Governing Equations and Non-Dimensional Parameters

The aerodynamics of biological flight can be modeled in the framework of unsteady, Navier-Stokes equations around a body and two or four wings. Nonlinear physics with multiple variables (velocity and pressure) and time-varying geometries are among the aspects of primary interest.

Scaling laws are useful to reduce the number of parameters, to clearly identify the physical flow regimes, and to offer guidelines to establish suitable models for predicting the aerodynamics of biological flight. Three main dimensionless parame­ters in the flapping flight scaling are (i) the Reynolds number (Re), which represents the ratio of inertial and viscous forces; (ii) the Strouhal number (St), which describes the relative influence of forward versus flapping speeds in forward flight; and (iii) the reduced frequency (k). The reduced frequency measures unsteadiness by compar­ing the spatial wavelength of the flow disturbance to the chord [186]. Together with geometric and kinematic similarities, the Reynolds number, the Strouhal number, and the reduced frequency are sufficient to define the aerodynamic similarity for a rigid wing.

3.2.1 Reynolds Number

The Reynolds number represents the ratio between inertial and viscous forces. Given a reference length Lref and a reference velocity Uref, one defines the Reynolds number Re as

U fL f

Re = ref ref j (3-6)

V

where v is kinematic viscosity of the fluid. In flapping wing flight, considering that flapping wings produce the lift and thrust, either a mean chord length cm or a wing length R is commonly used as the reference length, whereas the body length is typically used in swimming animals. Note that the definition of mean chord length is an averaged chord length in the spanwise direction. The reference velocity Uref is the

free-stream velocity in forward flight, but it is defined differently in hovering flight. In hovering, the mean wingtip velocity may be used as the reference velocity, which is also written as Uref = 2Ф fR, where Ф is the wing-beat amplitude (measured in radians). Therefore, the Reynolds number for a 3D flapping wing, Ree, in hovering flights can be cast as

. UrefLref 2Ф fRcm 2Ф fR2 4_

f3 V V V AR

where the aspect ratio AR as described in Chapter 1 has been introduced in a form of AR = 4R2/S, with the wing area being the product of the wingspan (2R) and cm. Note that the Reynolds number here is proportional to the wing-beat amplitude, the flapping frequency, and the square of the wing length, R2, but is inversely pro­portional to the AR of the wing. In insect flights, the wing-beat amplitude and the aspect ratio of the wing do not vary significantly, whereas the flapping frequency increases as the insect size is reduced. In general these characteristics result in Re ranging from ^(Ю1) to 0(104). In addition, given a geometrically similar wing model that undergoes flapping hovering with the same wing-beat amplitude, the product of fR2 can preserve the same Reynolds number. This implies that a scaled-up but low- flapping-frequency wing model can be built mechanically to mimic insect flapping flight based on the aerodynamic similarity. In fact, robotic model-based studies [199] [201] have been established on such a basis provided that the second parameter, the reduced frequency, can be satisfied simultaneously.

The Reynolds number can also be defined with an alternative reference length and/or reference velocity. For example, using the wing length R as the reference length and the wing velocity, Uref = 2n fr2R, where r2 is the radius of the second moment of wing area (approximately 0.52 for the hawkmoth, Manduca sexta [247] [248]), the Reynolds number Re^ is proportional with (ФfR2)/v and is not depen­dent on the aspect ratio of the wing. Note that the reference velocity here is almost half of that at the wingtip.

For a 2D hovering wing, the Reynolds number can be defined by the maximum plunging velocity given by Eq. (3-8a) or by the mean plunging velocity, Eq. (3-8b), with a factor of n/2 difference. Both definitions have their benefits, but in this chapter, we use Eq. (3-8b) as the Reynolds number for a 2D hovering motion unless stated otherwise.

UrefLref 2n fhacm

Re f 2 = V = V ’ (3-8a)