DOE/NASA/1059-79/4NASA TM-79275

{NASA-TM-79275) MODIFIED POWER LAW N80-13623

EQUATIONS Fu_ VERTICAL WIND PBOFILES (NASA)

: : 13 p EC A02/MF A01 CSCL 10AUnclas

' " ,, G3/4q _6296

, ;. MODIFIED POWER LAW EQUATIONS

FOR VERTICAL WIND PROFILES

D. A. Spera and T. R. RichardsNational ,a.eronaut,csand Space AdministrationLewis Recearch Center

Work performed forU.S. DEPARTMENT OF ENERGYEnergy TechnologyDistributed Solar Technology Division

Prepared forWind Characteri,,':icsand WandEnergy Siting Conferencesponsored by the U.S. Department of Energy, the American

Meteorological Society, and the Pacific Northwest LaboratoryPortland, Oregon, ,June 19-21, 1979

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1980005367

https://ntrs.nasa.gov/search.jsp?R=19800005367 2018-06-23T13:23:03+00:00Z

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DOEINASAii 0.59-i914NASATM-79275

MODIFIEDPOWERLAWEQUATIONS

': FORVERTICALWINDPROFILES

D. A. SperaandT. R. RichardsNationalAeronauticsandSpaceAdministrationLewisResearchCenterCleveland.Ohio 4413.5

WorkperformedforU.S. DEPARTMENTOFENERGYEnergyTechnologyDistributedSolarTechnologyDivisionWashington,D.C. 20545UnderinteragencyAgreementE(49-26)-1059

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WindCharacteristicsandWindEnergySiting CoDferencesponsoredby the U.S. Departmentof Energy,the AmericanMeteorologica]Society,andthe PacificNorthwestLabo,'atoryPortland,Oregon,June 19-21,1979

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MODIFIED POWER LAW EQUATIONSFOR VERTICAL WIND PROFILES

D. A. Spera and T. R. Richards

National Aeronautics and Space AdministrationLewis Research Center

Cleveland, Ohio, 44135

ABSTRA_

Equations are presented for calculating power law exponents from windspeed and surface roughness data. Results axe evaluated by comparison withwind profile date measured at a variety of sites.

INTRODUCTION

The power law equation is a simple yet useful model of the verticalwind profile which was first proposed by Hellman (1916), according to Simiuand Scanlan (1978). The general from of this equation is

V2 = Vl{Z2/Zl )_ (1)

_Tjwhich _i and V2 are simultaneous steady wind speeds over level ter-rain at elevations zI and z2, respectively. The exponent _ is deter-

_ined experimentally. For example, early work by Von Xarman (1921} showedthat under certain conditions _. is equal to I/7, indicating a correspon-dence between wind profiles and fluid flow over flat plates (Schlicting,1968). In the general case, however, 0L is a highly variable quantity.Sisterson and Frenzen (1978) measured wind profile exponents which changedfrom 1/7 during the day t_ 1/2 at night, over the same terrain. Golding(1955) describes _ as an exponent which varies with height, time of day,season of the year, nature o£ the terrain, wind speed, and temperature.:bst investigators agree that a constant value of _ is an oversimplifi-cation, and that 0_ must be treated as a statistical parameter.

In spite of the variable nature of the exponent _, Equation (1) issimple enough in form to permit adding some of the factors listed by Golding

without losing the engineering convenience of the power law. Two of thesefactors will be considered here: (1) the nature of the terrain, tn terms of

its surface roughness, and (2) the wind speed.

Up until now, the effects of wind speed and surface roughness have onlybeen considered separately. As shown in Figure I, several functional rela-

*.ionshipshave been proposed for the variation of the exponent (_ with windspeed. These functions are the constant 117 law, a step function (Fales,1967), a linear functim_ for win_ speeds exceeding 27 meters per second

(ASCE, 1961), a pvwer fu_ction (Fichtl _ Smith, 1977), and a logarithmicfunction (Justu: _ Mikhs_l, 1976). Ali of the variable functions show a

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decrease in the exponent with increasing wind speed, but none includes theeffect of surface roughness. Frost et al (1978) and Justus (1978) giveequivalent values of the exponent g and the well-known surface roughnesslength Zo, but without specifying the wind speed.

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Thus, information on the separate effects of wind speed and surfaceroughness is available, but a wind profile model which combines these twoeffocts is lacking. The objective of this paper, then, is to present equa-tions relating the mean value of /_. to both the surface roughness and thesteady wind speed. Other factors are assumed to be responsible for statis-tical variations about this mean value.

The equations presented in this paper are exploratory in nature and: should be reevaluated with more data, when such data are available.i

SUMMARYOF EQUATIONS

The Justus-Mikhail equation shown graphically in Figure 1 was selected

as a starting point for this study because of its relative simplicity andthe fact that it has been shoan to be compatible with wind distributiondata. This equation for the wind profile exponent _ in Equation (1) canbe written as follows:

= 0.37 - 0.203 log Vr (2)

in which Vr is the wind speed at a reference evaluation of 10 meters.

Equation (2) can be generalized to include a surface roughness para-=eter and statistical variations. Assume _. is a random variable which is

;;ormally distribt_ted about a mean value _ with standard deviation _'.Assume further that both _ and I/" are functions of the steady (non-

gusting) wind speed and _. is also dependent on the surface roughness.iiaen, a proposed equation for _ , based on Equation (2) is

1 - log Vl/lOg Vh (3a}

1 - 0(o log CZl/Zr}/log gh

and

_'o = (Zo/Zr)0"2 (3b)

in which

_o surfac e roughness exponent

Vh homogeneous wind speed (_ _ 0), m/s

V1 steady wind speed at elevation Zl, m/s

Zr reference elevation, 10m

Zo surface roughness length, m

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Equation (3b) is _ e_irical relationship _ich will be discussedlater. _e standard deviation _(Vl) must be determined from site data.

Equations (3) are s_o_ gr_hicaJly in Fibres 2 _d 3, The te_ain! descriptions in Figure 3 follow Frost et al (1978} and Justus (1978).

Weibull distribution par_eters C _d k and the wind profile para-

meters _o _d Yh are r_lated, as sho_ by the following equations:

• in _ich

P(VI_V) probability that the steady wind speed _ elevationz 1 will exceed V

C1 Weibull scale factor at elevation z 1, m/s

k 1 Weibull shoe factor at elevation z 1

Weibull distribution patterers at an elevation z 2 cm be calculatedfrom those at elevation z 1 by means of the following equations:

_C,1C2 = Cl(Z2/Zl) (Sa_

I - log el/lOg Vh(Sb)

-c,l --o 1 - _o log (Zl/Zr)/log Vh

1 - _o log (Zl/Zr)/log Vh(Sc)

k2 = kl 1 - _o log (z21Zr)/log Vh

The Justus-t:ikhail equation, (2), is a special case of Equation (3a)

with 0Co equal to 0.37 and Vh equal to 67 meters per second. Equationsequivalent to Equations (3a) and (S) with these specific v_lues of b_o and

Vh were also used by Justus et al (1976) to calculate reference wind speeds.

CALCULATED RND OBSERVED WIND PROFILES

Mean Wind Speed Profiles

Wind profile data (WSSI, 1976 to 1978) from wind turbine sites or poten-tial sites selected by the Department of Energy were used for a preli_inaryvvaluation of Equations (3). Table I lists calculatea roughness exponents

_ for five such sites and also for the Justus-Mikhail reference site.

A homogeneous wind speed of 67 meters per second was assumed for all sites

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in this prelininary evaluation. Figure 4 illustrates the procedure forcalculating go, using data for Clayton, N_4. Exponents calculated frommonthly average wind speeds at several elevations are plotted versus thelogarithm of the reference elevation wind speed, Vr. A straight linethrough the centroid of these data points and the homogeneous point(Vr = Vh = 67 m/s and _. = O) defines _o, at a wind speed of one meter persecond.

The roughness exponents in Table I vary from 0.1 to almost O.S, inqualitative agreement with the site characteristics. Higher do valuesoccur at rougher sites and lower values at smoother sites. The Justus-Mikhail constant of 0.57 appears to be somewhat too large to be used as anaverage roughness exponent. However, more profile data would have to beanalyzed before a better reference value can be established. Based on thesepreliminaz7 calculations, an average _o of about 0.50 is indicated.

Equation 5[b) is an empirical relationship between 0_o and Zo, whichis the conventional measure o£ surface roughness. As shown in Figure 5, theexponent of 0.2 was obtained by curve-fitting equivalent values of 0_ andZo from the literature (Frost, et al, 1978, and Justus, 1978). Calculated

values of 0_o from Table I then established the coefficient as (zr) -0-2.

Peak _ind Speed Profiles

As shown in Figure 2, the proposed profile model is based on the assump-*.ion that wind profiles become uniform or "homogeneous" at high wind speeds,_¢or all values of surface roughness. This concept represents an extrapola-tion of the Justus-Mikhail equation as originally proposed, and additionalv.orification is required. To do this, profile exponents were calculated formonthly peak winds measured at 11 DOE sites (WSSI, 1976 to 1978), to deter-mine if there was a correlation between high winds and low profile exponents.

Typical results of the peak wind analysis are shown in Figure 4, forthe Clayton NM, site. Values of O_ calculated from monthly peak windswhich averaged 23 meters per second were found to be significantly lowerthan exponents calculated from monthly average winds of about 6 meters per.'ccond. The assumed homogeneous wind speed of 67 meters per second is con-sistent with the Clayton peak wind data.

Table II summarizes the analysis of peak wind profile data. This tablelists mean peak wind speeds, mean profile exponents, and standard deviations,for each of II DOE sites. A co._posite average is also given which indicatesthat mean profile exponents less than 0.07 can be expected at wind speedshigher than 22 meters per second. To predict extreme wind loads, the "meanplus 3_'" value of _ would be used, which is 0.27 for the composite averagesite. II_is compares closely wJ th the ASCE recommended value of 0.50 shown inFigure 1_ for a wind speed of 22 meters per second. Thus, equations [3a) and(3b) are representative of the mean vatucs of the wind profile exponent_ andshould not be used in extreme winds ana]ysis. More conservative values shouldbe used.

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Mean profile exponents in Table II are shown graphically in Figure 6.Also shown in this figure are mean profile exponents calculated from monthlyaverage wind speeds measured at five DOE sites (see Table I). A band repre-senting predicted values of _ for flat, open terrain is also shown forcomparison. At low wind speeds surface roughness is an important considera-tion in determining _(. Exponents for the two rough sites (Pltun Brook andBlock Island) fall abcve the band, and that for a smooth hillside site(Culebra) falls well below. At high wind speeds surface roughness is nolonger a significant variable and mean exponents for all sites are low.

CONCLUDING REMARKS

Equations have been presented and evaluated for a wind profile modelwhich incorporates both roughness and wind speed effects while retaining thebasic simplicity of the Hellman power law. Moreover, these equations recog-

nize the statistical nature of wind profiles and are compatible with exist-ing anal)rticalmodels and recent wind profile data.

Predictions of energ; output based on the proposed profile equa-tions are l0 percent to 20 percent higher than those made with the 1/7 powerlaw. In addition, correlation between calculated and observed blade loads

is significantly better at hSgher wind speeds when the proposed wind profilemodel is used instead of a constant power model.

It is recommended that statistical analysis of site wind data include a

bivariate distribution of profile exponent versus wind speed. This wouldpermit more accurate calculation of the parameters in the proposed model anJpermit estimates of standard deviation as a function of wind speed. Statis-tical information of this type would be extremely useful to wind turbinedesiglers for predicting wind load spectra, fatigue life o£ components, andenergy output.

REFERENCES

American Society of Civil Engineers (1961): "Wind Forces on Structures,"Trans., 126, Part II, 1124-1198.

Fales, E. N. (1967): "Windmills," Standard Handbook for Mechanical Engin-, eers, 7th Ed., Baumeister and Marks, Eds. McGraw Hill (New Yerk),

9-13.

Fichtl, G. H., and Smith, O. E. (1977): "Wind," Terrestrial Environment

_Climatic) Criteria Guidelines for Use in Aerospace Vehicle Develop-ment, 1977 Revision, J. W. Kaufman, Ed., NASA TM-;'8118, 8.15-8.17.

Frost, W., Long, B. H., and Turner, R. E. (1978): "Engineering Handbookon the Atmospheric Environmental Guidelines for Use in Wind TurbineGenerator Development," NASA TP-i359, 3.15.

golding, E. W. (1955): The Generation of Electricit 7 by Wind Power(London), 80.

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Hellman, 6. (1916): "Uber die Bewegung der Luft in den untersten Schichtender Atmosphere," Meteorol. Z., 34, 273.

Justus, C. G., and Mxkhail, A. (1976): "Height Variation of Wind Speedand Wind _,stribution Statistics," Gcophys. Res. Letters, 3, 261-264.

Justus, C. G., Hargraves, W. R., and Mikhail, A. (1976): "ReferenceWind Speed Distributions and Height Profiles for Wind Turbine Designand Performance Evaluation Applications," ERDA Tech. Report ORO/5!08-76/4, A-21 to A-Z3.

Justus, C. G. (1978): Winds and Wind System Performance, Franklin Inst.Press (Philadelphia), 56-65.

\,Schlicting, H. {1968): Boundary La_er.The0ry, 6yh Ed., McGraw-}lill

(New York), 564.

Simiu, E., and Scanlan, R. H. (1978): Wind Effects on Structures, Wiley(New York), 47.

Sisterson, D L., and Frenzen, P. (1978): "Nocturnal Boundary-LayerWind Maxima and the Problem of Wind Power Assessment," Env. Sci.Tech., 12, 218-21.

Von Karman, T. (1921): Z. Angew. Math. Mech., 1, 239.

Western Scientific Services, Inc. (1976 to 1978): "Candidate WindTurbine Generator Site Meteorological Monitoring Program MonthlyReports," (unpublished).

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Table I. - Surface roughness exponents for five candidate windturbine sites,

Surface _roughness a

i Site Type of Terrain exponent, [_o

Plum Brook, OH flat; high woods 0.48Block Island, RI hilly; medium woods 0.44Reference (Justus and relatively flat 0.37

Mikhail, 1976)

Clayton, NN flat; open 0.30Russell, KS flat; open 0.30

' Culebra, PR shoreline hillside; 0.I0bushes

E = _o(l - log Vr/log Vh), with Vh = 67 m/sa

Table II. - Wind profile exponents for monthly peak winds atcandidate wind turbine sites.

Mean peak Wind profile exponent,_No. of speed at

Site readings 10m, m/s , Hkan' Std.0 _dev''

Amarillo, TX 26 22.3 0.053 0,045Block Island, RI 30 19.5 .I14 .052Clayton, NH 24 22.8 .054 .044

Cold Bay, AK 9 24.1 .042 .044Culebra, PR 30 17.4 .0]0 .078Huron, SD 28 20.6 .094 .056Kingsley Dam, NA 29 24.6 .061 .O73

. Medicine Bow, WY 5 29.0 .077 .027Point Arena, CA 23 20.1 .072 .144Russell, KS 29 23.4 .059 .040

San Gorgonio, CA 33 23.5 .088 .053

Composite Site 266 22.2 0.067 0.068

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0I 2 q 6 8 I0 20 qO 60 80 100

WINDSPEEDATI0M,V_,M/S

Figurel - Various recommendationsfor the relationshipbetweenwind profileexponent and wind speed.

0.6 OL-Olo(1-_5L06v,)

0,5

MEANWIND O.qPROFILEEXPONENT,

E 0.3

0I 2 q 6 8 ]D 20 qO 60 80 100

WINDSPEEDAT 10M,VR,MIS

Figure 2 - Graph of proposedequation for wind proflleexponentin terms of surface roughnessand wind speed.

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1,0 D

0.8- /

SUBURBIA-._/,

// "'-CITY

0.6-SURFACE PALMETTO-,.,y/,(.

HIGHGRASS-_ _'"Y//v"_HIGNROUGHNESS gOODS

EXPONENT, OA-- LogGRASS,STEPP_

0_, i _WN GRASS-_, i._w_._"-- LOWW_DS,L SAND_ _"_,_(_ --FALLOWFIELD

10"5 10"4 10-3 10.2 I0"1 1 10

SURFACEROUGHNESS_NGTH.Zo,M

Figure 3 - Graph of relationshipbetween surface roughnessexponent _o'surface roughnesslength, zo and terrain description.

• - MONTHLYAVERAGEWINDDATA

0.4-- • MONTHLYPEAKWINDDATA j" WSSI,CENTROIDO_MONTHLYAVENGES 1976-78

0.)

_• /r _ - 0.30(I-0.55LOGV_)0.2-

/

WINS . •

PROFILE ,,"EXPONENT, 0.i "

0 I I I I 1 'l _" I .-_',L I I

2 _ 6 8 10 20• • _0 60 80100

W;NDSPEE])AT 10 M,VR,WS-0.1

Figure4 - Comparisonof observed and calculatedwind profileexponents for Clayton, New Mexico site.

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[_ } NEUTRALSTABILITY0 V- S-lOWS

: '_Jl: A _oN,_ j

I: .qO Zn

WIND

,0_ I ! I ] I

10.q 10-3" 10-2 10"1 1 10

SURFACEROUGHNESSLENGTH,Zo: Mu.

Figure 5 - Curve-fit of data relating wind profile exponent _(.to surface roughness length.

O,S iA - BLOCKISLAND,I_I•- PLUMBROOK,OHe- CLAYTON,N_

0,_ -- _'- RUSSELL,KSD- CULEBRA,PR

.. MONTHLYAVERAGE G-OTHER(TABLE2)

WINDEXPONENI,

0,2 '.i_ MONTHLYPEAK

FLAT,OPEN--//__i!i:_!_'_WI_S--

l I I I I I_1 . I _'ll_ i 10

1 2 q f) 8 10 20 qO r_ 80 10o

W'NOSPEEDAT10' _, VM,WS

Figure 6 - Comparison of observed and calculated wtnd p_oftleexponents for vartous sttes.

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