An Approximate Method for the Evaluation of the Normal Force Acting on a Flexible Plate Normal

Transcription

1 Journal of Aled Flud Mechancs, ol. 9, No. 4, , 016. Avalable onlne at ISSN , EISSN OI: /acadub.jafm An Aroxmate Method for the Evaluaton of the Normal Force Actng on a Flexble Plate Normal to the Wnd Flow J. F. Hu 1 and W. X. Wang 1 eartment of Aeronautcs and Astronautcs, Engneerng School, Kyushu Unversty, Fukuoka ty, Fukuoka, , Jaan Research Insttute for Aled Mechancs, Kyushu Unversty, Kasuga, Fukuoka, , Jaan orresondng Author Emal: hu-junfeng@ram.kyushu-u.ac.j (Receved February, 015; acceted Setember 15, 015) ABSTRAT The urose of ths aer s to develo an aroxmate method for the evaluaton of the normal force actng on a flexble late normal to the wnd flow and the deformaton of the late. A theoretcal modellng s frstly roosed to redct the relatonsh between the normal drag coeffcent of a rgd curved-late and the confguraton of the late wth the ad of a seres of numercal analyses of structure and flud dynamcs. Then, based on the theoretcal modellng, an aroxmate method for the evaluaton of the normal force actng on the late and the deformaton of the late s constructed usng only the teraton of structure mechancs analyss, nstead of conventonal comlex teratons of flud-structure coulng analyss. Smulaton tests for 3 flexble lates wth dfferent lengths and dfferent materal modul are conducted. Also a comaratve smulaton test of a 3 flexble late used n a revous exerment s erformed to further confrm the valdty and accuracy of the aroxmate method. Numercal results obtaned from the aroxmate method agree well wth those obtaned from the flud dynamcs analyss as well as the results of the revous wnd tunnel exerment. Keywords: Wnd flow; Flexble late; Flud-structure nteracton; Normal force; Aroxmate method. NOMENLATURE A _ theory area of the late drag coeffcent of a rgd flat-late normal to the flow drag coeffcent of a rgd curved-late based on theoretcal modellng N F normal force coeffcent of a rgd curved- late obtaned from F smulaton real normal force coeffcent of a flexble N real late normal to the flow _ normal force coeffcent of a rgd N theory ds E theory f L ( ) curved-late based on theoretcal modellng an arbtrary nfntesmal element of rgd curved-late Young s modulus total normal force actng on an the late total normal force actng on a rgd curved-late obtaned from theoretcal modellng correcton functon length of the late average ressure actng on the late real averaged ressure actng on a flexble late average ressure actng on a rgd -real s T ux uy W N T theory curved-late obtaned from theoretcal modellng arbtrary locaton on the curved late thckness of the late x comonent of dslacement y comonent of dslacement wdth of the late wnd velocty velocty comonent normal to the curved surface velocty comonent along the tangental drecton of the curved surface correcton coeffcent chord angle of the arbtrary nfntesmal element along the curved late chord angle of the curved late real real chord angle of a flexble late normal to the flow Posson s rato densty of ar

2 J. F. Hu and W. X. Wang / JAFM, ol. 9, No. 4, , INTROUTION A flexble late normal to the wnd flow s a tycal and classc roblem of flud-structure nteracton (FSI), of drect relevance to many natural henomena. For nstance, leaves of lant, flexble fber, and lates roll u n a hgh wnd to reduce the drag and avod damage as reorted by ogel ( ), Alben et al. (00, 004), Schouveler and Boudaoud (006), and Gosseln et al. (010). Furthermore, ths roblem s also of relevance to many ractcal alcatons such as commercal lates and turbne blades (Maher et al. (007); Lu 009; Hoogedoorn et al. (010)). Many research efforts (e.g., Fage and Johansen (197), Kya and Are (1977), hen and hung (1988), Knsely (1990), Okajma (1990), Letchford (001), Shmada and Ishhara (00), and Breuer et al. (003)) have been contrbuted to the nvestgaton of the dynamc behavors of the flow around a rgd-late, such as drag coeffcent, lft coeffcent, Strouhal numbers, velocty fluctuaton behnd the late, and vortex behavors. Relatvely few studes (ogel (1989, 009), Alben et al. (00, 004), Schouveler and Boudaoud (006), and Gosseln et al. (010), ambell and Paterson, (011); Lee and Lee, (01)) have been focused on the evaluaton of the wnd ressure actng on a flexble late and the deformaton of the late. On the other hand, the wnd ressure actng on a deformable late and the deformaton of the late are mortant arameters n the strength desgn of the late and ts suortve structure n ractcal alcatons. In general, t s extremely dffcult to derve an analytcal soluton for the evaluaton of the wnd ressure actng on a flexble late because of the strong nonlnearty n the coulng of the flud flow and the late deformaton. Rgorous evaluaton requres comlex flud-structure coulng analyss although many numercal methods have been develoed as revewed by Hou et al. (01) and egroote (013), such as the arbtrary Lagrangan- Euleran (ALE) fnte element method (Km et al. (007), Peskn (00)) and the Boltzman -Lattce method (Fe ng and Mchaeldes (004), Lee et al. (01)). omlex teraton rocedures of numercal calculatons related to the alternatve flud and structure analyses are cumbersome and error-rone n the rgorous FSI numercal analyss. Therefore, develong a relatvely smle numercal method to evaluate the wnd ressure actng on a deformable late s qute useful n varous ractcal alcatons. In ths aer, we are nterested to develo an aroxmate numercal method for the evaluaton of the normal force actng on a flexble late normal to the wnd flow and the deformaton of the late. The averaged ressure s defned by the normal force dvded by the late area. A theoretcal modellng s roosed to aroxmately descrbe the relatonsh between the normal drag coeffcent of a rgd curved-late n the flow and ts confguraton wth the ad of a seres of regular numercal calculatons of flud dynamcs and structure mechancs. Based on the theoretcal curve of the normal drag coeffcent and the confguraton of the rgd curved-late, an aroxmate method for the evaluaton of the normal force actng on the late and the deformaton of the late s constructed usng only the teraton of structure mechancs analyss, nstead of conventonal comlex teratons of flud-structure coulng analyss. Smulaton tests for 3 flexble lates wth dfferent lengths and dfferent materal modul are conducted. Also a comaratve smulaton test of a 3 flexble late used n a revous exerment s erformed to further confrm the valdty and accuracy of the aroxmate method.. APPROXIMATE METHO In ths secton, a theoretcal formulaton to descrbe the relatonsh between the normal drag coeffcent of a rgd curved-late and the confguraton of the late s frst derved. Then, based on the theoretcal formulaton, an aroxmate numercal method s constructed to evaluate the normal force actng on a flexble late and the deformaton of the late usng only the teraton of structure mechancs analyss, nstead of the conventonal comlex flud-structure coulng analyss..1 Theoretcal Modellng A flexble late wth uer end fxed normal to the wnd flow s consdered, as shown n Fg. 1(a). The length, wdth, and thckness are denoted by L, W, and T, resectvely. The late subjected to wnd ressure tends z W y L T ux y T N O (a) ( ) sn x ( ) cos N T (b) ds Fg. 1. The schematc of a flexble late normal to wnd flow. uy x 1560

3 J. F. Hu and W. X. Wang / JAFM, ol. 9, No. 4, , 016. to bend towards the flow drecton to reduce drag. ashed lne denotes the late before deformaton. The dslacements of free end are denoted by ux and uy. The deformed confguraton s descrbed by the chord angle between the vertcal lne and the lne connectng the fxed end and the free end of the late. In general, ths s a comlex flud-structure coulng roblem, the ressure actng on the late s comlcatedly dstrbuted on the late surfaces and fluctuates around ts mean value over tme (Fage and Johansen (197), hen and hung (1988), and Lee and Lee (01)). In resent study, as the frst aroxmaton, t s assumed that the deformed late n unform and steady wnd flow s under a quas-statc state and that the mean ressure s unformly dstrbuted on the late surface. Therefore, the deformed late under the quas-statc state n the wnd flow can be consdered as an equvalent rgd curved-late. Furthermore, the effect of the vscous drag along the tangental drecton of the late surface due to the flud vscosty on the deformaton of the late s consdered to be very slght comared to the wnd ressure normal to the surface so that only the normal force s consdered n the resent study. In addton, for the sake of smlcty, the effect of gravty on the deformaton of late s not consdered hereafter. Based on the above assumtons, consder the wnd flow actng on an arbtrary nfntesmal element ds of a rgd curved-late, as shown n Fg. 1(b). s the wnd velocty, _N and _T are the velocty comonents normal to the curved surface and along the tangental drecton of the curved surface, and s s the Lagrangan coordnate defned along the mdlane of the late from ts fxed end to ts free end, resectvely. Therefore, the total normal force roduced by the wnd flow wth velocty on the curved late can be exressed by F N theory 1 0 L ( cos ) W ds (1) Where 3 ( 1.5 kg / m ) s the densty of ar, s the tangental angle of the curved late at locaton s, and s the drag coeffcent of an equvalent rgd flat-late (L W T) normal to the flow. can be fnd from books of flud mechancs (e.g. Whte 1998) for regular 3 lates or can be obtaned from flud dynamc analyss usng a comutatonal flud dynamcs ( F) codes. Generally, ( ( F, s)) s a functon of the normal force N theory and the locaton s along the curved late, and ths equaton s actually an ntegral equaton of fludstructure coulng. It s dffcult to obtan the exact soluton. In order to exress the total normal force n an exlct formulaton, we assume that theory the total normal force exressed by 1 theory ( ) A A L W theory can be aroxmately ) cos f (, (-a) (-b) Where A denotes the area of the late, denotes the chord angle of the curved late (Fg. 1(a)) and s used to characterze the confguraton of the curved late, and f ( ) s a correcton functon whch s used to correct the errors caused by the above smlfcaton because Eq. ( -a) s not a rgorous soluton of the ntegral equaton of Eq. (1). Then the average ressure can be calculated by _ theory theory actng on the curved late F ( ) 1 ( ) cos N f (3) A From Eq. (3), the normal drag coeffcent of the curved late can be exressed by _ theory ( ) ( ) N theory ( ) (4) A f ( ) cos And the conventonal drag coeffcent s exressed by F d ( ) d theory ( ) 0.5 A (5) ( ) cos 0.5 A ( N theory ) cos 3 f ( ) cos Observng Eq. (4) and Fg. 1(a), t s recognzed that 0 and / corresond to the two secal cases of a rgd flat-late normal and arallel to the wnd flow, resectvely. Therefore, accordng to flud mechancs, the normal drag coeffcent N theory ( ) should satsfy N theory( 0) and ( / ) 0. The correcton functon f ( ) Ntheory should satsfy f ( 0) 1 and f ( / ). As a result, f ( ) s assumed here by f ( ) 1 sn (6) Where arameter s a constant whch s determned from the comarson between the curve N theory ( ) obtaned from Eq. (4) and the curve N Ex. ( ) obtaned from exerment or N F. ( ) obtaned from a seres of F calculatons of rgd curved-lates wth chord angles (0 / ; 1,,, k ). In the resent study, F calculatons are emloyed to determne the curve N F. ( ). The geometres of rgd curved-lates wth chord angles ( 1,,..., k) used n the F calculatons are determned from a seres of structure calculatons of a flexble late subjected to a seres of unform ressures as follows.. etermnaton of f ( ) In order to determne the correcton functon f ( ), a seres of numercal calculatons of structure 1561

4 J. F. Hu and W. X. Wang / JAFM, ol. 9, No. 4, , 016. mechancs and F calculatons are conducted usng commercally avalable codes of MS Marc010 and ANSYS Fluent 13.0, resectvely. Frstly, a flexble late subjected to a seres of unform ressures are conducted to determne a seres of related reconfguratons of the late. Large deformaton, namely, the geometrcal nonlnearty of the deformaton s consdered. The ressure aled to the flexble late s gven by below equaton 1, (0 max ; 1,,, k ) (7) Fg.. Flowchart of the theoretcal modelng. Where ( 1,,, k ) denote a seres of gven wnd veloctes, s the maxmum velocty max secfed accordng to the desgn requrement of the flexble late, and s the drag coeffcent of a rgd flat-late normal to the flow as mentoned before. Then, alyng ( 1,,, k ) of Eq. (7) to a gven flexble late and conductng the analyss of structure mechancs, we can obtan a seres of self-smlar geometres ( 1,,, k ) of curved lates. Accordng to the exermental facts reorted n many references as mentoned above, t s well known that the value of the real normal drag coeffcent N of a real flexble flat-late normal to the flow s always smaller than because the deformaton of the flexble late reduces the drag force. Therefore, for a gven, the real averaged ressure real actng on a flexble late s always lower than calculated by Eq. (7) because N real. In other words, the gves the uer bound of real for a gven. Smlarly, the chord angle of the deformed late, obtaned from the above structure analyss related to, s also not equal to the real chord angle for a gven real. The real chord angle s always smaller than real and 0 real because 0 real for a gven. Hence, also gves the uer bound of for a gven real. Secondly, we use these curved lates wth chord angles ( 1,,, k ) as a seres of rgd curved-lates n the F calculatons to solve the corresondng normal drag coeffcents N F ( ) ( 1,,, k ). That s, we obtan a curve of N F ( ) ( 1,,, k ) related to a seres of rgd curved-lates wth chord ( 1,, k ) angles,. On the other hand, accordng to Eq. (4) and Eq. (6) wth 0, we can obtan the theoretcal formulaton of normal drag coeffcen ( ). In consequence, N theory 0 we obtan two curves of N F ( ) and ). Plottng these two curves together N theory ( 0 and comarng them wth each other reveal the deference between these two curves. Fnally, select a roer value of through a rocess of tral and error to make the dfference between these two curves as small as ossble. That s, select a to make N theory ( ) ) cos ( f ( ) (8) N F Then, Eq. (4) becomes a theoretcal equaton to redct the normal drag coeffcent ( ) for N theory a known rgd curved-late wth chord angle. The flow chart of the resent theoretcal modellng s descrbed n Fg...3 Algorthm to Solve the Average Pressure Actng on a Flexble Plate Normal to the Flow Assume that the geometry and materal roertes of a flexble late are known, the velocty of unform and steady wnd nflow s, and the late 0 normal to the flow s at a quas-statc state. Then, the real average ressure n the sense of FSI can be exressed by real _ theory (), bend (9-1) f the followng equaton s satsfed. _ theory () bend, _ bend (9-) where, _bend denotes the ressure aled to the flexble late n the structural analyss, bend s the corresondng chord angle of the deformed late, and _ theory () bend s the theoretcal average ressure obtaned from Eq. (3). It s noted that the theoretcal average ressure s aroxmately equal to the average ressure 156

5 J. F. Hu and W. X. Wang / JAFM, ol. 9, No. 4, , 016. obtaned from F analyss of a rgd curved-late wth chord angle based on the recedng bend Fg. 3. Flowchart of the calculaton sequence of the resent aroxmate method. theoretcal modellng. In other words, Eq. (9-1) means that the real ressure equals the theoretcal ressure obtaned from Eq. 3 when the ressure aled to the flexble late n the structure calculaton equals the theoretcal ressure actng on the rgd curved-late wth the chord angle bend. Then, based on Eq. (9), an algorthm s develoed to solve the real ressure re al, chord angle real, and normal drag coeffcent N real () real for a flexble late normal to the wnd flow of usng only teratve smulatons of 0 structure analyss as follows. s an arbtrary 0 gven wnd velocty. The flowchart of calculaton rocedures of the aroxmate method s llustrated n Fg. 3. At the frst teraton, nonlnear bendng calculaton of the flexble late subjected to a gven ressure s conducted. The ntal ressure s calculated by Eq. (6), that s, 1 1_ bend 0 (10) Where the drag coeffcent of a rgd late normal to the wnd flow s obtaned from F analyss for the rgd late wth the same geometry as the flexble late. Then, a reconfguraton 1bend of the flexble late s obtaned from the bendng analyss. Insertng ths 1 bend nto Eq. (3) and Eq. (4) yelds 1_ theory () 1 bend and N 1_ theory () 1 bend, resectvely. Accordng to the recedng subsectons, 1_bend and 1 bend gve the uer bounds of the real ressure and real reconfguraton Therefore, 1_theory 1 bend It s obvous that 0 real 0 re al, resectvely. obtaned from Eq. (4) usng gves the lower bound of the real ressure. 1_bend s not equal to 1_theory. Thus, we move to the second teraton usng 1_theory as the bendng load aled to the flexble late as follows. _ bend 1_ theory (11) Smlar to the calculaton rocedures at the frst teraton, bend s obtaned from the bendng calculaton, and then _ theory () bend and N _ theory () bend can be obtaned from Eq. (3) and Eq. (4). It s observed that 1bend bend 0 N 1 theory ()() 1bend N theory bend 1_ bend _ theory () bend _ bend (1) Assume that _ theory () bend s stll not equal to _bend and that the dfference between the both values s stll sgnfcant. Thus, we move to the thrd teraton usng 3 _ bend _ theory (13) as the ressure aled to the flexble late n the bendng calculaton of the thrd teraton. Smlar to recedng teratons, s obtaned from 3bend bendng calculaton, and then a set of ), and ) are obtaned 3 _ theory ( 3 bend N 3 _ theory ( 3 bend usng Eq. (3) and Eq. (4), resectvely. It s observed that 1 bend 3bend bend N 1theory( 1 bend) N3theory( 3 bend) Ntheory( ( ) 1_ bend 3_ bend 3_ theory 3bend _ bend bend ) (14) We must move to the next teraton f the dfference between 3_ bend and 3 _ theory ( 3 bend ) s stll sgnfcant. Smlar to the recedng teratons, teratvely reeat the comutatonal rocedures as descrbed above untl the dfference between the theoretcal ressure k _ theory () k bend obtaned from Eq. (3) and the ressure k _ bend aled to the flexble late n the bendng calculaton at the k- th teraton s equal to or less than a gven small value. In ths study, the teratve calculaton s comleted when ths dfference satsfes the followng crteron. 1563

6 J. F. Hu and W. X. Wang / JAFM, ol. 9, No. 4, , 016. k _ theory k _ theory k _ thoery k _ thoery k _ bend k1_ theory 0.5% (15) Therefore, based on Eq. (15) the real ressure actng on the gven flexble late normal to the wnd flow of s defned by followng equaton. 0 0 real k _ theory (16) 3. NUMERIAL SIMULATION TESTS In order to demonstrate the valdty of the roosed aroxmate method, numercal smulaton tests for several 3 flexble lates wth dfferent geometres and dfferent materal modul were conducted usng the aroxmate method. Furthermore, smulaton for a 3 flexble late studed by Gosseln et al. 010 was also carred out for a comarson. 3.1 Smulaton Tests for Fve Flexble Plates A 3 model of flexble lates normal to the wnd flow s dected n Fg. 4. The uer end of the model s clamed and the lower end s free. The length, wdth and thckness of the late model are denoted by L, W and T, resectvely. Fve lates wth dfferent lengths or dfferent materal constants are used n the smulaton tests. Table 1 and Table gve the geometres of fve Fg. 4. A 3- smulaton model. lates and the materal constants of three knds of materals. PP, PE and PET are the abbrevatons of olyroylene, olyethylene and olyethylene terehthalate. Followng the comutatonal rocedures descrbed n Secton, the determnaton of the theoretcal curve of N theory ()() N F s frstly carred out based on the theoretcal modellng wth the ad of a seres of structural and F analyses. Then, teratve calculatons of bendng analyss are conducted to solve the real averaged ressure actng on the flexble late and the deformaton of the late for a seres of gven wnd veloctes accordng to the algorthm of the aroxmate method. Both of structure and F analyses are ordnary calculatons wth defnte boundary condtons and easly to be erformed, whch are relatvely smle comared to comlex conventonal teratve fludstructure coulng calculatons. Table 1 Geometres of fve flexble thn lates Plate L (mm) W (mm) T (mm) PP PP PP PE PET Table Materal constants of three materals Young s Materal Modulus Posson s rato ensty (10 (GPa) 3 kg/m 3 ) PE PP PET In the F analyss, the undsturbed nflow s assumed as unform and the flud s assumed to be ncomressble. The comutaton doman of F analyss s 4L 4L 11W wth the orgn of the coordnates located at the center of the late, as shown n Fg. 5. The nlet, to, and bottom boundary walls are set at the dstance of 1L from the center of the late, resectvely. The outlet s set at 30L downstream from the late center. A unform velocty flow s secfed at the nlet, zero ressure s secfed at outlet, sl condton s secfed along the to and bottom boundary walls, and nosl condton s secfed on the late surface. The large eddy smulaton (LES) method s utlzed n the F analyss. Accordng to ANSYS Fluent 13.0 User s Gude (010), n resent F analyss, t he Reynolds number Re, the turbulence ntensty I, the turbulent knetc energy k, and the turbulent length scale l are calculated as follows: Re L / (17) 0.15 I 0.16(Re) (18) k 1.5() I (19) l = 0.07L (0) Where (= Pa s) s the vscosty coeffcent of ar at 0 degrees elsus. The effect of the number of grds on the calculaton accuracy s nvestgated frstly usng the geometry of PP

7 J. F. Hu and W. X. Wang / JAFM, ol. 9, No. 4, , 016. late wth dfferent numbers of grds from 1,133,500 to 5,69,000. The late s assumed to be rgd and normal to the wnd flow. The drag coeffcent s calculated for each case of grd number. As the result, the number of grds around,500,000 s adoted for all the lates based on the consderatons of accetable comutatonal accuracy and cost. Fg. 5(b) demonstrates a tycal grd dstrbuton around the rgd curved-late of PP-300 and the total number of grds of the comutatonal doman s,66,000. Fg. 6. Smulaton model for the comaratve test. The comutatonal doman for the comaratve smulaton test s a square secton duct wth m n wdth and 4 m n length accordng to the exerment of Gosseln et al. (010). The boundary condtons on the nlet, outlet, late surface and boundary walls are the same as descrbed n Fg. 5(a). The calculaton rocedures of the smulaton test are the same as those descrbed n subsecton RESULTS AN ISUSSION Numercal results of smulaton tests are resented n the followng fgures. Effects of the number of grds on the drag coeffcent of a 3 vertcal rgd late of 500 mm n length, 100 mm n wdth, and 1 mm n thckness are descrbed n Fg. 7. It s seen that the drag Fg. 5. F comutatonal doman (a) and grd around the late (b). 3. A omaratve Smulaton Test A comaratve smulaton test of a 3 flexble late used n a revous wnd tunnel exerment by Gosseln et al. (010), as shown n Fg. 6, s also conducted based on the resent aroxmate method to confrm the valdty of the aroxmate method. The length and wdth of the late are 100 mm and 35 mm, resectvely. The flexural rgdty of the late s 10-6 Nm. The Young s modulus E = 3.1 GPa and Posson s rato 0. 3 are used. In order to kee the same flexural rgdty as that used n the exerment of Gosseln et al. (010), the thckness h of the late s mm determned from the followng equaton. Eh (Nm) (1) 1(1 ) Fg. 7. Effect of the number of grds on the calculaton results. coeffcent vares slowly and tends to converge a stable value when the number of grds s larger than,000,000. As the result, the grd number around,500,000 was adoted for the F analyses of all lates based on the consderatons of accetable comutatonal accuracy and cost. Followng the analyss rocedures as descrbed n secton, the theoretcal curve ( ) wth N _ theory = 0.5 s determned wth the ad of structure and F analyses of fve flexble lates lsted n Table 1. The curve and the normal force coeffcents obtaned from the F analyses of fve lates are resented n Fg

8 J. F. Hu and W. X. Wang / JAFM, ol. 9, No. 4, , 016. longest late of PP-500. The normal force actng on the shortest late of PP-100 has the largest value at the wnd veloctes hgher than m/s and the longest late Fg. 8. Theoretcal curve of normal force coeffcent vs. chord angle of rgd curved-lates. The normal force coeffcents obtaned from the F analyses of fve lates are dslayed by fve knds of marked onts and the sold curve s the theoretcal curve. It s obvous that the theoretcal curve agrees well wth the results obtaned from the F analyses of fve lates wth dfferent length or dfferent modulus, although relatvely large dfference can be found for large chord angle beyond 60 degree. Therefore, t s reasonable that usng the roosed theoretcal modelng method to redct the relatonsh between the normal force coeffcent and the chord angle of rgd curvedlate. We can calculate the normal force actng on a rgd curved-late based on the theoretcal curve f we know the chord angle of the curved-late. In the case of very large chord angle whch means large bendng deformaton of the late, the theoretcal curve overestmates the normal force. For a more accurate redcton of the normal force n the chord angle range of larger than 60 degree, a dfferent arameter may be selected. That s, the theoretcal curve may be defned n two ranges of chord angle: 0 60 and 60 90, selctng dfferent for dfferent range. Ths ssue s a further research subject whch s ongong now. Results of the normal forces actng on the flexble lates and the deformatons of the lates are resented versus to wnd velocty n followng fgures. Effects of the late length on the normal force actng on the three flexble lates at varous wnd veloctes are dected n Fg. 9. The F analyss results of normal force actng on the three PP lates wth lengths of 100 mm. 300 mm, and 500 mm are comared wth those obtaned from the resent aroxmate method. Here, the results of F analyses are obtaned from the analyss of 3 rgd curved-lates whch have the same geometres obtaned from the aroxmate method. An enlarged draft for the data at low wnd veloctes s also resented below. Marked onts denote the F results and sold, dashed and dotted curves denote the results of aroxmate method. It s seen that, when the wnd velocty s lower than 5 m/s, the longest late of PP-500 gves the largest normal force because of ts largest area normal to the wnd flow. However, as the wnd velocty ncreases, the normal forces actng on the relatvely short lates of PP-100 and PP-300 ncrease more radly and successvely exceed the normal force actng on the Fg. 9. Effects of the late length on the normal force actng on the flexble lates at dfferent wnd veloctes. of PP-500 gves the smallest one. Furthermore, t s seen that both results obtaned from the resent aroxmate method and the F analyss are n good agreement, esecally n the case of PP-100 late. In the cases of PP-300 and PP-500 lates, the dfference between both results ncreases wth the ncrease of wnd velocty. These features reflect the effects of the bendng deformatons of flexble lates on the normal force actng on the lates. Longer late s easly bended by the wnd flow than shorter one at the same wnd velocty. Larger bendng chord angle leads to lower normal force actng on the bended late. For a more accurate redcton of the normal force actng on the flexble late wth a large chord angle, a dfferent arameter may be selected, as mentoned above. In addton, the average ressure actng on the lates can be calculated by the normal force dvded by the late area, referrng to Eq. (3b). Effects of materal modulus on the normal force actng on the three flexble lates wth the same geometry but dfferent modul are descrbed n Fg. 10. The normal forces actng on a rgd late havng the same geometry are also dected n the fgure for a comarson. Smlar to Fg. 9, the results obtaned from the resent aroxmate method agree well wth those obtaned from the F analyses. The normal force actng on the PET-300 whch has the hghest modulus ncreases quckly. In contrast, the normal force actng on the PE-300 whch has the lowest modulus ncreases slowly. Plate wth hgh modulus s hard to be bended by wnd flow comared to the late wth low modulus. Large bendng deformaton reduces the normal force actng on the bended late. These results are 1566

9 J. F. Hu and W. X. Wang / JAFM, ol. 9, No. 4, , 016. n consstence wth the nature henomena frequently observed from the bended trees under hgh wnd. dected for a contrast. The drag values obtaned from the resent aroxmate method are well consstent wth the revous exerment results. These comaratve smulaton results further confrm the valdty and accuracy of the resent aroxmate method. 5. ONLUSION Fg. 10. Effect of materal modulus on the normal force actng on the flexble lates at varous wnd veloctes. The chord angles of fve bended flexble lates at varous wnd veloctes, obtaned from the resent aroxvmate method, are descrbed n Fg. 11. It s clear that the flexble late wth low modulus or large length has large chord angle whch means large bendng deformaton. Fg. 11. hord angles of fve flexble lates at varous wnd veloctes. Based on the theoretcal modellng, the constructon of the aroxmate method, and the numercal smulaton tests, ths study leads to followng conclusons. An exlct theoretcal formulaton for the redcton of the curve of the normal drag coeffcent of a rgd curved-late versus to ts chord angle s derved through a theoretcal moldellng wth the ad of a seres of ordnary nonlnear structure mechancs and F analyses.. The formulaton s smle and gves good accuracy of rdcton. Further study s needed to mrove the redcton accuracy by selectng dfferent correcton arameters for dfferent chord angle ranges. An aroxmate method to evaluate the normal force actng on a flexble flat-late normal to the wnd flow and the deformaton of the late s develoed usng the resent theoretcal formulaton and the teraton of only structure analyses nstead of comlex flud-structure coulng analyss. Results of the numercal smulaton tests for the lates wth dfferent geometres and materal modul demonstrate the valdty and accuracy of the aroxmate method. Moreover, the comaratve smulaton test of a 3 flexble late used n a revous wnd tunnel exerment further confrm the accuracy of the resent aroxmate method. Therefore, t s consdered that the resent aroxmate method s relatvely smle comared to conventonal comlex flud-structure coulng analyss and s useful for the evaluaton of the normal force actng on the flexble late normal to the wnd flow and the deformaton of the late n the ractce alcatons. REFERENES Fg. 1. omarson of drag force obtaned from the revous exerment and the resent aroxmate method. Fnally, the results of drag force obtaned form the comaratve smulaton are comared wth the revous exermental results ( Gosseln et al. (010)) n Fg. 1. The drag force of a rgd late obtaned from the revous exerment s also Alben, S., M. Shelley and J. Zhang (00). rag reducton through self-smlar bendng of a flexble body. Nature 40(6915), Alben, S., M. Shelley and J. Zhang (004). How flexblty nduces streamlnng n a twodmensonal flow. Physcs of Fluds 16(5), Breuer, M., N. Jovcc and K. Mazaev (003). omarson of ES, RANS and LES for the searated flow around a flat late at hgh ncdence. Internatonal Journal for Numercal 41, ambell, R. L. and E. G. Paterson (011). Flud 1567

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