afpt (version 1.1.0.4)

fDfPfunctions: Coefficient for thrust dependency of drag and power factors

Description

Computes the thrust requirement dependency factor for drag and power factors in flapping flight based on reduced frequency (kf) and strokeplane angle (phi).

Usage

fD.ind(kf, phi)
fD.pro0(kf, phi)
fD.pro2(kf, phi)
fP.ind(kf, phi)
fP.pro0(kf, phi)
fP.pro2(kf, phi)

Value

Numeric value

Arguments

Using \(f\) for wingbeat frequency, \(b\) for wingspan, and \(U\) for air speed:

kf

reduced frequency (\(k_f = \frac{2 \pi f b}{U}\)); valid range between 1 and 6

phi

strokeplane angle in radians; valid range between 0 and 0.87 rad (50 deg)

Author

Marco Klein Heerenbrink

Details

Flapping of the wings alters the drag components on the wing. A drag component in flapping flight can be related to the drag component in non-flapping flight as \(D = k_D D^\prime\). The factor \(k_D\) depends on reduced frequency \(k_f\), strokeplane angle \(\phi\) and the thrust-to-lift ratio \(T/L\): \(k_D = 1 + f_D(k_f,\phi) \frac{T}{L}\). Functions fD.ind,fD.pro0 and fD.pro2 compute \(f_D(k_f,\phi)\) for induced drag, zero lift profile drag and lift dependent profile drag, respectively.

Similarly, the flapping power components can be computed as: \(P = k_P D^\prime U\), again with \(k_P = 1 + f_P(k_f,\phi) \frac{T}{L}\). Functions fP.ind,fP.pro0 and fP.pro2 compute \(f_P(k_f,\phi)\) for induced power, zero lift profile power and lift dependent profile power, respectively.

References

Klein Heerenbrink, M., Johansson, L. C. and Hedenström, A. 2015 Power of the wingbeat: modelling the effects of flapping wings in vertebrate flight. Proc. R. Soc. A 471, 2177 tools:::Rd_expr_doi("10.1098/rspa.2014.0952")

See Also

computeFlappingPower

Examples

## reduced frequency
kf <- 2*pi*4/10 # 4 Hz at 10 m/s

## strokeplane angle
phi <- 20*pi/180 # 20 degrees

## thrust ratio
TL <- 0.2

## induced drag factor:
fDind <- fD.ind(kf,phi)
kDind <- 1 + fDind*TL
print(kDind)
#   [1] 1.623659

## zero lift drag factor:
fDpro0 <- fD.pro0(kf,phi)
kDpro0 <- 1 + fDpro0*TL
print(kDpro0)
#   [1] 1.014899

## lift dependent profile drag factor:
fDpro2 <- fD.pro2(kf,phi)
kDpro2 <- 1 + fDpro2*TL
print(kDpro2)
#   [1] 1.511107

## induced power factor:
fPind <- fP.ind(kf,phi)
kPind <- 1 + fPind*TL
print(kPind)
#   [1] 1.996891

## zero lift power factor:
fPpro0 <- fP.pro0(kf,phi)
kPpro0 <- 1 + fPpro0*TL
print(kPpro0)
#   [1] 1.076046

## lift dependent profile power factor:
fPpro2 <- fP.pro2(kf,phi)
kPpro2 <- 1 + fPpro2*TL
print(kPpro2)
#   [1] 1.811983