Governing Equations for Wing Structures

As an illustration, consider the hummingbird wing shown in Figure 3.5a. A major research interest is to probe the coupled dynamics between the fluid flow and the flexible wing. The fluid flow creates pressure and viscous stresses, which cause the wing to deform. The wing, in turn, affects the fluid flow structure via the shape change, resulting in a moving boundary problem [417].

Wingspan

4.5′

Fuselage length

4.5′

Takeoff weight

45 g

Engine

Maxon Re10

Propeller

U-80 (62 mm)

RC receiver

PENTA with customized half-wave antenna

Maximum mission radius

0.9 miles

Video transmitter

SDX-22 70 mw

Camera

CMOS camera (350 lines resolution)

Table 4.1. General specification for the university of Florida MAV

Partitioned analyses have been very popular in the area of computational fluid – structure interactions/computational aeroelasticity. A main motivating factor in adopting this approach is that one can develop and use state-of-the-art fluid and structure solvers and recombine them with minor modifications to allow for the coupling of the individual solvers. The accuracy and stability of the resulting cou­pled scheme depend on the selection of the appropriate interface strategy, which depends on the type of application. The key requirements for any dynamic coupling scheme are (i) kinematic continuity of the fluid-structure boundary, which leads to the mass conservation of the wetted surface, and (ii) dynamic continuity of the fluid-structure boundary, which accounts for the equilibrium of tractions on either sides of the boundary. This leads to the conservation of linear momentum of the wetted surface. Energy conservation at the fluid-structure interface requires both of these two continuity conditions to be satisfied simultaneously. The subject of fluid and structural interactions is vast. Recent reviews by Friedmann [418], Livne [419], and Chimakurthi et al. [420] offer substantial information and references of interest.

To take wing deformation into account, it is necessary to solve governing equa­tions of wing structures. There are various wing structural models, and the choice of models depends on the problems of interest. In this section we consider the wing structure as a beam, a membrane, an isotropic flat plate, and a shell. Governing equations of each structural model are presented in each subsection.