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Solutions to the compact debris flight equations
J. Wind Eng. Ind. Aerodyn. 138 (2015) 69–76
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Journal of Wind Engineering and Industrial Aerodynamics journal homepage: www.elsevier.com/locate/jweia
Solutions to the compact debris flight equations Nigel B. Kaye n Glenn Department of Civil Engineering, 114 Lowry Hall, Clemson University, Clemson, SC 29634, USA
art ic l e i nf o
a b s t r a c t
Article history: Received 16 April 2014 Received in revised form 6 October 2014 Accepted 1 January 2015 Available online 23 January 2015
Analytic, approximate analytic, and numerical solutions to the compact debris flight equations are presented. Analysis shows that, after release pffiffiffiffiffi from rest, the slope of the particle trajectory adjusts from an initial slope of Ω to a final slope of Ω where Ω is the inverse of the Tachikawa number. For Ω o 1 the trajectory steepens whereas for Ω 4 1 the path becomes less steep over the full flight distance. However, for all values of Ω the trajectory initially steepens before adjusting to its steady trajectory slope. The final steady straight line trajectory is shown to project back to a virtual release height that is analytical solutions for the flight calculated numerically and shown to be a function of Ω. Approximate pffiffiffiffiffi distance required to achieve the final steady-state slope ( Ω) are presented and show that the transition height is a function of Ω for small values of Ω but is independent of Ω for larger values. The transition height is shown to be very large for a broad range of physically realistic conditions. Contour plots are presented that summarize the change in trajectory, horizontal flight distance, horizontal and vertical velocity, and kinetic energy as a function of vertical distance traveled and Ω. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Windborne debris Tachikawa number Analytic solutions.
1. Introduction The flight of wind-borne compact debris during severe storms is of significant interest to the wind engineering community due to the severe damage that can be caused by such debris upon impact. A recent example of such damage was the destruction of a façade of the Hyatt Hotel in down town New Orleans due to roof gravel from an adjacent building being blown off during hurricane Katrina. It is therefore important to understand how far a piece of compact debris will travel for a given wind speed. The compact debris equations of motion have long been established (Tachikawa, 1983, 1988). However, there are no general analytic solutions to these equations so flight calculations must be done numerically which inhibits their general use. The compact debris flight equations are mainly used in probabilistic models to assess risk that use statistical input and wind conditions. They are of limited use in predicting the flight dynamics of a given piece of debris as the size of the debris, the debris initial velocity, and the ambient wind field are all unknown. In general, the equations have been solved for a piece of compact debris released from rest at some specified height in a steady uniform wind field (Holmes, 2004; Baker, 2007). The goal with this approach is to understand the behavior of the system of equations under idealized conditions to provide a framework for interpreting the results of more sophisticated models. This approach has been extended to examine the role of ambient turbulence on the
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http://dx.doi.org/10.1016/j.jweia.2015.01.001 0167-6105/& 2015 Elsevier Ltd. All rights reserved.
particle flight path (Holmes, 2004; Karimpour and Kaye, 2012a; Moghim and Caracoglia, 2014). Karimpour and Kaye (2012a) showed that ambient turbulence will slightly increase the flight distance and can be accounted for by using the Root Mean Squared horizontal wind speed in calculations. Karimpour and Kaye (2012a) also examined the role of input uncertainty on the debris flight path and showed that, for some set of randomly distributed particle sizes, using the mean particle size in flight calculations will underestimate the mean particle flight path. Some special cases of the debris flight equations can be solved analytically. Holmes (2004) presented a solution for the case where vertical air resistance is ignored. Baker (2007) showed that, for long enough flight times, a piece of compact debris will reach a steady velocity in which it travels horizontally at the wind speed (U) and vertically at its terminal velocity (wT ). Therefore, provided the flight duration is long enough that the initial adjustment to steady-state can be ignored, the flight distance (X) for a particle released from rest at a height (H) above the ground, can be approximated by X HU=wT :
ð1Þ
However, no limitations on the use of this equation, or discussion of, in general, how large H must be for this to be a valid approximation, have been presented in the literature. The solution of the compact debris equations has implications for a range of wind engineering applications. For example, theoretical estimates of flight distance and velocity could allow engineers to develop appropriate impact mitigation designs. Calculations of debris kinetic energy can be used to develop test standards for impact
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resistant cladding. Finally, the flight path of a piece of compact debris has implications for the scaling of wind tunnel blow-off tests up to full scale. The remainder of the paper is structured as follows. The compact debris flight equations are presented in Section 2 in both dimensional and non-dimensional form along with the solutions for the near field and far field trajectory. The near field adjustment to the far field steady-state trajectory and the distance required to achieve that steady-state trajectory are discussed in Section 3. Full numerical solutions for a very broad range of Tachikawa number are presented graphically in Section 4. Conclusions are presented in Section 5.
2. Model development The two-dimensional compact debris flight equations have been presented numerous times in the literature (Holmes, 2004; Baker, 2007). Compact debris is any object for which all length to width ratios are approximately one (as opposed to rod like debris which has one long and two short dimensions and plate like debris which has two long dimensions and one short dimension). Additionally, compact debris has a negligible lift coefficient and negligible rotational inertia. As such, the compact debris flight equations can be developed from the aerodynamic drag equation and the two dimensional equations of motion for a particle in a gravitational field. Consider a particle moving horizontally with velocity u and vertically with velocity w o0 (taking up as positive) in a steady uniform wind field of horizontal velocity U and zero vertical velocity (see Fig. 1a). The resulting drag force acts in the direction of the relative velocity while the weight force acts vertically downward (see Fig. 1b). Ignoring the buoyancy force acting on the particle then the resulting equations for the time variation of vertical and horizontal particle velocity are given by d x du ρC D A ¼ ðU uÞ ¼ 2m dt 2 dt 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðU uÞ2 þ w2
ð2Þ
and d z dw ρC D A ¼ ð wÞ ¼ dt 2m dt 2 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðU uÞ2 þ w2 g
ð3Þ
and d ζ dω ¼ ¼ ð ωÞ dτ2 dτ 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 μ þ ω2 Ω
ð5Þ
where the non-dimensional variables are given by u w C D rA C D rA C D rUA 2mg ;ζ ¼z ;τ ¼t and Ω ¼ m¼ ;o¼ ;w¼x U U 2m 2m 2m C D rAU 2 ð6Þ Here, Ω is proportional to the inverse of the Tachikawa number (Holmes et al., 2006). Note that this is a slightly different nondimensional scheme than that used by Baker (2007) as the drag coefficient is included in the non-dimensionalization in order to keep the equations and resulting solutions tidier. For a particle released from rest the initial conditions are zero horizontal and vertical particle velocity (μ ¼ ω ¼ 0 at τ ¼ 0). Therefore, the velocity diagram has only the horizontal wind speed U (see Fig. 2a) and the forces acting on the particle are the weight acting vertically down and the drag acting horizontally (see Fig. 2b). For small times after the initial release the equations of motion can be approximated by ignoring the particle velocity terms, giving d χ dμ d ζ dω ¼ ¼ 1 and 2 ¼ ¼ Ω: dτ dτ 2 dτ dτ 2
2
ð7Þ
Therefore, the small time velocities are given by
μ τ and ω Ωτ;
ð8Þ
leading to an initial trajectory slope of S¼
ω 2mg ¼ Ω ¼ μ C D ρAU 2
ð9Þ
equal to the initial force direction as shown in Fig. 2(b). This is similar to the result of Baker (2007), though again with the CD contained in the non-dimensional parameter Ω. In the limit of large time a steady trajectory is achieved in which the time derivatives in (4) and (5) are zero. This leads to the result that pffiffiffiffiffi ð10Þ μ ¼ 1 and ω ¼ Ω: That is, the debris travels horizontally at the wind speed (Fig. 2c). Therefore, the only drag force is in the vertical direction and is exactly balanced by the particles weight (Fig. 2d). As such,
where x and z are the horizontal and vertical coordinates, ρ is the density of air, m is the mass of the particle, A is the cross sectional area of the particle (assumed constant), CD is a drag coefficient (assumed constant), and g is the gravitational acceleration constant. These equations can be re-written in non-dimensional form as 2 d χ dμ ¼ ¼ 1μ dτ2 dτ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 μ þ ω2
ð4Þ
Fig. 1. (a) Velocity diagram for a particle showing the wind speed U, particle velocity (u; w) and velocity of the air relative to the particle V. (b) Force diagram showing the drag force acting in the direction of the relative velocity and the weight force acting down.
Fig. 2. Particle velocity diagrams (a,c) and force diagrams (b,d) for a piece of compact debris upon release (a,b) and at the large time limit steady state (c,d). The initial trajectory slope of Ω is shown on the force diagram (b) and the final pffiffiffiffi trajectory slope of Ω is shown on the velocity diagram (c).
N.B. Kaye / J. Wind Eng. Ind. Aerodyn. 138 (2015) 69–76
Initial Slope Final slope
−2
Slope
−10
and ignoring the wind speed leads to d ζ dω dω ¼ ¼ω ¼ ð ωÞjωj Ω dτ2 dτ dζ 2
ð12Þ
which, given that ω o 0, (11) can be re-written as
0
−10
71
Steeper
1 dω2 ¼ ω2 Ω: 2 dζ
2
−10
−2
10
10
0
2
10
Ω Fig. 3. Log–log scale plot of numerical calculations of the initial (thin line) and final (thick line) trajectory slope as a function of Ω showing exact agreement with the theoretical predictions of (9) and (11). The slopes are negative and as such the slope magnitude increases as you move down the vertical axis.
the vertical velocity is the particle terminal velocity (Fig. 2c) and the slope of the steady-state trajectory is given by pffiffiffiffiffi pffiffiffiffiffi ω Ω S¼ ¼ ð11Þ ¼ Ω: μ 1 This is the same result established by Baker (2007). Numerical simulations over a broad range of Ω were run and the slopes of the initial and final trajectories were calculated. The numerical solutions were generated by integrating (4) and (5) subject to the initial conditions μ ¼ ω ¼ 0 at τ ¼ 0. The integration was done using the built in ODE solvers in MATLABs. The results are shown in Fig. 3 and agree exactly with (9) and (11). Agreement with the analytical result in (11) demonstrates the accuracy of the numerical integration technique and agreement with (9) illustrates the accuracy of the approximate near field trajectory solution. The figure illustrates that, for Ω o 1 the magnitude of the initial slope is less than the magnitude of the final slope and that, on average, the trajectory steepens during the flight. The opposite is true for Ω 41.
Eq. (10) can be integrated, subject to the initial condition ω ¼ 0 at ζ ¼ 0, to give qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð14Þ ω ¼ Ω 1 e2ζ : Therefore, particle will exponentially approach its terminal ffi pffiffiffiffithe velocity Ω. The flight distance required to achieve 95% of the particle terminal velocity is ζ 95% ¼ 1:2 which is independent of Ω. For very small Ω, the wind speed is considerably larger than the particle terminal velocity and the flight is largely horizontal. In this case, the vertical velocity terms and Ω can be ignored and (4) becomes 2 2 d χ dμ ¼ ¼ 1μ dτ2 dτ
The initial trajectory, given by (8) and (9), and the final trajectory, given by (10) and (11), are easy to calculate and provide a useful check on numerical simulations. However, they are only useful once the limits on their applicability have been established. Therefore, it is important to establish how long the initial trajectory slope can be approximated by –Ω after what ffi pffiffiffiffiand distance the trajectory slope approaches Ω. Analytic and numerical results that place limits on these solutions are presented in this section along with details on the evolution of the flight path between these limits.
3.1. Adjustment distance to reach steady-state trajectory The flight distance required for a piece of debris to achieve the steady-state velocity and trajectory given in (9) and (10) cannot be established analytically. However, some analytical progress can be made by considering the extreme cases of very small and very large Ω. For very large Ω the flight is dominated by vertical acceleration due to gravity and the wind speed and horizontal velocity are small compared to the vertical velocity. As such, in this limit the horizontal velocity Eq. (4) can be ignored and only the vertical Eq. (5) need be considered. Re-writing (5) in terms of the rate of change of velocity along the (approximately) vertical flight path
ð15Þ
This is analogous to the case of zero vertical drag considered by Holmes (2004). Following the same solution procedure presented by Holmes (2004) leads to Z Z dμ 2 τ¼ ð16Þ 2 ¼ ψ dψ 1μ where the variable ψ used in the substitution is ψ ¼ 1 μ. Integration from an initial condition of μ ¼ 0 at τ ¼ 0 leads to
μ¼
dχ τ ¼ : dτ 1 þ τ
ð17Þ
Therefore, the horizontal velocity will attain 95% of its final velocity (μ ¼ 1) when τ ¼ 19. Eq. (17) can be integrated a second time to give
χ ¼ τ lnð1 þ τÞ: 3. Flight trajectory development
ð13Þ
ð18Þ
Substituting τss 19 into (18) leads to a horizontal travel distance of χ 16 for the particle to achieve 95% of its final velocity. Assuming that the vertical velocity adjusts rapidly such that the vertical velocity can be approximated by its terminal velocity at all times, then the vertical distance over which the horizontal velocity adjusts is given by pffiffiffiffiffi pffiffiffiffiffi ζ 95% Ωτss 19 Ω: ð19Þ Therefore, for small Ω, the adjustment height is a function of Ω whereas, for large Ω the adjustment height approaches a constant independent of Ω. The small Ω (19) and large Ω (14) results can be seen in Fig. 4(a) which shows the numerically calculated slope of the trajectory at different vertical distances below the release height as a function of Ω. The steady-state trajectory is reached when the slope contour becomes a vertical line indicating that the slope no longer changes with vertical location. For small Ω the adjustment height increases with increasing Ω as predicted by (19). However, for large Ω the adjustment height approaches a constant, that is, it is independent of Ω, as predicted by (14). This can be seen more clearly in Fig. 4(b) which shows the slope scaled on the final pffiffiffiffi ffi steady-state slope (S^ ¼ S= Ω) where the lower contour decreases with increasing Ω approaching a constant in the limit of large Ω. Fig. 4(b) also shows that the initial slope is larger than the steady state slope for Ω 4 1 and smaller than the steady-state slope for Ω o 1 as shown in Fig. 3. Note that in this analysis the choice of 95% is arbitrary. However, the result that the adjustment height is
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N.B. Kaye / J. Wind Eng. Ind. Aerodyn. 138 (2015) 69–76
−1
−1
−10
−10
−1 −1
−2 −3
ζ
−4
0
−10
−1.5
ζ
0
−10
−5 −6
1
−10
−7
−2 1
−10
−2.5
−8 10
−1
0
10
10
1
10
−1
10
Ω
0
1
10
Ω
Fig. 4. (a) Log–log scale contour plot of the flight trajectory slope (s ¼ ω=μ) in Ω; ζ space. (b) Log–log scale contour plot of the flight trajectory slope scaled on the steadypffiffiffiffi state slope (S^ ¼ S= Ω).
only a function of Ω for small Ω would be the same regardless of what fraction of the terminal velocity was used in the analysis. It is worth considering what the non-dimensional adjustment distances plotted in Fig. 4 represent in dimensional space. To illustrate this we consider three examples from the literature. The first two are the example cases presented by Holmes (2004), namely an 8 mm stone sphere with U ¼ 20 m/s, and an 80 mm wooden sphere with U ¼ 30 m/s. The third example is that of a 1 mm grain of sand in a wind tunnel blowing at a velocity of U ¼ 6 m/s, typical of the critical blow off velocity tests of Karimpour and Kaye (2012b). The dimensional and non-dimensional parameters for these three examples are given in Table 1 calculated taking the drag coefficient to be C D ¼ 0:5. For the examples in Table 1, the adjustment height is taken to be ζ ¼ 2:5 based on the numerical result for Ω ¼ 1 shown in Fig. 4. It is interesting to note that all three examples have an Ω value of approximately one. This is consistent with the flight initiation model of Wills et al. (2002) who posited that a piece of loose debris will start to move when the initiating force (say drag or lift) balances the fixing force that scales on the weight of the particle. Assuming turbulent flow, the drag force scales on the wind speed squared and the particle cross sectional area (F I ρAU 2 ). Motion initiation will occur when the ratio of the initiating force and the particle weight is approximately one. That is, flight will begin when F I =W ρAU 2 =mg 1=Ω Oð1Þ:
ð20Þ
This is a similar result to the scaling presented for roof gravel blow off by Karimpour and Kaye (2012b). Therefore, for compact debris, the flight initiation conditions will likely result in compact debris flight occurring within a fairly narrow range of Ω around Ω 1. However, this is a local condition for motion initiation. Actual flight conditions could vary significantly in regions where there are large velocity gradients such as in the flow separation region over a roof. In the examples presented the reference length scale (Lref ) is very large (much larger than the particle diameter) and, therefore, the vertical distance taken for the particle to reach its steady-state trajectory slope (zss ) takes place over extremely long distances. This is due to the large density difference between the particle and the air. For example, an 8 mm piece of roof gravel would have to fall the 87 m, or roughly the height of a 25 floor building, before achieving a steady state. Further, a sand grain released from rest would fall 11.4 m before achieving its steady trajectory. Even the largest boundary layer wind tunnels are not this tall and, as such, are unable to achieve a steady state trajectory for even a small grain of sand.
Table 1 List of parameters for the example debris particles. Material Mass
Stone Wood Sand
Diameter (mm)
0.54 g 8 134 g 80 1.4 mg 1
U Lref ¼ C2m Ω ¼ C 2mg 2 D ρA D ρAU (m/s) (m)
(s)
zss (m)
20 30 6
1.75 2.9 076
87 217 11.4
0.86 0.95 1.2
35 87 4.6
T ref ¼ C D2m ρAU
3.2. Virtual release height As shown above, following an initial adjustment pffiffiffiffiffi the debris particle will travel in a straight line of slope S ¼ Ω. Therefore, in the far field, the flight trajectory can be described by a straight line that appears to come from a height given by the release height plus a virtual release height offset ζ v where if ζ v is positive then the virtual release height is above the actual release height and is below the actual release height if ζ v o 0. This is shown schematically in Fig. 5. The horizontal flight distance for large ζ is, therefore, given by
ζ ζ χ ¼ pffiffiffiffiffiv ð21Þ Ω where ζ is measured from the release height and is positive upwards. pffiffiffiffiffi For Ω o 1 the initial slope is less than the final slope (Ω o Ω), whereas for Ω 4 1 the initial slope is greater than the final slope. Intuitively, one might therefore expect that, for Ω o 1, the virtual release height would be above the actual release height (ζ v 4 0), as shown in Fig. 5(a). Conversely, for Ω 41 the initial trajectory slope is greater than the final slope and one would expect the virtual release height to be below the actual release height (ζ v o 0) as shown in Fig. 5(b). To test this hypothesis a series of 600 numerical flight simulations over 6 orders of magnitude in Ω were conducted and the virtual release height offset was calculated by fitting a line through the flight trajectory late in the flight. A plot of the calculated virtual release height offset as a function of Ω is shown in Fig. 6(a). For Ω 4 1 the virtual release height is negative, that is, the trajectory projects back to below the actual release height as shown in Fig. 5(b) and as hypothesized in the previous paragraph. Conversely, for Ω{1 the virtual release height is above the actual release height (see Fig. 5a). However, for 0:3 o Ω o1, the virtual release height is below the actual release height, that is, ζ v o 0. Therefore, for this range of Ω, the trajectory slope must initially steepen beyond the steady state slope and then approach the
N.B. Kaye / J. Wind Eng. Ind. Aerodyn. 138 (2015) 69–76
73
Fig. 5. (a) Schematic trajectory for a particle with Ω{1 showing the initial and final horizontal and vertical velocities, trajectory (solid line), and projected trajectory (dashed line) originating at the virtual release height ζ v above the actual release height. (b) Schematic trajectory for a particle with Ω4 1 showing same details but with the virtual release height below the actual release height.
1
ζ
v
0.5 0 −0.5 −1
−2
10
10
0
2
10
Ω Fig. 6. (a) Log-linear scale plot of the calculated virtual release height offset correction (ζ v ) as a function of Ω. (b) Schematic diagram for 0:3 o Ωo 1 showing the initial and final horizontal and vertical velocities, trajectory (solid line), and projected trajectory (dashed line) originating at the virtual release height ζ v below the actual release height.
steady-state trajectory from a steeper rather than shallower slope. This implies that the trajectory has an inflection. This is shown schematically in Fig. 6(b). Also, Fig. 6(a) shows that for large Ω, ζ v approaches a constant whereas, for small Ω, ζ v remains as a function of Ω. This qualitatively identical to the relationship between Ω and the vertical distance required to achieve a steady-state trajectory slope as shown in Fig. 4(b).
3.3. Near release flight trajectory Given the large fall heights required for typical debris to achieve their steady-state trajectory slope, and the unexpected behavior observed in the virtual release height model for 0:3 o Ω o1 it is clear that the near field trajectory behavior has a significant influence on the entire flight path. We therefore seek to quantify the range over which the initial trajectory, (8) and (9), is a valid approximation for the flight path, and to examine how the flight path adjusts away from this initial trajectory toward the steady state trajectory given by (11). The approximate initial solution (17) results from assuming that the debris horizontal and vertical velocities are approximately zero. This assumption will be valid provided qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 1μ 1 μ þ ω2 Ω Ω: 1 μ þ ω2 1 and ω ð22Þ Substituting (8) into (22), and taking (8) to be valid until the left hand sides of (22) are greater than 1.1 times the right hand
sides, leads to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 τÞ ð1 τÞ2 þ Ω2 τ2 1 ¼ 0:1
ð23Þ
and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Ωτ ð1 τÞ2 þ Ω2 τ2 ¼ 0:1:
ð24Þ
The solutions for the times, τ, that satisfies (23) and (24) were calculated numerically over the range 0:1 o Ω o 10. The results are shown in Fig. 7. The results indicate that there is a very limited time range over which the approximate initial trajectory (8) is valid (typically around τ ¼ 0:05). This is very small indeed given the time scales listed in Table 1 which were all of the order of a second. That is, the initial trajectory slope is only a valid approximation for a few hundredths of a second. Interestingly, with the exception of a small range of Ω (approximately 4 o Ω o 8) the time is limited by the vertical acceleration term (24). This seems to indicate that the vertical velocity may grow relatively more rapidly than the horizontal velocity for a given Ω. The results of Section 2 show that the trajectory of a particle will change from an initial slope Ω to a final steady-state slope pffiffiffiffiffi of Ω (see Eqs. (9) and (11) and Fig. 3). For the range of Ω considered in the contour plots, the slope transition occurs over a relatively constant distance of ζ 2:5 (see Fig. 4(b)). One can write an approximate function for the slope of the trajectory that has the appropriate slope limits as given by (9) and (11). One possible
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N.B. Kaye / J. Wind Eng. Ind. Aerodyn. 138 (2015) 69–76
equation is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω Ω2 αζΩ : S¼ μ 1 αζ
an expression for the rate of change of slope with time as dS d ω 1 dω ω dμ 1 dω dμ ¼ 2 S ¼ ¼ : dτ dτ μ μ dτ μ dτ μ dτ dτ
ð25Þ
However, the same analysis approach used to calculate the small time velocity (8) and slope (9) breaks down when trying to estimate dS=dτ at the release point. This is because the slope is undefined for the initial conditions μ ¼ ω ¼ 0. That is, the initial trajectory slope is only defined in the limit of τ-0 þ . Numerical calculations of the slope for very small times indicate that
where α is a fitting parameter which controls the rate of adjustment from the initial to the final slope. Eq. (25) has the correct pffiffiffiffiffi limits of S ζ ¼ 0 ¼ Ω and limlarge ζ S ζ ¼ Ω. Trial and error testing showed that α ¼ 20 provided the appropriate transition distance. A contour plot of the approximate slope as a function of Ω and ζ is shown in Fig. 8(a). The main problem with (25) is that there are no local maxima in the contours as were seen in Fig. 3(a). This is illustrated more clearly in Fig. 8(b) in which there are clear local magnitude maxima for the numerically calculated Ω ¼ 0:5 and 1 slope plots. While the slopes estimate provided by (25) do not match particularly well in the near field (though they do match exactly at χ ¼ ζ ¼ 0 and then diverge), they only vary near the release point over short distances. A more realistic test of the quality of the slope approximation is provided by comparing the particle trajectories calculated numerically, by solution of (4) and (5), and approximately, by integration of the slope approximation (25). Trajectories calculated by these two methods are shown in Fig. 9 with the approximate solution in Fig. 9(a) and the numerical solution in Fig. 9(b). The approximate solution provides a reasonable description of the flight path in the far field for Ω ¼ 10; 1, and 0:1 and a poor representation for Ω ¼ 5 and 0:5. This is because, although the slope limits are correct, the resulting approximate trajectories have the wrong virtual (or apparent) release height (see Fig. 6a). Clearly the near field slope adjustment is important and requires further attention. Recalling that S ¼ ω=μ one can write 0.4
τ
dS Ω ¼ : dτ 2
ð27Þ
That is, the slope of the trajectory initially steepens for all Ω even if the steady-state slope is less steep than the initial slope, as for Ω 4 1. This results can be seen qualitatively by examining small velocity changes in the vertical (δω ¼ Ωδτ) and horizontal (δμ ¼ δτ) directions that occur over a small time (δτ). Substituting these small changes into (4) and (5) leads to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 dμ ð28Þ 1 δτ þ ðΩδτÞ2 ¼ 1 δτ dτ and d ζ dω ¼ ¼ ð ωÞ dτ2 dτ 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 δτ þ ðΩδτÞ2 Ω
ð29Þ
Writing the term under the square root as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 KðδτÞ ¼ 1 2δτ þ ð1 þ Ω Þδτ2
ð29Þ
leads to dμ dω ¼ K 1 δτ and ¼ Ω 1 K δτ : dτ dτ
ð30Þ
Regardless of the values of Ω and δτ K must be greater than zero. Further, for small enough values of δτ 2δτ 4 ð1 þ Ω2 Þδτ2 : ð31Þ
(23) (24)
0.3
ð26Þ
τ
min
0.2
Therefore, the value of K for small δτ is in the range 0 o K o 1. Taking the leading order terms in (30) leads to
0.1
dμ dω ¼ K o1 and ¼ Ω: dτ dτ
0 −1 10
10
0
ð32Þ
Therefore, for small times when released from rest, the rate at which the horizontal velocity increases will initially decrease whereas the rate of change of the vertical velocity will stay the same. As such, the trajectory will initially steepen for all Ω. Given this result, there must be an inflection in the trajectory for all Ω 41. Further, the negative ζ v calculated for 0:3 o Ω o 1,
1
10
Ω Fig. 7. Time for which (17) is a reasonable approximation for the particle velocity.
−1
−10
−1
−10
−1
ζ
−3 −4 1
−10
ζ
−10
1
Steeper
−10
−5 −6
10
0
−10
−2
0
−1
10
0
Ω
10
1
2
−10 1 −10
0
−10
−1
−10
slope
Fig. 8. (a) Log–log scale plot of the approximate slope during flight calculated using (25) with α ¼ 20. (b) Log–log scale plot of slope from full numerical solutions of Eqs. (4) and (5) (thick lines) and Eq. (25) (thin lines), as a function of ζ for (from left to right) Ω ¼ 10; 5; 1; 0:5; 0:1.
N.B. Kaye / J. Wind Eng. Ind. Aerodyn. 138 (2015) 69–76
−1
−1
−10
−10 Ω=10 Ω=5 Ω=1 Ω=0.5 Ω=0.1
ζ
Ω=0.1 Ω=0.5 Ω=1 Ω=5 Ω=10
0
−10
ζ
0
−10
1
1
−10
−10
2
−10 −1 10
75
2
10
0
χ
10
1
10
2
−10 −1 10
0
10
χ
10
1
10
2
Fig. 9. Log–log scale plot of the trajectories of compact debris for Ω ¼ 10; 5; 1; 0:5; 0:1. (a) Approximate solutions (25) and (b) numerical solutions.
where intuitively one might expect a positive value, is due to the trajectory initially steepening beyond its steady-state slope, inflecting, and then approaching the steady value. In fact, it is possible for debris with Ω o 1 to have a trajectory that steepens beyond its steady slope and inflect while still having a positive ζ v provided the trajectory never drops pffiffiffiffiffi below a line drawn from the release point and of slope S ¼ Ω. Numerical calculations of the flight of debris for 201 different value of Ω, logarithmically spaced in the range 0:1 r Ω r10, found inflections in all the trajectories. This result is consistent with the earlier result that the deviation of the trajectory from its initial path, see Eqs. (8) and (9), is almost always dominated by the vertical acceleration equation as shown in Fig. 7.
steady-state values such that ϖ and κ both approach one in the limit of large flight distance. The plots in Fig. 10 indicate that, as expected, the adjustment to steady horizontal velocity is faster for smaller Ω while the adjustment to steady vertical velocity is faster for higher Ω. The kinetic energy adjustment to its maximum steady value is largest for intermediate values of Ω 1. That is, for values of Ω typical of windborne loose laid debris, the time taken to reach the steady-state (maximum) kinetic energy is close to a maximum and will only be reached for release heights greater than z 9m=C D ρA or approximately 160 m for the example of an 8 mm stone presented in Table 1.
5. Conclusions 4. Numerical solutions While it would be possible to extend the approximate algebraic model of (25) to achieve the correct virtual release height (Fig. 6a), this would rapidly become algebraically complex and selfdefeating. Rather, a graphical solution for the horizontal flight distance (X H ) as a function of the release height (H) and Ω is presented. For a given release height H the non-dimensional release height can be calculated,
η¼
HC D ρA ; 2m
ð33Þ
along with the flight distance scaled on the release height
λ¼
X H χ ðζ ¼ ηÞ ¼ H η
ð34Þ
where λ ¼ λðη; ΩÞ. A contour plot of this function is presented in Fig. 10(a) which allows the calculation of the flight distance of any particle released from rest at any height under any steady uniform wind conditions. Fig. 10(a) also shows that, as the flight path increases the ratio of the flight distance to the release height approaches a constant (vertical contours). This occurs once the vertical drop is large enough that the virtual release height can be neglected (that is, when ζ c ζ v ) and the flight distance can be approximated by (1). In fact, all the major flight parameters can be represented in scaled non-dimensional contour plots. Contour plots of horizontal velocity, μ, scaled vertical velocity pffiffiffiffiffi ð35Þ ϖ ¼ ω= Ω; and scaled particle kinetic energy
κ ¼ ðμ2 þ ω2 Þ=ð1 þ ΩÞ
ð36Þ
(see Baker 2007) are presented in Fig. 10c,d respectively. The vertical velocity and particle kinetic energy are rescaled on their
Analytic, approximate analytic, and numerical solutions to the compact debris flight equations have been presented. The work of Baker (2007) on the near field trajectory and far field steady flight slope has been extended to examine the trajectory slope adjustment during flight. The flight slope will, on average, increase during flight for Ω o 1 and decrease for Ω 4 1, though the trajectory initially steepens for all Ω. The distance over which this transition occurs was investigated analytically and it was shown that, for small Ω the adjustment height is a function of Ω, whereas for larger Ω the adjustment height approaches a constant. This was confirmed through numerical solutions of the debris flight equations over a broad range of Ω (Fig. 4b). An approximate algebraic function of the slope as a function of distance below the release point was presented (25). However, the approximation is not a particularly good representation of the slope, particularly for Ω 1 as the slope initially steepens and there is a clear inflection in the flight path for such intermediate Ω (Fig. 8b). In fact, all trajectories initially steepen and the trajectories of all flight paths have inflections. The change in slope during flight means that, in the steady state linear trajectory phase of the flight, the particle travels along a line that projects back to a virtual release height, ζ v , that is a function of Ω (Fig. 6a). Given the complexity of the approximate algebraic solutions presented, it is simpler to represent the full set of solutions to the compact debris flight equations over a broad range of Ω in a series of non-dimensional contour plots in [Ω; η] space. Plots were presented for the flight distance scaled on the release height, horizontal velocity, vertical velocity scaled on the terminal velocity, and kinetic energy scaled on the terminal kinetic energy (Fig. 10). While many of the results presented are somewhat abstract, they do have some practical implications for wind engineering. First, as illustrated in Table 1, the steady flight solution presented by Baker (2007) is typically only applicable for extreme events
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−1
−1
−10
6
3 10
0.6
4
η
10
0.8
5
0
η
10
0
−10
0.4
1
2 0.2
1 1
2
10 −1 10
10
0
10
−10 −1 10
1
Ω
10
0
10
1
Ω −1
−1
−10
−10
0.8
0.8 0.6
η
η
0.6 0
−10
1
−10 −1 10
0
10
0
1
10
0
−10
0.4
0.4
0.2
0.2
0
1
−10 −1 10
10
0
1
10
0
Ω
Ω
pffiffiffiffi Fig. 10. Log–log scale contour plots in [Ω; η] space of (a) flight distance λ, (b) horizontal velocity μ, (c) scaled vertical velocity ϖ ¼ ω= Ω, and (d) scaled particle kinetic energy κ ¼ ðμ2 þ ω2 Þ=ð1 þ ΩÞ. Note that the color bar for ϖ (c) is skewed compared to the color bars in (b) and (d). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
involving very high release heights. The second interesting point raised by the examples in Table 1 is that for realistic flight conditions, Ω is quite close to one. This is in keeping with the work of Wills et al. (2002) who proposed a simple scaling argument for motion initiation in which the motion inducing force (either lift or drag) scales on the square of the wind speed and the particle's area. The ‘fixing force’ scales on the weight of the particle. Motion will commence when the inducing force is slightly larger than the fixing force. Given a drag or lift coefficient of order 100, their model implies that motion will be initiated when Ω Oð100 Þ. Therefore, their model predicts that most wind driven release and flight events will occur at moderate Ω 1. Acknowledgments This material is based upon work supported by the National Science Foundation, United States of America under Grant no. 1200560. Any opinions, findings, and conclusions or recommendations expressed in the material are those of the author and do not necessarily reflect the views of the NSF. The author would also like
to thank Dr. Greg Kopp for many interesting discussions on this topic during the preparation of this manuscript.
References Baker, C.J., 2007. The debris flight equations. J. Wind Eng. Ind. Aerodyn. 95, 329–353. Holmes, J.D., 2004. Trajectories of spheres in strong winds with application to wind-borne debris. J. Wind Eng. Ind. Aerodyn. 92, 9–22. Holmes, J.D., Baker, C.J., Tamura, Y., 2006. Tachikawa number: a proposal. J. Wind Eng. Ind. Aerodyn. 94, 41–47. Karimpour, A., Kaye, N.B., 2012a. On the stochastic nature of compact debris flight. J. Wind Eng. Ind. Aerodyn. 100, 77–90. Karimpour, A., Kaye, N.B., 2012b. The critical velocity for aggregate blow-off from a built-up roof. J. Wind Eng. Ind. Aerodyn. 107–108, 83–93. Moghim, F., Caracoglia, L., 2014. Effect of computer-generated turbulent wind field on trajectory of compact debris: a probabilistic analysis approach. Eng. Struct. 59, 195–209. Tachikawa, M., 1983. Trajectories of flat plates in uniform flow with application to wind-generated missiles. J. Wind Eng. Ind. Aerodyn. 14, 443–453. Tachikawa, M., 1988. A method for estimating the distribution range of trajectories of wind-borne missiles. J. Wind Eng. Ind. Aerodyn. 29, 175–184. Wills, J.A.B., Lee, B.E., Wyatt, T.A., 2002. A model of wind-borne debris damage. J. Wind Eng. Ind. Aerodyn. 90, 555–565.
Contents lists available at ScienceDirect
Journal of Wind Engineering and Industrial Aerodynamics journal homepage: www.elsevier.com/locate/jweia
Solutions to the compact debris flight equations Nigel B. Kaye n Glenn Department of Civil Engineering, 114 Lowry Hall, Clemson University, Clemson, SC 29634, USA
art ic l e i nf o
a b s t r a c t
Article history: Received 16 April 2014 Received in revised form 6 October 2014 Accepted 1 January 2015 Available online 23 January 2015
Analytic, approximate analytic, and numerical solutions to the compact debris flight equations are presented. Analysis shows that, after release pffiffiffiffiffi from rest, the slope of the particle trajectory adjusts from an initial slope of Ω to a final slope of Ω where Ω is the inverse of the Tachikawa number. For Ω o 1 the trajectory steepens whereas for Ω 4 1 the path becomes less steep over the full flight distance. However, for all values of Ω the trajectory initially steepens before adjusting to its steady trajectory slope. The final steady straight line trajectory is shown to project back to a virtual release height that is analytical solutions for the flight calculated numerically and shown to be a function of Ω. Approximate pffiffiffiffiffi distance required to achieve the final steady-state slope ( Ω) are presented and show that the transition height is a function of Ω for small values of Ω but is independent of Ω for larger values. The transition height is shown to be very large for a broad range of physically realistic conditions. Contour plots are presented that summarize the change in trajectory, horizontal flight distance, horizontal and vertical velocity, and kinetic energy as a function of vertical distance traveled and Ω. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Windborne debris Tachikawa number Analytic solutions.
1. Introduction The flight of wind-borne compact debris during severe storms is of significant interest to the wind engineering community due to the severe damage that can be caused by such debris upon impact. A recent example of such damage was the destruction of a façade of the Hyatt Hotel in down town New Orleans due to roof gravel from an adjacent building being blown off during hurricane Katrina. It is therefore important to understand how far a piece of compact debris will travel for a given wind speed. The compact debris equations of motion have long been established (Tachikawa, 1983, 1988). However, there are no general analytic solutions to these equations so flight calculations must be done numerically which inhibits their general use. The compact debris flight equations are mainly used in probabilistic models to assess risk that use statistical input and wind conditions. They are of limited use in predicting the flight dynamics of a given piece of debris as the size of the debris, the debris initial velocity, and the ambient wind field are all unknown. In general, the equations have been solved for a piece of compact debris released from rest at some specified height in a steady uniform wind field (Holmes, 2004; Baker, 2007). The goal with this approach is to understand the behavior of the system of equations under idealized conditions to provide a framework for interpreting the results of more sophisticated models. This approach has been extended to examine the role of ambient turbulence on the
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http://dx.doi.org/10.1016/j.jweia.2015.01.001 0167-6105/& 2015 Elsevier Ltd. All rights reserved.
particle flight path (Holmes, 2004; Karimpour and Kaye, 2012a; Moghim and Caracoglia, 2014). Karimpour and Kaye (2012a) showed that ambient turbulence will slightly increase the flight distance and can be accounted for by using the Root Mean Squared horizontal wind speed in calculations. Karimpour and Kaye (2012a) also examined the role of input uncertainty on the debris flight path and showed that, for some set of randomly distributed particle sizes, using the mean particle size in flight calculations will underestimate the mean particle flight path. Some special cases of the debris flight equations can be solved analytically. Holmes (2004) presented a solution for the case where vertical air resistance is ignored. Baker (2007) showed that, for long enough flight times, a piece of compact debris will reach a steady velocity in which it travels horizontally at the wind speed (U) and vertically at its terminal velocity (wT ). Therefore, provided the flight duration is long enough that the initial adjustment to steady-state can be ignored, the flight distance (X) for a particle released from rest at a height (H) above the ground, can be approximated by X HU=wT :
ð1Þ
However, no limitations on the use of this equation, or discussion of, in general, how large H must be for this to be a valid approximation, have been presented in the literature. The solution of the compact debris equations has implications for a range of wind engineering applications. For example, theoretical estimates of flight distance and velocity could allow engineers to develop appropriate impact mitigation designs. Calculations of debris kinetic energy can be used to develop test standards for impact
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resistant cladding. Finally, the flight path of a piece of compact debris has implications for the scaling of wind tunnel blow-off tests up to full scale. The remainder of the paper is structured as follows. The compact debris flight equations are presented in Section 2 in both dimensional and non-dimensional form along with the solutions for the near field and far field trajectory. The near field adjustment to the far field steady-state trajectory and the distance required to achieve that steady-state trajectory are discussed in Section 3. Full numerical solutions for a very broad range of Tachikawa number are presented graphically in Section 4. Conclusions are presented in Section 5.
2. Model development The two-dimensional compact debris flight equations have been presented numerous times in the literature (Holmes, 2004; Baker, 2007). Compact debris is any object for which all length to width ratios are approximately one (as opposed to rod like debris which has one long and two short dimensions and plate like debris which has two long dimensions and one short dimension). Additionally, compact debris has a negligible lift coefficient and negligible rotational inertia. As such, the compact debris flight equations can be developed from the aerodynamic drag equation and the two dimensional equations of motion for a particle in a gravitational field. Consider a particle moving horizontally with velocity u and vertically with velocity w o0 (taking up as positive) in a steady uniform wind field of horizontal velocity U and zero vertical velocity (see Fig. 1a). The resulting drag force acts in the direction of the relative velocity while the weight force acts vertically downward (see Fig. 1b). Ignoring the buoyancy force acting on the particle then the resulting equations for the time variation of vertical and horizontal particle velocity are given by d x du ρC D A ¼ ðU uÞ ¼ 2m dt 2 dt 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðU uÞ2 þ w2
ð2Þ
and d z dw ρC D A ¼ ð wÞ ¼ dt 2m dt 2 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðU uÞ2 þ w2 g
ð3Þ
and d ζ dω ¼ ¼ ð ωÞ dτ2 dτ 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 μ þ ω2 Ω
ð5Þ
where the non-dimensional variables are given by u w C D rA C D rA C D rUA 2mg ;ζ ¼z ;τ ¼t and Ω ¼ m¼ ;o¼ ;w¼x U U 2m 2m 2m C D rAU 2 ð6Þ Here, Ω is proportional to the inverse of the Tachikawa number (Holmes et al., 2006). Note that this is a slightly different nondimensional scheme than that used by Baker (2007) as the drag coefficient is included in the non-dimensionalization in order to keep the equations and resulting solutions tidier. For a particle released from rest the initial conditions are zero horizontal and vertical particle velocity (μ ¼ ω ¼ 0 at τ ¼ 0). Therefore, the velocity diagram has only the horizontal wind speed U (see Fig. 2a) and the forces acting on the particle are the weight acting vertically down and the drag acting horizontally (see Fig. 2b). For small times after the initial release the equations of motion can be approximated by ignoring the particle velocity terms, giving d χ dμ d ζ dω ¼ ¼ 1 and 2 ¼ ¼ Ω: dτ dτ 2 dτ dτ 2
2
ð7Þ
Therefore, the small time velocities are given by
μ τ and ω Ωτ;
ð8Þ
leading to an initial trajectory slope of S¼
ω 2mg ¼ Ω ¼ μ C D ρAU 2
ð9Þ
equal to the initial force direction as shown in Fig. 2(b). This is similar to the result of Baker (2007), though again with the CD contained in the non-dimensional parameter Ω. In the limit of large time a steady trajectory is achieved in which the time derivatives in (4) and (5) are zero. This leads to the result that pffiffiffiffiffi ð10Þ μ ¼ 1 and ω ¼ Ω: That is, the debris travels horizontally at the wind speed (Fig. 2c). Therefore, the only drag force is in the vertical direction and is exactly balanced by the particles weight (Fig. 2d). As such,
where x and z are the horizontal and vertical coordinates, ρ is the density of air, m is the mass of the particle, A is the cross sectional area of the particle (assumed constant), CD is a drag coefficient (assumed constant), and g is the gravitational acceleration constant. These equations can be re-written in non-dimensional form as 2 d χ dμ ¼ ¼ 1μ dτ2 dτ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 μ þ ω2
ð4Þ
Fig. 1. (a) Velocity diagram for a particle showing the wind speed U, particle velocity (u; w) and velocity of the air relative to the particle V. (b) Force diagram showing the drag force acting in the direction of the relative velocity and the weight force acting down.
Fig. 2. Particle velocity diagrams (a,c) and force diagrams (b,d) for a piece of compact debris upon release (a,b) and at the large time limit steady state (c,d). The initial trajectory slope of Ω is shown on the force diagram (b) and the final pffiffiffiffi trajectory slope of Ω is shown on the velocity diagram (c).
N.B. Kaye / J. Wind Eng. Ind. Aerodyn. 138 (2015) 69–76
Initial Slope Final slope
−2
Slope
−10
and ignoring the wind speed leads to d ζ dω dω ¼ ¼ω ¼ ð ωÞjωj Ω dτ2 dτ dζ 2
ð12Þ
which, given that ω o 0, (11) can be re-written as
0
−10
71
Steeper
1 dω2 ¼ ω2 Ω: 2 dζ
2
−10
−2
10
10
0
2
10
Ω Fig. 3. Log–log scale plot of numerical calculations of the initial (thin line) and final (thick line) trajectory slope as a function of Ω showing exact agreement with the theoretical predictions of (9) and (11). The slopes are negative and as such the slope magnitude increases as you move down the vertical axis.
the vertical velocity is the particle terminal velocity (Fig. 2c) and the slope of the steady-state trajectory is given by pffiffiffiffiffi pffiffiffiffiffi ω Ω S¼ ¼ ð11Þ ¼ Ω: μ 1 This is the same result established by Baker (2007). Numerical simulations over a broad range of Ω were run and the slopes of the initial and final trajectories were calculated. The numerical solutions were generated by integrating (4) and (5) subject to the initial conditions μ ¼ ω ¼ 0 at τ ¼ 0. The integration was done using the built in ODE solvers in MATLABs. The results are shown in Fig. 3 and agree exactly with (9) and (11). Agreement with the analytical result in (11) demonstrates the accuracy of the numerical integration technique and agreement with (9) illustrates the accuracy of the approximate near field trajectory solution. The figure illustrates that, for Ω o 1 the magnitude of the initial slope is less than the magnitude of the final slope and that, on average, the trajectory steepens during the flight. The opposite is true for Ω 41.
Eq. (10) can be integrated, subject to the initial condition ω ¼ 0 at ζ ¼ 0, to give qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð14Þ ω ¼ Ω 1 e2ζ : Therefore, particle will exponentially approach its terminal ffi pffiffiffiffithe velocity Ω. The flight distance required to achieve 95% of the particle terminal velocity is ζ 95% ¼ 1:2 which is independent of Ω. For very small Ω, the wind speed is considerably larger than the particle terminal velocity and the flight is largely horizontal. In this case, the vertical velocity terms and Ω can be ignored and (4) becomes 2 2 d χ dμ ¼ ¼ 1μ dτ2 dτ
The initial trajectory, given by (8) and (9), and the final trajectory, given by (10) and (11), are easy to calculate and provide a useful check on numerical simulations. However, they are only useful once the limits on their applicability have been established. Therefore, it is important to establish how long the initial trajectory slope can be approximated by –Ω after what ffi pffiffiffiffiand distance the trajectory slope approaches Ω. Analytic and numerical results that place limits on these solutions are presented in this section along with details on the evolution of the flight path between these limits.
3.1. Adjustment distance to reach steady-state trajectory The flight distance required for a piece of debris to achieve the steady-state velocity and trajectory given in (9) and (10) cannot be established analytically. However, some analytical progress can be made by considering the extreme cases of very small and very large Ω. For very large Ω the flight is dominated by vertical acceleration due to gravity and the wind speed and horizontal velocity are small compared to the vertical velocity. As such, in this limit the horizontal velocity Eq. (4) can be ignored and only the vertical Eq. (5) need be considered. Re-writing (5) in terms of the rate of change of velocity along the (approximately) vertical flight path
ð15Þ
This is analogous to the case of zero vertical drag considered by Holmes (2004). Following the same solution procedure presented by Holmes (2004) leads to Z Z dμ 2 τ¼ ð16Þ 2 ¼ ψ dψ 1μ where the variable ψ used in the substitution is ψ ¼ 1 μ. Integration from an initial condition of μ ¼ 0 at τ ¼ 0 leads to
μ¼
dχ τ ¼ : dτ 1 þ τ
ð17Þ
Therefore, the horizontal velocity will attain 95% of its final velocity (μ ¼ 1) when τ ¼ 19. Eq. (17) can be integrated a second time to give
χ ¼ τ lnð1 þ τÞ: 3. Flight trajectory development
ð13Þ
ð18Þ
Substituting τss 19 into (18) leads to a horizontal travel distance of χ 16 for the particle to achieve 95% of its final velocity. Assuming that the vertical velocity adjusts rapidly such that the vertical velocity can be approximated by its terminal velocity at all times, then the vertical distance over which the horizontal velocity adjusts is given by pffiffiffiffiffi pffiffiffiffiffi ζ 95% Ωτss 19 Ω: ð19Þ Therefore, for small Ω, the adjustment height is a function of Ω whereas, for large Ω the adjustment height approaches a constant independent of Ω. The small Ω (19) and large Ω (14) results can be seen in Fig. 4(a) which shows the numerically calculated slope of the trajectory at different vertical distances below the release height as a function of Ω. The steady-state trajectory is reached when the slope contour becomes a vertical line indicating that the slope no longer changes with vertical location. For small Ω the adjustment height increases with increasing Ω as predicted by (19). However, for large Ω the adjustment height approaches a constant, that is, it is independent of Ω, as predicted by (14). This can be seen more clearly in Fig. 4(b) which shows the slope scaled on the final pffiffiffiffi ffi steady-state slope (S^ ¼ S= Ω) where the lower contour decreases with increasing Ω approaching a constant in the limit of large Ω. Fig. 4(b) also shows that the initial slope is larger than the steady state slope for Ω 4 1 and smaller than the steady-state slope for Ω o 1 as shown in Fig. 3. Note that in this analysis the choice of 95% is arbitrary. However, the result that the adjustment height is
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N.B. Kaye / J. Wind Eng. Ind. Aerodyn. 138 (2015) 69–76
−1
−1
−10
−10
−1 −1
−2 −3
ζ
−4
0
−10
−1.5
ζ
0
−10
−5 −6
1
−10
−7
−2 1
−10
−2.5
−8 10
−1
0
10
10
1
10
−1
10
Ω
0
1
10
Ω
Fig. 4. (a) Log–log scale contour plot of the flight trajectory slope (s ¼ ω=μ) in Ω; ζ space. (b) Log–log scale contour plot of the flight trajectory slope scaled on the steadypffiffiffiffi state slope (S^ ¼ S= Ω).
only a function of Ω for small Ω would be the same regardless of what fraction of the terminal velocity was used in the analysis. It is worth considering what the non-dimensional adjustment distances plotted in Fig. 4 represent in dimensional space. To illustrate this we consider three examples from the literature. The first two are the example cases presented by Holmes (2004), namely an 8 mm stone sphere with U ¼ 20 m/s, and an 80 mm wooden sphere with U ¼ 30 m/s. The third example is that of a 1 mm grain of sand in a wind tunnel blowing at a velocity of U ¼ 6 m/s, typical of the critical blow off velocity tests of Karimpour and Kaye (2012b). The dimensional and non-dimensional parameters for these three examples are given in Table 1 calculated taking the drag coefficient to be C D ¼ 0:5. For the examples in Table 1, the adjustment height is taken to be ζ ¼ 2:5 based on the numerical result for Ω ¼ 1 shown in Fig. 4. It is interesting to note that all three examples have an Ω value of approximately one. This is consistent with the flight initiation model of Wills et al. (2002) who posited that a piece of loose debris will start to move when the initiating force (say drag or lift) balances the fixing force that scales on the weight of the particle. Assuming turbulent flow, the drag force scales on the wind speed squared and the particle cross sectional area (F I ρAU 2 ). Motion initiation will occur when the ratio of the initiating force and the particle weight is approximately one. That is, flight will begin when F I =W ρAU 2 =mg 1=Ω Oð1Þ:
ð20Þ
This is a similar result to the scaling presented for roof gravel blow off by Karimpour and Kaye (2012b). Therefore, for compact debris, the flight initiation conditions will likely result in compact debris flight occurring within a fairly narrow range of Ω around Ω 1. However, this is a local condition for motion initiation. Actual flight conditions could vary significantly in regions where there are large velocity gradients such as in the flow separation region over a roof. In the examples presented the reference length scale (Lref ) is very large (much larger than the particle diameter) and, therefore, the vertical distance taken for the particle to reach its steady-state trajectory slope (zss ) takes place over extremely long distances. This is due to the large density difference between the particle and the air. For example, an 8 mm piece of roof gravel would have to fall the 87 m, or roughly the height of a 25 floor building, before achieving a steady state. Further, a sand grain released from rest would fall 11.4 m before achieving its steady trajectory. Even the largest boundary layer wind tunnels are not this tall and, as such, are unable to achieve a steady state trajectory for even a small grain of sand.
Table 1 List of parameters for the example debris particles. Material Mass
Stone Wood Sand
Diameter (mm)
0.54 g 8 134 g 80 1.4 mg 1
U Lref ¼ C2m Ω ¼ C 2mg 2 D ρA D ρAU (m/s) (m)
(s)
zss (m)
20 30 6
1.75 2.9 076
87 217 11.4
0.86 0.95 1.2
35 87 4.6
T ref ¼ C D2m ρAU
3.2. Virtual release height As shown above, following an initial adjustment pffiffiffiffiffi the debris particle will travel in a straight line of slope S ¼ Ω. Therefore, in the far field, the flight trajectory can be described by a straight line that appears to come from a height given by the release height plus a virtual release height offset ζ v where if ζ v is positive then the virtual release height is above the actual release height and is below the actual release height if ζ v o 0. This is shown schematically in Fig. 5. The horizontal flight distance for large ζ is, therefore, given by
ζ ζ χ ¼ pffiffiffiffiffiv ð21Þ Ω where ζ is measured from the release height and is positive upwards. pffiffiffiffiffi For Ω o 1 the initial slope is less than the final slope (Ω o Ω), whereas for Ω 4 1 the initial slope is greater than the final slope. Intuitively, one might therefore expect that, for Ω o 1, the virtual release height would be above the actual release height (ζ v 4 0), as shown in Fig. 5(a). Conversely, for Ω 41 the initial trajectory slope is greater than the final slope and one would expect the virtual release height to be below the actual release height (ζ v o 0) as shown in Fig. 5(b). To test this hypothesis a series of 600 numerical flight simulations over 6 orders of magnitude in Ω were conducted and the virtual release height offset was calculated by fitting a line through the flight trajectory late in the flight. A plot of the calculated virtual release height offset as a function of Ω is shown in Fig. 6(a). For Ω 4 1 the virtual release height is negative, that is, the trajectory projects back to below the actual release height as shown in Fig. 5(b) and as hypothesized in the previous paragraph. Conversely, for Ω{1 the virtual release height is above the actual release height (see Fig. 5a). However, for 0:3 o Ω o1, the virtual release height is below the actual release height, that is, ζ v o 0. Therefore, for this range of Ω, the trajectory slope must initially steepen beyond the steady state slope and then approach the
N.B. Kaye / J. Wind Eng. Ind. Aerodyn. 138 (2015) 69–76
73
Fig. 5. (a) Schematic trajectory for a particle with Ω{1 showing the initial and final horizontal and vertical velocities, trajectory (solid line), and projected trajectory (dashed line) originating at the virtual release height ζ v above the actual release height. (b) Schematic trajectory for a particle with Ω4 1 showing same details but with the virtual release height below the actual release height.
1
ζ
v
0.5 0 −0.5 −1
−2
10
10
0
2
10
Ω Fig. 6. (a) Log-linear scale plot of the calculated virtual release height offset correction (ζ v ) as a function of Ω. (b) Schematic diagram for 0:3 o Ωo 1 showing the initial and final horizontal and vertical velocities, trajectory (solid line), and projected trajectory (dashed line) originating at the virtual release height ζ v below the actual release height.
steady-state trajectory from a steeper rather than shallower slope. This implies that the trajectory has an inflection. This is shown schematically in Fig. 6(b). Also, Fig. 6(a) shows that for large Ω, ζ v approaches a constant whereas, for small Ω, ζ v remains as a function of Ω. This qualitatively identical to the relationship between Ω and the vertical distance required to achieve a steady-state trajectory slope as shown in Fig. 4(b).
3.3. Near release flight trajectory Given the large fall heights required for typical debris to achieve their steady-state trajectory slope, and the unexpected behavior observed in the virtual release height model for 0:3 o Ω o1 it is clear that the near field trajectory behavior has a significant influence on the entire flight path. We therefore seek to quantify the range over which the initial trajectory, (8) and (9), is a valid approximation for the flight path, and to examine how the flight path adjusts away from this initial trajectory toward the steady state trajectory given by (11). The approximate initial solution (17) results from assuming that the debris horizontal and vertical velocities are approximately zero. This assumption will be valid provided qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 1μ 1 μ þ ω2 Ω Ω: 1 μ þ ω2 1 and ω ð22Þ Substituting (8) into (22), and taking (8) to be valid until the left hand sides of (22) are greater than 1.1 times the right hand
sides, leads to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1 τÞ ð1 τÞ2 þ Ω2 τ2 1 ¼ 0:1
ð23Þ
and qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Ωτ ð1 τÞ2 þ Ω2 τ2 ¼ 0:1:
ð24Þ
The solutions for the times, τ, that satisfies (23) and (24) were calculated numerically over the range 0:1 o Ω o 10. The results are shown in Fig. 7. The results indicate that there is a very limited time range over which the approximate initial trajectory (8) is valid (typically around τ ¼ 0:05). This is very small indeed given the time scales listed in Table 1 which were all of the order of a second. That is, the initial trajectory slope is only a valid approximation for a few hundredths of a second. Interestingly, with the exception of a small range of Ω (approximately 4 o Ω o 8) the time is limited by the vertical acceleration term (24). This seems to indicate that the vertical velocity may grow relatively more rapidly than the horizontal velocity for a given Ω. The results of Section 2 show that the trajectory of a particle will change from an initial slope Ω to a final steady-state slope pffiffiffiffiffi of Ω (see Eqs. (9) and (11) and Fig. 3). For the range of Ω considered in the contour plots, the slope transition occurs over a relatively constant distance of ζ 2:5 (see Fig. 4(b)). One can write an approximate function for the slope of the trajectory that has the appropriate slope limits as given by (9) and (11). One possible
74
N.B. Kaye / J. Wind Eng. Ind. Aerodyn. 138 (2015) 69–76
equation is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω Ω2 αζΩ : S¼ μ 1 αζ
an expression for the rate of change of slope with time as dS d ω 1 dω ω dμ 1 dω dμ ¼ 2 S ¼ ¼ : dτ dτ μ μ dτ μ dτ μ dτ dτ
ð25Þ
However, the same analysis approach used to calculate the small time velocity (8) and slope (9) breaks down when trying to estimate dS=dτ at the release point. This is because the slope is undefined for the initial conditions μ ¼ ω ¼ 0. That is, the initial trajectory slope is only defined in the limit of τ-0 þ . Numerical calculations of the slope for very small times indicate that
where α is a fitting parameter which controls the rate of adjustment from the initial to the final slope. Eq. (25) has the correct pffiffiffiffiffi limits of S ζ ¼ 0 ¼ Ω and limlarge ζ S ζ ¼ Ω. Trial and error testing showed that α ¼ 20 provided the appropriate transition distance. A contour plot of the approximate slope as a function of Ω and ζ is shown in Fig. 8(a). The main problem with (25) is that there are no local maxima in the contours as were seen in Fig. 3(a). This is illustrated more clearly in Fig. 8(b) in which there are clear local magnitude maxima for the numerically calculated Ω ¼ 0:5 and 1 slope plots. While the slopes estimate provided by (25) do not match particularly well in the near field (though they do match exactly at χ ¼ ζ ¼ 0 and then diverge), they only vary near the release point over short distances. A more realistic test of the quality of the slope approximation is provided by comparing the particle trajectories calculated numerically, by solution of (4) and (5), and approximately, by integration of the slope approximation (25). Trajectories calculated by these two methods are shown in Fig. 9 with the approximate solution in Fig. 9(a) and the numerical solution in Fig. 9(b). The approximate solution provides a reasonable description of the flight path in the far field for Ω ¼ 10; 1, and 0:1 and a poor representation for Ω ¼ 5 and 0:5. This is because, although the slope limits are correct, the resulting approximate trajectories have the wrong virtual (or apparent) release height (see Fig. 6a). Clearly the near field slope adjustment is important and requires further attention. Recalling that S ¼ ω=μ one can write 0.4
τ
dS Ω ¼ : dτ 2
ð27Þ
That is, the slope of the trajectory initially steepens for all Ω even if the steady-state slope is less steep than the initial slope, as for Ω 4 1. This results can be seen qualitatively by examining small velocity changes in the vertical (δω ¼ Ωδτ) and horizontal (δμ ¼ δτ) directions that occur over a small time (δτ). Substituting these small changes into (4) and (5) leads to qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 dμ ð28Þ 1 δτ þ ðΩδτÞ2 ¼ 1 δτ dτ and d ζ dω ¼ ¼ ð ωÞ dτ2 dτ 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 δτ þ ðΩδτÞ2 Ω
ð29Þ
Writing the term under the square root as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 KðδτÞ ¼ 1 2δτ þ ð1 þ Ω Þδτ2
ð29Þ
leads to dμ dω ¼ K 1 δτ and ¼ Ω 1 K δτ : dτ dτ
ð30Þ
Regardless of the values of Ω and δτ K must be greater than zero. Further, for small enough values of δτ 2δτ 4 ð1 þ Ω2 Þδτ2 : ð31Þ
(23) (24)
0.3
ð26Þ
τ
min
0.2
Therefore, the value of K for small δτ is in the range 0 o K o 1. Taking the leading order terms in (30) leads to
0.1
dμ dω ¼ K o1 and ¼ Ω: dτ dτ
0 −1 10
10
0
ð32Þ
Therefore, for small times when released from rest, the rate at which the horizontal velocity increases will initially decrease whereas the rate of change of the vertical velocity will stay the same. As such, the trajectory will initially steepen for all Ω. Given this result, there must be an inflection in the trajectory for all Ω 41. Further, the negative ζ v calculated for 0:3 o Ω o 1,
1
10
Ω Fig. 7. Time for which (17) is a reasonable approximation for the particle velocity.
−1
−10
−1
−10
−1
ζ
−3 −4 1
−10
ζ
−10
1
Steeper
−10
−5 −6
10
0
−10
−2
0
−1
10
0
Ω
10
1
2
−10 1 −10
0
−10
−1
−10
slope
Fig. 8. (a) Log–log scale plot of the approximate slope during flight calculated using (25) with α ¼ 20. (b) Log–log scale plot of slope from full numerical solutions of Eqs. (4) and (5) (thick lines) and Eq. (25) (thin lines), as a function of ζ for (from left to right) Ω ¼ 10; 5; 1; 0:5; 0:1.
N.B. Kaye / J. Wind Eng. Ind. Aerodyn. 138 (2015) 69–76
−1
−1
−10
−10 Ω=10 Ω=5 Ω=1 Ω=0.5 Ω=0.1
ζ
Ω=0.1 Ω=0.5 Ω=1 Ω=5 Ω=10
0
−10
ζ
0
−10
1
1
−10
−10
2
−10 −1 10
75
2
10
0
χ
10
1
10
2
−10 −1 10
0
10
χ
10
1
10
2
Fig. 9. Log–log scale plot of the trajectories of compact debris for Ω ¼ 10; 5; 1; 0:5; 0:1. (a) Approximate solutions (25) and (b) numerical solutions.
where intuitively one might expect a positive value, is due to the trajectory initially steepening beyond its steady-state slope, inflecting, and then approaching the steady value. In fact, it is possible for debris with Ω o 1 to have a trajectory that steepens beyond its steady slope and inflect while still having a positive ζ v provided the trajectory never drops pffiffiffiffiffi below a line drawn from the release point and of slope S ¼ Ω. Numerical calculations of the flight of debris for 201 different value of Ω, logarithmically spaced in the range 0:1 r Ω r10, found inflections in all the trajectories. This result is consistent with the earlier result that the deviation of the trajectory from its initial path, see Eqs. (8) and (9), is almost always dominated by the vertical acceleration equation as shown in Fig. 7.
steady-state values such that ϖ and κ both approach one in the limit of large flight distance. The plots in Fig. 10 indicate that, as expected, the adjustment to steady horizontal velocity is faster for smaller Ω while the adjustment to steady vertical velocity is faster for higher Ω. The kinetic energy adjustment to its maximum steady value is largest for intermediate values of Ω 1. That is, for values of Ω typical of windborne loose laid debris, the time taken to reach the steady-state (maximum) kinetic energy is close to a maximum and will only be reached for release heights greater than z 9m=C D ρA or approximately 160 m for the example of an 8 mm stone presented in Table 1.
5. Conclusions 4. Numerical solutions While it would be possible to extend the approximate algebraic model of (25) to achieve the correct virtual release height (Fig. 6a), this would rapidly become algebraically complex and selfdefeating. Rather, a graphical solution for the horizontal flight distance (X H ) as a function of the release height (H) and Ω is presented. For a given release height H the non-dimensional release height can be calculated,
η¼
HC D ρA ; 2m
ð33Þ
along with the flight distance scaled on the release height
λ¼
X H χ ðζ ¼ ηÞ ¼ H η
ð34Þ
where λ ¼ λðη; ΩÞ. A contour plot of this function is presented in Fig. 10(a) which allows the calculation of the flight distance of any particle released from rest at any height under any steady uniform wind conditions. Fig. 10(a) also shows that, as the flight path increases the ratio of the flight distance to the release height approaches a constant (vertical contours). This occurs once the vertical drop is large enough that the virtual release height can be neglected (that is, when ζ c ζ v ) and the flight distance can be approximated by (1). In fact, all the major flight parameters can be represented in scaled non-dimensional contour plots. Contour plots of horizontal velocity, μ, scaled vertical velocity pffiffiffiffiffi ð35Þ ϖ ¼ ω= Ω; and scaled particle kinetic energy
κ ¼ ðμ2 þ ω2 Þ=ð1 þ ΩÞ
ð36Þ
(see Baker 2007) are presented in Fig. 10c,d respectively. The vertical velocity and particle kinetic energy are rescaled on their
Analytic, approximate analytic, and numerical solutions to the compact debris flight equations have been presented. The work of Baker (2007) on the near field trajectory and far field steady flight slope has been extended to examine the trajectory slope adjustment during flight. The flight slope will, on average, increase during flight for Ω o 1 and decrease for Ω 4 1, though the trajectory initially steepens for all Ω. The distance over which this transition occurs was investigated analytically and it was shown that, for small Ω the adjustment height is a function of Ω, whereas for larger Ω the adjustment height approaches a constant. This was confirmed through numerical solutions of the debris flight equations over a broad range of Ω (Fig. 4b). An approximate algebraic function of the slope as a function of distance below the release point was presented (25). However, the approximation is not a particularly good representation of the slope, particularly for Ω 1 as the slope initially steepens and there is a clear inflection in the flight path for such intermediate Ω (Fig. 8b). In fact, all trajectories initially steepen and the trajectories of all flight paths have inflections. The change in slope during flight means that, in the steady state linear trajectory phase of the flight, the particle travels along a line that projects back to a virtual release height, ζ v , that is a function of Ω (Fig. 6a). Given the complexity of the approximate algebraic solutions presented, it is simpler to represent the full set of solutions to the compact debris flight equations over a broad range of Ω in a series of non-dimensional contour plots in [Ω; η] space. Plots were presented for the flight distance scaled on the release height, horizontal velocity, vertical velocity scaled on the terminal velocity, and kinetic energy scaled on the terminal kinetic energy (Fig. 10). While many of the results presented are somewhat abstract, they do have some practical implications for wind engineering. First, as illustrated in Table 1, the steady flight solution presented by Baker (2007) is typically only applicable for extreme events
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N.B. Kaye / J. Wind Eng. Ind. Aerodyn. 138 (2015) 69–76
−1
−1
−10
6
3 10
0.6
4
η
10
0.8
5
0
η
10
0
−10
0.4
1
2 0.2
1 1
2
10 −1 10
10
0
10
−10 −1 10
1
Ω
10
0
10
1
Ω −1
−1
−10
−10
0.8
0.8 0.6
η
η
0.6 0
−10
1
−10 −1 10
0
10
0
1
10
0
−10
0.4
0.4
0.2
0.2
0
1
−10 −1 10
10
0
1
10
0
Ω
Ω
pffiffiffiffi Fig. 10. Log–log scale contour plots in [Ω; η] space of (a) flight distance λ, (b) horizontal velocity μ, (c) scaled vertical velocity ϖ ¼ ω= Ω, and (d) scaled particle kinetic energy κ ¼ ðμ2 þ ω2 Þ=ð1 þ ΩÞ. Note that the color bar for ϖ (c) is skewed compared to the color bars in (b) and (d). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
involving very high release heights. The second interesting point raised by the examples in Table 1 is that for realistic flight conditions, Ω is quite close to one. This is in keeping with the work of Wills et al. (2002) who proposed a simple scaling argument for motion initiation in which the motion inducing force (either lift or drag) scales on the square of the wind speed and the particle's area. The ‘fixing force’ scales on the weight of the particle. Motion will commence when the inducing force is slightly larger than the fixing force. Given a drag or lift coefficient of order 100, their model implies that motion will be initiated when Ω Oð100 Þ. Therefore, their model predicts that most wind driven release and flight events will occur at moderate Ω 1. Acknowledgments This material is based upon work supported by the National Science Foundation, United States of America under Grant no. 1200560. Any opinions, findings, and conclusions or recommendations expressed in the material are those of the author and do not necessarily reflect the views of the NSF. The author would also like
to thank Dr. Greg Kopp for many interesting discussions on this topic during the preparation of this manuscript.
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