Strouhal Number and Reduced Frequency

The Strouhal number (St) is well known for characterizing the vortex dynamics and shedding behavior for flows around a stationary cylindrical object, such as the von Karman vortex street behind a cylindrical object, and for characterizing induced unsteady flows about 2D airfoils undergoing pitching and plunging motions. At cer­tain Strouhal numbers, the pitching and plunging airfoils produce forward thrust,
and the vortices in the wake have a flow structure that is similar to the von Karman vortex street, but with reversed direction of vorticity. Such vortex structures are named reverse von Karman vortices [179]. In general, for flapping flight, the dimen­sionless parameter St describes the dynamic similarity between the wing velocity and the characteristic velocity; it is usually defined as

St — fLref — 2 fha (3_9)

St ■ ^ref ■ U ■ (39)

This definition offers a measure of propulsive efficiency in flying and swimming animals. For natural flyers and swimmers in cruising conditions, it was found that the Strouhal number is within a narrow region of 0.2 < St < 0.4 [186] [ 249]. For hovering motions, the Strouhal number has no specific meaning because the reference velocity is also based on the flapping velocity.

In addition to the Strouhal number, an important dimensionless parameter that characterizes the unsteady aerodynamics of pitching and plunging airfoils is the reduced frequency, defined in Eq. (1-19). In hovering flight, for which there is no for­ward speed, the reference speed Uref is defined as the mean wingtip velocity 2Ф fR, and the reduced frequency can be reformed as

k _ 2n fLref n

= 2Uref = ФAR ^

where the aspect ratio AR is introduced here again as in Eq. (1-7). For the special case of 2D hovering airfoils, here the reference velocity Uref is the maximum translational flapping velocity, and the reduced frequency is defined as

k 2n fLref cm

2Uref 2ha

which is the inverse of the normalized stroke amplitude. Based on the definition of the reference velocity and reduced frequency, the airfoil kinematics Eqs. (3-4) and (3-5) can be rewritten as

h(t jT) = ha sin(2kt/T + у)

a(t/T) = a0 + aa sin(2kt/T)

where t/Tis a dimensionless time, which is non-dimensionalized by a reference time T. Another interpretation of the reduced frequency is that it gives the ratio between the fluid convection time scale, T, and the motion time scale, 1/f.

In the case of forward flight, another dimensionless parameter is the advance ratio, J. In a 2D framework, J is defined as

which is related to St because J = 1 /(nSt). In Eq. (3-14), the reference velocity (Uref) is the forward flight velocity (U).

With the reduced frequency the 3D wing kinematics as illustrated in Eqs. (3-1), (3-2), and (3-3) can be further reformed as

3

ф(/T) = [фт cos(2nkt/T) + фт sin(2nkt/T)], (3-15)

3

в (t/T) = [6cn cos(2nkt/T) + esn sin(2nkt/T)], (3-16)

3

a(t/T) = [acn cos(2nkt/T) + asn sin(2nkt/T)], (3-17)

where t/T is a dimensionless time, which is non-dimensionalized by a reference time T, resulting in a dimensionless period of п /k.

If we choose cm, Uref, and 1 /f as the length, velocity, and time scales, respec­tively, for non-dimensionalization, then the corresponding momentum equation for constant density fluid yields

(3-18)

where * designates a dimensionless variable. The reduced frequency and Reynolds number appear in the momentum equation, whereas the Strouhal number comes up in the kinematics.

Morphological and flight parameters for the fruit fly (Drosophila melanogaster), bumblebee (Bombus terrestris), hawkmoth (Manduca sexta), and hummingbird (Lampornis clemenciae) are summarized in Table 3.1. For all these flyers, the flapping frequency is around 20-200 Hz, and the flight speed is about several m/s, yielding a Reynolds number from 102 to 104 based on the mean chord and the forward flight speed. In this flight regime, the unsteady, inertia, pressure, and viscous forces are all important.