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An estimate of the interaction of windmills in widespread arrays
Templin, R. J.
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https://nrc-publications.canada.ca/eng/view/object/?id=94d3fbff-cb95-4967-8926-ebb4494d1578 https://publications-cnrc.canada.ca/fra/voir/objet/?id=94d3fbff-cb95-4967-8926-ebb4494d1578L TR-Lft-
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National Research
Council Canada
Conseil national
de recherches Canada
NLIMITED
APR
3 1979
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NATIONAL AERONAUTICAL ESTABLISHMENT
LABORATORY TECHNICAL REPORT
LTR- LA- 171
AN ESTIMATE OF THE INTERACTION
OF WINDMILLS IN WIDESPREAD ARRAYS
R.
J.
TEMPLINRAPPORT TECHNIQUE DE LABORATOIRE
ｬＮｾｏｱｬｾ＠
ETABLISSEMENT AERONAUTIQUE NATIONAL
DECEMBER 1974
NATIONAL AERONAUTICAL ESTABLISHMENT ETABLISSEMENT AERONAUTIQUE NATIONAL PAGES PAGES FIG.
23
5REPORT
RAPPORT
DIAG . LABORATORY/ LABORATOIRE TABLES TABLES FOR POUR REF'ERENCE RUERENCEInternal
LTR-
LA - 171
An Estimate of the Interaction
of Windmills
in Widespread Arrays
SUBMITTED BY
R
J
T
1 .
PRtsENTt ｐａｒＭＭＭｾｾﾷｾﾷｾｾ･ｾｭｾｐｾｬｾｮ＠ _ _ _ LABORATORY HEAD CHEF DE LABORATOIRE APPROVEDF R
Th
t ａｐｐｒｏｕｖｅＭＭｾｾﾷｾﾷＭｾｾｕｾｲ｟ｓｾ｟｟｣ｏ｟ｮ＠ _ _ _ __ OIR[CTOR OIRlCti[URg:
ｾＺ＠
__
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_9_7_4 LAB. ORDER COMM. LAB ·-Ｖｬｾ］＠
B 1 E R _ _ _ 4:...;Q::....:...l...:.l_-...=,9 COPY NO. CO'!E NR . _ _ _ _ _---...
CONTENTS
List of Symbols
1. Introduction
2. Theory
3. Computation of Power Reduction Due to Sheltering
3.1 The ｒｯｵｾｨｮ･ｳｳ＠ of Natural Terrain
3.2 Relative Power at 100 ft. Altitude
I+. Conclusion
5. References
B h k s
k
sos
u
X z a K p T T 0 LIST OF SYMBOLStot al windmill frontal area (ft.2)
a constant i n logarithmic velocity profile equation refere nce height of wi ndmill discs (ft.)
combined ro ughness height of windmill array (ft.) ro u g hn es s height of original terrain (ft.)
te rrai n surface area (ft. 2 ) mean wind velocity (ft./sec. 2 )
me a n wind velocity at height h (ft./sec.)
mean wind velocity at terrain roughness height (ft./sec.)
friction velocity
ｾ＠
(ft./sec.)g radient wind velocity (ft./sec.)
2.5
l n(k
so/ k )
s+
8.5
height above ground level (ft.)
exponent in Davenport's power-law velocity profiles wi nd shear layer depth (ft.)
v o n Karman's constant
w i ndm i l l density parameter A/S air dens i ty (slug/ft. 3 )
friction drag ( l b . / f t . 2 )
SUMMARY
A theoretical est i mate is made of the effect of windmil l spacing in a large-area array on the mean wind velocity-hei g ht pro f ile within the array, and on the reduction of power available due to mutual interference. The results indicate that if the spacing between individual power un i ts is more than about 30 rotor d i ameters the power r eduction per machine wiJ. l not exceed 5 to 10 percent, but as s pac i ng is further reduced there is a rapidly increas-i ng increas-i nterference, wincreas-ith a correspondincreas-ing loss increas-in avaincreas-ilable
energy. At about 30 diameters spacing a large-area array
converts wind energy to shaft power at a rate · that is only a few percent of the average rate of conversion from solar to wi nd ener g y over the same surface area, and thus should not a f fect significantly the wind environment, or the solar-to-wind energy balance.
1.0 INTRODUCTION
The density of air is roughly 1/1000 of that of water, and thus kinetic energy in natural winds is disappoint-ingly dilute. Although in theory i t is there for the taking, large machinery is required for its conversion to useful shaft
power. For example, a windmill of about 100 ft. diameter is
required to generate an average power of about 50 kilowatts
jn a good, windy location. A conventional hydro-electric
generating plant of modest size by present standards, say 1000 megawatts, is thus the equivalent of about 20,000 100-ft. wind-mills.
The economic feasibility of wind energy conversion on this sort of scale has not yet been proven, but is being
studied in several countries. Research on promising new wind
turbine concepts is also in progress (see for example, Reference 1). An ohvious question arises early in any consideration of
large-scale systems: how close together can large windmills be spaced,
in order to minimize the cost of interconnection, without
incurring unacceptable losses in power due to mutual aerodynamic
interference? In this note a method of analysis is described,
and is used to estimate the effect of windmill "density" on the mean wjnd shear layer velocity profile within an array of
indefinitely large extent, and hence the effect on the power available from each machine.
2.0 THEORY
A high-speed vertical axis rotor of the type des-cribed in Reference 1 generates most of its useful power in a f . a r .. i ] y narrow horizontal band centred about the turbine equator .
Although all wi nd tunnel measurements so far have been carried out i n un ifor m flow, i t is assumed in the following analysis
that the maximum power delivered in a shear layer i s proportioned to the cube of the local wind velocity at a height h above ground level, correspond i ng to the height at . the centre of the rotor, i . e . at the rotor equator. The wind tunnel measurements have also shown that, when maximum shaft power is being delivered, the rotor drag coefficient, based on total swept area and ambient wind dynamic pressure, is close to unity.
In a nalysing the effect of a large number of turbine
rot or s on the local wind velocity profile several simplifying
assumptions have been made. The rotors are treated as an array
of large-scale roughness elements added to a terra i n surface
that is already rough. The situation is sketched in Figure 1.
T he geometric pattern of the turbines in the array is not
specified, because i t i s assumed that they are sparsely distr ib -uted so that their i ndividual wakes have been smeared out by large-scale shear layer turbulence well within the average spac i ng between adjacent units. This does not imply that the local veloc i ty felt by each individual turbine has been restored
to i ts ori gi nal undisturbed value, and in fact i t is the
purpose of th ts analysis to estimate the mean velocity deficit
resulting from the drag of the entire array. The extent of the
turb i ne array has been taken to be indefinitely large, both in
the windward a nd in the cross-wind direction. Since the array
fetch length is thus unspecified, i t is not possible to estimate
the effect i t has on thickening the shear layer. This
two separate, and opposin g assumptions, as explained in Section
3 . 2 .
It is assumed that the wind velocity profi l e is g iven b y the usual l o g -law relation, which has been shown to be app l i c a b l e f or sm a l l scale flows such as those in rou g h pipes and o ve r rou g h p l ates (Reference 2 ) and in the natural surface w i nd shear l ay er (Reference 3). This law may be expressed in the form g i ven by Schlichtin g (Reference 2)
l.
ln z+
B
K k (1)
s
Whe r e U i s the local mean wind velocity at hei g ht z above the sur f ace and U* i s the so-called fr i ction velocity, defined to be
ｾ＠ T is the surface friction force per unit area and p is air
density. K is the von Karman "constant" and is usually taken to
h a ve a numeric a l value of 0.40. The reference hei g ht k is the
s
effective hei g ht of the surface roughness. B is a constant, pro-v i ded that the roughness Reynolds number U* k is g reater than a
--=-u:--'s=- .
certain cr i tic a l value. In the case of natural surface winds the rou g hness Reynolds number is much greater than this critical va l ue.
E quation (1) has been found to be in good a g reement w i th w i nd shear layer and with rough surface bounda r y layer
veloc i ty measu r ements from a height z close to the roughness height k , up to a fairly lar g e fraction of the shear layer
s
thickness, provided that the layer is in thermally neutral
equil-ibrium. This l ast cond i tion tends to be approximately satisfied
at hi g h wind speeds such as those of interest here.
The "const a nt" B is not a universal constant, since i ts numerical value depends in part on the definition of the
shown that the velocity profiles obtained in the classical
experiments of flow in roughened pipes by Nikuradse can be fitted accurately by equation (1) with B
=
8.5, when the roughnessheight k is defined to be the actual size of the uniform sand s
grains used by Nikuradse. In his experiments the sand grains
were packed tightly together in a single layer, and the ratio of
k to the actual roughness height, defined by Schlichting to be
s
unity for Nikuradse's sand roughness, will in fact vary when roughness spacing is changed, and also with the shape of the individual roughness elements.
Schlichting's empirical form of the log-law relation is thus:
u
z=
2.5 lnk
+
8.5u*
s( 2)
It is assumed here that equation (2) accurately describes the shear layer velocity profile over natural terrain
(for which the effective roughness height k is assumed to be
so
known) and also the velocity profile after the wind turbine
array has been installed. In the latter case the roughness
height k is assumed to have been increased by the presence of s
the wind turbines, and our task is essentially complete if we can estimate the ratio k /k
8 so
The addition of the windmills to the original terrain will reduce local average wind velocities, and hence will reduce the friction drag over the terrain surface between
the windmills. Let us make the somewhat arbitrary assumption
that the terrain friction varies as the square of the local
terrain sur f ace friction as , 0 • With no windmills installed,
equation (2) will g ive the value for the velocity Uk at the so
terrain roughness height if we set z
=
k s uksou
=
8.5*
u*=J_!f
But, Puz
t 0 "' kso 8.52 nncl hence, k so ( 3)and we are assuming that, when the windmill array is present, equation (3) will g ive the friction over the terrain between
the windmills. The total friction drag per unit terrain area is
t
0 plus a term that takes account of the windmill drag.
For an array of windmills having a total rotor swept area A, the added drag is
(4)
where Uh is the local mean wind velocity at ｴｾ･＠ effective rotor heir;ht h. This assumes a rotor drag coefficient of unity, as
discussed previously. If the windmill array occupies a terrain
surface of area S, then the windmill drag per unit of ｴ･ｾｲ｡ｩｮ＠
area is D .e._ uh 2 ;., (5)
s
2where I, is the windmill "density"
A/S.
The total terrain surface friction per unit area,
T , i S T
=
-D+
Tos
u 2;.,
u2
£. P kso (6)
2 h
+
8. 52nut, by definition T =
pu;
where U* is now the friction velocity of the combined mean velocity pro f ile over the windmill landscape.
Thus by pU* , 2
Equation (2) will give the two velocity ratios by
substituting first z
=
h and then z kso uh
Ｈｾｊ＠
2.5 ln
ｾｨ＠
k )
2.5 ln+
8.5 so u* so ｘｾ＠+
8.5 u2.5
ｬｮＨｫｾＺＩＫ＠
8.5
and kso u* 1 A 2 Whenｾ＠
.5ln
ｾＺＺ＠
8 : : : : ) • :b: : :
r
t:
d
[t
i
01
:7
｣ｾＺ＠ ｾ＠
b:
c:
r
8 (8 )
k so This equation, a quadratic in terms of ln ks
can be
solved for the roughness height ratio k /k as a function of the
s so
windmill area density parameter
A
and the windmillheight-to-terrain roughness ratio h/k .
so The ratio k /k s so is the ratio of the "new" effective roughness height (for windmills
+
original terrain) to the original roughness height of the terrain only. From i t we can calculate, using equation (2), the change in the velocity profile due to the windmill array.In view of the crudity of the assumptiorn that have been made, some sort of comparison with experimental data is
desirable. In Appendix A the theory has been used to estimate
the roughness height of bluff roughness elements as a function of roughness density for comparison with measurements made by
Schlichting on a variety of simulated rivet-head roughness arrays. As a further check, the theory is used to predict the drag
partit i on between roughness elements and the surface between elements, and the predictions are compared with available data
for a variety of shapes from Reference 3. The comparisons are
approximate only, but indicate a rough agreement between experiment and theory.
3.0 COMPUTATION OF POWER REDUCTION DUE TO SHELTERING
3.1 The Roughness of Natural Terrain
Numerical solution of equation
(8)
to obtain thecombined roughness height k of a windmill array of density
A
8
requires that the windmill reference height h, and the natural
terrain roughness height k be specified.
so
Jn order to he able to compare the effects of two different types of natural terrain upon windmill performance,
two of the velocity profiles proposed by Davenport in Reference
4
were selected. He proposed a set of three standard wind profiles
for use in civil engineering structural design, and these have
now become widely accepted. They are considered to be
represent-ative of natural wind shear layers over open country, rough
wooded country and in city centres, and are described by a power law relationship as
where
u
follows:
u
UG
mean wind velocity at height z
gradient wind speed at the top of the shear layer
shear layer thickness power law exponent.
( 9)
The two profiles we have chosen for reference pur-poses are Davenport's profiles for open country (o = 9 0 0 f t . , a = 0.16) and for rough wooded country (o = 1300 f t . , a = 0.28). Since the theory presented here is based upon the assumption that the velocity profile is given by the logarithmic law, as defined · in equation (2), i t was necessary to convert Davenport's profiles,
by means of a curve-fitting technique, into "equivalent" logarithmic profiles.
If we speed at height
o,
define the gradient wind speed UG to be the then over a terrain of roughness height k
so, equation (2) gives
and at any other height
u
u*
Hence
u
UG
Obviously, and thus the attempt to
2.5 ln ko
+
8.5 so z < 0, 2.5 ln kz+
8.5 so 2.5 ln z+
8.5 k 2.5 ln 0+
8. 5 k so the form of (10) is force them to agree(10)
different from ( 9) , involves some arbitrary choices. We have chosen to force a fit such iliat the values of
the velocity ratio U/UG given by (9) and (10) are equal, or very nearly so, at a standard reference height z
=
100 f t . , and also to provide as close agreement as possible at heights well below 1 0 0 f t . The results of the curve-fitting process are summarized in Table I below.Table I
Type of Terrain Power law eq. ( 9) Log law eq. (10)
0 Ct 0 k so
- -
- -
- -
-Open Country 900 f t . 0.16 1,250 ft. 5 f t .
Rough Wooded Country 1,300 f t . 0.28 10,000 f t . 40 ft. Since the log-law value of shear layer thickness i s substantially larger than the power law value, especially in the rough wooded country case, an alternative f i t , with a much lower value of o and a higher value of k , which s t i l l retained
so
of windmill relative power calculations. The effect of wind-mill density on relative power was found to be insensitive to
this change. A graphical comparison of the Davenport power-law
and the fitted log-law profiles is shown in Figure 2.
3 . 2 Relative Power at 100ft. Altitude
An altitude h - 100 ft. above ground was chosen as a reference windmill height for most of the calculations described
here. It was assumed that the wind power available is proportional
to the cube of the wind velocity Uh at the reference height, and
a dimensionless relative power (Uh/UG) 3 is the quantity computed.
For the combined roughness height ks' the gradient wind
UG
isgiven by equation ( 2) : UG 2.5 ln 0
+
8.5 u* k s and s i milarly, uh 2.5 ln h+
8.5 u* k Hence original s= (
2. 5 2.5 ln I ln h k s 0 k s+
8. 5)3+
8. 5 k h so =(2.5 ln kso+
2:5 ln ks 2.5 lnk
+
8.5 s (11)It is assumed that the reference height h, and the
terrain surface roughness height k are specified, and
so
the simple theory developed in section 2.0 provides a method for
k
so
calculating ln ｾﾷ＠ The wind shear layer depth
oo
over theoriginal terrain surface is also to be specified for the two
remains undetermined, because i t is likely that if the wind-mi l l array f etch l en g th is la r ge, the shear layer depth will be i nc r e a sed above its orig i nal value o o. For this reason two a lternat i ve c a ses have been considered in the computation of r el a t i ve power. F i r s t , i t was assumed that fue shear layer thick-n ess I s thick-not ithick-n c r ea s e d by th e withick-ndm i l l array, athick-nd hethick-nce 6 = 6 o
In e quatl o n (11) , whl c h th e n becom es
ｾｾＩＧ＠
2.5 ln h + X k 3 so 2. 5 ln 0 0 + X k (12) so whe r e x 2.5 ln k so + 8. 5 (13) k sIn the limit of an indefinitely long windmill fetch, i t seemed reasonable to assume that the shear layer depth ratio o / oo would not increase beyond the ratio of roughness heights with, and without windmills, i.e. beyond the ratio k /k
8 so. Thus in equation (11) we set k 0 s
ＨｾＩＧ＠
C's
2.5 T he quant i ty x, 0 0 k so, and the equation becomes
ln h + k X ) 3 (14) so ln 0 0 + 8. 5 k so
defined in (13) above, can be obtained directly by solution of equation (8), which may be rewr i tten:
;,
2 ln k h
+
xJ
3+
so
1
It is a quadratic in x, with the positive solution,
x = - b + \ ] b 2 -4ac (15)
where b 2.5 A ln h k so
ｾ＠
(2 .5
lnｾｽ＠
-1. so cThere are of course two possible solutions to the
quadratic equation. Since we are considering cases where the
windmill height h is always at least as great as the original
terrain roughness height k , b is always positive.
so Since a is
positive, and c is negative for small A, the+ sign in (15) yields a positive value of x and the alternative negative sign would
yield a negative value. Now x is proportional to the computed
wind speed at a height kso' the top of the terrain roughness, and only the positive root has physical meaning.
Using equation (15) to obtain x, and equations (12) and (14) for the alternative forms of the relative power, the effect of windmill spacing density A has been computed for open and he a vily wooded terrain. The results are shown in Figure 3 for a windmill reference height of 100 ft.
I t will be seen immediately that, regardless of the windmlll dens i ty, the power available is much greater, for a given gradient wind speed, over open terrain than over heavily wooded areas. This is in spite of the fact that the centre of
the windmill discs is 2.5 times the terrain roughness height in
the l a t t e r case . There is of course nothing new about this result; i t has long been realized that windmill performance is extremely sensitive to choice of location.
Figure 3 also indicates that there is a rather sharp loss in power i f the windmill spacing parameter A, the ratio of
windmill disc area to terrain area, exceeds a value of roughly This corresponds to an average spacing of about 30
diameters between adjacent windmills in the array. At larger
values of A the extent of the loss depends upon which of the two shear layer thickness assumptions is adopted, but the thresh-old value of
A
i t s e l f is not greatly affected. Although the power available at low values ofA
is much greater over smooththan over rough terrain, the loss in power is also much greater
when windmills are closely packed. The windmill array then
be-comes i t s own "forest" and the importance of the original terrain roughness diminishes.
I t may be of some interest to note that the shape of the curves in Figure 3 suggests that, for each terrain type and wind-mill height, there is a maximum power that can be extracted per
unit of terrain area. As the windmill density
A
is increased, theincrease in windmill area eventually becomes proportionately less than the loss in available power per windmill (i.e. the product
of (Uh/UG) 3 and A passes through a maximum). However, this occurs
for values of A that are of the order of 0.01 or greater, which are about an order of magnitude beyond the "knee" of the relative power curves, and therefore undoubtedly unacceptable from an
economic point of view.
Close spacing of large windmill arrays may be unacceptable
for other reasons as well. One of the potential attractions of
wind energy conversion is that i t is often assumed to have no unfavourable environmental effects, apart from possible asthetic
problems. However, the average rate at which solar radiation is
two watts per square meter of earth's surface area (different sources suggest somewhat different values), and i t must be presumed th a t thls sets a kind of upper limit upon the
avail-ability of wind energy. Calculations based upon the data
presented in Figure 3 indicate that the maximum possible rate of extraction of wind energy in a large, closely spaced windmill array is of the order of 0.4 watts per square meter of terrain area, that is, about 20% of the solar-to-wind conversion rate. On the other hand, at more realistic values of the windmill
density (say A
=
0.001), the total shaft power available perunit of terrain area is not more than about 5 percent of the
solar-to-wind energy supply. It is considered unlikely that this
extraction rate would significantly affect the wind environment
or the solar-to-wind energy balance. In more familiar units,
this limited rate of power conversion works out to be about 200 to 300 k i lowatts per ｾｵ｡ｲ･＠ mile over flat open country under average wind conditions.
I t should be emphasized that the method of analysis described here is limited to arrays of windmills that are of indefinitely large extent in the windward and in the cross-wind
directions. Much c l oser spacings than those suggested above
(i.e closer than about 30 diameters) may be acceptable, both for economic and environmental reasons, in small windmill groups, or in s i ngle rows set across the prevailing wind direction.
4.0 CONCLUSION
In very extensive windmill power-generating systems, such as are now being reassessed in various countries, there
appears to be a limit to the density of windmill spacing that
would be acceptable in large, two-dimensional arrays. In this
note an approximate method is developed for estimating the
mutual sheltering and hence the power loss per windmill. Since
no other estimate, and no directly applicable experimentil data could be found ln the .literature, the results presented here are
unsubstantiated. In Appendix A, comparisons are made between
the theory and certain experimental measurements that are not s t r i c t l y applicable to the windmill problem as such. With these limitations in mind, the following conclusions are suggested.
1. Windmill height in relation to terrain roughness
height is of prime importance in determining available wind power, since power is proportional to the cube of the local wind speed. The choice of suitable locations is thus of primary importance.
2. When the ratio of total windmill disc area to terrain surface area in a large-scale array exceeds ｾｰｰｲｯｸｩｭ｡ｴ･ｬｹ＠
0.001 (equivalent to a mean spacing of about 30 diameters between adjacent windmi l ls) the power available per machine begins to f a l l off rapidly, and i t is suggested therefore that this density is about the maximum that would be economically acceptable in a large system.
3. At a disc-to-terrain area ratio of 0.001, the
power available in typical wind conditions is about 200 to 300
kilowatts per square mile over flat open country. This rate of
extraction is about 5 percent of the average rate at which solar radiation is converted into wind kinetic energy, and so should not react unfavourably upon the natural wind envirnoment.
4. In a country such as Canada with extensive open
-areas and windy coastlines, and a low population density, the theoretical supply of wind energy is comparable ｷｩｴｾ＠ the total national energy needs for the foreseeable future, even if
extraction rates were limited to values as low as 200 kw/sq. mile, The question of feasibility therefore is not one of supply, but of economics. However, in smaller industrialized countries, with high population densities, i t would seem that wind energy
conversion can not offer a solution to their energy problem.
5. The present theory deals only with windmill arrays that are of unlimited extent in two dimensions. Considerably closer spacings than that mentioned above may be economically and environmentally acceptable in small windmill groups, or in single or double rows set across the prevailing wind direction.
•
REFERENCES
1. South, P. and Rangi, R. The Performance and Economics of
2.
the Vertical-Axis Wind Turbine Developed at the National
Research Council, Ottawa, Canada. Paper delivered at the
Annual Meeting of the Pacific Northwest Region of the
American Society of Agricultural Engineers, Calgary, Alberta, October 10 - 12, 1973.
Schlichting, H. Boundary Layer Theory.
1968. McGraw Hill Book Co.
Sixth Edition,
3. Wooding, R.A., Bradley, E.F., and Marshall, J.K. Drag
Due to Regular Arrays of Roughness Elements of Varying
Geometry. Boundary-Layer Meteorology 5 (1973) pp. 285-308.
4. Davenport, A.G. The Relationship of Wind Structure to Wind
Loading. Proceedings of Conference on Wind Effects on
Buildings and Structures, held at the National Physical
APPENDIX A
Limited Comparisons Between Theory and Experiment
The theory developed in section 2.0 considers the addition of relatively large, but widely spaced "roughness
elements" to an already rough surface when the effective height h of the added elements is large in comparison with the surface effective roughness height k and provides a basis for
so'
estimating the combined roughness height k , as a function of s
the density of the added roughness. No directly comparable experimental measurements have been found, but an attempt has been made to compare some of the implications of the theory with available experimental measurements on roughness elements of various shapes and densities added to smooth surfaces.
A
comparison is first made between theoretical values of effective roughness height as a function of roughness density with measurements made by Schlichting for variousarrangements of simulated rivet head roughness.
Secondly, since the development of equation (8)
depended upon a certain hypothesis regarding the partition of drag between the windmills and the original terrain surface, i t is useful to compare the theory with measurements of drag partition for various roughness arrays. Measurements by several investigators were obtained from Reference 3.
The results of the first comparison are shown in
Figure
4.
The theoretical curves, shown as solid lines, were obtained by numerical solution of equation (8) to yield thedensity
A.
Schlichting's experimental results (from Reference 2) ratio k /h as a function of h/k , and of the roughness elementare shown as data points denoting his different roughness element shapes. The roughness density A is, as previously defined, the ratio of the total roughness element frontal area to the total flat surface area. The comparison is not strictly equivalent, for three reasons:
(a) In the theory the height h is the physical height from the surface to the centre of the windmill disc area, whereas in plotting Schlichting's data h was taken to be the
height to the top of his roughness elements. For a more accurate comparison all of the experimental points should be raised some-what in the graph, so that h would correspond more nearly to some sort of centroid of frontal area.
(b) The theory deals with "roughness elements" that are raised clear of the surface. Schlichting's roughness elements were attached to the surface.
(c) The theory considers the roughness elements
as being added to an already rough surface, whereas Schlichting's surface was initially smooth. Thus the comparison becomes less valid as the theoretical value of h/k so is reduced.
About all that can be said for this comparison is that the experimental and theoretical trends are similar. From Schlichting's data i t is obvious that the roughness element shape is an important factor. For example, his spherical seg-ments probably have a low value of the individual element drag coefficient, which we have assumed in the theory to be unity, when based on element frontal area.
At very low values of
A,
the theoretical combined roughness height k must eventually approach k , and for thisreason the theoretical values of k /h in Figure 4 can not be
s
expected to agree with the experimental data at small
A,
particularly when h/k is also small.
so
Reference 3 contains a collection of measurements
of the partition of drag between roughness elements of various
shapes and the surface between the elements. The measurements
were made by several investigators, and in most cases were
carried out in wind tunnels. Figure 5 (adapted from Figure 4
of Reference 3) shows a shaded region representing these
measure-ments. The abscissa of the graph is the quantity ａｾＬ＠ where
A
is, as previously defined, the ratio of the frontal area of the
roughness elements to the flat surface area, and ｾ＠ is a roughness
shape parameter that is a function of the roughness element
height to its dimension parallel to the flow. The value of ｾ＠
is defined in Reference 3 to be unity for roughness elements
for which this height-to-length ratio is unity. In plotting
the calculated curves from the present theory,
ｾ＠
has been takento be unity.
The ordinate in Figure 5 is the ratio of the
rough-ness element drag to the total drag. It is easily calculated
from the present theory by noting that in equation (8) the two
terms on the right-hand side of the equation are proportional respectively to the roughness element drag and to the drag of
Since the sum of
the surface between roughness elements.
these terms is unity, the first term is our required ratio:
ＮＺＺｒＺＺＮＮＺｯＺＺＺＮＮｵＺＺ［｟［［［ｧｾＮＮＮＺｨｾｮＮＮＮＺＺ･ＺＮＮＮＮＺｳＺＺＺＮＮｳ］ＭＮＮＮＺＺＮｅ］Ｍﾷ＠
=-1-=e-=m=-e=-n=--=t---'D'---r_a...wg = _ 2A
[z .
5 1 0Total Drag
(If-X so ks
o)
+
8 5 ] 2 -k- . sThis ratio is, in the present theory, a function of the roughness element density
A
and of h/k so , the ratio of the added roughness element height to the roughness height of the original surface, and since ilie experimental data,represented by the shaded area in Figure
4,
were obtained for roughness elements added to smooth surfaces, agreement between experiment and theory is hardly to be expected for low values of h/kso Even so, the agreement, particularly for values of
A
which are of order of magnitude 10- 3 is not particularly good. Since i t is in this region of the density parameter that the calculated windmill sheltering effects begin to become important i t is obvious that the present theory requires further experimental
confirmation. This could possibly be obtained in wind tunnel
experiments in which model windmill arrays (or some aerodynamic equivalent) are combined with simulated terrain features, and in which the natural wind shear layer is adequately reproduced.
