holmes 2006 journal of wind engineering and industrial aerodynamics 1

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Journal of Wind Engineering

and Industrial Aerodynamics 94 (2006) 21–39

0167-6105/$ -

doi:10.1016/j

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Investigations of plate-type windborne debris—PartII: Computed trajectories

J.D. Holmesa,�, C.W. Letchfordb, Ning Linb

aJDH Consulting, P.O. Box 269, Mentone, Victoria, 3194, AustraliabWind Science and Engineering Research Center, Texas Tech University, Lubbock, TX 79409-1023, USA

Received 30 December 2004; received in revised form 7 October 2005; accepted 7 October 2005

Available online 28 November 2005

Abstract

In Part I, trajectories of plates, carried by strong winds, were studied experimentally by wind-

tunnel and full-scale tests. The application is to windborne debris occurring in severe windstorms

such as hurricanes. In this paper (Part II), a numerical model of square plate trajectories is described

and compared with experimental data from Tachikawa, and that described in Part I. Generally, good

to excellent agreement is found; lift forces induced by the Magnus effect were found to be significant

in determining trajectories.

r 2005 Elsevier Ltd. All rights reserved.

Keywords: Debris; Hurricane; Plate; Wind loads

1. Introduction

Windborne debris is known as a major source of damage in strong wind events such ashurricanes. Using the classification scheme proposed by Wills et al. [1], generic debris typescan be classified as either ‘compact’, ‘sheets/plates’, or ‘rods’. Although the types of testmissile used in impact tests have traditionally been of the ‘compact’ or ‘rod’ type [2], thereis also evidence of considerable damage produced by ‘plate’-type objects—for example,concrete roof tiles during Hurricane ‘Charley’ in 2004.

Published work on windborne debris was reviewed in a previous paper by Holmes [3].This included the pioneer work by Tachikawa [4,5] who studied the trajectories of small

see front matter r 2005 Elsevier Ltd. All rights reserved.

.jweia.2005.10.002

nding author.

dresses: [email protected] (J.D. Holmes), [email protected] (C.W. Letchford).

www.elsevier.com/locate/jweia

ARTICLE IN PRESSJ.D. Holmes et al. / J. Wind Eng. Ind. Aerodyn. 94 (2006) 21–3922

flat plates, and other objects, in a wind tunnel. Wang and Letchford [6] also studied flighttrajectories of small ‘sheet’ objects in a wind tunnel.This paper is a companion to Part I [7], in which new experimental data on the

trajectories of plates of various aspect ratios are described. The present paper describesnumerical solutions of the flight characteristics of plates of square planform, followinginitiation of flight. Unlike compact objects, such as spheres [3], which are driven only bydrag forces, the trajectories of plates in strong winds are subject to drag and lift forces, andalso pitching moments, for motion in a vertical plane.

2. Aerodynamic forces and moments on square plates

2.1. Normal force coefficient

The dimensions of a square planform plate, and the angle of attack, a, of the relativewind are shown in Fig. 1. Flachsbart [8] measured the force coefficient on a square flatplate as a function of the angle of attack. These data were reported by Hoerner [9], and isalso the basis of an ESDU Data Item [10]. Similar measurements were carried out byTachikawa [11], and more recently by the present authors at Monash University and TexasTech University.At an angle of attack of about 401 the plate ‘stalls’, and the normal force coefficient

shows a sharp peak before falling to a near-constant ‘stalled’ value of 1.1–1.2. The assumedline, as shown in Fig. 2, consisting of linear segments shows quite good agreement with theexperimental data.The model is as follows:

CN ¼ 1:7ða=40Þ for ao40�, (1a)

CN ¼ 1:15 for 40�pao140�, (1b)

CN ¼ 1:7ð180� aÞ=40 for 140�pap180�. (1c)

chord, l

span, l

α

c

centre of pressure

thickness, h

Fig. 1. Dimensions and angle of attack for a square plate.

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0.0

0.4

0.8

1.2

1.6

2.0

0 45 90 135 180angle of attack (deg.)

CN

assumedmeasured-Monashmeasured-TTUmeasured-Tachikawameasured-Flachsbart

Fig. 2. Normal force coefficient for a square flat plate.

J.D. Holmes et al. / J. Wind Eng. Ind. Aerodyn. 94 (2006) 21–39 23

The normal force is resolved into drag and lift components, but an additional dragcoefficient of 0.1 has been added to allow for the skin friction component. Thus

CD ¼ 0:1þ CN sin ðaÞ, (2a)

CL ¼ CN cos ðaÞ. (2b)

2.2. Centre of pressure position

The distance of the centre of pressure position, c, from the centre of the plate, shown inFig. 1, determines the pitching moment on a plate. For a pitching moment coefficientdefined as

CM ¼M

ð1=2ÞrU2‘3, (3)

the relationship between the moment coefficient, the normal force coefficient, and thecentre of pressure position is

CM ¼ CNðc=‘Þ. (4)

Fig. 3 shows the assumed model of the relative centre of pressure position, c/‘, as afunction of angle of attack. This is a segmented model adjusted to fit the availableexperimental data, with the following values of c/‘:

ðc=‘Þ ¼ 0:3� 0:22ða=38Þ for the angle of attack; ap38�, (5a)

ðc=‘Þ ¼ 0:08 cos ½2ða� 38Þ� for 38�oao82:5�, (5b)

ðc=‘Þ ¼ 0:0 for 82:5�pao97:5�, (5c)

ðc=‘Þ ¼ �0:08 cos ½2ð142� aÞ� for 97:5�oap142�, (5d)

ðc=‘Þ ¼ �0:3þ 0:22½ð180� aÞ=38� for 142�oap180�. (5e)

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-0.4

-0.3

-0.2

-0.1

0.0

0.1

0.2

0.3

0.4

0 45 90 135 180

angle of attack (deg.)

c/l

assumedmeasured-Monashmeasured-TTUmeasured-Tachikawameasured-Flachsbart

Fig. 3. Centre of pressure position for a square flat plate.

J.D. Holmes et al. / J. Wind Eng. Ind. Aerodyn. 94 (2006) 21–3924

2.3. Alternative models of static force and moment coefficients

Linear segmented models of the normal force coefficient variation, and centre ofpressure position variation with angle of attack were previously adopted by the authors inRef. [12]. These have been improved to better reflect the available experimental data for asquare plate, as described in Sections 2.1 and 2.2.Alternative models of the drag, lift and normal force coefficient, and centre of pressure

position, have also been proposed by Baker [13]. However, these have not been comparedwith experimental data, and do not reflect the stall region at around 401 angle of attack.

2.4. Magnus effect forces

Tachikawa [4] carried out measurements of aerodynamic lift and drag forces on auto-rotating plates, and was able to determine the Magnus effect forces due to the angularrotational velocities. The effects of rotation on lift forces are significant; however, theMagnus components of drag and moment are small and less well defined. FollowingTachikawa [4], the following bilinear function was assumed for the component of liftcoefficient due to the rotational velocity, o ¼ dy=dt:

CLr ¼ 0:42ð2:5o=o0Þ for o=o0o0:2, (6a)

CLr ¼ 0:42ð0:375þ 0:625o=o0Þ for o=o0X0:2, (6b)

where the steady-state angular velocity, o0, is given by [4]

o0 ¼0:64U

‘, (6c)

where U is the wind speed.

ARTICLE IN PRESSJ.D. Holmes et al. / J. Wind Eng. Ind. Aerodyn. 94 (2006) 21–39 25

The total lift coefficient is then given by the sum of CL from Eq. (2b), and CLr from Eqs.(6a) or (b). Magnus effect components for the drag force and pitching moment were notincluded in the numerical calculations in this paper.

3. Equations of motion

Fig. 4 shows the total angle of attack, a, of the relative wind to a plate. It consists of thesum of two components: b—the angle of attack of the relative wind induced by the verticalmotion of the sheet, with respect to the horizontal, and y—the angle of the sheet to thehorizontal, i.e. the initial angle of attack at the start of the flight, plus the angle the sheethas rotated under the action of the rotational moments.

The horizontal acceleration of the plate is given by

d2x

dt2¼

raðCD cos b� CL sin bÞ½ðU � umÞ2þ v2m�

2rmh. (7)

The vertical acceleration of the plate is given by

d2z

dt2¼

raðCD sin bþ CL cos bÞ½ðU � umÞ2þ v2m�

2rmh� g. (8)

The angular acceleration is given by

d2ydt2¼

raCMA‘½ðU � umÞ2þ v2m�

2I, (9)

β

um

vm β

vm

U-um

U

θ

Fig. 4. Relative angle of attack of the wind for a translating, rotating plate. (Note: positive vm is upwards—the

negative direction is shown above).


where ra is the density of air, rm is the density of the plate material, U is the wind speed, umis the horizontal velocity of the plate, vm is the vertical velocity of the plate, h is thethickness of the plate, ‘ is the side dimension of the plate (square here), A is the plan area(‘2), I is the mass moment of inertia, CD, CL and CM are the drag, lift and momentcoefficients, x is the horizontal distance travelled, z is the vertical distance travelled, g is theacceleration due to gravity, and t is time.These equations can be solved numerically using small time steps, for the horizontal,

vertical, and angular velocities and displacements. The normal force coefficient and centreof pressure positions obtained for static plates at various angles of attack, described inSections 2.1 and 2.2, can be used to obtain drag, lift and moment coefficients in the aboveequations. Additional lift forces due to the Magnus effect are obtained using theTachikawa model, described in Section 2.4.

4. Computed trajectories and comparisons with experimental data

4.1. Comparison with Tachikawa’s experiments

Tachikawa [4] carried out a series of free-flight trajectory tests on small flat plates in auniform flow in a wind tunnel. The various downstream positions of the plate in the earlypart of the trajectories were photographed using a stroboscope. An important observationwas the differing trajectories and rotational behaviour, depending on the initial angle ofattack, a0, at release.Numerical calculations were made to compare with the square plate trajectories reported

by Tachikawa. The previous linear segmented models of normal force coefficient andcentre of pressure position were used in these calculations [12], but the results are littlechanged if the model shown in Figs. 2 and 3 are used instead. Examples of the comparisonsare given in Figs. 5 and 6. The plate was made of plastic with side length of 40mm, andthickness 2mm. The Tachikawa parameter K (see Appendix A) was equal to 2.3. The

Fig. 5. Comparison of computed trajectories (upper) with Tachikawa’s recordings [4] (lower). Initial angle of

attack: 151. (Note: vertical scale of Tachikawa’s measurements is same as horizontal scale).

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Fig. 6. Comparison of computed trajectories (upper) with Tachikawa’s [4] recordings (lower). Initial angle of

attack: 451. (Note: vertical scale of Tachikawa’s measurements is same as horizontal scale).


agreement of the rotational positions with the experimental recordings is very goodin both Figs. 5 and 6. In the latter case, the plate first rotates clockwise, and then anti-clockwise in the experiments; this behaviour is also reproduced in the numericalcalculations. The vertical displacements in the calculations, in both cases, are slightlylower than those recorded in the experiments (note that the vertical displacements areplotted to a different scale to the horizontal displacements in the numerical calculations inFigs. 5 and 6).

4.2. Influence of the Magnus effect

The numerical calculations for the Tachikawa plate used in the previous section werealso used to investigate the influence of the Magnus effect terms on the calculations. Theassumed models for normal force coefficient and centre of pressure position in Figs. 2 and3, respectively, were used. Some examples are shown in Fig. 7.

In the case of the 151 initial angle, the angular velocities are large and always in theclockwise direction—in that case the Magnus effect generates significant additionalpositive (upwards) lift forces and hence less negative vertical displacements. For thea0 ¼ 451, angular velocities are small, and in both clockwise and anti-clockwise directions,resulting in negligible differences. For the 1501 initial angle, the rotational velocities anddisplacements are anti-clockwise for motion from left to right [4], and the Magnus effectgives negative lift; hence the calculated vertical displacements are more negative when theMagnus effect is included.

For initial angles between 451 and 1501 (not shown), the angular velocities are small, andtrajectories with and without the Magnus effect forces included differ little from eachother.

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Comparison of trajectories - 15 degree initial angle

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

0 2 4

Horizontal displ. (m)

Ver

tical

dis

pl. (

m)


-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

0 2 4


Ver

tical

dis

pl.

(m)


-2.0

-1.5

-1.0

-0.5

0.00 2


Ver

tical

dis

pl.

(m)

1 3 5

531

1 3

No MagnusEffect

With MagnusEffect

No MagnusEffect

With MagnusEffect

No MagnusEffect

With MagnusEffect

Fig. 7. Influence of Magnus effect lift term on calculated trajectories. Plastic plate (Tachikawa),

40mm� 40mm� 2mm. U ¼ 9:2m=s, K ¼ 2:3.



4.3. Comparison with the Texas Tech experiments

A large number of experimental runs were made in the wind tunnel at Texas TechUniversity to determine flight trajectories of square plates and are reported in thecompanion paper [7]. Most of these were released at an initial angle of attack, a0, of 01. Arange of wind speeds was used in each case.

Some examples of the comparisons between experiment and calculations are shown inFigs. 8–10, for the same plate released at 01 angle of attack, for three different wind speeds.Calculations were carried out with and without the Magnus effect on lift forces included(Section 2.4). From these figures, it can be seen that the Magnus effect has a large influenceon the computed vertical displacements, but that there is little influence on the calculatedhorizontal displacements and resultant plate velocities. However, it is clear that Magnuseffect lift forces should be included in the numerical calculations of plate trajectories. Theagreement between the computed trajectories (Magnus effect included) and theexperimental data is good, although the horizontal displacements are slightly under-estimated in every case.

Figs. 11 and 12 show computed data and calculations for a plate released at initialangles, a0, of 151 and 1351, respectively. Magnus effect lift forces are included, according toEqs. (6). The calculations give reasonable agreement with the experimental data forvertical displacements, horizontal displacements and resultant plate velocities.

5. Discussion

Since the wind flow in the atmospheric boundary layers of severe windstorms such ashurricanes, is highly turbulent, especially near the ground, the effect of turbulence on thetrajectories of debris items, such as plates, needs to be considered. The effect on trajectorieswas studied in the previous paper on spheres (Holmes [3]) and found to introduce a largedegree of variability into trajectories under given mean wind conditions. However, mostflights of windborne debris in full scale last only a second or two and are determined by thegust speed during, and particularly at the beginning of the flight. Thus, the variability intrajectories noted in [3] is primarily due to the varying gustiness in the atmosphericboundary layer. Thus the wind speed used in full-scale trajectory calculations canconveniently be taken as a 2–3 s gust speed, assumed constant over the time of thetrajectory.

The effects of turbulence on the aerodynamic coefficients used in trajectory calculationsalso need to be considered. Atmospheric turbulence has length scales much larger than thedimensions of typical debris objects, and hence, it is expected that coefficients determinedin nominally smooth flow in a wind tunnel should be appropriate for full-scale trajectorycalculations. However, model objects with trajectories that finish in the floor boundarylayers of wind tunnels, may experience turbulence with scales that influence theaerodynamic forces acting (this was taken account of in a previous study of thetrajectories of cubes [12]).

Appendix A describes a non-dimensional analysis of the trajectories of plates. It isapparent from this that the main parameter, apart from the initial angle of attack, a0,determining the trajectories of plates is the Tachikawa parameter, K, equal toraU2=2rmhg, that represents the ratio between aerodynamic forces and gravity forces.This parameter, in slightly different forms, is the governing parameter for trajectories of

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Trajectory of basswood plate (plate10 - 0deg- 15.6 m/s)

-0.6

-0.4

-0.2

0

0.2

0 2


0

2

4

6

8

10

12

0 0.2 0.4 0.6 0.8


0

1

2

3

4

0 0.2 0.4 0.6 0.8

horizontal displacement (m)

vert

ical

dis

plac

emen

t (m

)

time (secs)

resu

ltant

spe

ed (

m/s

)

time (secs)

horiz

onta

l dis

plac

emen

t (m

)

1 3

123Calculatedno Magnus Effect



Fig. 8. Comparison of computed trajectories with Texas Tech data. Plate 10, basswood, 75mm� 75mm� 9mm.

U ¼ 15:6m=s, K ¼ 3:0.


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-0.6

-0.4

-0.2

0

0.2

0 4


vert

ical

dis

plac

emen

t (m

)

Trajectory of basswood plate (plate 10 - 0deg- 21.4 m/s)

0

5

10

15

20

0 0.5 1 1.5

time (secs)

resu

ltant

spe

ed (

m/s

)


0

1

2

3

4

5

6

0 0.5 1

time (secs)

horiz

onta

l dis

plac

emen

t (m

)

2 6

1

2

3

Calculatedno MagnusEffect

1

2

3


1

2

3


Fig. 9. Comparison of computed trajectories with Texas Tech data. Plate 10, basswood, 75mm� 75mm� 9mm.

U ¼ 21:4m=s, K ¼ 5:6.


ARTICLE IN PRESS


-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 4horizontal displacement (m)

vert

ical

dis

plac

emen

t (m

)


0

5

10

15

20

25

0 0.2 0.4 0.6

time (secs)

resu

ltant

spe

ed (

m/s

)

Trajectory of basswood plate (plate 10 - 0deg- 25.6m/s)

0

1

2

3

4

5

6

0 0.2 0.4

time (secs)

horiz

onta

l dis

plac

emen

t (m

)

2 6

0.6

123Calculatedno MagnusEffect



Fig. 10. Comparison of computed trajectories with Texas Tech data. Plate 10, basswood,

75mm� 75mm� 9mm. U ¼ 25:6m=s, K ¼ 8:0.


other generic forms of debris (e.g. [3]). Its reciprocal has also been suggested by Baker [13]as a suitable parameter. The effect of varying K can be seen in Figs. 8–10, in which theTachikawa parameter varies between 3 and 8.

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Trajectory of plywood plate (Plate8 - 15 deg)

0

0.1

0.2

0.3

0.4

0.5

0 4horizontal displacement (m)

vert

ical

dis

plac

emen

t (m

)

Trajectory of plywood plate (Plate8- 15 deg)

0

2

4

6

8

10

12

14

0 0.2 0.4 0.6 0.8time (secs)

resl

utan

t spe

ed (

m/s

)

Trajectory of plywood plate (Plate8- 15 deg)

0

1

2

3

4

5

6

0 0.2 0.4 0.6 0.8time (secs)

horiz

onta

l dis

plac

emen

t (m

)

2 6

2-15-12-15-22-15-3Calculated

2-15-12-15-22-15-3Calculated

2-15-12-15-22-15-3Calculated

Fig. 11. Comparison of computed trajectories with Texas Tech data. Plate 8, plywood, 75mm� 75mm� 3mm,

a0 ¼ 151, U ¼ 15:4m=s, K ¼ 5:9.


Figs. 13 and 14 present some results for plates with zero initial angle of attack, in a non-dimensional form. In Fig. 13, numerical calculations of non-dimensional horizontal platevelocity (um/U) are plotted as a function of the non-dimensional horizontal displacement(xg/U2), for two values of the Tachikawa parameter, K. The weak dependency on theFroude number, Fr‘, (defined in Eq. (A.4)), for the higher value of K, is shown in thisfigure. A higher value of K corresponds to a lighter plate, and, not surprisingly, to a morerapid increase in velocity with increasing distance.

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Trajectory of plywood plate (Plate 8- 135 deg)

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

0 0.5 1 1.5


vert

ical

dis

plac

emen

t (m

)


0

2

4

6

8

10

0 0.1 0.2 0.3 0.4

time (sces)

resu

ltant

spe

ed (

m/s

)


0

0.5

1

1.5

0 0.1 0.2 0.3 0.4

time (secs)

horiz

onta

l dis

plac

emen

t (m

)

2-135-12-135-22-135-3Calculated

1-135-11-135-21-135-3Calculated

1-135-11-135-2

1-135-3Calculated

Fig. 12. Comparison of computed trajectories with Texas Tech data. Plate 8, plywood, 75mm� 75mm� 3mm,

a0 ¼ 1351, U ¼ 14:6m=s, K ¼ 5:3.


Fig. 14 shows a comparison of the computed variations of non-dimensional horizontalplate velocity (um/U) against non-dimensional horizontal displacement (xg/U2), withexperimental values, for four different values of the Tachikawa parameter K. Although thegeneral trends are similar, the experimental data seem to indicate a more rapid increase ofum/U towards 1.0, than do the computations.Non-dimensional presentations of data such as those in Figs. 13 and 14, are potentially

of practical value to facilitate prediction of horizontal impacts speeds of plate-type debris

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Square Flat plate 0 degrees K=6.7

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Square Flatplate 0 degrees K=17.6

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

u m/U

u m/U

Fr2= 100Fr2= 200Fr2= 300

Fr2= 100Fr2= 200Fr2= 300

x

x

Fig. 13. Non-dimensional horizontal plate velocity versus non-dimensional horizontal displacement. K ¼ 6:7,upper; K ¼ 17:6, lower. Varying values of Fr2‘ .


on walls of buildings of fixed distance, x, downstream from a source of debris. As anexample of this, consider a concrete roof tile of dimensions 300� 300� 15mm3 with amass of 3.0 kg. In a (gust) wind speed of 60m/s, the value of the Tachikawa parameter, K,is 6.7. Fig. 14 can be used to estimate the horizontal impact speed when it hits a buildingwall 20m away (horizontally) from its release point; in that case the non-dimensionaldisplacement is 20� 9.8/602 ¼ 0.0544. From the graph for K equal to 6.6–6.8 in Fig. 14,um/U is approximately 0.5, thus giving a horizontal impact velocity of one half of the windspeed—i.e. 30m/s.

6. Conclusions

Numerical calculations of the trajectories of plates of square planform have been made,and compared with experimental data from two wind tunnels. Generally, good toexcellent, agreement has been shown.

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Trajectory of squareplate (K=3.0)

0

0.2

0.4

0.6

0.8

1

0.0 0.2 0.4 0.6 0.8

xg/ (U2)

u m/U

Trajectory of squareplate (K=6.6-6.8)

0

0.2

0.4

0.6

0.8

1

1.2

0.0 0.2 0.4 0.6 0.8

xg/U2

u m/U

Trajectory of squareplate (K=11.8)

0

0.2

0.4

0.6

0.8

1

1.2

0.0 0.2 0.4 0.6 0.8

xg/ (U2)

u m/U

Trajectory of squareplate (K=17.3-17.9)

0

0.2

0.4

0.6

0.8

1

1.2

0.0 0.2 0.4 0.6 0.8

xg/U2

u m/U

#2 Vw=6.1m/s #5 Vw=15.6m/s #7 Vw=5.6m/s #8 Vw=10.9m/s #9 Vw=14.6m/s #10 Vw=15.6m/s #13 Vw=10.0m/s Calculated

#1 Vw=26.0m/s #3 Vw=18.8m/s #4 Vw=10.2m/s #5 Vw=23.1m/s #7 Vw= 8.3m/s #8 Vw=16.4m/s

#9 Vw=22.0m/s #10 Vw=23.1m/s #11 Vw=10.2m/s #12 Vw=23.0m/s #13 Vw=15.1m/s Calculated

#2 Vw=14.8m/s #4 Vw=16.5m/s #7 Vw=13.6m/s #11 Vw=16.5m/s Calculated

#4 Vw=13.5m/s #8 Vw=21.5m/s #11 Vw=13.6m/s #13 Vw=19.9m/s Calculated

Fig. 14. Non-dimensional horizontal plate velocity versus non-dimensional horizontal displacement—compar-

ison of numerical calculations with experiments, for varying values of K.


The numerical models incorporate the angular displacements and velocities, andMagnus effect lift forces. Simple segmented models of normal force coefficient, centre ofpressure position, and Magnus effect lift seem to be quite adequate for engineeringpredictions of trajectories.The Tachikawa parameter, K, is a governing non-dimensional parameter for

determining trajectories of windborne debris of all generic types, including the plate type.Non-dimensional presentations of horizontal plate velocity against horizontal displace-ment (e.g. Figs. 13 and 14), have practical value to allow prediction of horizontal impactsspeeds of full-scale plate-type debris on walls of buildings of fixed distance, x, downstreamfrom a debris source.

Acknowledgements

The research in this paper was supported through the John P. Laborde endowed Chairat Louisiana State University held by the first author, and by the L.S.U. Sea GrantProgram. The first author also acknowledges Professor Bill Melbourne and Dr. JohnCheung of Monash University, Australia, for allowing access to the 450 kW wind tunnelfor force measurements on a flat plate. Useful discussions with Professor Chris Baker(University of Birmingham) and Associate Professor Elizabeth English (LSU) are alsoacknowledged.


Appendix A. Dimensional analysis

A.1. Non-dimensional parameters

The trajectories of windborne missiles are conveniently expressed in non-dimensionalterms.

It is clear from the analysis in the main text that the average horizontal velocity of asquare plate at the end of a flight path is a function of the following variables:

um ¼ f ðra;U ;m; ‘; I ;CD;CL;CM;x; g; a0Þ, (A.1)

where, ra is the density of air, U is the wind speed, m is the mass of the plate, ‘ is the sidedimension of the plate, I is the mass moment of inertia, CD, CL and CM are the drag, liftand moment coefficients, x is the horizontal distance travelled (or the distance to the wallof a building at impact), g is the acceleration due to gravity, a0 is the initial angle of attack.

There are a number of different ways of reducing the above variables into non-dimensional quantities.

Tachikawa [4] defined a non-dimensional number defining the trajectories of all types ofwindborne objects. For a plate, this can be written as

K ¼raU

2‘2

2mg¼

raU2

2rmhg, (A.2)

where h is the thickness of the plate, and rm is the density of the material of the plate. Thisparameter represents a ratio of aerodynamic forces to gravity forces on the plate.

K can also be written as follows:

K ¼1

2

rarm

� �‘

h

� �U2

‘g

� �,

i.e. K is the product of a density ratio, a ratio of dimensions, and the square of a Froudenumber, Fr‘, based on ‘.

The ratio In was also defined by Tachikawa:

In ¼I

m‘2. (A.3)

For a square plate, In is fixed and equal to 1/12.Two Froude numbers, based on the characteristic plate length, ‘, and the horizontal

distance travelled, x, can be obtained as

Fr‘ ¼Uffiffiffiffiffig‘p , (A.4)

Frx ¼Uffiffiffiffiffiffigxp . (A.5)

The drag, lift and moment coefficients are themselves functions of the angle of attackduring a trajectory.

Hence, for a square plate, Eq. (A.1) can be written as

um

U

� �¼ F ðK ;Fr‘;Frx; a0Þ. (A.6)


Baker [13] has defined some alternative parameters

O ¼mg

ð1=2ÞraAU2, (A.7)

F ¼ð1=2ÞraA‘

m, (A.8)

D ¼m‘2

I, (A.9)

where A is the planform area, equal to ‘2 for a square plate.O is the reciprocal of the Tachikawa parameter K. F was described as a ‘buoyancy

parameter’ [13]—it is effectively a density ratio. D is the reciprocal of In used by Tachikawa[4], and is equal to 12 for a square plate.For a rectangular plate with span B, and chord D, ‘ in the above relations should be

replaced by B, and an additional non-dimensional parameter, the aspect ratio, B=D

included.

A.2. Non-dimensional forms of the equations of motion

Baker [13] expressed the equations of motions in a dimensionless form using thefollowing dimensionless variables for horizontal, vertical and angular displacements, andtime:

x ¼x

‘

� �F; z ¼

z

‘

� �F; y ¼ yF; t ¼

tU

‘

� �F: (A.10)

Alternative variables, suggested by Tachikawa [4], have been used in the present paper:

x ¼xg

U2

� �; z ¼

zg

U2

� �; y; t ¼

tg

U

� �. (A.11)

For both schemes, the horizontal and vertical non-dimensional missile velocities aredefined as

u ¼um

U

� �; v ¼

vm

U

� �.

Using the variables defined in Eq. (A.11), the equations of motion (7), (8) and (9) can bere-written in non-dimensional form as follows [4]:

d2x

dt2¼ KðCD cos b� CL sin bÞ½ð1� uÞ2 þ v2�, (A.12)

d2z

dt2¼ KðCD sin bþ CL cos bÞ½ð1� uÞ2 þ v2� � 1, (A.13)

d2y

dt2¼ DKFr2‘ ½ð1� uÞ2 þ v2�, (A.14)

where cos b ¼ ð1� uÞ=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½ð1� uÞ2 þ v2�

qand sin b ¼ v=

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½ð1� uÞ2 þ v2�

q.


The solutions to these equations are consistent with the non-dimensional relationship ofEq. (A.6). The non-dimensional horizontal missile speed at impact depends primarily onthe Tachikawa parameter, K, and the non-dimensional displacement, x ¼ ðxg=U2Þ. It alsodepends on the initial angle of attack of the plate, a0, since this will determine the values ofthe drag and lift coefficients during the trajectories. Noting Eq. (A.14), there is also a weakdependency of the non-dimensional horizontal missile speed on the Froude number, Fr‘,due to the effect of the rotational displacement on the force coefficients. Note that the non-dimensional horizontal displacement x ¼ ðxg=U2Þ is also the reciprocal of the square of aFroude number based on the displacement x, as defined in Eq. (A.5).

References

[1] J.A.B. Wills, B.E. Lee, T.A. Wyatt, A model of wind-borne debris damage, J. Wind Eng. Ind. Aerodyn. 90

(2002) 555–565.

[2] American Society for Testing Materials, Standard specifications for performance of exterior windows,

curtain walls, doors and storm shutters impacted by windborne debris in hurricanes, ASTM E 1996-02, 2002.

[3] J.D. Holmes, Trajectories of spheres in strong winds with application to windborne debris, J. Wind Eng. Ind.

Aerodyn. 92 (2004) 9–22.

[4] M. Tachikawa, Trajectories of flat plates in uniform flow with applications to wind-generated missiles,

J. Wind Eng. Ind. Aerodyn. 14 (1983) 443–453.

[5] M. Tachikawa, A method for estimating the distribution range of trajectories of windborne missiles, J. Wind

Eng. Ind. Aerodyn. 29 (1988) 175–184.

[6] K. Wang, C.W. Letchford, Flying debris behaviour, 11th International Conference on Wind Engineering,

Lubbock, Texas, June 2–5, 2003.

[7] N. Lin, C.W. Letchford, J.D. Holmes, Investigations of plate-type windborne debris, Part I, Experiments in

full scale and wind tunnel, submitted for publication.

[8] O. Flachsbart, Messungen an ebenen und gewolbten Platten, Ergebnisse der AVA. IV, 1932.

[9] S.F. Hoerner, Fluid-dynamic drag, Hoerner Fluid Dynamics (1965).

[10] ESDU, Fluid Forces and Moments on Flat Plates. Data Item 70015, Engineering Science Data Unit,

London, 1970.

[11] M. Tachikawa, Trajectories and velocities of typhoon-generated missiles, Part I, Aerodynamic characteristics

of flat plates and equations of motion, Trans Architect Inst Japan 314 (1981) 1–10.

[12] J.D. Holmes, E.C. English, C.W. Letchford, Aerodynamic forces and moments on cubes and flat plates, with

application to windborne debris, Fifth International Colloquium on Bluff-body Aerodynamics and

Applications, Ottawa, Canada, July 11–15, 2004.

[13] C.J. Baker, Solutions of the debris equations, Sixth UK Conference on Wind Engineering, Cranfield,

September 15–17, 2004.

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