domains are examined. An assessment of the formulation through numerical results is included.

Dynamics of Microstructured Shells with Thickness Extension

The analysis of the natural frequencies of a plate model is carried out in this paper starting from a model of shells which incorporates the eect of thickness extension and that is derived from the Virtual Work Theorem using material coordinates in the deformed configuration. Moreover, the shell is regarded as a micro-structured body whose fibers are free to rotate and distend. Finally, introducing proper internal constraints, and suitable stress resultant definitions, the equilibrium equations are reduced, in the framework of properly modified Reissner-Mindlin kinematical assumptions, to ones formally equivalent to that that can be obtained in the framework of properly modified Kirchho-Love hypotheses, but with additional equations describing the equilibrium in the fiber direction. Using a numerical approach based on a finite dierence scheme, it is shown how the natural frequencies of the Reissner-Mindlin model reduce when the Kirchho-Love constraints are retained. In particular, results indicate that, in the limit in which the Kirchho-Love hypotheses tend to become valid, the numerical frequencies of the pure Reissner-Mindlin model are aected by some round-o error, whereas in the case corresponding to the formulation adopted, where the solution is sought in terms of transversal displacement and dierence between the RM and the KL rotation of the fiber, the numerical solution reproduces exactly the analytical solution, and, notably, this behavior is emphasized at the highest frequencies. Therefore, in the case in which the transverse shear is treated independently one obtains good results, whereas, in the case in which the transverse shear has to be obtained as the dierence of the derivative of the transverse displacement and of the fiber rotation, the numerical solution introduces numerical errors due to the closeness of the present model to the KL kinematical hypotheses.