ON THE HEIGHT OF MAXIMUM SPEED-UP IN ATMOSPHERIC BOUNDARY LAYERS OVER LOW HILLS

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1 ON THE HEIGHT OF MAXIMUM SPEED-UP IN ATMOSPHERIC BOUNDARY LAYERS OVER LOW HILLS Cláudio C. Pellegrini FUNREI Departamento de Ciências Térmicas e dos Fluidos Praça Frei Orlando 17, São João del-rei, MG, pelle@serv.com.ufrj.br Gustavo C. R. Bodstein COPPE/UFRJ Departamento de Engenaria Mecânica C.P. 6853, , Rio de Janeiro, RJ. gustavo@serv.com.ufrj.br Abstract In tis paper, we present a study on te eigt of maximum speed-up, l, for flows over low ills under neutral atmospere. We consider te four most well known expressions to calculate l, due to Jackson and Hunt (JH), Jensen (JEN), Claussen (CL) and Beljaars and Taylor (BT). In te analysis, we present a formal demonstration of te fact tat l can, in fact, be calculated as te inner layer dept, were inertia and turbulent forces balance. Te need for suc a demonstration as received little attention by researcers over te years. We also propose a new value for te constant in CL s expression and confirm tat JEN s expression gives better results tan JH s one. Regarding tis fact, we suggest tat JH s expression sould definitively be substituted by JEN s or CL s wit te proposed constant. Key-words: inner-layer dept, maximum speed-up, flow over ills, atmosperic boundary layers. 1. INTRODUCTION Tere as been a remarkable interest over te years on estimating te eigt above te ground were wind speed-up is a maximum in te atmosperic boundary layer (ABL) over low ills. Te idea is strongly appealing for wind power specialists and for tose wo want to calculate wind loads on various kinds of structures. Many expressions to calculate tis eigt, often denoted by l, ave been proposed since te idea appeared. Te most well known expressions come from te pioneering work of Jackson and Hunt (1975) and from later works by Jensen et al., (1984) Claussen (1988) and Beljaars and Taylor (1989) (ereafter JH, JEN, CL and BT, respectively). Te expressions obtained by tese autors, respectively, read ( / L ) ln( l / ) = κ ( / L ) ln ( l / ) = κ ( / L ) ln( l / ) = n ( / L ) ln ( l / ) = l, (1) l, () l const., (3) l const., (4)

2 were L is te alf-lengt of te ill, defined following JH as te distance from te illtop to te upstream point were te elevation is alf its maximum ; is te rougness lengt and k is te von Karman s constant, adopted as.39, as suggested by a recent work by Frenen and Voguel (1995). A comparative study of te relative merits of te four expressions can be found in Walmsley and Taylor (1996). If we divide te four expressions by and rewrite te constants in (3) and (4) as = κ L ( l ) = κ L ) = C 1 κ L n ( l ) = Cn κ L C 1k and C nk, respectively, we get: l ln(l ), (5) l ln, (6) l ln(l, (7) l ln, (8) were l l and L L. Based on only one experimental result, CL suggests tat C1 k =.9, wic means tat C 1 =. 59 for κ=.39. After comparison wit model results, BT suggests tat n=1.4 to 1.6, depending on te turbulence closure assumed, and tat C n κ =.6 to.55 (also depending on closure), wic yields C n = 3. 6 to In is work, JH divides te ABL in two regions. In te first, more external region, te effects of inertia dominate and in te second, more internal, turbulent forces ave to be considered too. In te same work, tey identify l as te dept of te inner layer. Altoug tis idea was adopted in most of te works tat followed, none of te works we ad access to demonstrate tat l, calculated as te inner layer dept (were inertia and turbulent forces balance), was also te eigt of maximum speed-up (ereafter called l max, to avoid confusion). All works considered restricted temselves to verify tat te field data for l max confirmed te proposed expressions for l. In a review paper, Taylor et al. (1987) state tat l is probably best considered as a scale eigt for te inner layer rater tan te eigt at wic someting specific occurs. Te autors, owever, follow JH s ypotesis and compared te results of teir expression for l wit field data for l max. In a more recent paper, Beljaars and Taylor (1989) say tat since te inner-layer dept, l, as been introduced by means of order of magnitude considerations, its practical definitions is somewat arbitrary. Apart from tat, a lot of discussion is found in te literature about te relative merits of expressions (1)-(3). Te following points were summarised from Walmsley and Taylor (1996): independent of wic expression is used, l is always considered to be te eigt of maximum speed-up; values predicted by te JH expression are too ig wen compared to field results and no reasonable adjustment of can fix te problem; values predicted by te JEN expression agree very well wit observed values in te wole range of variation of L ; CL s expression gives better agreement to observed values tan JEN s at te specific value of L wit wic it was calibrated; Model results suggest a value for n between 1 and in te BT expression; More observational data is required to solve definitively te question. In tis work, we present a new deduction for te JEN s expression obtained troug sligtly modified order of magnitude arguments applied to te ypotesis tat l is te eigt were inertia and turbulence forces balance in te ABL. To our present knowledge, tis

3 deduction is bot new and simpler tan te previous ones. We also present, for te first time, a formal demonstration tat l, calculated tis way is, in fact, te eigt of maximum speed-up. We compare our results to JH s, CL s and observational data sowing tat JEN s expression agrees better wit observational data tan JH s and tat te constant in CL s expression can be calibrated to agree well wit field data. We also ratify Walmsley and Taylor s conclusion tat more observation is needed at some ranges of te parameter L.. DEFINITIONS Consider one isolated D ill in te middle of an oterwise flat terrain, of constant rougness and under a neutrally stratified atmospere. For our purposes, we consider a ill to be a topograpical variation wit caracteristic lengt about 5 Km and eigt less tan 5m. A ill is called low wen it slope never exceeds o. Fig. 1 illustrates te main features of a typical low ill. Te vertical co-ordinate, is defined as te eigt above te local terrain rater tan te vertical eigt above sea level. u () u () u () u(x,) Streamline L / Fig.1. Definitions of, L, u, u and. In te case under study, we assume tat te vertical profile of te oriontal mean wind is essentially logaritmic far from te ill. Hereafter we refer to tis profile as u (), and te location upwind of te illtop (HT) were it is found as te reference site (RS). Te RS profile suffers te influence of te ill in suc a way tat it is modified by a speed-up quantity u(x, ) and becomes u (x,) at a given point over te ill. Tus: u(x, ) u () u(x, ), (9) were u is positive at HT, because te flow is accelerated to satisfy te continuity equation. If we divide te speed-up by te RS velocity, we ave te relative speed-up, S: u(x, ) S(x, ) 1. (1) u (x, ) Te eigt were u is maximum, l max, is defined as

4 l ( u = u ). (11) max max Conversely, we can write u max u(x, lmax ). In te next section, we develop an order of magnitude analysis of te governing equations to obtain an expression for l max. We also establis te co-ordinate system most appropriate for our purposes. 3. ORDER OF MAGNITUDE ANALYSIS To obtain te expression for te eigt of maximum speed-up we proceed in two steps: first we obtain te expression for te inner layer dept, l, and second, we sow tat tis dept is really te maximum speed-up eigt, l max. In order to establis a co-ordinate system suitable for te task, we follow te work of Kaimal and Finnigan (1994), wic recommends te use of te streamline co-ordinates for distorted flows of tis type. Te work of Finnigan (1983) gives te mean mass conservation and te mean x-momentum equations for D flows in tis system, respectively, as w =, (1) 1 p ' ' w' u' w' u' w' T u = g x Vx. (13) ρ L R T a In eqs. (1) and (13), x is te direction parallel to te streamlines and u and u are te mean and turbulent velocities in tis direction, respectively. Te direction normal to te streamlines is, and w and w are te corresponding velocities. Te termodynamic mean pressure is denoted by p, te mean density by ρ, te mean temperature by T, te reference mean temperature by T, te x-component gravity acceleration by g x and te x-component mean viscous force by V x. R and L a are flow lengt scales and tey are related to te mean variables troug R = u ( Ω ) and L a = u ( ), were Ω is te mean vorticity component in te direction in te original Cartesian co-ordinate system. Supposing te existence of a region were inertia and turbulence terms balance we write u u' w'. (14) To evaluate expression (14) we assume tat x L and tat inertia and turbulence terms balance in te region were l. We also assume tat u u wic means tat u << u. Finally, we assume tat u' w' u *, were u * is te friction velocity. Wit tese assumptions we ave u ( l) u*. (15) L l

5 To transform tis order of magnitude relation in an equality, we introduce an unknown function C of order one, suc tat u ( l ) L = C u / l and, tus, l u* = C. (16) u ( l) L Tis relation is presumably valid for all incident wind profiles. For a logaritmic profile of te form u u* = (1/ κ)ln( l ), we can write l/ L = C κ ln ( l / ) wic, upon dividing and multiplying by, gives * = C κ L l ln(l ). (17) Tis equation is identical to tat of JEN except for te constant C, to be determined. In te mainframe of an order of magnitude analysis, tis can only be accomplised troug a comparison wit observational data. We do tat in te next section. First, owever, we sall sow tat l, calculated from (17), is indeed te maximum speed-up eigt. Consider te mass conservation and x-momentum equations in Cartesian co-ordinates. Tey are essentially te same as eqs. (1) and (13), except for te curvature terms containing R and L a in eq. (13). Returning to te ypotesis tat te inertia and turbulence terms balance in te inner region, we ave u w u' w'. (18) As we now want to obtain results about te eigt were u = u u into (18) to obtain u is maximum, we substitute u u w u' w', (19) after considering tat u () = and tat u << u. If we are to obtain te eigt were u is maximum, we must impose tat u = at = lmax (in Cartesian coordinates). Differentiating (19) wit respect to allows us to substitute tis condition in te resulting equation and find u w u w u' w', () at = l. Substituting for te orders of te individual terms yields max ( u u wu u* l ) L ( l) ( l). (1)

6 Te magnitude order for w can be obtained from te mass conservation equation as w u ( lmax ) L. Substituting tis expression on eq. (1), multiplying by ( lmax ) and introducing a function C 3 to obtain an equality, we get ( l ) u = max ( u u ) C3u* L. () Recalling tat u >> u and returning to te streamline co-ordinate system, were lmax is simply equal to l, we finally get l L u* = C3. (3) u ( l ) max Eq. (3) is identical to eq. (16), except for te constant to be determined. As te constant value is entirely arbitrary, we can set C = C3, terefore proving tat l= l max, indeed. In te next item, we calculate te value of C troug comparison wit field data and test te overall capability of expression (17). 4. COMPARISON WITH OBSERVATIONAL DATA Many field studies provide te observational data needed for our purposes. Te most popular data can be obtained from te work of Taylor et al. (1987) and Copin et al. (1994). Te available measurements for l are represented in fig. () togeter wit te results of eqs. (17), (5), (6) and (7). l ln (l ) l ln (l ) 5 C =.4 Const.= Const.= 1 5 C =.41 1 Eq. (17) Eq. (7) Eq. (6) Eq. (5) Obs. data L k 5 1 Obs. data 5 1 (.a) (.b) Fig.. Non-dimensional eigt of maximum speed-up.

7 Equation (17) was tested for some values of C and we verified tat good agreement is obtained wit C =. for te HT. All measurements were made at te HT and, terefore, it was not possible to assess te x dependence of C. Te result, owever, confirms JEN s equation, expression (6). We also obtained te best fit value of C =. 4, for eq. (17), wic as never been proposed before. We believe, owever, tat more field data is necessary before we can state tat tis value is definitive. Our tests also sowed tat te agreement between observation and JH s expression, eq. (5), is acceptable only for te Bungendore Ridge (BR) results (Bradley, 1983), not represented in fig. (.b). Tis conclusion is confirmed by oter workers, e.g. Mickle et al. (1988) and Taylor and Walmesley (1996). Te BR results deserve some attention, neverteless. According to Taylor et al. (1987), te value varied between. and.5 m during te BR experiment. Tose limits, wit l estimated as 5m, (following Taylor et al., 1987) correspond to te values tat were well represented by eq. (5). In addition, te speed-up vertical profile presented a very broad maximum, from te first measurement point up to te eigt of 8m. Te point represented in figs. (.a) and (.b) were calculated supposing tat l=1 m and adopting =.35 m as an average value. Tis point agrees very well wit eq. (17). Fig. (.b) sows a plot of te field data against eqs. (5) and (7), for te same experimental data used on fig. (.a). Agreement is good in tis case too, except in te BR case. Comparison of eqs. (5) and (7) sows tat CL s expression differs from JH s only by a constant. In fact, before CL proposed is expression, Teunissen et al. (1987) ad suggested tat a different value for te constant could correct its prediction ability. CL proposed C 1 =. 59 based on one field result (1 o wind direction case of ASK). Based on fig. (.b), we propose a value of C 1 =.41, wic seems to fit te observational data better, as a wole. To our present knowledge, tis result is also new. 5. CONCLUSIONS In tis paper, we present a study on te eigt of maximum speed-up for flows over low ills under neutral atmospere. Te flow was assumed to be two-dimensional and te upwind velocity profile was considered to be logaritmic. Furtermore, te determination of te function C was made from observational data obtained over te HT; so te result is restricted to tis site. In our analysis, we sow tat l, calculated as te inner layer dept, is indeed te eigt of maximum speed-up, l max. We also propose a value for te function C 1 in CL s expression, eq. (6), and confirm tat JEN s expression gives better results tan JH s. Regarding tis fact, we suggest tat JH s expression sould definitively be substituted for JEN s or CL s (wic seem to work just as well). We also believe tat te BT s results ratify tat te best expression lies between CL s and JEN s, as proposed ere. One ting wort of note is tat te simple demonstration we present ere, sowing tat l=l max, as apparently passed unnoticed over te years. We speculate tat tis is probably due te fact tat comparison between predictions for l max and observations of l as always sowed good agreement, in most cases. It is also wort noting tat te demonstration of te equality l=l max could ave been used itself as a new form of obtaining l. It uses te classical ypotesis about te relative order of inertia and turbulence terms and introduces te requirement tat u =, wic guarantees tat l=l max. Furtermore, it makes no use of turbulence closure models and allows

8 to a re-calibration of te function C in case it is required by new observational data available. ACKNOWLEDGEMENTS Te autors would like to acknowledge te financial support from CNPq, troug grant No /97-5 and FAPERJ, grant No. E-6/171.84/99. REFERENCES Beljaar, A. C. M. and Taylor, P. A.: 1989, On te inner-layer scale eigt of boundary layer flow over low ills, Boundary Layer Meteorology, 49, Bradley, E. F.: 1983, Te influence of termal stability na angle of incidence on te acceleration of wind up a slope, J. Wind Eng. and Indust. Aerodyn., 15, Coppin, P. A., Bradley, E. F., Finnigan, J. J.: 1994, Measurements of flow over an elongated ridge and its termal stability dependence, Boundary Layer Meteorology, 69, Claussen, M.:1988, On te inner layer scale eigt of boundary layer flow over low ills, Boundary Layer Meteorology, 44, Finnigan, J. J.: 1983, A streamline co-ordinate system for distorted turbulent sear flows, J. Fluid Mec., 13, Frenen, P. and Voguel, C. A.: 1995, On te magnitude and apparent range of variation of von Karman constant in te atmosperic surface layer, Boundary Layer Meteorology, 7, Jackson, P. S., and Hunt, J. C. R.: 1975, Turbulent wind flow over a low ill, Quart. J. Roy. Meteorol. Soc.,16, Jensen, N. O., Petersen, E. L., Troen, I.: 1984, Extrapolation of mean wind statistics wit special regard to wind energy applications, Rep. WCP 86, World Meteorol. Organ., Geneva, 85 pp. Kaimal, J. C. and Finnign, J. J.: 1994, Atmosperic boundary layer flows: teir structure and measurement, Oxford Univ. Press, New York, 89 pp. Mickle, R. E., Cook, N. J., Hoff, A M., Jensen, N. O., Salmon, J. R., Taylor, P.A., Tetlaff, G. and Teunissen, H.W.:1988, Te Askervein ill project: vertical profiles of wind and turbulence, Boundary Layer Meteorology, 43, Taylor, P. A., Mason, P. J and Bradley, E. F.: 1987, Boundary layer flow over low ills (a review), Boundary Layer Meteorology, 39, Teunissen, H. W., Sokr, M. E., Bowen, A. J., Wood, C. J. and Green, D. W. R.: 1987, Askervein ill project: wind tunnel simulations at tree lengt scales, Boundary Layer Meteorology, 4, 1 9. Walmsley, J. L and Taylor, P. A.: 1996, Boundary layer flow over topograpy: impacts of te Askervein study, Boundary Layer Meteorology, 78, 91 3.