1
Determination of Maximum Mechanical Energy Ef
fi
ciency in
2
Energy Galloping Systems
1
3
J. Meseguer
1; A. Sanz-Andrés
2; and G. Alonso
34 Abstract:Transverse galloping is a type of aeroelastic instability characterized by large amplitude, low frequency, normal to wind oscillations 5 that appear in some elastic two-dimensional bluff bodies when subjected to afluidflow, provided that theflow velocity exceeds a threshold 6 critical value. Such an oscillatory motion is explained because of the energy transfer from theflow to the two-dimensional bluff body. The 7 amount of energy that can be extracted depends on the cross section of the galloping prism. Assuming that the Glauert-Den Hartog quasi-8 static criterion for galloping instability is satisfied in afirst approximation, the suitability of a given cross section for energy harvesting is eval-9 uated by analyzing the lateral aerodynamic force coefficient,fitting a function with a power series in tana(abeing the angle of attack) to 10 available experimental data. In this paper, a fairly large number of simple prisms (triangle, ellipse, biconvex, and rhombus cross sections, as well 11 as D-shaped bodies) is analyzed for suitability as energy harvesters. The influence of thefitting process in the energy harvesting efficiency 12 evaluation is also demonstrated. The analysis shows that the more promising bodies are those with isosceles or approximate isosceles cross 13Q : Asections.DOI:10.1061/(ASCE)EM.1943-7889.0000817.©2014 American Society of Civil Engineers.
14 Author keywords:Galloping; Energy harvesting; Wind tunnel.
15 Introduction
16 The potential use of different aeroelastic phenomena to extract en-17 ergy from an incidentflow, and through the oscillations of the ex-18 cited structure, convert those oscillations into piezoelectric or 19 electromagnetic energy, is receiving the attention of more and more 20 researchers (St. Clair et al. 2010;Peng and Chen 2012). As a con-21 sequence, numerous studies have been published recently on the use 22 offluid-bodies instabilities interactions (the number of papers has 23 remarkably grown in the last 2 years). The focus of these papers has 24 been either on the particularities of oscillating systems (transverse 25 galloping, wake galloping,flutter, and vortex shedding) or on the use 26 of these oscillating systems for energy harvesting.
27 Besides the pioneer work of Glauert (1919), Den Hartog (1965), 28 Novak (1969), Novak and Tanaka (1974), and Luo et al. (1988), 29 among others, some relevant publications have appeared more re-30 cently concerning transverse galloping instability (Naudascher and 3119 Rockwell 1994;Luo et al. 1998;Barrero-Gil et al. 2010; Païdoussis 32 et al. 2010,Sewatkar et al. 2012,Alonso and Meseguer 2014;Ibarra 33 et al. 2014), whereas the use of transverse galloping for energy 34 harvesting purposes has been analyzed by Sirohi and Mahadik (2011, 35 2012), Abdelkefiet al. (2012a,2013a), Abdelkefiet al. (2013c,2014), 36 and Zhao et al. (2012).
37 However, transverse galloping is not the onlyflow-induced os-38 cillating phenomenon in energy harvesting devices, and some
39 efforts have been devoted to using other instability sources, such as
40 wake galloping (Jung et al. 2009;Akaydin et al. 2010;Jung and
41 Lee 2011;Abdelkefiet al. 2013b,c), vortex-induced vibrations
42 (Abdelkefiet al. 2012b;Grouthier et al. 2012;Hobbs and Hu 2012;
43 Peng and Chen 2012;Mackowski and Williamson 2013;Mehmood
44 et al. 2013), andflutter of streamlined bodies (Peng and Zhu 2009;
45 Tang et al. 2009;Zhu et al. 2009;Erturk et al. 2010;Bryant et al.
46 2011;Doaré and Michelin 2011;Zhu 2011;Abdelkefiet al. 2012c,d;
47 Abdelkefiand Nuhait 2013;Bae and Inman 2014).
48 Most of these studies focused on the analysis of the suitability of
49 some specific mechanisms to efficiently harvest the energy through
50 different energy converters (piezoelectric, electromagnetic, etc.). In
51 contrast, some attempts have been made to develop methodologies
52 to evaluate the energy harvesting efficiency of two-dimensional
53 bluff bodies from a purely aeroelastic point of view.
54 In this sense, Barrero-Gil et al. (2010) developed a simple
ap-55 proach to evaluate the efficiency of the transverse galloping of bluff
56 bodies as an energy harvesting mechanism. To do this, following an
57 already published methodology (Ng et al. 2005), afirst
approxi-58 mation of the transverse galloping phenomenon can be explained
59 by a one-degree of freedom (the transverse displacement,z) linear
60 differential equation. The forcing aerodynamic coefficientczis
61 expanded in a power series of the transverse velocity, dz=dt,
62 retaining for simplicity only, the twofirst terms in theczseries
63 expansion, that is,cz5a1ðdz=dtÞ=U1a3½ðdz=dtÞ=U3. Barrero-Gil
64 et al. (2010) demonstrates that, within this approximation, the
65 maximum harvesting efficiency ishmax5 2a2
1=ð6a3Þ. The criterion
66 developed is applied to assess the energy harvesting efficiency of
67 several two-dimensional bodies (square, isosceles triangle, and
68 D-section) whose aerodynamic coefficients are taken from the
69 existing literature. The conclusion drawn is that D-shaped,
two-70 dimensional bodies are the most appropriate ones for energy
har-71 vesting devices based on transverse galloping, because they provide
72 the highesthmaxvalue among the considered bodies.
73 Further experiments conducted at IDR/UPM on families of
74 bodies with different cross sections provided a better insight on the
75 transverse galloping phenomenon, and allowed for improvement of
76 the approach initiated by Barrero-Gil et al. (2010) to predict the
1Professor, IDR/UPM, E.T.S.I. Aeronáuticos, Univ. Politécnica de
Madrid, E-28040 Madrid, Spain. E-mail: j.meseguer@upm.es
2
2Professor, IDR/UPM, E.T.S.I. Aeronáuticos, Univ. Politécnica de
Madrid, E-28040 Madrid, Spain. E-mail: angel.sanz.andres@upm.es 3Professor, IDR/UPM, E.T.S.I. Aeronáuticos, Univ. Politécnica de
Madrid, E-28040 Madrid, Spain (corresponding author). E-mail: gustavo .alonso@upm.es
Note. This manuscript was submitted on May 20, 2013; approved on May 2, 2014; published online onnnnnnn. Discussion period open until
nnnnnnnnnnnnnnnnnn; separate discussions must be submitted for in-dividual papers. This paper is part of theJournal of Engineering
77 energy harvesting efficiency of these structures by also providing 78 a larger set of experimental aerodynamic data to validate the meth-79 odology. In particular, the transverse galloping stability of triangles 80 (Alonso et al. 2005,2007;Alonso and Meseguer 2006), ellipses 81 (Alonso et al. 2010), biconvex, and rhombi (Alonso et al. 2009) were 82 thoroughly determined by wind tunnel testing of cylinders with 83 respective cross-sectional geometry and variable relative thickness. 84 By properly using the new and extended set of experimental 85 aerodynamic data, it is possible to further develop and improve the 86 methodology presented in Barrero-Gil et al. (2010) to evaluate the 87 transverse galloping energy efficiency of bluff bodies.
88 A brief summary of the theoretical foundation already published 89 in Barrero-Gil et al. (2010) is provided in the“Mathematical Model” 90 section, and a new and improved way to estimateh
maxis presented.
91 The section “Analysis of the Energy Harvesting Properties of 92 Several Bluff Bodies”is devoted to the application of theh
max
93 criterion to a large amount of new cross-sectional shapes. Con-94 clusions are outlined in the“Conclusions”section.
95 Mathematical Model
96 Theoretical foundations of galloping are well established and un-97 derstood (Den Hartog 1965; Novak 1969, 1972). The simplest 98 model to analyze the translational galloping behavior of a bluff body 99 consists of a damped spring-mounted cylindrical body under the 100 action of an incoming smooth flow of velocityU, parallel to the 101 x-axis (Fig.1). Other relevant magnitudes are the cylinder mass per 102 unit length,m, the mechanical damping ratio,z, the natural circular 103 frequency of oscillations,vn, thefluid density,r, and a characteristic 104 transversal length of the galloping two-dimensional body (as the 105 normal to wind length of its cross section,bm). Then, assuming the 106 body is slender enough to assure two-dimensional flow, the dy-107 namics of the system is driven by
m
d2z dt2þ2zvn
dz dtþv
2
nz
¼ fzðaÞ ¼1 2rU
2b
mczðaÞ (1)
108 wherez5vertical displacement;t5time;f
z5instantaneous two-109 dimensional lateral force (parallel to thez-axis);cz5instantaneous 110 two-dimensional lateral force coefficient; vn 5 undamped reso-111 nance frequency; andz5damping coefficient.
112 In a body reference frame, across-wind oscillation of the structure
113 periodically changes the angle of attack of the incident wind. Such
114 a variation in the angle of attack produces variation in the
aero-115 dynamic forces acting on the structure (Fig.1), which, in turn, affects
116 the dynamic response of the structure. Therefore, transverse
117 galloping can be explained as although the incident velocityUis
118 uniform and constant, because of the lateral oscillation of the body,
119 in a body reference system, the total velocity changes both its
mag-120 nitude and direction with time. Consequently, the body angle of attack
121 also changes with time, hence, the aerodynamic forces acting on it.
122 Consider that a structure at rest is oriented at a given angle of
123 referencea0with respect to the incidentflow (Alonso et al. 2005;
124 Alonso and Meseguer 2006). When the body oscillates along
125 the z-axis direction within an uniform flow with velocity U,
126 the relative velocity between the fluid and the body is Ur
127
5U21ðdz=dtÞ21=2
5U11tan2a1=2, where the angle of
at-128 tack caused by the oscillation is tana5ðdz=dtÞ=U; therefore,
129
Ur5U=cosa. Drag,dðaÞ, and lift,lðaÞ, forces are, respectively,
130
dðaÞ5ð1=2ÞrUr2bmcdðaÞ and lðaÞ5ð1=2ÞrU2rbmclðaÞ, where 3 131
cdðaÞ stands for the drag coefficient,clðaÞ is the lift coefficient,
132 ris thefluid density, andbmis the characteristic transversal length
133 of the cylinder, as already stated.
134 The projection of those forces in the z-axis direction is fzðaÞ
135
5edðaÞsinaelðaÞcosa, and then Eq.(1)reads as
m
d2z dt2þ2zvn
dz dtþv
2
nz
¼21
2rU 2
rbm½cdðaÞsinaþclðaÞcosa
¼21
2rU 2b
mcdðaÞtanaþclðaÞ cosa ¼1
2rU 2b
mczðaÞ
(2)
136 To solve Eq.(2), the functionczðaÞcan be expanded in a power
137 series of tana(Barrero-Gil et al. 2010;Ng et al. 2005) byfitting the
138 function czðaÞ, measured, for instance, in wind tunnel tests, to
139 a polynomial in tanaas
czðaÞ ¼P‘
j51
a2j21tan2j21a (3)
140 Only the antisymmetric part ofczðaÞaroundczða0Þcontributes to
141 galloping. Therefore, only the antisymmetric part ofczðaÞis used to
142 obtain the polynomial Eq.(3). As long asais small enough, the linear
143 term coefficienta1 reduces to the function dcl=da1cdappearing
144 in the Glauert-Den Hartog quasi-static criterion for galloping.
145 Considering the movement is quasi-steady, because tana1,
146 the lateral force coefficient becomesczðaÞ≅eðcda1clÞ1Oða2Þ
147
5 2ðcd1dcl=daÞa1Oða2Þ; hence,a15eðcd1dcl=daÞ.
148 The usual dimensionless variables in aeroelastic analysis are now
149 introduced:y5z=bm,t5vnt,mp5m=ðrb2mÞ, andUp5U=ðvnbmÞ
150 in Eq.(2), and substitutingczðaÞby its series expansion in Eq.(3),
151 Eq.(2)yields
d2c dt2þ2z
dc dtþc¼
Up2 2mp
P‘
j51
a2j21 1 Up
dc dt 2j21
(4)
152 4 wherey5dimensionless vertical displacement;t5dimensionless
153 time;mp5ratio of the mean density of the body to the density of the
154 surroundingfluid; andUp5reduced velocity.
155 Once the problem formulation has been established (the
pre-156 ceding differential equation with suitable boundary and initial
157 conditions), an approximation to the problem solution can be
Fig. 1.Two-dimensional body in galloping oscillation:clðaÞ,cdðaÞ, lift and drag coefficients, respectively;czðaÞ, vertical force coefficient; D, damper; dz=dt, vertical velocity caused by galloping motion; S, spring;U, upstream unperturbedflow velocity;a, angle of attack,a0, reference angle
158 obtained by applying, for instance, the procedure already presented 159 in Barrero-Gil et al. (2010). The method consists of the application of 160 the Krylov-Bogoliubov method to this differential equation to obtain 161 the time variation of the dimensionless galloping amplitude. To
162 solve the problem, it is assumed that the body motion can be
163 expressed as c5ApðtÞcos½t1fðtÞ, whereApðtÞ and fðtÞ are
164 functions slowly varying witht. Then, the dimensionless amplitude
165 becomes
dAp dt ¼2
1 2p ð 2p 0 "
22zdc dtþ
Up2 2mp
P
j51 a2j21 Up2j21
dc dt
2j21#
sinðtþfÞdt
¼zAp2Up2 4mp a1
Ap Upþ
3 4a3
Ap Up
3
þ5 8a5
Ap Up
5
þ35 64a7
Ap Up
7
þ 63 128a9
Ap Up 9 þ/ " # (5)
166 The steady amplitude of the oscillations is given by the real and 167 positive roots of dAp=dt50. Independent of the number of terms 168 retained in Eq.(5), one of the roots is alwaysAp50. The analytical 169 calculation of the other roots of Eq. (5)becomes more and more 170 complicated as the number of terms increases. If onlya1anda3are 171 retained, the efficiency, or conversion factor, defined by the ratio of 172 the power extracted by the structure from theflow per unit length 173 and the total power in the flow per unit length, becomes h
max
174 5 2a2
1=ð6a3Þ, as already stated.
175 The power that can be obtained in a cycle from transverse gal-176 loping oscillations,P
g, is obtained from the expression
Pg¼1
T ðT
0 fzdz
dtdt¼
rU2b 2T
ð T
0 czdz
dtdt (6)
177 whereT 5oscillation period.
178 In contrast, the power associated with the unperturbed flow is 179 Pf5rU3b
m=2, and therefore, the energy harvesting efficiency is 180 h5Pg=Pf .
181 The direct integration of Eq. (6), after the introduction of the 182 dimensionless variables, to obtainPgas performed in Barrero-Gil 183 et al. (2010), results in a value for the efficiency of
h¼1 2
a1 Ap Up 2 þ3 4a3 Ap Up 4 þ5 8a5
Ap Up 6 þ35 64a7 Ap Up 8 þ 63 128a9 Ap Up 10
þ/ (7)
184 A different approach can be introduced here for the determination 185 of the efficiency in a simpler way. The idea is to evaluate the power
186 P
g, not the aerodynamic force used in Eq.(6),fz, but the dissipative 187 term in Eq.(1), that is, the left-hand side of Eq.(1). Eq.(6)forPg 188 then becomes
Pg¼m
T ð T
0 d2z dt2
dz
dtdtþ2zvn ðT
0 dz dt
2 dtþv2n
ðT
0 zdz
dtdt 2 6 4 3 7 5
¼2mzvn
T ðT 0 dz dt 2
dt (8)
189 In writing the last term of Eq.(8), it has been considered that,zbeing 190 a periodic function, the integrals of both ðd2z=dt2Þ3ðdz=dtÞ and
191 z3ðdz=dtÞare zero in a cycle.
192 Then, because it has been assumed thatz5AðtÞcos½vnt1fðtÞ,
193 and the introduction of the dimensionless variables already
194 having been defined, gives c5ApðtÞcos½t1fðtÞ, and dz=dt
195
5vnbmdc=dt5 2vnbmApsinðt1fÞ, where it has been considered
196 that the time variation of bothApðtÞandfðtÞis so slow that they
197 can be considered constant in a cycle. Therefore
Pg¼z
mv3nb2m p ð 2p 0 dc dt 2 dt
¼zmv3nb2m p
ð 2p
0
f2ApðtÞsin½tþfðtÞg2dt¼zmv3nb2mAp2 (9)
198 Thus, the expression giving the efficiency now results in
h¼2zmp
Up Ap Up 2 (10) 199 wherezmpis proportional to the Scruton number. Obviously, this
200 last expression states that the only energy that is harvested out of
201 the flow by the galloping motion is the energy dissipated by
202 the structure.
203 The analytical calculation of the maximum efficiency becomes
204 too complex when more than thefirst two terms in the polynomial
205 approximation toczðaÞ, Eq.(3), are considered. In that case, the
206 problem solution is better achieved by a numerical method. The
207 procedure used in the present work when more than two terms are
208 used in Eq.(3)is the following:
209 1. First, the coefficientsa2je1of the series expansion in Eq.(3)are
210 calculated byfitting this expression to available experimental
211 data. Note that the values of thefitted coefficients depend on
212 the number of terms considered in Eq.(3).
213 2. Once the coefficients a2je1 are known, the corresponding
214 functions dAp=dt, Eq.(5), are evaluated. It must be pointed
215 out that the parameters involved in Eq.(5)areAp,Up, andzmp;
216 therefore, in this process, two of them arefixed (namely,Ap
217 andzmp), whereas the third one,Up, is considered to be an
218 independent variable. The result depends on the values chosen
219 forApandzmp(additional comments on this point are given
220 later in this work).
221 3. The roots of dAp=dt50 are then calculated; thus, the critical
223 zmpare obtained (formally, the smaller value of the criticalUp 224 is the one of interest).
225 4. Then, the efficiency is determined by using either Eq.(7)or 226 Eq.(10)as a function ofUp,Ap, andzmp.
227 Analysis of the Energy Harvesting Properties of 2285 Several Bluff Bodies
229 In the simplest approach, when only thefirst two terms (a1anda3) 2306 are considered in the series (3), the maximum efficiency occurs when
231 h
max5 2a21=ð6a3Þ. This expression gives a rough approximation to
232 the quality of a given two-dimensional body to extract energy 233 through transverse galloping oscillations. Despite the limited ap-234 plication of this approximation, it seems clear that the more 235 promising bodies for power extraction are those whose cross sec-236 tions provide large positive values of the coefficienta1 and small 237 negative values ofa3. In afirst approximation (small values of the 238 angle of attack), the coefficienta1can be identified with the Glauert-239 Den Hartog parameter for galloping, and it must be positive (the 240 body must be stalled), whereasa3has to be negative, because this 241 parameter represents the capacity of the system to keep the amplitude 242 of the galloping oscillations betweenfinite boundaries (limit cycle 243 oscillations).
244 Note also that when only two terms are considered, the maximum 245 efficiency does not depend on the parameters involved in Eq.(10). 246 This expression is not applicable, however, when more terms are 247 retained in Eq.(5).
248 Using sufficient terms in the series expansion of the force co-249 efficientczðaÞis important for properly predicting the energy har-250 vesting efficiency of a given structure. In addition, when this force 251 coefficient is obtained from experimental data, the process offitting 2527 these data determines the accuracy of the results. Thefitting process 253 is affected by the value of the reference anglea0selected [formally, 254 only the antisymmetric part ofczðaÞobtained from experimental 255 data are considered] and by the number of data points taken around 256 the central one. In thefitting process of a given equation to a set of 257 experimental data, thefirst condition that should be fulfilled is that 258 the equation must be appropriate to reproduce the main features of 259 the experimental data (in the case of a power series, this is generally 260 achieved by retaining enough terms in the truncated series), oth-261 erwise, very different conclusions can be reached from the con-262 clusions that the experimental data show.
263 The main purpose of this work is to investigate how the ap-264 proximation chosen to derive the force coefficient from the experi-265 mental data for calculating the energy harvesting efficiency 266 determines the accuracy of the results. The criteria derived for that 267 approximation, together with the simpler approach to evaluate the 268 harvesting efficiency presented in the“Mathematical Model” sec-269 tion, substantially improves the method presented in Barrero-Gil 270 et al. (2010). This improved method is applied to bodies with dif-271 ferent cross sections.
272 The analysis is performed first on D-shaped cross sections, 273 whose efficiency is derived in Barrero-Gil et al. (2010) by choosing 274 a cubic polynomial for the force coefficient approximation from 275 experimental data available in the literature. To contrast these results 276 and apply the new findings to the energy harvesting efficiency 277 evaluation, new wind tunnel tests were performed at IDR/UPM on 278 a D-shaped cross-section body.
279 A prism with the cross section (Fig. 2) was built up, and its 280 aerodynamic properties were measured in the A9 wind tunnel at 281 IDR/UPM. This wind tunnel is an open circuit, closed test chamber 282 whose cross section is 1.8-m high and 1.5-m wide. Turbulence 283 intensity is approximately 2.5% at 20 m=s wind velocity. The model 284 span was 0.8 m, and there were twoflat plates parallel to theflow at
285 the ends of the prism to assure two-dimensional behavior.
Aero-286 dynamic loads were measured through a six-component
strain-287 gauge balance (ATI DELTA, ATI Industrial Automation, Apex, 8
288 North Carolina), whereas theflow velocity was measured with a
289 Pitot tube (model 3.3.311, Airflow Systems, Dallas, Texas), connected
290 to a Schaewitz Lucas pressure transducer. The D-shaped model
291 was mounted on a rotating platform NEWPORT RV120-PP-HL
292 (Newport Corporation, Irvine, California), which, in turn, was
293 mounted on the strain-gauge balance. The rotating platform can set
294 the model angle of attack within 60:1°accuracy. The balance has
295 a maximum measurement uncertainty of 1.25%. No provisions for
296 blockage corrections of the measured loads have been undertaken, as
297 even in the worst case, the model front area (including end plates) is
298 less than 8% of the test sectionflow area.
299 Aerodynamic forces were measured from a05 290°toa0
300
5 190°at a 1°step. At each reference angle, the loads were sampled
301 at 200 Hz during 30 s. The results obtained are shown in Fig.2, in the
302 same plot of the experimental data reported by Barrero-Gil et al.
303 (2010). In Fig. 2, the symbols represent the experimental data
304 reported by Barrero-Gil et al. (2010) (circles) and the data measured
305 at IDR/UPM facilities (rhombi). The lines correspond to fitted
306 equations in the experimental data reported by Barrero-Gil et al.
307 (2010) (circles) by retaining only thefirst two terms in Eq.(3)(solid
308 lines). The influence of the number of experimental points in the
309 fitting process is as follows: line 1 represents thefitting result when
310 only nine experimental points (white circles) are used; line 2
rep-311 resents thefitting result when 11 experimental points (white and gray
312 circles) are considered, and line 3 is obtained when the whole set of
313 experimental points reported by Barrero-Gil et al. (2010) is
con-314 sidered in thefitting process [15 experimental points (white, gray,
315 and black circles)]. Line 4 is the one given in Barrero-Gil et al.
316 (2010),fitting with only two terms.
317 The dependence on the considered number of experimental
318 points used tofit lateral force coefficients to data presented in Fig.2
319 can be found in Table1.
320 There are some differences between the two sets of experimental
321 data, which can be explained by keeping in mind the experiments
322 described herein, that the noncircular part of the D-shaped body
323 contour is not exactly a flat surface, but is formed by two planes, 324 creating an angle of 170° between them (instead of 180°). This 325 slightly modifies the aerodynamic behavior at large values of the 326 angle of attack (in this case, the relative thickness is 1.84 instead of 2). 327 The force coefficient curve is then fitted by taking different 328 numbers of experimental points measured during the IDR/UPM 329 wind tunnel tests to assess the influence of this parameter, and 330 thus, thefitting curves using the 9, 11, and 15 experimental points 331 are also represented in Fig. 2. Finally, the fitting curve given in 3329 Barrero-Gil et al. (2010) is also included in Fig.2. Aclose look at the 333 curves in Fig.2indicates how important it is to have the right choice 334 of the approximation toczðaÞfrom the experimental data; specifi-335 cally, if the range of experimental data points is large (this applies to 336 D-shaped cross sections but also to other bodies), then the curvature 337 variation of the function should not be so simple. As shown in Fig.2, 338 the real behavior of the experimental data is characterized by an 339 almost zero slope at the origin, but if the data interval taken for the 340 curvefitting is too large, the extreme points of the interval are those 341 driving the value of the slope of thefitting curve. In this case, it is 342 possible to obtain afitting curve with a slope at the origin larger than 343 that shown by the experimental data, which means an overestimation 344 of the energy harvesting efficiency. The curvature of the curves at the 345 origin is also changed.
346 The new approach for the evaluation of the energy harvesting 347 efficiency is also applied to other two-dimensional bodies that may 348 be unstable and experience transverse galloping oscillations. During 349 the past several years, a research program devoted to studying the 350 transverse galloping characteristics of prismatic bodies with simple 351 basic cross-sectional shapes has been carried out at IDR/UPM. 352 Within this program, parametric experimental studies of bodies 353 like isosceles triangles, ellipses, biconvex aerofoils, and rhombi
354 (Fig.3) have been carried out by wind tunnel tests. The parameters
355 that define the members of a given family of bluff bodies are the
356 relative thickness,t, which is defined as the ratio of the maximum
357 thickness,bm, of the reference body cross section to its chord,c. The
358 reference body cross section is the one corresponding toa050
359 (Fig.3). SimilarczðaÞcurves for these bodies, as the one depicted in
360 Fig.2, can be found in Alonso and Meseguer (2006), in Alonso et al.
361 (2005,2007) in the case of triangular cross section bodies, in Alonso
362 et al. (2009) for biconvex cross section bodies, in Alonso et al.
363 (2009) and in Ibarra et al. (2014) for rhomboidal cross-section
364 prisms, and in Alonso et al. (2010) for elliptical cross-section
365 beams. In all the preceding cases, the dependence ofczðaÞon the
366 parameters defining the section geometry is considered.
367 Obviously, to extract energy from galloping oscillations, the
368 configuration must develop instability, which according to
pub-369 lished information, only occurs at given values of the reference
370 anglea0, which is different for each family of bodies. The
differ-371 ent stability maps for the previously listed geometries have been
372 published elsewhere. Information concerning triangular prisms can
373 be found in Alonso et al. (2005,2007,2012), Alonso and Meseguer
374 (2006), and Iungo and Buresti (2009); elliptical bodies are
consid-375 ered in Alonso et al. (2010), and both biconvex and rhomboidal
376 cross-section bodies are analyzed in Alonso et al. (2009).
377 The corresponding values of the function czðaÞ5½cdðaÞtana
378
1clðaÞ=cosahave been calculated from the available wind tunnel
379 tests data on the aerodynamic coefficients of the bodies under
380 consideration. These data arefitted to polynomials in tana, thus
381 giving the values of the coefficients a2je1. Then, the maximum
382 efficiencies are obtained according to Eq.(10)and the new approach
383 presented in the“Mathematical Model”section.
384 The geometry that shows the maximum efficiency now needs to
385 be determined, as well as which geometry (represented by its relative
386 thickness) also shows the maximum efficiency for each family of
387 bodies. Fig.4shows the variation with the relative thickness of the
388 maximum efficiency of the different families of bluff bodies. The
389 results represented in Fig. 4 correspond to the case Ap52 and
390 zmp51ða0 ≅180°Þ, but, as has already been mentioned, these
391 results will be different if other values of the parameters concerned
392 are considered. Such behavior is shown in Fig.5, where the variation
393 of the maximum efficiency with the dimensionless amplitude Ap,
394 in the case of isosceles triangular bodies with main vertex angle
395 b530°, is shown. There are two families of curves in this plot.
396 Those that are almost vertical straight lines come from Eq.(7)for
397 the giving values ofUp, whereas the curved lines are obtained from
398 Eq.(10), assuming somefixed values ofzmp.
399 From this plot, it can be realized that onceUpisfixed, the
am-400 plitude of oscillationsApfor maximum efficiency isfixed, because
Fig. 3.Two-dimensional prism families whose energy harvesting suitability under galloping oscillations are considered: (a) triangle; (b) ellipse; (c) biconvex; (d) rhombus
Table 1.Variation of thea1anda3Coefficients Appearing in Eq.(3)of the Lateral Force Acting on D-shaped Bodies
17
Line number
Number of experimental
points a1
18 a3
1 11 20:159 20:758
2 9 20:305 20:394
3 15 20:601 0.043
4 The same as line 3, but
(0,0) excluded
20:79 0.19
401 the maximum position is obtained by differentiation of Eq.(7), and 402 zmpdoes not have an influence. The maximum efficiency grows as 403 the parameterzmpgrows [as deduced from Eq.(10)], and similar 404 conclusions can be obtained by considering any other order of the 405 parameters involved. The reference configuration (Ap52,zmp51)
406 used to calculate the maximum efficiency of the different bluff
407 bodies is represented by a circle in Fig.5.
408 By keeping in mind Eqs. (7) and(10), each geometrical
con-409 figuration can be represented by a single point if the self-similar
410 variablesAp=Up and hmaxUp=ð2zmpÞare used instead ofAp and
411 hmax. In these self-similar variables, Fig.5reduces toAp=Up50:77
412 andhmaxUp=ð2zmpÞ50:35.
413 The values of the self-similar variables change with the body
414 shape, but it is remarkable that, within the validity range of this
415 simplified model, the transverse galloping energy harvesting
prop-416 erties of a given two-dimensional body can be summarized in
417 a single point in theAp=UpversushmaxUp=ð2zmpÞ plane.
418 Concerning Fig.4, in the case of triangular cross-section prisms,
419 which are identified by the angle b of the main vertex, tanb
420
5bm=ð2cÞ, there are several instable regions in the a0-bplane
421 (Alonso et al. 2005,2007,2009,2010,2012;Alonso and Meseguer
422 2006;Iungo and Buresti 2009). One of these regions is in the vicinity
423 ofa05180°, whereas another one appears at approximately a0
424
560°. Obviously, to obtain energy from the galloping oscillations,
425 the body under consideration must undergo instability, which means
426 that at least the Glauert-Den Hartog criterion must be satisfied.
427 Taking this fact into consideration, only those configurations that
428 clearly meet this criterion (a0≅180°) have been considered in
429 Fig.4.
430 According to Fig.4, triangular cross-section bodies ata05180°
431 seem to be the more promising ones for energy harvesting from the
432 galloping phenomenon.
433 Conclusions
434 An improved approach to estimate the energy harvesting efficiency
435 from the transverse galloping of bluff bodies using experimental
436 data has been presented. The importance of properly choosing the
437 process offitting the polynomial curve of the aerodynamic force
438 coefficient to the experimental set of data has been demonstrated
439 with existing experimental data from bodies with different cross
440 sections and with data obtained from new experiments. It has been
441 proven that, unless the aerodynamic force coefficient experimental
442 curve is extremely simple, experimental data cannot befitted by
443 a cubic polynomial, and higher powered terms must be retained in
444 the series expansion (3)to reproduce the real behavior. 10
445 The inappropriate election of the polynomial used tofit the
ex-446 perimental data would give an unrealistic and an even too optimistic
447 capability of the given body to gather energy from windflow, or,
448 in contrast, lead to disregarding the given two-dimensional body
449 shape. This is based on a wrong estimation of the energy harvesting
450 capability, which seems to be the case of isosceles triangular
cross-451 section bodies in Barrero-Gil et al. (2010).
452 Another important conclusion is that, although there are many
453 cross-section shapes that show transverse galloping instability in
454 a range of angles of attacka0, from the point of view of energy
455 harvesting, only a few of them seem to be appropriate to give
456 reasonable values of the maximum efficiency associated with the
457 extraction of energy from the incident wind. Values of the maximum
458 efficiency of different families of prisms are provided and compared.
459 The most promising configuration seems to correspond to the
re-460 verse triangular cross-section prisms, and if the study is confined
461 to isosceles triangles (identified by the anglebof the main vertex),
462 the maximum efficiency is reached for triangles with a vertex angle
463 close tob560°.
464 Further experimental investigations concerning triangular bodies
465 (Ibarra et al. 2014) demonstrated that inverted isosceles triangles
466 develop galloping instabilities atUp≅ 35; in the case of isosceles
Fig. 4.Maximum efficiency,hmax, versus relative thickness,t, of the
different families of prisms shown in Fig. 3; triangles at a0≅180°
(triangles); rhombi (rhombi), biconvex (squares), and ellipses (circles); solid circles represent results of D-shaped bodies; vertical bars indicate the estimated error
15
Fig. 5. Variation of the maximum efficiency for energy harvesting, hmax, with the dimensionless amplitude of galloping oscillations,Ap, of a
two-dimensional bluff body whose cross section is an isosceles triangle with a main vertexb530°; results correspond toa05180°; numbers
on the curves indicate the values of the reduced velocity,Up(vertical lines) or the product of the reduced mass and the damping ratio,zmp; the circle (Ap52,zmp51) represents the reference configuration selected to compare the efficiency of different galloping bluff bodies (Fig.4
467 triangles with a main vertexb560°, whereUpstands for the re-468 duced velocity,Up5U=ðvncÞ, whereUis the air speed,vnis the 469 first natural body frequency, andcis a characteristic length. Before 470 galloping occurs, at lower wind velocities, Up≅3, a vortex-induced 471 vibration (VIV) episode appears. However, this VIV episode is of 472 less interest from the point of view of wind energy harvesting, 473 because VIV takes place at low air velocity (thus, the wind energy to 474 be gathered is small) and because of the small amplitude of the body 475 oscillations.
476 In galloping oscillations, these two effects are enlarged. In the 477 experiments reported by Ibarra et al. (2014), the galloping phe-478 nomenon appears forUp.35, whereas VIV oscillations occur at 479 Up≅3, as already stated, which implies the galloping wind kinetic 480 energy is more than 100 times greater than the one associated with 481 vortex shedding. The oscillation amplitudes are highly dissimilar, 482 and the ratio between galloping amplitude oscillation, AG, and 483 vortex-induced one, AV, can be up to AG=AV$4. In galloping 484 experimentation, the allowed maximum oscillation amplitudes must 485 be limited, and the experimental sequences should be stopped when 486 such maxima of the oscillation amplitudes are reached, to avoid any 487 damage to the setup.
488 From a practical point of view, the advantages of energy har-489 vesting by using inverted isosceles triangles are probably limited 490 by the drawbacks of large amplitude oscillations and the potential 491 destructive effects of galloping instabilities, but the aspects of these 492 problems are out of the scope of this paper.
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