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The Deaves and Harris ABL model applied to heterogeneous terrain
JOURNALOF
lml nmd Journal of Wind Engineering and Industrial Aerodynamics 66 (1997) 197 214
ELSEVIER
The Deaves and Harris ABL model applied to heterogeneous terrain Nicholas J. Cook Wind Engineering Consultant, 10 Arretine Close, St Albans AL3 4JL, UK
Received 1 March 1996; accepted 10 March 1997
Abstract The mean velocity profile given by the Deaves and Harris model for strong winds is compared with the log-law and the power-law models for terrain with roughness changes and is shown to predict accurately the directional exposure characteristics of UK meteorological stations. The ease of application of the Deaves and Harris roughness-change method to the power-law model is also demonstrated.
1. Introduction The Deaves and Harris (D&H) model for the characteristics of the atmospheric boundary layer (ABL) in strong winds was developed in two stages. In the first stage, a model for the ABL in equilibrium over flat uniform roughness was developed under the auspices of the Construction Industry Research and Information Association (CIRIA) and was published in 1978 [1]. In the second stage, the model was developed to account for multiple step changes in surface roughness, under the auspices of the U K Building Research Establishment (BRE). This development was done to extend the applicability to the kinds of heterogeneous terrain found in nature, and was published in 1981 [2,3]. The D & H model was subsequently adapted into U K design guidance [4,5], the new U K wind loading code [6], the Australian and New Zealand codes and in the E S D U D a t a Items. The D & H model is chiefly known for its "logarithmic with parabolic defect" mean velocity profile equation: ----
,i,n
K
--+5.75 Z 0
-1.88
-1.33
(;,)3
+0.25
,
(1)
where V is the mean wind speed at height z above ground, K = 0.40 is yon K a r m a n ' s constant, u, is the friction velocity, Zo is the aerodynamic roughness length and h is the equilibrium boundary layer height. 0167-6105/97/$17.00 ~) 1997 Elsevier Science B.V. All rights reserved. PII S0 1 67-6 1 05(97)00034-2
198
VJ ('ook.,l. Wiml Eng. Ind..4erodv;t
66 H997) 197 214
But the model also predicts the equilibrium boundary layer height, h:
;,
Lt,
g/,
(2)
where / is the Coriolis parameter, and parameters describing the in-wind component of turbulence, including the turbulence intensity, I,,, and spectral density function, b;,. The extension of the D&H model to step changes in roughness introduced the concept of a transition region between the outer original boundary layer and a new internal boundary layer, giving equations for the height of the top and bottom boundary of this region. The internal boundary layer, below the transition zone, is assumed to be in local equilibrium as described by the local friction velocity,
;/,,inner : {
ln{zo.om~.rZoi ...... [
't,.i ....... = u, ......... 1
0.42 + In m,~
(3) "
where the parameter m0 is given in turn by the solution to mo=
0.32X :o.i .......{In mo
11
(4)
where X is the distance downwind of the change of roughness, imwr refers to the new inner boundary layer and outer refers to the previous overlaying boundary layer.
2. Comparison with log-law and power-law models 2.1. Preamble
Although the D&H model has been widely accepted in the UK, Australia and New Zealand, becoming incorporated into the respective codes of practice, much of Europe has continued to use the log-law model, while North America and Japan have continued to use the earlier power-law model. Comparisons between these models have been made and arguments for and against have been rehearsed many times. The arguments are reprised again below, but ensuring for the first time that a number of important conditions for valid comparison have been met properly. 2.2. Similari O, theory
Similarity theory predicts that
!:i . ! In
t as= --0
t5)
N.J. Cook/J. Wind Eng. Ind. Aerodyn. 66 (1997) 19~214
199
and V ~ VG and
dV dz ~0
as z ~ h ,
(6)
where VG is the geostrophic wind speed, representing the lower and upper boundary conditions, respectively, to the ABL. 2.3. Log-law model The log-law model assumes that Eq. (5) holds over a suffÉciently useful depth near the surface, giving the log-law mean velocity profile equation V
u, i n ( i " ]
\zo/
K
(7)
It is clear that the log-law velocity profile meets the lower boundary condition, but has no upper boundary at all. It is a two-parameter model, with u, providing the velocity scale (slope of log-linear plot) and z0 providing the length scale (intercept of log-linear plot). It is accepted that the log law is a poor model for the mean velocity profile at large heights, typically for z > 200 m. This causes a difficulty in translating the wind speed over one form of roughness, e.g. open country, to that over another, e.g. urban, because the mean wind speed at the top of the boundary layer cannot be used. In practice, such translations are made through empirical "exposure factors" based on measurements made near the ground, typically at z = 10 m, over both types of roughness. To extend the height range of the log-law model up to about z = 300 m, ESDU introduced a linear deviation term to their 1976 Data Item [7]: =
in
,8)
where the value of K was empirical and varied over the range 4 < K < 7. This linear term, which represents the deviation observed at moderate heights in the neutrally stable ABL, should not be confused with the linear deviation observed in non-neutral conditions which is more pronounced and occurs closer to the ground. 2.4. Power-law model The power-law model uses an empirical formula for the mean velocity profile of the form:
<9) g~ef = \ Z r e f /
where Vref is the mean wind speed at some arbitrary reference height Zref and ~ is the exponent of the power-law.
200
\'.,I. Cook./. ~4:ind EnN. lnd. Aerodvp~. 66 (1997) 197 214
The power-law profile does not meet the lower boundary conditions and also has no upper boundary. Despite its appearance, it is effectively a two-parameter model, since the parameters Vr~r and Zr~.~combine as the factor Vret / Z~.r to scale the velocity, while the exponent sets the shape of the profile (slope of log log plot). That is, for any given terrain, the power-law model defines a mean velocity profile of fixed shape which passes through the reference values. Unlike the log-law model, the power law remains a good model for the mean velocity profile at moderately large heights, which is the major reason for its preference. although it becomes poorer very close to the ground. The power-law profile fits best over the range of moderate heights 30 in < z < 300 m. It is recognised that fitted values of the exponent, z~, vary with wind speed and height-range of fit, but depend principally on the roughness of the ground surface. In practice, the exponent, ~, is often regarded as equivalent to the aerodynamic roughness length, zo. Although it does not recognise the upper boundary, translation between terrain of different roughness is performed through commonality of the wind speed at a notional "'top" to the profile, the "gradient height" zg. When the progress of this concept is traced through its many years of application, particularly through the many papers of A.G. Davenport, it is seen that the value initially adopted for "gradient height" was quite small, zg v 300 m, but rises through the years as more was discovered about the characteristics of the ABL until, presently, values as large as zg ~ 1200 m are commonly used. There is no doubt that the power-law model is convenient to use, particularly in wind tunnel modelling applications, for a number of reasons. These include ease of plotting and fitting when calibrating simulations {simply plot the wind speed against height on log log paper and draw a straight line through the points), ease of integration with height and independence from linear scaling. This latter property stems from the self-preserving shape of the power-law profile. In the early days of "velocity-profile-only" simulations, this independence from linear scale meant that a given terrain class could be represented by a single wind tunnel simulation over a wide range of linear scale factors. Contemporary simulation methods recognise the importance of the turbulence length scales, through which a unique linear scale factor is derived for any power-law-based simulation. The power-law profile equation can be ~'disguised" in terms of the log-law parameters, zo and u,, thus, u,
. ,
( I
()1
but this is merely cosmetic. The "'extra" parameter C is constrained by the reference parameters to be
so that the scaling factor remains Vref/2¢e l- and the log-law parameters have no distinctive role in the model. However, the formulation of Eq. (10) facilitates comparison with the log-law model in terms of thc non-dimensional velocity V/u,.
N.J. Cook/,L Wind Eng. Ind. Aerodyn. 66 (1997) 197 214
201
2.5. Deaves and Harris model
The mean velocity profile D & H model is given by Eq. (1) and this meets the boundary conditions at both top and bottom of the ABL. The advantage of the D & H model over the others is that it extends the accurate representation of the log-law at small z, through the range of best accuracy for the log-linear and power-law models at moderate z, continuing up to the top of the ABL at z = h. This is the only model of the three that "recognises" the top of the ABL. This means that translation between terrain of different roughness can be performed by commonality of the geostrophic, Va, wind speed at a height above z = h. However, a critical difference of the D & H model is that it has three scaling parameters, zo and u, inherited from the log-law model, and the additional length parameter, h, the ABL height. The value of h is a function of wind speed and latitude, through Eq. (2). Therefore, the D & H mean velocity profile is not a single curve for any given location, but is a family of curves dependent on wind speed, It is, therefore, a condition for valid comparison of the log-law and power-law models that the wind speed is specified so that the appropriate member of the family of D & H profiles is selected. Before moving on to make the comparisons, it is convenient to list in Table 1 the design values of the terrain-dependent parameters for extreme winds in the UK, taken from Ref. [4]. It is a fortunate coincidence that the increase in design extreme wind speed northwards across the U K is almost exactly matched by the increase in Coriolis parameter, f, enabling a constant design value of ABL height, h, to be adopted for each terrain type. In this table, VB refers to the "basic mean wind speed" which corresponds to the 1-in-50-year return mean wind speed at z = 10 m over fiat uniform terrain of Category 2. This varies across the U K from V8 = 20 m/s in Southern England to VB = 30 m/s in the Orkney Islands, i.e. with a "typical" or median value of VR = 25 m/s. Of course, at wind speeds outside this design range account should be taken of the changing equilibrium ABL height, h. The characteristic of the D & H model that stands out most is the prediction for the equilibrium height of the ABL in the design extreme storm which ranges from h = 2210 m over the sea to h = 3250 m over cities (although there are currently no Table I Values of terrain-dependent parameters for the U K Roughness category
0
1
2
3
4
5
Description
Sea
Rough rural 0.I 0.0748 2770 0.861 0.20 0.18
City
0.003 0.0597 2210 1.211 0.12 0.12
Typical rural 0.03 0.0689 2550 1.000 0.16 0.16
Town
Aerodynamic roughness, Zo (m) Friction velocity, u,/Va ABL height, h (m) Exposure factor, SE = Vlo/VB Exponent ~ for 0 < z < 50 m Exponent ~ for 0 < z < 3 0 0 m
Flat rural 0.01 0.0642 2380 1.108 0.14 0.14
0.3 0.0811 3000 0.711 0.24 0.22
0.8 0.0877 3250 0.554 0.32 0.27
2()2
N.,I. Cook,J. Wind Eng lnd. ,4eroctvn. 66 (1997) 197 214
40.00
- x - 5 m/s - o - 10 m/s 20 m/s --n- 30m/s Log law Power law
35.00
30.00
/
/d
~ .-I~
J
25.00
20.00
Olllll | ,a,w,..~IIir
15.00 10
100
1000
10000
z (m) Fig. I. ('ornparison for % = 0.03 m in log-law formal.
cities of sufficient size to achieve equilibriuml. These values are more than twice the currently accepted values used with the power-law model. It is probably this characteristic, above all others, that makes proponents of full-depth wind tunnel simulations of the ABL reluctant to adopt the D&H model, since this implies that the wind tunnel simulations need to be twice as deep than at present 1. In more frequently occurring conditions, VR = 10 m/s, the predicted height is close to 1000 m which matches well with observations. The relationship between wind speed, latitude and ABL depth in conditions of neutral stability is a subject that merits further study and for which sufficient data may already be available in the meteorological record. 2.6. The comparisons 2.6.1. Comparison/br Catego O, 2 terrain The D&H, log-law and power-law models are compared for the UK datum ground roughness, Category 2, over a range of wind speed in Figs. 1 and 2, in each of their respective plotting format conventions. The comparison is based on the UK design
LIn many establishments wherc the D&H model has been adopted as standard, the "'part-depth" simulation method, in which only the lower layers of the ABL arc accurately represented and the majority of the ABL at the top (upper two-thirds) is specilically excluded, is the standard simulation method, and this approach is unaffected by the larger values of h.
A(J. Cook/J. Wind Eng. Ind. Aerodyn. 66 (1997) 197 214
203
40
30
J
I
20
- x - 5 m/s --o- 10 m/s - a - 20 m/s 30m/s Log law - - - - Power law
f[I 10
100
I 1000
10000
z (m) Fig. 2. Comparison for Zo = 0.03 m in power-law format.
parameters in Table 1, where Zo = 0.03 m is taken to correspond to ~ = 0.16. The values of wind speed correspond to VB = Vret at Zret = 10 m, i.e. the values of the log-law and power-law models are matched at the meteorological standard height of z = 10 m. Note how the D & H model forms a family of profiles with increasing depth with increasing wind speed. D & H and power-law models deviate progressively above the log-law from moderate heights. The match between the D & H and power-law models is excellent over the design range of wind speed, 20 m/s < VB < 30 m/s, but becomes progressively poorer at lower wind speeds. However, Fig. 2 shows that all the D & H profiles plot as reasonable straight lines in the power-law format, except approaching the upper boundary where dV/dz =~0, indicating that the models still compare well, but with different values of power-law exponent, ~.
2.6.2. Comparisons at the median UK design wind speed The models are again compared, this time for the median U K design wind speed, V8 = 25 m/s, over a range of ground roughness in Fig. 3, again based on the design parameters in Table 1. The values of the log-law and power-law models for "sea" Category 0 (z0 =0.003 m, ~ =0.12) and "typical rural" Category 2 (z0 =0.03 m, = 0.16) are again matched at z = 10 m, but it seemed more appropriate to match the models for the rougher "town" Category 4 (z0 = 0.3 m, ~ = 0.22) somewhat higher at z = 30 m, since the meteorological standard height would be close to the rooftops. These categories correspond to the "sea", "land" and "town" categories of BS6399 Part 2 [6].
,~'..1. (bok,,/. Wind l'~ng. Ind. Aero~tvn. 66 (1997) 197 214
204
The match between the D&H and power-law models is again excellent, except for z > 500 m in the least rough "sea" category where the power-law profile diverges below the D&H profile at a lower height than before.
2.6.3. Conclusions
The mean velocily profile of the D&H model is seen to compare well with the power-law profile over a wide range of surface roughness and wind speed. The D&H profile is better than the power-law profile close to the ground and near the top of the ABL because it meets the upper and lower boundary conditions predicted by similarity theory.
45 40 35 30
25 20 15 10
5 100
10
1000
10000
z (m) Fig. 3. ( ' o m p a r i s o n for median UK design ~ i n d speed in log-la~s format.
1able 2 Values ol power-law e x p o n e n t m a t c h e d to D & H proliles Aerodynamic roughness zo (m)
1" ~,z', matched at z,~f (m)
I,]~ 5 m/s l(; = 10,8 m,s
I,~ 10 m:s V~ - 22.8 m/s
1,~ - 20 m/s V(; = 47.9 m/s
I/, - 30 m/s t'{~ - 74.0 m/s
0.003 0.01 0.03 o. 1 0.3 0.8
10 lO I0 10 30 50
O. 16 0.18 0.20 0.24 0.27 0.32
O. 135 ().1~ O. 175 0.21 (!.24 0.28
0.125 0.14 0.165 O. 19 0.22 0.27
0.12 0.135 0.155 0.185 0.20 0.25
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205
However, the equivalence between the aerodynamic roughness parameter, Zo, of the D&H model and exponent, ~, of the power-law model is dependent upon wind speed. This equivalence is given in Table 2 in terms of the UK "basic wind speed", V~, and the corresponding geostrophic wind speed, V6.
3. Roughness changes 3.1. Roughness change method 3.1.1. Derivation The D&H method equations for a single roughness change, Eqs. (3) and (4), are quite inconvenient to implement: Eq. (4) because it is non-linear and requires to be solved by iteration, and Eq. (3) because it relates the friction velocity in the new inner layer to that in the outer layer. Why should this latter case be inconvenient? It requires that multiple roughness changes are addressed in sequence in the downwind direction, i.e. from far upwind towards the required site location, because it predicts the friction velocity for each successive inner layer in terms of the previous layer. It would be more convenient to predict the friction velocity from the values for the same roughness at equilibrium, i.e. to work from the site in the upwind direction, until the next roughness change is sufficiently far away to be insignificant. This latter approach is used in the design implementation of the D&H model developed by the author [4], implemented as a computer program [5] and used subsequently, in simplified form, in the 1995 UK code of practice for wind loads [-6]. In this approach, a value of basic mean wind speed, VB, is progressively factored by a series of"S-factors" to reach the desired design value. This mimics the structure of many wind loading codes of practice. The basic mean wind speed corresponds to equilibrium conditions at z = 10 m flat uniform roughness, Zo = 0.03 m, equivalent to typical UK open countryside. The four "S-factors" relevant in this context are the "exposure factor" SE, the "height factor" Sz, the "fetch factor" Sx, and the "direction factor" So: SE --
Sz-
V{z = 10 m} VB
[ln(2550/0.03) + 2.79]ln(10/z0) , [ln(h/zo) + 2.79] In(10/0.03)
V{z}
ln(z/zo) + 5.75(z/h) - 1.875(z/h) 2 - 4(z/h)3/3 + (z/h)4/4
V{z = 10 m}
ln(10/z0)
(12)
(13)
Sx{j~i}
U*,equilibriumU*'l°calln(zo,JZo,i)] ln(lO/zo,i) SE,j -
1
(14)
0.42 + l n m o ] ln(10/Zo,~) SEi'
where V{z = 10m} is the equilibrium mean wind speed at z = 10 m above uniform terrain of the site roughness, V{z} is the equilibrium mean wind speed at height z above uniform terrain of the site roughness, i and j refer to the ith and jth surface roughness upwind of the site, and m0 is given by the solution of Eq. (4).
206
\~.,I. Cook'.l. Wind Eng. Ind. Aerodvn. 66 (1997) 197 214
Table 3 Values of direction factor, S(.),tot the UK from Ref. [4J Wind direction, O 0 30 (~ll 9(t 120 150 1N0 210 240 Direction factor. S¢_, 0 . S l 0.76 0.76 I).77 0.76 0.83 0.89 (t.97 1.05
270 .04
300 330 0.95 0.86
The direction factor, So, is d e t e r m i n e d empirically from o b s e r v a t i o n s [8] a n d is given for the U K -~ in T a b l e 3. In Eqs. (12) (141. the values of the basic p a r a m e t e r s ( C a t e g o r y 2 in T a b l e 1) have been retained, instead of being i n c o r p o r a t e d into global constants, to allow their derivation to be traced. The p e n a l t y for t r a n s f o r m i n g the roughness change f o r m u l a is increased c o m p l e x i t y of form 3. The practical solution to this in design guidance [4] a n d codes of practice [6] has been to t a b u l a t e values of the factors, or to p r e - c o m p u t e the results for a range of s t a n d a r d cases a n d present these as r e a d y - r e c k o n e r tables [9]. Values of Fetch factor, Sx, for three typical changes of roughness are p l o t t e d against fetch, X, in Fig. 4. The fetch is now the distance upwind, from the site to the change of roughness, a n d is therefore the distance over which the roughness has acted on the wind. N o t e that the effect of a s m o o t h to r o u g h change is to increase the local friction velocity a b o v e the e q u i l i b r i u m value, while the effect of a r o u g h to s m o o t h change is to decrease the local friction velocity, and in both cases for this change, to decay g r a d u a l l y back t o w a r d s the equilibrium value with increasing fetch. Use in guidance or codes of s t a n d a r d roughness categories having values of a e r o d y n a m i c roughness, zo, differing by equal fractional p o w e r s of ten (e.g. Categories 0 4 in T a b l e 1) allows further simplification with a small loss of precision. It is found that Sxl~sea --* town I ~ Sxttsea - , l a n d } × Sx{land ~ t o w n } ~ S~{sea --,land}.
(15)
It is a p p a r e n t from Fig. 4 that, a l t h o u g h a building 5 km inside a coastal town (sea ~ t o w n ) m a y be fully i m m e r s e d in the '~town" internal ABL, the wind speed will be a b o u t 10% higher than if the town were well inland (land --*town), because of the increased wind speed in the overlying o u t e r layer. This has implications for the accuracy of codes of practice and design guidance that d o not differentiate between coastal and inland towns. A c c o u n t i n g p r o p e r l y for the roughness changes found with real sites requires i m p l e m e n t a t i o n for multiple roughness changes.
2 Directional factors were determined in Rei. [8] for the design risk of excccdancc in each sector and. therefore, an overall risk greater than the design risk. The direction factor, So, uses these values factored to make the risk for all directions equal to the design risk, but distributed equally by direcLion. BS6399 uses these values factored to give a maximum of unity lat ~'-)= 240 ), so gives a risk somewhere between ~'design risk in directional sector" and "design risk for all directions". The appropriate way of assigning the design risk by direction is still a matter of debale. Unfortunately, Eq. (14) defeated both the typesetter and the technical editor of Ref. E4] and readers may wish to take the opportunity to correct Eq. (9.20) of this reference.
NJ. Cook/~ Wind Eng. Ind. Aerodyn. 66 (1997) 197 214
207
1.6 ..............................
l-llff
............. l . . . . . I. . . . . . . . . . .
--e-- Land
1.5 ~
1.4
-
-
o
-
to town
Sea to town
1.3
.9o o li= U.
1.2 1.1 1
14.
0.9 0.8 0.7 0.6 0.1
10
100
Fetch X (km) Fig. 4. Fetch factors for three typical roughness changes.
3.1.2. Implementation for multiple roughness changes It can be seen that after a sequence of roughness changes n ~ m --, ... --*c --, b --, a from the furthermost upwind roughness "n" to the site roughness "a", the mean wind speed at any height, z, in the innermost "a" layer above the site will be given by: V{z,a} = Sx{n ~ m} ... Sx{c ~ b}Sx{b ~ a}Sz{a}SE{a}SoV,,
(16)
and in the next-level "b" layer by
V{z,b} = Sx{n --* m} ,.. Sx{c --* b}Sz{b}Sz{b}SoV,,
(17)
or, in general terms, for the ith level layer by i
V{z,i} = I~ Sx{j}Sz{i}SE{i}SoV,"
(18)
j=n
This leaves the requirement to determine which individual internal boundary layer controls the wind speed at any given height above ground. A simple and satisfactory approach is to assume that the transition region between layers has zero thickness, so that the velocity profile of an inner layer intersects with the profile of the overlying layer. In a smooth ~ rough change, the wind speed at any height will be the smaller value of the inner and outer profiles, and in a rough ~ smooth change it will be the larger value. This is demonstrated in Fig. 5 for a single smooth --* rough change at VB = 25 m/s.
20g
'V,.I. ('ook ,]. Wind Eng. Ind. Aerodvn, 66 (1997) 197 214
6O
50
- - -
40 30 ~
20' t-
Power
law, town
D&H, sea-town
Power law, sea-town
I I
10 10
100
I
1000
Height above ground, z (m) Fig. 5. Example of smglc smooth +rough~,ua
,town) roughnesschangc.
Fig. 5 is presented in the power-law format to demonstrate that the D&H roughness change method works perfectly well with power-law profiles and therefore provides an effective method for extending the power-law model to include roughness changes. The transition profile has the same slope (exponent) as the equilibrium "rough" profile, but is displaced upwards (scaled by Sx) so that it intersects with the equilibrium "'smooth" profile. The expedient of assuming zero thickness transition layers leads apparently to the prediction of discontinuities in the values of 1" and dV/dz at the intersection of the profiles. It should be noted that these are artifacts of the method and, in reality, there will be a smooth transition. Note that the wind speed axis is in absolute units (m/s), since the three profiles each have a different value of friction velocity. The wind speed could have been normalised against the common value of geostrophic wind speed, Vo, but this would not necessarily be known when implementing the power-law model alone.
3.2. Comparisons 3.2.1. Method qlcomparison Although field data on the ABL velocity profile to any significant height is very sparse, there is a wealth of observations near the surface from anemograph sites. In the UK there are over 100 anemograph stations that have been recording the hourly mean wind speed continuously since 1970. The majority of these anemometers are
N.~L Cook/J. Wind Eng. Ind Aerodyn. 66 (1997) 197 214
209
sited at z = 10 m above open level ground, according to the W M O standard, but some have been raised to compensate for local obstructions to a "10 m effective height", while others are well above the ground for reasons of necessity (London Telecom Tower, z = 195 m). These anemometers have all been assigned a value of "effective height" by the U K Meteorological Office. The "effective height" is the height at which the anemometer would be set to give the same wind speed if the exposure was "standard". This concept is flawed because it limits the assessment of exposure to the terrain in the immediate vicinity and assumes this exposure is uniform by direction. In reality, each anemometer is exposed differently by direction. If the record from each anemometer is divided into convenient 30%wide sectors by direction, then data are available for comparing 12 different exposures at the same site with the corresponding observations from that site. The method of comparison used here is based on observations of the extreme wind speeds. The rationale for this is given in Refs. [8,10] but, in brief, the extreme wind climate of the U K is dominated by Atlantic depressions with tracks which pass over the whole UK. The increase in extreme wind speed from south-east to north-west across the UK is principally due to an increase in depression frequency and not to a change in character. Accordingly, it is expected that the directional characteristics of extreme winds will be constant across the UK. The average from a large number of anemometer sites (50 sites in Ref. [10]) distributed across the UK is expected to eliminate variations in directional exposure between individual sites and, therefore, to represent this climatic variation by direction. Comparisons on this basis have been made before, [4,8]. The comparisons here differ in two respects: • roughness changes have been computed automatically from a database of the surface roughness of every 1 km square in the UK, so are more detailed and accurate in the medium to far field, and • climatic variations by direction have been removed and the comparisons made in terms of the "gain factor" for the anemometer, by direction. This latter transformation is achieved by re-arranging Eq. (18) into 12i Sx{ j} Sz{ a} SE{ a} = SovVB " ~=,
(19)
The expression on the left, which is the "gain factor" estimated by the D & H roughness-change method, may be compared with the expression on the right, which is the "gain factor" estimated from the observations. Here, V is the value of the l-in-50-year return hourly mean wind speed from the analysis of Ref. [10]. There is no method for obtaining direct measurements of VB, SO the value is taken from the best overall fit to Eq. (19); thus,
E V~ -
360 / / - - a
~
~
360 ~ V/S(o)
/
(20)
210
N..L (?ook/J. Wind Eng. Ind. Aerodvn. 66 (1997) 197 214
Because these two estimates of "~gain lactor'" are forced to an overall fit, the comparison must be on the deviations from this fit by direction, i.e. the comparison is relative, not absolute. The advantages of this method are that 12 comparisons can be made for each site and that objective exposure-corrected estimates of Vt~ are obtained as a by-product for improving the UK base map of design wind speed. 3.2.2. Results,lot ten coastal sites in the UK
Ten anemograph stations in the UK that were near the coast, and therefore with exposures differing significantly by direction, were selected for comparison. Five are in effectively flat terrain and five are influenced by significant topography. These are listed below with the National Grid reference and site characteristics: • • • • • • • • • •
Aberporth Benbecula Dyce Hurn Kinloss Leuchars Machrihanish Manston Mount Batten Wick
SN242521 NF764555 NJ878124 SZ117978 NJ067627 NO457202 NR663224 TR335666 SX492527 ND363524
NW-facing coast W-facing coast, small island E-facing coast S-facing coast N-facing coast NE-facing coast W-facing coast SE-facing coast complex coastline E-facing coast
ttat flat mountains to west flat mountains to south flat between two hills hilly terrain summit of hill fiat
The influence of the topography has not been included in the D & H estimates in four of the five affected sites. The exception is the Mount Batten site where the acceleration over the hill has been assessed using the method of BS6399 Part 2 [6] and included in the D & H estimate. The directional comparisons for live of these stations and also for Valley (SH309757) are presented as polar plots in Fig. 6. While this format enables subjective comparison and allows deviations to be related to site characteristics, a more objective assessment of quality of fit is needed. An objective measurement is given by the directional variance of the differences between the observations and the roughness change model. This directional variance is compared between the assumption of uniform exposure {"effective height" model) and the D & H roughness change model in Fig. 7 for all ten sites. The variance from the assumption of uniform exposure is a measure of the deviation of the observations from the expected climatic variation by direction. The variance from the D & H roughness change model is a measure of how well the D & H roughness-change model accounts for these deviations. 3.3. Conclusions
It is tempting to assign reasons for the observed deviations in the corresponding examples of Fig. 6, e.g. that the deviation for H u m at O = 0 : is due to an overassessment of the effective roughness of permanent woodland bordering the site, but such post-analysis justifications are less than objective. Further improvements might
N.J. Cook/J. Wind Eng. Ind. Aerodyn. 66 (1997) 197-214 " " ~ D & H estir~tes l . ~ Observations
Benbecula
~ D & H estimates " o - Ob~rvations
.
.
.
Hum . .
Fig. 6. Comparison of directional variation of V/(So VB) for six coastal sites.
211
,\;.J. Cook./. Wind L'~g. hid. 4crodvt~. 66 (1997) 197 214
212
Aberporth
I
Benbecula
I
I
I ] :~ '
' .........
IIUniform exposure assumed
~
IrlDeaves & Harris r°ughness cl ~
modl
Hurn Leuchars
I ...... i!:1'1!7
wl~k
I I I
Dyce
II111111 "111 Ii': Machrihanlsh . . . . . III ................................ I'illl I Ki.,os,
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
I¸
I
iii
.
Manston
I
I
II I
| -
I III wll ~
Mt Baden 0
I
I I .................
............
0.005
0 01
0 015
0 02
0025
0.03
0 035
004
Directional variance Fig. 7. ('omparisoll of directional \~[li;.[ilcc o[" II (~C-I ~ B)for ten coastal sites.
be obtained from a more accurate assessment of near-site roughness changes than can be obtained from the 1 kin resolution data base. Keeping to an objective view, it is clear from the upper group of "fiat" sites in Fig. 7 that the D & H roughnesschange model makes signiticant reductions to the directional variance, and it is therefore better to apply the I)&H model than to assume uniform exposure. The reductions m variance are less in the case of the sites affected by topography, where the uncorrected topographic influences dominate the directional "gain factor" and produce higher variance. This is particularly evident in the examples of Dyce and Valley in Fig. 6. Dyce illustrates how the presence of the mountains to the west "steers" the wind into a preferential north south axis along the coastal plain. "Steering" increases the frequency of occurrence of certain wind directions and so increases the value of extreme wind speeds in those directions by a statistical, rather than directly physical, process. There is currently no methodology for assessing this efl'ect. When the site is on a hill and the principal mffuence of topography is acceleration of the wind speed, for which there are a number of efl'ective assessment methods. When the method of BS6399 Part 2 [6] is applied to Mount Batten, the total variance from a combination of complex roughness changes and topography is dramatically reduced.
N.Z Cook~J. Wind Eng. Ind. Aerodyn. 66 (1997) 197 214
213
4. Concluding comments The purpose of this paper has been to demonstrate objectively the close equivalence of the D & H and power-law models for the mean velocity profile and the ease by which the D & H roughness-change model can be applied to a power-law profile. The concept of a single "effective height" for standard anemographs is shown to be flawed due to differing exposures by wind direction. The power-law profile will obviously continue to be widely used, but for the convenience of its form alone. Clearly, there is no longer any scope for rejecting the D & H model on grounds of validity, since it would be necessary to reject the equivalent power-law too! However, it is hoped that the dependence of the exponent, ~, on wind speed will be recognised more widely and the equivalence defined in Table 2 adopted. Nevertheless, it is likely that the larger values of height, h, that are predicted for the ABL by the D & H model and its dependence on wind speed and latitude will continue to be debated until confirmation is obtained from the meteorological record. When further comparisons are made of the D & H model with observations, it should be noted that all the observations will relate to real terrain, so that the comparisons should be made with the corresponding multiple roughness-change profile and not, as before, with the equilibrium profile. It is also clear that the effects of topography are dominant for sites in hilly terrain. A number of reliable methods exist to account for topography, including the twodimensional model in BS6399 Part 2 [6] and three-dimensional numerical models such as MSMicro Ell], supported in the U K by Ordnance Survey's commercial topographic database products. The latter gives reliable estimates of "steering" as well as speed. It is relatively straightforward to incorporate the changes in wind direction caused by "steering" into parent wind information, changing windrose frequencies and the corresponding Weibull parameters. There is currently no method for incorporating "steering" directly into design extreme wind data and the development of such a method would be most welcome.
References Ill D.M. Deaves, R.I. Harris, A mathematical model of the structure of strong winds, CIRIA Report 76, Construction Industry Research and Information Association, London, 1978. [2] R.I. Harris, D.M. Deaves, The structure of strong winds, Wind Engineering in the Eighties, Proc. CIRIA Conf., 12/13 November 1980, Paper 4, Construction Industry Research and Information Association, London, 1981. [3] D.M. Deaves, Computations of wind flow over changes in surface roughness, J. Wind Eng. Ind. Aerodyn. 7 (1981) 65 94. [4] N.J. Cook, The designer'sguide to wind loading of building structures, Part 1, Butterworths, London, 1985. E5] N.J. Cook, B.W. Smith, M.V. Huband, BRE program STRONGBLOW: user's manual, BRE Micro.computer package, Building Research Establishment, Garston, 1985. E6] British Standard, BS6399 Loading for buildings. Part 2, Code of practice for wind loads, British Standards Institution, London, 1995.
214
N.,/. ('ook:,L Wind Eng. Ind. Aerodvn. 66 f1997) 197 214
[7] ESDU, Strong winds in the atmospheric boundary layer, Part 1: mean-hourly wind speeds, Data Item 72026, EDSU International, London, 1976. [8] N.J. Cook. Note on directional and seasonal assessment ol extreme winds for design, J. Wind Eng. Ind. Aerodyn. 12 [1983) 365 372. [9] N.J. Cook, The assessment of design wind speed data: manual workshects with ready-reckoner tables, Building Research Establishment, Garston. 1985. [ 10] N.J. Cook, M.J. Prior, Extreme wind climate of the United Kingdom. J. Wind Eng. Incl. Aerodyn. 26 (19871 371 389. [11] J.k. Walmsley. ,hR. Salmon. P.A. l-t~ylor, On the application of a model of boundary layer flow over low hills to real terrain, Bound. Layer Meteorol. 23 41982i 17 46.
lml nmd Journal of Wind Engineering and Industrial Aerodynamics 66 (1997) 197 214
ELSEVIER
The Deaves and Harris ABL model applied to heterogeneous terrain Nicholas J. Cook Wind Engineering Consultant, 10 Arretine Close, St Albans AL3 4JL, UK
Received 1 March 1996; accepted 10 March 1997
Abstract The mean velocity profile given by the Deaves and Harris model for strong winds is compared with the log-law and the power-law models for terrain with roughness changes and is shown to predict accurately the directional exposure characteristics of UK meteorological stations. The ease of application of the Deaves and Harris roughness-change method to the power-law model is also demonstrated.
1. Introduction The Deaves and Harris (D&H) model for the characteristics of the atmospheric boundary layer (ABL) in strong winds was developed in two stages. In the first stage, a model for the ABL in equilibrium over flat uniform roughness was developed under the auspices of the Construction Industry Research and Information Association (CIRIA) and was published in 1978 [1]. In the second stage, the model was developed to account for multiple step changes in surface roughness, under the auspices of the U K Building Research Establishment (BRE). This development was done to extend the applicability to the kinds of heterogeneous terrain found in nature, and was published in 1981 [2,3]. The D & H model was subsequently adapted into U K design guidance [4,5], the new U K wind loading code [6], the Australian and New Zealand codes and in the E S D U D a t a Items. The D & H model is chiefly known for its "logarithmic with parabolic defect" mean velocity profile equation: ----
,i,n
K
--+5.75 Z 0
-1.88
-1.33
(;,)3
+0.25
,
(1)
where V is the mean wind speed at height z above ground, K = 0.40 is yon K a r m a n ' s constant, u, is the friction velocity, Zo is the aerodynamic roughness length and h is the equilibrium boundary layer height. 0167-6105/97/$17.00 ~) 1997 Elsevier Science B.V. All rights reserved. PII S0 1 67-6 1 05(97)00034-2
198
VJ ('ook.,l. Wiml Eng. Ind..4erodv;t
66 H997) 197 214
But the model also predicts the equilibrium boundary layer height, h:
;,
Lt,
g/,
(2)
where / is the Coriolis parameter, and parameters describing the in-wind component of turbulence, including the turbulence intensity, I,,, and spectral density function, b;,. The extension of the D&H model to step changes in roughness introduced the concept of a transition region between the outer original boundary layer and a new internal boundary layer, giving equations for the height of the top and bottom boundary of this region. The internal boundary layer, below the transition zone, is assumed to be in local equilibrium as described by the local friction velocity,
;/,,inner : {
ln{zo.om~.rZoi ...... [
't,.i ....... = u, ......... 1
0.42 + In m,~
(3) "
where the parameter m0 is given in turn by the solution to mo=
0.32X :o.i .......{In mo
11
(4)
where X is the distance downwind of the change of roughness, imwr refers to the new inner boundary layer and outer refers to the previous overlaying boundary layer.
2. Comparison with log-law and power-law models 2.1. Preamble
Although the D&H model has been widely accepted in the UK, Australia and New Zealand, becoming incorporated into the respective codes of practice, much of Europe has continued to use the log-law model, while North America and Japan have continued to use the earlier power-law model. Comparisons between these models have been made and arguments for and against have been rehearsed many times. The arguments are reprised again below, but ensuring for the first time that a number of important conditions for valid comparison have been met properly. 2.2. Similari O, theory
Similarity theory predicts that
!:i . ! In
t as= --0
t5)
N.J. Cook/J. Wind Eng. Ind. Aerodyn. 66 (1997) 19~214
199
and V ~ VG and
dV dz ~0
as z ~ h ,
(6)
where VG is the geostrophic wind speed, representing the lower and upper boundary conditions, respectively, to the ABL. 2.3. Log-law model The log-law model assumes that Eq. (5) holds over a suffÉciently useful depth near the surface, giving the log-law mean velocity profile equation V
u, i n ( i " ]
\zo/
K
(7)
It is clear that the log-law velocity profile meets the lower boundary condition, but has no upper boundary at all. It is a two-parameter model, with u, providing the velocity scale (slope of log-linear plot) and z0 providing the length scale (intercept of log-linear plot). It is accepted that the log law is a poor model for the mean velocity profile at large heights, typically for z > 200 m. This causes a difficulty in translating the wind speed over one form of roughness, e.g. open country, to that over another, e.g. urban, because the mean wind speed at the top of the boundary layer cannot be used. In practice, such translations are made through empirical "exposure factors" based on measurements made near the ground, typically at z = 10 m, over both types of roughness. To extend the height range of the log-law model up to about z = 300 m, ESDU introduced a linear deviation term to their 1976 Data Item [7]: =
in
,8)
where the value of K was empirical and varied over the range 4 < K < 7. This linear term, which represents the deviation observed at moderate heights in the neutrally stable ABL, should not be confused with the linear deviation observed in non-neutral conditions which is more pronounced and occurs closer to the ground. 2.4. Power-law model The power-law model uses an empirical formula for the mean velocity profile of the form:
<9) g~ef = \ Z r e f /
where Vref is the mean wind speed at some arbitrary reference height Zref and ~ is the exponent of the power-law.
200
\'.,I. Cook./. ~4:ind EnN. lnd. Aerodvp~. 66 (1997) 197 214
The power-law profile does not meet the lower boundary conditions and also has no upper boundary. Despite its appearance, it is effectively a two-parameter model, since the parameters Vr~r and Zr~.~combine as the factor Vret / Z~.r to scale the velocity, while the exponent sets the shape of the profile (slope of log log plot). That is, for any given terrain, the power-law model defines a mean velocity profile of fixed shape which passes through the reference values. Unlike the log-law model, the power law remains a good model for the mean velocity profile at moderately large heights, which is the major reason for its preference. although it becomes poorer very close to the ground. The power-law profile fits best over the range of moderate heights 30 in < z < 300 m. It is recognised that fitted values of the exponent, z~, vary with wind speed and height-range of fit, but depend principally on the roughness of the ground surface. In practice, the exponent, ~, is often regarded as equivalent to the aerodynamic roughness length, zo. Although it does not recognise the upper boundary, translation between terrain of different roughness is performed through commonality of the wind speed at a notional "'top" to the profile, the "gradient height" zg. When the progress of this concept is traced through its many years of application, particularly through the many papers of A.G. Davenport, it is seen that the value initially adopted for "gradient height" was quite small, zg v 300 m, but rises through the years as more was discovered about the characteristics of the ABL until, presently, values as large as zg ~ 1200 m are commonly used. There is no doubt that the power-law model is convenient to use, particularly in wind tunnel modelling applications, for a number of reasons. These include ease of plotting and fitting when calibrating simulations {simply plot the wind speed against height on log log paper and draw a straight line through the points), ease of integration with height and independence from linear scaling. This latter property stems from the self-preserving shape of the power-law profile. In the early days of "velocity-profile-only" simulations, this independence from linear scale meant that a given terrain class could be represented by a single wind tunnel simulation over a wide range of linear scale factors. Contemporary simulation methods recognise the importance of the turbulence length scales, through which a unique linear scale factor is derived for any power-law-based simulation. The power-law profile equation can be ~'disguised" in terms of the log-law parameters, zo and u,, thus, u,
. ,
( I
()1
but this is merely cosmetic. The "'extra" parameter C is constrained by the reference parameters to be
so that the scaling factor remains Vref/2¢e l- and the log-law parameters have no distinctive role in the model. However, the formulation of Eq. (10) facilitates comparison with the log-law model in terms of thc non-dimensional velocity V/u,.
N.J. Cook/,L Wind Eng. Ind. Aerodyn. 66 (1997) 197 214
201
2.5. Deaves and Harris model
The mean velocity profile D & H model is given by Eq. (1) and this meets the boundary conditions at both top and bottom of the ABL. The advantage of the D & H model over the others is that it extends the accurate representation of the log-law at small z, through the range of best accuracy for the log-linear and power-law models at moderate z, continuing up to the top of the ABL at z = h. This is the only model of the three that "recognises" the top of the ABL. This means that translation between terrain of different roughness can be performed by commonality of the geostrophic, Va, wind speed at a height above z = h. However, a critical difference of the D & H model is that it has three scaling parameters, zo and u, inherited from the log-law model, and the additional length parameter, h, the ABL height. The value of h is a function of wind speed and latitude, through Eq. (2). Therefore, the D & H mean velocity profile is not a single curve for any given location, but is a family of curves dependent on wind speed, It is, therefore, a condition for valid comparison of the log-law and power-law models that the wind speed is specified so that the appropriate member of the family of D & H profiles is selected. Before moving on to make the comparisons, it is convenient to list in Table 1 the design values of the terrain-dependent parameters for extreme winds in the UK, taken from Ref. [4]. It is a fortunate coincidence that the increase in design extreme wind speed northwards across the U K is almost exactly matched by the increase in Coriolis parameter, f, enabling a constant design value of ABL height, h, to be adopted for each terrain type. In this table, VB refers to the "basic mean wind speed" which corresponds to the 1-in-50-year return mean wind speed at z = 10 m over fiat uniform terrain of Category 2. This varies across the U K from V8 = 20 m/s in Southern England to VB = 30 m/s in the Orkney Islands, i.e. with a "typical" or median value of VR = 25 m/s. Of course, at wind speeds outside this design range account should be taken of the changing equilibrium ABL height, h. The characteristic of the D & H model that stands out most is the prediction for the equilibrium height of the ABL in the design extreme storm which ranges from h = 2210 m over the sea to h = 3250 m over cities (although there are currently no Table I Values of terrain-dependent parameters for the U K Roughness category
0
1
2
3
4
5
Description
Sea
Rough rural 0.I 0.0748 2770 0.861 0.20 0.18
City
0.003 0.0597 2210 1.211 0.12 0.12
Typical rural 0.03 0.0689 2550 1.000 0.16 0.16
Town
Aerodynamic roughness, Zo (m) Friction velocity, u,/Va ABL height, h (m) Exposure factor, SE = Vlo/VB Exponent ~ for 0 < z < 50 m Exponent ~ for 0 < z < 3 0 0 m
Flat rural 0.01 0.0642 2380 1.108 0.14 0.14
0.3 0.0811 3000 0.711 0.24 0.22
0.8 0.0877 3250 0.554 0.32 0.27
2()2
N.,I. Cook,J. Wind Eng lnd. ,4eroctvn. 66 (1997) 197 214
40.00
- x - 5 m/s - o - 10 m/s 20 m/s --n- 30m/s Log law Power law
35.00
30.00
/
/d
~ .-I~
J
25.00
20.00
Olllll | ,a,w,..~IIir
15.00 10
100
1000
10000
z (m) Fig. I. ('ornparison for % = 0.03 m in log-law formal.
cities of sufficient size to achieve equilibriuml. These values are more than twice the currently accepted values used with the power-law model. It is probably this characteristic, above all others, that makes proponents of full-depth wind tunnel simulations of the ABL reluctant to adopt the D&H model, since this implies that the wind tunnel simulations need to be twice as deep than at present 1. In more frequently occurring conditions, VR = 10 m/s, the predicted height is close to 1000 m which matches well with observations. The relationship between wind speed, latitude and ABL depth in conditions of neutral stability is a subject that merits further study and for which sufficient data may already be available in the meteorological record. 2.6. The comparisons 2.6.1. Comparison/br Catego O, 2 terrain The D&H, log-law and power-law models are compared for the UK datum ground roughness, Category 2, over a range of wind speed in Figs. 1 and 2, in each of their respective plotting format conventions. The comparison is based on the UK design
LIn many establishments wherc the D&H model has been adopted as standard, the "'part-depth" simulation method, in which only the lower layers of the ABL arc accurately represented and the majority of the ABL at the top (upper two-thirds) is specilically excluded, is the standard simulation method, and this approach is unaffected by the larger values of h.
A(J. Cook/J. Wind Eng. Ind. Aerodyn. 66 (1997) 197 214
203
40
30
J
I
20
- x - 5 m/s --o- 10 m/s - a - 20 m/s 30m/s Log law - - - - Power law
f[I 10
100
I 1000
10000
z (m) Fig. 2. Comparison for Zo = 0.03 m in power-law format.
parameters in Table 1, where Zo = 0.03 m is taken to correspond to ~ = 0.16. The values of wind speed correspond to VB = Vret at Zret = 10 m, i.e. the values of the log-law and power-law models are matched at the meteorological standard height of z = 10 m. Note how the D & H model forms a family of profiles with increasing depth with increasing wind speed. D & H and power-law models deviate progressively above the log-law from moderate heights. The match between the D & H and power-law models is excellent over the design range of wind speed, 20 m/s < VB < 30 m/s, but becomes progressively poorer at lower wind speeds. However, Fig. 2 shows that all the D & H profiles plot as reasonable straight lines in the power-law format, except approaching the upper boundary where dV/dz =~0, indicating that the models still compare well, but with different values of power-law exponent, ~.
2.6.2. Comparisons at the median UK design wind speed The models are again compared, this time for the median U K design wind speed, V8 = 25 m/s, over a range of ground roughness in Fig. 3, again based on the design parameters in Table 1. The values of the log-law and power-law models for "sea" Category 0 (z0 =0.003 m, ~ =0.12) and "typical rural" Category 2 (z0 =0.03 m, = 0.16) are again matched at z = 10 m, but it seemed more appropriate to match the models for the rougher "town" Category 4 (z0 = 0.3 m, ~ = 0.22) somewhat higher at z = 30 m, since the meteorological standard height would be close to the rooftops. These categories correspond to the "sea", "land" and "town" categories of BS6399 Part 2 [6].
,~'..1. (bok,,/. Wind l'~ng. Ind. Aero~tvn. 66 (1997) 197 214
204
The match between the D&H and power-law models is again excellent, except for z > 500 m in the least rough "sea" category where the power-law profile diverges below the D&H profile at a lower height than before.
2.6.3. Conclusions
The mean velocily profile of the D&H model is seen to compare well with the power-law profile over a wide range of surface roughness and wind speed. The D&H profile is better than the power-law profile close to the ground and near the top of the ABL because it meets the upper and lower boundary conditions predicted by similarity theory.
45 40 35 30
25 20 15 10
5 100
10
1000
10000
z (m) Fig. 3. ( ' o m p a r i s o n for median UK design ~ i n d speed in log-la~s format.
1able 2 Values ol power-law e x p o n e n t m a t c h e d to D & H proliles Aerodynamic roughness zo (m)
1" ~,z', matched at z,~f (m)
I,]~ 5 m/s l(; = 10,8 m,s
I,~ 10 m:s V~ - 22.8 m/s
1,~ - 20 m/s V(; = 47.9 m/s
I/, - 30 m/s t'{~ - 74.0 m/s
0.003 0.01 0.03 o. 1 0.3 0.8
10 lO I0 10 30 50
O. 16 0.18 0.20 0.24 0.27 0.32
O. 135 ().1~ O. 175 0.21 (!.24 0.28
0.125 0.14 0.165 O. 19 0.22 0.27
0.12 0.135 0.155 0.185 0.20 0.25
N.J. Cook/J. Wind Eng. Ind. Aerodyn. 66 (1997) 197 214
205
However, the equivalence between the aerodynamic roughness parameter, Zo, of the D&H model and exponent, ~, of the power-law model is dependent upon wind speed. This equivalence is given in Table 2 in terms of the UK "basic wind speed", V~, and the corresponding geostrophic wind speed, V6.
3. Roughness changes 3.1. Roughness change method 3.1.1. Derivation The D&H method equations for a single roughness change, Eqs. (3) and (4), are quite inconvenient to implement: Eq. (4) because it is non-linear and requires to be solved by iteration, and Eq. (3) because it relates the friction velocity in the new inner layer to that in the outer layer. Why should this latter case be inconvenient? It requires that multiple roughness changes are addressed in sequence in the downwind direction, i.e. from far upwind towards the required site location, because it predicts the friction velocity for each successive inner layer in terms of the previous layer. It would be more convenient to predict the friction velocity from the values for the same roughness at equilibrium, i.e. to work from the site in the upwind direction, until the next roughness change is sufficiently far away to be insignificant. This latter approach is used in the design implementation of the D&H model developed by the author [4], implemented as a computer program [5] and used subsequently, in simplified form, in the 1995 UK code of practice for wind loads [-6]. In this approach, a value of basic mean wind speed, VB, is progressively factored by a series of"S-factors" to reach the desired design value. This mimics the structure of many wind loading codes of practice. The basic mean wind speed corresponds to equilibrium conditions at z = 10 m flat uniform roughness, Zo = 0.03 m, equivalent to typical UK open countryside. The four "S-factors" relevant in this context are the "exposure factor" SE, the "height factor" Sz, the "fetch factor" Sx, and the "direction factor" So: SE --
Sz-
V{z = 10 m} VB
[ln(2550/0.03) + 2.79]ln(10/z0) , [ln(h/zo) + 2.79] In(10/0.03)
V{z}
ln(z/zo) + 5.75(z/h) - 1.875(z/h) 2 - 4(z/h)3/3 + (z/h)4/4
V{z = 10 m}
ln(10/z0)
(12)
(13)
Sx{j~i}
U*,equilibriumU*'l°calln(zo,JZo,i)] ln(lO/zo,i) SE,j -
1
(14)
0.42 + l n m o ] ln(10/Zo,~) SEi'
where V{z = 10m} is the equilibrium mean wind speed at z = 10 m above uniform terrain of the site roughness, V{z} is the equilibrium mean wind speed at height z above uniform terrain of the site roughness, i and j refer to the ith and jth surface roughness upwind of the site, and m0 is given by the solution of Eq. (4).
206
\~.,I. Cook'.l. Wind Eng. Ind. Aerodvn. 66 (1997) 197 214
Table 3 Values of direction factor, S(.),tot the UK from Ref. [4J Wind direction, O 0 30 (~ll 9(t 120 150 1N0 210 240 Direction factor. S¢_, 0 . S l 0.76 0.76 I).77 0.76 0.83 0.89 (t.97 1.05
270 .04
300 330 0.95 0.86
The direction factor, So, is d e t e r m i n e d empirically from o b s e r v a t i o n s [8] a n d is given for the U K -~ in T a b l e 3. In Eqs. (12) (141. the values of the basic p a r a m e t e r s ( C a t e g o r y 2 in T a b l e 1) have been retained, instead of being i n c o r p o r a t e d into global constants, to allow their derivation to be traced. The p e n a l t y for t r a n s f o r m i n g the roughness change f o r m u l a is increased c o m p l e x i t y of form 3. The practical solution to this in design guidance [4] a n d codes of practice [6] has been to t a b u l a t e values of the factors, or to p r e - c o m p u t e the results for a range of s t a n d a r d cases a n d present these as r e a d y - r e c k o n e r tables [9]. Values of Fetch factor, Sx, for three typical changes of roughness are p l o t t e d against fetch, X, in Fig. 4. The fetch is now the distance upwind, from the site to the change of roughness, a n d is therefore the distance over which the roughness has acted on the wind. N o t e that the effect of a s m o o t h to r o u g h change is to increase the local friction velocity a b o v e the e q u i l i b r i u m value, while the effect of a r o u g h to s m o o t h change is to decrease the local friction velocity, and in both cases for this change, to decay g r a d u a l l y back t o w a r d s the equilibrium value with increasing fetch. Use in guidance or codes of s t a n d a r d roughness categories having values of a e r o d y n a m i c roughness, zo, differing by equal fractional p o w e r s of ten (e.g. Categories 0 4 in T a b l e 1) allows further simplification with a small loss of precision. It is found that Sxl~sea --* town I ~ Sxttsea - , l a n d } × Sx{land ~ t o w n } ~ S~{sea --,land}.
(15)
It is a p p a r e n t from Fig. 4 that, a l t h o u g h a building 5 km inside a coastal town (sea ~ t o w n ) m a y be fully i m m e r s e d in the '~town" internal ABL, the wind speed will be a b o u t 10% higher than if the town were well inland (land --*town), because of the increased wind speed in the overlying o u t e r layer. This has implications for the accuracy of codes of practice and design guidance that d o not differentiate between coastal and inland towns. A c c o u n t i n g p r o p e r l y for the roughness changes found with real sites requires i m p l e m e n t a t i o n for multiple roughness changes.
2 Directional factors were determined in Rei. [8] for the design risk of excccdancc in each sector and. therefore, an overall risk greater than the design risk. The direction factor, So, uses these values factored to make the risk for all directions equal to the design risk, but distributed equally by direcLion. BS6399 uses these values factored to give a maximum of unity lat ~'-)= 240 ), so gives a risk somewhere between ~'design risk in directional sector" and "design risk for all directions". The appropriate way of assigning the design risk by direction is still a matter of debale. Unfortunately, Eq. (14) defeated both the typesetter and the technical editor of Ref. E4] and readers may wish to take the opportunity to correct Eq. (9.20) of this reference.
NJ. Cook/~ Wind Eng. Ind. Aerodyn. 66 (1997) 197 214
207
1.6 ..............................
l-llff
............. l . . . . . I. . . . . . . . . . .
--e-- Land
1.5 ~
1.4
-
-
o
-
to town
Sea to town
1.3
.9o o li= U.
1.2 1.1 1
14.
0.9 0.8 0.7 0.6 0.1
10
100
Fetch X (km) Fig. 4. Fetch factors for three typical roughness changes.
3.1.2. Implementation for multiple roughness changes It can be seen that after a sequence of roughness changes n ~ m --, ... --*c --, b --, a from the furthermost upwind roughness "n" to the site roughness "a", the mean wind speed at any height, z, in the innermost "a" layer above the site will be given by: V{z,a} = Sx{n ~ m} ... Sx{c ~ b}Sx{b ~ a}Sz{a}SE{a}SoV,,
(16)
and in the next-level "b" layer by
V{z,b} = Sx{n --* m} ,.. Sx{c --* b}Sz{b}Sz{b}SoV,,
(17)
or, in general terms, for the ith level layer by i
V{z,i} = I~ Sx{j}Sz{i}SE{i}SoV,"
(18)
j=n
This leaves the requirement to determine which individual internal boundary layer controls the wind speed at any given height above ground. A simple and satisfactory approach is to assume that the transition region between layers has zero thickness, so that the velocity profile of an inner layer intersects with the profile of the overlying layer. In a smooth ~ rough change, the wind speed at any height will be the smaller value of the inner and outer profiles, and in a rough ~ smooth change it will be the larger value. This is demonstrated in Fig. 5 for a single smooth --* rough change at VB = 25 m/s.
20g
'V,.I. ('ook ,]. Wind Eng. Ind. Aerodvn, 66 (1997) 197 214
6O
50
- - -
40 30 ~
20' t-
Power
law, town
D&H, sea-town
Power law, sea-town
I I
10 10
100
I
1000
Height above ground, z (m) Fig. 5. Example of smglc smooth +rough~,ua
,town) roughnesschangc.
Fig. 5 is presented in the power-law format to demonstrate that the D&H roughness change method works perfectly well with power-law profiles and therefore provides an effective method for extending the power-law model to include roughness changes. The transition profile has the same slope (exponent) as the equilibrium "rough" profile, but is displaced upwards (scaled by Sx) so that it intersects with the equilibrium "'smooth" profile. The expedient of assuming zero thickness transition layers leads apparently to the prediction of discontinuities in the values of 1" and dV/dz at the intersection of the profiles. It should be noted that these are artifacts of the method and, in reality, there will be a smooth transition. Note that the wind speed axis is in absolute units (m/s), since the three profiles each have a different value of friction velocity. The wind speed could have been normalised against the common value of geostrophic wind speed, Vo, but this would not necessarily be known when implementing the power-law model alone.
3.2. Comparisons 3.2.1. Method qlcomparison Although field data on the ABL velocity profile to any significant height is very sparse, there is a wealth of observations near the surface from anemograph sites. In the UK there are over 100 anemograph stations that have been recording the hourly mean wind speed continuously since 1970. The majority of these anemometers are
N.~L Cook/J. Wind Eng. Ind Aerodyn. 66 (1997) 197 214
209
sited at z = 10 m above open level ground, according to the W M O standard, but some have been raised to compensate for local obstructions to a "10 m effective height", while others are well above the ground for reasons of necessity (London Telecom Tower, z = 195 m). These anemometers have all been assigned a value of "effective height" by the U K Meteorological Office. The "effective height" is the height at which the anemometer would be set to give the same wind speed if the exposure was "standard". This concept is flawed because it limits the assessment of exposure to the terrain in the immediate vicinity and assumes this exposure is uniform by direction. In reality, each anemometer is exposed differently by direction. If the record from each anemometer is divided into convenient 30%wide sectors by direction, then data are available for comparing 12 different exposures at the same site with the corresponding observations from that site. The method of comparison used here is based on observations of the extreme wind speeds. The rationale for this is given in Refs. [8,10] but, in brief, the extreme wind climate of the U K is dominated by Atlantic depressions with tracks which pass over the whole UK. The increase in extreme wind speed from south-east to north-west across the UK is principally due to an increase in depression frequency and not to a change in character. Accordingly, it is expected that the directional characteristics of extreme winds will be constant across the UK. The average from a large number of anemometer sites (50 sites in Ref. [10]) distributed across the UK is expected to eliminate variations in directional exposure between individual sites and, therefore, to represent this climatic variation by direction. Comparisons on this basis have been made before, [4,8]. The comparisons here differ in two respects: • roughness changes have been computed automatically from a database of the surface roughness of every 1 km square in the UK, so are more detailed and accurate in the medium to far field, and • climatic variations by direction have been removed and the comparisons made in terms of the "gain factor" for the anemometer, by direction. This latter transformation is achieved by re-arranging Eq. (18) into 12i Sx{ j} Sz{ a} SE{ a} = SovVB " ~=,
(19)
The expression on the left, which is the "gain factor" estimated by the D & H roughness-change method, may be compared with the expression on the right, which is the "gain factor" estimated from the observations. Here, V is the value of the l-in-50-year return hourly mean wind speed from the analysis of Ref. [10]. There is no method for obtaining direct measurements of VB, SO the value is taken from the best overall fit to Eq. (19); thus,
E V~ -
360 / / - - a
~
~
360 ~ V/S(o)
/
(20)
210
N..L (?ook/J. Wind Eng. Ind. Aerodvn. 66 (1997) 197 214
Because these two estimates of "~gain lactor'" are forced to an overall fit, the comparison must be on the deviations from this fit by direction, i.e. the comparison is relative, not absolute. The advantages of this method are that 12 comparisons can be made for each site and that objective exposure-corrected estimates of Vt~ are obtained as a by-product for improving the UK base map of design wind speed. 3.2.2. Results,lot ten coastal sites in the UK
Ten anemograph stations in the UK that were near the coast, and therefore with exposures differing significantly by direction, were selected for comparison. Five are in effectively flat terrain and five are influenced by significant topography. These are listed below with the National Grid reference and site characteristics: • • • • • • • • • •
Aberporth Benbecula Dyce Hurn Kinloss Leuchars Machrihanish Manston Mount Batten Wick
SN242521 NF764555 NJ878124 SZ117978 NJ067627 NO457202 NR663224 TR335666 SX492527 ND363524
NW-facing coast W-facing coast, small island E-facing coast S-facing coast N-facing coast NE-facing coast W-facing coast SE-facing coast complex coastline E-facing coast
ttat flat mountains to west flat mountains to south flat between two hills hilly terrain summit of hill fiat
The influence of the topography has not been included in the D & H estimates in four of the five affected sites. The exception is the Mount Batten site where the acceleration over the hill has been assessed using the method of BS6399 Part 2 [6] and included in the D & H estimate. The directional comparisons for live of these stations and also for Valley (SH309757) are presented as polar plots in Fig. 6. While this format enables subjective comparison and allows deviations to be related to site characteristics, a more objective assessment of quality of fit is needed. An objective measurement is given by the directional variance of the differences between the observations and the roughness change model. This directional variance is compared between the assumption of uniform exposure {"effective height" model) and the D & H roughness change model in Fig. 7 for all ten sites. The variance from the assumption of uniform exposure is a measure of the deviation of the observations from the expected climatic variation by direction. The variance from the D & H roughness change model is a measure of how well the D & H roughness-change model accounts for these deviations. 3.3. Conclusions
It is tempting to assign reasons for the observed deviations in the corresponding examples of Fig. 6, e.g. that the deviation for H u m at O = 0 : is due to an overassessment of the effective roughness of permanent woodland bordering the site, but such post-analysis justifications are less than objective. Further improvements might
N.J. Cook/J. Wind Eng. Ind. Aerodyn. 66 (1997) 197-214 " " ~ D & H estir~tes l . ~ Observations
Benbecula
~ D & H estimates " o - Ob~rvations
.
.
.
Hum . .
Fig. 6. Comparison of directional variation of V/(So VB) for six coastal sites.
211
,\;.J. Cook./. Wind L'~g. hid. 4crodvt~. 66 (1997) 197 214
212
Aberporth
I
Benbecula
I
I
I ] :~ '
' .........
IIUniform exposure assumed
~
IrlDeaves & Harris r°ughness cl ~
modl
Hurn Leuchars
I ...... i!:1'1!7
wl~k
I I I
Dyce
II111111 "111 Ii': Machrihanlsh . . . . . III ................................ I'illl I Ki.,os,
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
I¸
I
iii
.
Manston
I
I
II I
| -
I III wll ~
Mt Baden 0
I
I I .................
............
0.005
0 01
0 015
0 02
0025
0.03
0 035
004
Directional variance Fig. 7. ('omparisoll of directional \~[li;.[ilcc o[" II (~C-I ~ B)for ten coastal sites.
be obtained from a more accurate assessment of near-site roughness changes than can be obtained from the 1 kin resolution data base. Keeping to an objective view, it is clear from the upper group of "fiat" sites in Fig. 7 that the D & H roughnesschange model makes signiticant reductions to the directional variance, and it is therefore better to apply the I)&H model than to assume uniform exposure. The reductions m variance are less in the case of the sites affected by topography, where the uncorrected topographic influences dominate the directional "gain factor" and produce higher variance. This is particularly evident in the examples of Dyce and Valley in Fig. 6. Dyce illustrates how the presence of the mountains to the west "steers" the wind into a preferential north south axis along the coastal plain. "Steering" increases the frequency of occurrence of certain wind directions and so increases the value of extreme wind speeds in those directions by a statistical, rather than directly physical, process. There is currently no methodology for assessing this efl'ect. When the site is on a hill and the principal mffuence of topography is acceleration of the wind speed, for which there are a number of efl'ective assessment methods. When the method of BS6399 Part 2 [6] is applied to Mount Batten, the total variance from a combination of complex roughness changes and topography is dramatically reduced.
N.Z Cook~J. Wind Eng. Ind. Aerodyn. 66 (1997) 197 214
213
4. Concluding comments The purpose of this paper has been to demonstrate objectively the close equivalence of the D & H and power-law models for the mean velocity profile and the ease by which the D & H roughness-change model can be applied to a power-law profile. The concept of a single "effective height" for standard anemographs is shown to be flawed due to differing exposures by wind direction. The power-law profile will obviously continue to be widely used, but for the convenience of its form alone. Clearly, there is no longer any scope for rejecting the D & H model on grounds of validity, since it would be necessary to reject the equivalent power-law too! However, it is hoped that the dependence of the exponent, ~, on wind speed will be recognised more widely and the equivalence defined in Table 2 adopted. Nevertheless, it is likely that the larger values of height, h, that are predicted for the ABL by the D & H model and its dependence on wind speed and latitude will continue to be debated until confirmation is obtained from the meteorological record. When further comparisons are made of the D & H model with observations, it should be noted that all the observations will relate to real terrain, so that the comparisons should be made with the corresponding multiple roughness-change profile and not, as before, with the equilibrium profile. It is also clear that the effects of topography are dominant for sites in hilly terrain. A number of reliable methods exist to account for topography, including the twodimensional model in BS6399 Part 2 [6] and three-dimensional numerical models such as MSMicro Ell], supported in the U K by Ordnance Survey's commercial topographic database products. The latter gives reliable estimates of "steering" as well as speed. It is relatively straightforward to incorporate the changes in wind direction caused by "steering" into parent wind information, changing windrose frequencies and the corresponding Weibull parameters. There is currently no method for incorporating "steering" directly into design extreme wind data and the development of such a method would be most welcome.
References Ill D.M. Deaves, R.I. Harris, A mathematical model of the structure of strong winds, CIRIA Report 76, Construction Industry Research and Information Association, London, 1978. [2] R.I. Harris, D.M. Deaves, The structure of strong winds, Wind Engineering in the Eighties, Proc. CIRIA Conf., 12/13 November 1980, Paper 4, Construction Industry Research and Information Association, London, 1981. [3] D.M. Deaves, Computations of wind flow over changes in surface roughness, J. Wind Eng. Ind. Aerodyn. 7 (1981) 65 94. [4] N.J. Cook, The designer'sguide to wind loading of building structures, Part 1, Butterworths, London, 1985. E5] N.J. Cook, B.W. Smith, M.V. Huband, BRE program STRONGBLOW: user's manual, BRE Micro.computer package, Building Research Establishment, Garston, 1985. E6] British Standard, BS6399 Loading for buildings. Part 2, Code of practice for wind loads, British Standards Institution, London, 1995.
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[7] ESDU, Strong winds in the atmospheric boundary layer, Part 1: mean-hourly wind speeds, Data Item 72026, EDSU International, London, 1976. [8] N.J. Cook. Note on directional and seasonal assessment ol extreme winds for design, J. Wind Eng. Ind. Aerodyn. 12 [1983) 365 372. [9] N.J. Cook, The assessment of design wind speed data: manual workshects with ready-reckoner tables, Building Research Establishment, Garston. 1985. [ 10] N.J. Cook, M.J. Prior, Extreme wind climate of the United Kingdom. J. Wind Eng. Incl. Aerodyn. 26 (19871 371 389. [11] J.k. Walmsley. ,hR. Salmon. P.A. l-t~ylor, On the application of a model of boundary layer flow over low hills to real terrain, Bound. Layer Meteorol. 23 41982i 17 46.