“The debris flight equations” by C.J. Baker

ARTICLE IN PRESS Journal of Wind Engineering and Industrial Aerodynamics 97 (2009) 151–154

Contents lists available at ScienceDirect

Journal of Wind Engineering and Industrial Aerodynamics journal homepage: www.elsevier.com/locate/jweia

Discussion

‘‘The debris flight equations’’ by C.J. Baker B. Kordi, Gregory A. Kopp  Boundary Layer Wind Tunnel Laboratory, Faculty of Engineering, University of Western Ontario, London, ON, Canada N6A 5B9

a r t i c l e in fo Article history: Received 5 April 2007 Received in revised form 16 August 2007 Accepted 7 October 2008 Available online 18 November 2008 Keywords: Windborne debris Plates Autorotation Aerodynamics

Baker’s paper provides an interesting analysis of the debris flight equations for compact and plate debris. The subject of this discussion is the autorotational aspects of wind-driven plates and their asymptotic limits. It seems reasonable to assume that single degree-of-freedom autorotating plates represent the asymptotic (steady) limit of wind-driven plates and Baker rightly incorporated the earlier work on autorotation by Iversen (1979) and Tachikawa (1983) into his analysis. Both Iversen and Tachikawa determined the ultimate, steady rotational speeds associated with thin pin-mounted plates, which Baker imposed as an upper bound in his calculations. This constraint gives rise to non-physical rotational speeds, as evidenced by the sharp changes in the slope of the resultant rotational speed versus time in, for example, Fig. 7(c) for an initial angle of 601 and in Fig. 8(c) for 601 and 901. Since the flight of plate debris is sensitive to initial conditions, as shown clearly in the paper, one wonders whether this non-physical model has a significant impact on the resulting flight trajectories. The purpose of the discussion is to examine this point further. While one can argue over the practical relevance of the asymptotic limit of plate flight speeds due to the length of time it takes to occur, the use of asymptotic limits is a common mathematical tool to check model equations and to further physical understanding. It is also important, in this case, to establish that the autorotational-based model does indeed capture the aerodynamic effects with sufficient accuracy in order to utilize such a model for establishing flight speeds and

DOI of original article: 10.1016/j.jweia.2006.08.001

 Corresponding author. Tel.: +1 519 661 3388; fax: +1 519 661 3339.

E-mail address: [email protected] (G.A. Kopp). 0167-6105/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.jweia.2008.10.001

trajectories. Here, we focus on two aspects: (i) Baker’s interpretation of Iversen’s and Tachikawa’s work for the present context, and (ii) our own interpretation and application of Lugt’s (1980) work on the autorotation of two-dimensional thin elliptic cylinders to ease Baker’s strict constraint on the upper bound of the rotational speed. Building on the work of Tachikawa (1983), Baker defined additional lift, CLA, and pitching moment, CMA, coefficients to account for autorotational effects to add to the quasi-steady lift, CL, and moment, CM, coefficients. Thus, CLA and CMA were defined in his Eqs. (33) and (34) as   o o ¯ ¯ ; 1X X1 (1) C LA ¼ kLA o o ¯m ¯m and

C MA

   8 o o ¯ ¯ > > ; 1  k > MA < o o ¯m ¯m    ¼ o o ¯ ¯ > > > kMA 1 þ ; : o o ¯m ¯m

o ¯ X0 o ¯m o ¯ 1p o0 o ¯m 1X

(2)

where we have added the limiting speeds for clarity. Note that o ¯ can be either positive or negative since plates can rotate in either direction, but that the maximum non-dimensional rotational velocity, o ¯ m , is defined to be positive only, so that all directional sense is carried by the sign of o ¯ in the equations. Based on Tachikawa’s data, Baker defined the maximum non-dimensional rotational velocity, o ¯ m ¼ þ0:64. Tachikawa’s experiment was conducted on a single degree-offreedom autorotating plate where the effective velocity was the upstream horizontal wind tunnel velocity. For flying debris, the

ARTICLE IN PRESS 152

B. Kordi, G.A. Kopp / J. Wind Eng. Ind. Aerodyn. 97 (2009) 151–154

relevant reference velocity for the aerodynamic coefficients is not the horizontal wind speed, U, but is rather the wind velocity relative to the plate, qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi (3) U rel ¼ ðU  uÞ2 þ v2 Our interpretation is that the non-dimensional steady asymptotic rotational velocity should be defined as

o ¯r¼

or l U rel

(4)

where or is the steady asymptotic rotational velocity. Note that or in Eq. (4) is normalized by Urel. To be consistent with Baker’s nondimensionalization, om and Urel are normalized by upstream horizontal wind velocity so that

o ¯r¼

o ¯m U¯ rel

(5)

where U¯ rel ¼ U rel =U is the non-dimensional relative velocity. We make this distinction because the governing equations of motion use the horizontal wind speed, U, as the normalizing velocity, while the asymptotic autorotational forces require Urel. Tachikawa’s autorotational speeds were obtained from experiments conducted on square and rectangular plates with different thickness ratios. Iversen (1979) examined a wider range of parameters both for free flight and autorotation, in particular, for aspect ratios in the range of 0.25 to 4 and thickness ratios in the range of 0.005–0.5. He concluded that if the non-dimensional moment of inertia of a plate, which is equivalent to 1=3F in Baker’s notation, is larger than 1, the ‘‘spin parameter’’, or tip speed ratio (which is o ¯ r =2), depends on its aspect ratio and thickness ratio. This resulted in the dimensionless steady asymptotic rotational speed, " # R o ¯ r ¼ 2ð0:329 ln t1  0:0246ðln t1 Þ2 Þ 2 þ ð4 þ R2 Þ1=2 "  0:76 #!2=3 R (6) 2 R þ 0:595 where R is the aspect ratio, R ¼ W/l, t is the thickness ratio, t ¼ h/l, l is the chord, h is the thickness, and W is the width of the plate. A comparison between Tachikawa’s experimental data and Iversen’s model is made in Fig. 1, which shows that the results are not inconsistent with each other, even though there are only four data points from Tachikawa included. Any discrepancies are most likely due to the effects of bearing friction, which have a significant effect on resulting speeds, and is a major point of discussion in Iversen’s paper. It should be mentioned that the dimensionless steady asymptotic rotational velocities, o ¯ r,

1.4

Iverson, R=1 Iverson, R=2 Tachikawa, R=1 Tachikawa, R=2

1.2 1

ωr

0.8 0.6 0.4 0.2 0 0

0.1

0.2

τ

0.3

0.4

0.5

Fig. 1. Plot of o ¯ r versus thickness ratio for plates with aspect ratios, R ¼ 1 and 2: , Eq. (6);

, Tachikawa (1983).

calculated from Eq. (6) for Baker’s ‘‘small’’ and ‘‘large’’ plates are equal to 0.70 and 0.96, respectively. Lugt (1980) solved the Navier–Stokes equations numerically for the flow around a two-dimensional autorotating thin elliptic cylinder and determined the variation that the aerodynamic torque exerted on the body during a cycle as a function of the rotational speed, at Re ¼ 200, with the Reynolds number based on the upstream (horizontal) velocity. He was able to demonstrate that prior to the point of stable autorotation, the aerodynamic torque is in favor of rotation, but if the point of stable autorotation is exceeded, the net torque counteracts rotation. Thus, stable autorotation occurs when the average moment coefficient, C¯ M , over every half revolution, is equal to zero. (Note that here we use the overbar to denote the average over the half cycle, rather than to indicate a non-dimensionalization.) If we compare CMA defined by Eq. (2) with Lugt’s numerical results (see his Fig. 3, for example), it is apparent that both have the same trend with rotational velocity up to the point of stable (i.e., steady) autorotation. Beyond the point of stable autorotation, there has not been any other work done, to our knowledge, except what is actually presented in Lugt (1980). So, if one assumes that CMA for three-dimensional plates follows the same trend as Lugt’s two-dimensional thin elliptic cylinders, one can allow for the possibility of plates rotating faster than the steady autorotational speed in the plate debris computations. It should be emphasized that for plates rotating faster than the steady autorotation speed, the moment coefficient opposes the rotation so that for free rotation, the asymptotic (steady) speed must be at the limit where the net torque during a cycle is zero (i.e., C¯ M ¼ 0). Thus, one could assume that the model for CMA is unchanged except that we now allow speeds outside the range 1po ¯= U¯ rel =o ¯ r p1 so that     8 o o ¯ =U¯ rel ¯ =U¯ rel ¯ =U¯ rel > > kMA 1  o ; X0 > < o o o ¯r ¯r ¯r     C MA ¼ (7) > o=U¯ rel o o ¯ =U¯ rel ¯ =U¯ rel > > kMA 1 þ ¯ ; o0 : o o o ¯r ¯r ¯r If we assume no additional lift for the overspeeding condition (since we have no information on this at present), then CLA becomes 8 o > ¯ =U¯ rel > > 41 kLA ; > > o ¯r > > > < o o ¯ =U¯ rel ¯ =U¯ rel 1p p1 C LA ¼ kLA (8) > o o ¯r ¯r > > > > o > ¯ =U¯ rel > > o1 : kLA ; o ¯r The constants kLA and kMA for a square plate are equal to about 0.4 and 0.12, respectively, as indicated by Baker. While this notation may appear to be cumbersome, it is required to incorporate the autorotational forces and torques which depend on the relative wind speed. Since the asymptotic relative wind speed is a result of the analysis which cannot be obtained a priori, it is perhaps not the best choice for normalization of the governing equations of motion, i.e., Baker’s choice of the horizontal wind speed, U, is the appropriate one. This necessitates the form of Eqs. (7) and (8). Applying the above-mentioned modifications (using Eqs. (7) and (8), and Eq. (6) for o ¯ r ), we solved the governing equations presented in Baker’s paper explicitly using a fourth-order Runge–Kutta scheme. Based on Eq. (6), o ¯ r was set prior to the calculations. Fig. 2 shows the resulting dimensionless horizontal, vertical, and rotational velocity, rotational displacements, trajectories, and relative velocity for initial angular displacements of 01, 301, 601, and 901 for Baker’s large sheet (the small sheet results are not shown here, for brevity). These should be compared to Baker’s

ARTICLE IN PRESS

u

B. Kordi, G.A. Kopp / J. Wind Eng. Ind. Aerodyn. 97 (2009) 151–154

v

1.4 1.2 1 0.8 0.6 0.4 0.2 0

0 deg 60 deg

0

5

30 deg 90 deg

10

0.4 0.2 0 -0.2 0 -0.4 -0.6 -0.8

153

0 deg 60 deg

5

30 deg 90 deg

10

15

t

15

t 2

1

0

2

4

8 10 12 14 16

6

y

ω

-2 0

x

0

0.5 0

5

10

-4

15

-0.5 0 deg 60 deg

-1

-6

30 deg 90 deg

-8

0 deg

30 deg

60 deg

90 deg

t 50

1.0

0 deg 60 deg

0.8 Urel

θ (Rad)

25 0 0 -25 -50

5 0 deg 60 deg

10

30 deg 90 deg

0.6 0.4

15

0.2

30 deg 90 deg

0.0

t

0

5

10

15

t

1

1.5 1 0.5 0 -0.5 -1 -1.5 0.01

ω

-0.5 -1 0.1 Ω

-1.5 0.01

1

0.1 Ω

0 -0.5

ω

2 1.5 1 0.5 0 0.01

1

1.5 1 0.5 0 -0.5 -1 -1.5 0.01

0 v

2 1.5 1 0.5 0 0.01

v

u

u

Fig. 2. Numerical solution of Baker’s ‘‘large’’ sheet debris with the revised aerodynamic parameters: (a) u, ¯ (b) v, ¯ and (c) o ¯ versus t¯ ; (d) the trajectories y¯ versus x¯ ; (e) y, and (f) U¯ rel versus t¯ .

-1 0.1 Ω

1

-1.5 0.01

0.1 Ω

0.1 Ω

1

0.1 Ω

1

Fig. 3. Effect of variation of O on the asymptotic values of u, ¯ v, ¯ and o ¯ for square plates with F ¼ 0.15, h/l ¼ 0.005: (a) y0 ¼ 301 (b) y0 ¼ 901; numerical solution.

Fig. 8. Generally, the results are similar to Baker’s, although there are some important differences. In Fig. 2(c), it can be seen that the large plate reaches its rotational asymptotic limit smoothly so that the behavior appears to be more realistic (this will still need to be confirmed by comparisons to experiment). Also, the rotational speed exceeds the steady asymptotic rotational speed, or, early in the flight trajectory for some of the initial angles of attack. Thus, overspeeding rotations do occur and could be practically important since this aids the resultant lift so that the plate actually lifts early in the trajectory. Such lifting has been observed in experiments (Lin et al., 2006; Visscher and Kopp, 2007). Baker showed in his Eq. (36) there should be two asymptotic horizontal plate speeds, u, ¯ one exceeding the horizontal wind

________,

Eqs. (9)–(11); ~,

speed and the other less than it. In contrast, he found that the vertical speed, v¯ has a single asymptotic limit, given by his Eq. (37). The asymptotic solutions given by Baker in his Eqs. (36)–(38) are correct; however, since the non-dimensional steady asymptotic rotational velocity, o ¯ r , changes with plate geometry (as determined by Iversen and shown in Eq. (6)) and on the relative wind speed (which is a function of u¯ and v), ¯ the interpretation used to obtain his Fig. 12 is not correct. To aid the discussion, we have repeated these equations here:

u¯ ¼ 1 



O

C¯ D

0:5 

C¯ D kLA

0:5

1þ

2 C¯ D 2

kLA

!0:75 (9)

ARTICLE IN PRESS 154

B. Kordi, G.A. Kopp / J. Wind Eng. Ind. Aerodyn. 97 (2009) 151–154

!0:75  0:5  1:5 2 C¯ D C¯ O v¯ ¼  1 þ 2D kLA C¯ D kLA

(10)

o ¯ ¼ o ¯ r  U¯ rel

(11)

switches between clockwise and counterclockwise directions of rotation with the signs as indicated in Eqs. (9) and (11). It can be seen that, unlike Baker’s results in the range of Oo0.1, the nondimensional rotational velocity reaches the asymptotic solution.

2

where O ¼ Mg=0:5rAU , which is the inverse of the Tachikawa number. Using Iversen’s model to get the actual magnitude for o ¯ r, the asymptotic limit of o ¯ can be written as

o ¯ ¼  2ð0:329 ln t1  0:0246ðln t1 Þ2 Þ "





#"

R

2 þ ð4 þ R2 Þ1=2 8 < O 0:5  C¯ 0:5 D

: C¯ D

kLA

0:76 #!2=3 R 2 R þ 0:595 !0:25 9 2 = C¯ D 1þ 2 ; k 

The authors gratefully acknowledge the support provided for this work by the Natural Sciences and Engineering Research Council (Canada) and the University of Western Ontario. G.A. Kopp also gratefully acknowledges the support provided by the Canada Research Chairs Program. References

(12)

LA

where Eqs. (9) and (10) are used to obtain the asymptotic value of Urel. The relationships in Eqs. (9), (10), and (12) are plotted in Fig. 3, together with the asymptotic results obtained from the numerical computations, for initial inclination angles of 301 and 901, and F ¼ 0.15, and h/l ¼ 0.005, like for Baker’s Fig. 12. The figure illustrates that the plate with the initial inclination of 301 always rotates in the clockwise direction (for flow from left to right), but the plate with the initial inclination of 901 ultimately

Iversen, J.D., 1979. Autorotating flat plate wings: the effect of the moment of inertia, geometry and Reynolds number. J. Fluid Mech. 92, 327–348. Lin, N., Letchford, C., Holmes, J.D., 2006. Investigation of plate-type windborne debris. Part I. Experiments in wind tunnel and full scale. J. Wind Eng. Ind. Aerodyn. 94, 51–76. Lugt, H.J., 1980. Autorotation of an elliptic cylinder about an axis perpendicular to the flow. J. Fluid Mech. 99, 817–840. Tachikawa, M., 1983. Trajectories of flat plates in uniform flow with application to wind-generated missiles. J. Wind Eng. Ind. Aerodyn. 14, 443–453. Visscher, B., Kopp, G.A., 2007. Trajectories of roof sheathing panels under high winds. J. Wind Eng. Ind. Aerodyn. 95, 697–713.