A Modified Logarithmic Law for Neutrally Stratified Flow over Low-Sloped Hills
CLAUDIO C. PELLEGRINI
Department of Thermal and Fluid Sciences (DCTEF), Federal University of So Jodao del-Rei (UFSJ), Sdo Joao del-Rey, Brazil
GUSTAVO C. R. BODSTEIN
Department of Mechanical Engineering (EE/COPPE), Federal University of Rio de Janeiro (UFRJ), Rio de Janeiro, Brazil
(Manuscript received 2 October 2003, in final form 6 September 2004) ABSTRACT
The study of the atmospheric boundary layer flow over two-dimensional low-sloped hills under a neutral atmosphere finds numerous applications in meteorology and engineering, such as the development of large-scale atmospheric models, the siting of wind turbines, and the estimation of wind loads on transmis-sion towers and antennas. In this paper, the intermediate variable technique is applied to the momentum equations in streamline coordinates to divide the flow into regions, with each characterized by the domi-nance of different terms. Using a simple mixing-length turbulence closure, a simplified form of the x momentum equation is solved for the fully turbulent region, resulting in a modified logarithmic law. The solution is expressed as a power series correction to the classical logarithmic taw that is valid for flat terrain. A new parameter appears: the effective radius of curvature of the hill. The modified logarithmic law is used to obtain new equations for the speedup, the relative speedup, the maximum speedup, and the height at which it occurs. A new speedup ratio is proposed to calculate the relative speedup at specific heights. The results are in very good agreement with the Askervein and Black Mountain field data.
1. Introduction
The ability to predict the atmospheric boundary layer (ABL) flow over hills has long been the subject of in-tense studies by meteorologists, environmentalists, and engineers. Of particular importance in practical appli-cations is the knowledge of the so-called flux-gradient relationships, which connect the fluxes of momentum, heat, and moisture at the surface to the velocity, tem-perature, and specific humidity gradients, respectively. Integration of these expressions produces the corre-sponding velocity, temperature, and specific humidity profiles. The most well known flux-profile relation is the logarithmic law for flows over flat, vegetated sur-faces, that is,
Ti(z) =-lIn( Z ) (1) where z is the displaced height above the ground, u* is the friction velocity, K iS the von Karman parameter,
and z0is the roughness length.
Corresponding author address: Prof. Gustavo C. R. Bodstein, Dept. of Mechanical Engineering-EE/COPPE, Federal University of Rio de Janeiro-UFRJ, Caixa Postal 68503, 21945-970, Rio de Janeiro, RJ-Brazil.
E-mail: gustavo@mecanica.coppe.ufrj.br
In the ABL flow over hills, meteorologists apply the flux-profile relationships in the development of large-scale atmospheric models. These models depend criti-cally on the parameterizations adopted for the ABL. Equation (1) has been widely used as a lower boundary condition, despite the fact that it is valid for flat terrain only. The reason is to avoid performing the integration all of the way down to the surface, where strong veloc-ity gradients require a fine computational mesh. Equa-tion (1) is also used in meso- and microscale models, for the same reasons. Other meteorological applications for flux-profile relations over hills are the direct pa-rameterization of the momentum flux and the develop-ment of high-order turbulence closure schemes. Finni-gan (1992) and Taylor and Lee (1984) list a number of other relevant meteorological and engineering applica-tions. The most important in the engineering field are the siting of wind turbines in regions of enhanced wind speed and the estimation of wind loads on towers, an-tennas, and buildings located on the top of hills.
Several descriptions of the flow over hills and com-plex terrain are available in the literature. An impor-tant part of this study started with the paper by Jackson and Hunt (1975), who divided the flow field into two sublayers using asymptotic expansion techniques, with each layer having different flow dynamics. The validity of their analysis was then extended to three dimensions
PELLEGRINI AND BODSTEIN
by Mason and Sykes (1979). Sykes (1980) also showed that it is necessary to include a third layer to impose the surface boundary condition consistently. Newley (1985) also contributed to the understanding of the problem in the form of a detailed discussion about the various height scales. Newley also modeled the problem nu-merically and compared his results very favorably with field data (see also Belcher et al. 1993). Two years later, Zeman and Jensen (1987) solved the turbulence equa-tions in streamline coordinates, establishing the role of each term and pointing out the importance of curvature effects.
The linear analysis initiated by Jackson and Hunt (1975) was further revised by Hunt et al. (1988a), who divided the flow field into four regions. The study of Hunt et al. (1988b) extended the results to stably strati-fied flow. The division proposed in these two papers has deeply influenced the current understanding of the ABL flow structure. Belcher et al. (1993) refined the analysis even further, proposing a smooth match be-tween the inner and outer regions. An excellent review of the study of the airflow over complex terrain is pre-sented by Wood (2000), who also reviewed numerical and observational studies.
Although Eq. (1) provides a simple description of the wind field near the ground, it represents very well the atmospheric flow over a flat terrain under neutral at-mosphere, even in the case of very rough surfaces. This fact suggests that the logarithmic law may be extended to cover flows over hills, but few attempts are reported in the literature. Taylor and Lee (1984) proposed an empirical exponential damping of the maximum speedup over the hilltop, which can be used to obtain a modified log law. Their result was later revised by Walmsley et al. (1989) and Weng et al. (2000). Later, Finnigan (1992), using asymptotic matching techniques, assumed the well-known buoyancy-curvature analogy and proposed the acceleration-curvature analogy to obtain another solution in the form of a modified log law. Finnigan's solution is an implicit function because it depends on the curvature Richardson number and on the curvature of the z axis in streamline coordinates. Comparison with observational data showed encourag-ing results.
In this work, a new modified logarithmic law is pro-posed for the mean wind velocity distribution in flows over low-sloped 2D hills, under neutral stratification conditions. The result appears in the form of a flux-profile relationship, obtained through an order-of-magnitude analysis of the x momentum equation, which is carried out by employing the intermediate variable technique (IVT). The analysis also suggests a new dy-namic subdivision of the flow field in regions charac-terized by the predominance of groups of forces. From the modified log law, new results for the flux-gradient relationship, the velocity speedup, the relative speedup, and the heights at which their maximum values occur are also derived. A new parameter, called the speed-up
FIG. 1. Idealization of airflow over a low hill, showing its radius
of curvature at the hilltop.
ratio, is proposed that allows the calculation of the rela-tive speedup at given heights to be performed. All re-sults are tested against field data for vegetated hills. Comparison with the data of Askervein Hill (Taylor and Teunissen 1987), for the case of short vegetation, and with the data of Black Mountain (Bradley 1980), for the case of tall vegetation, shows very good agree-ment. Because this work aims at obtaining a new modi-fied log law, the structure of turbulence is not consid-ered. As a consequence, the law proposed here is not an exact detailed solution for the velocity field in the en-tire ABL. It is an extension to the well-known Eq. (1) and serves essentially the same purposes.
2. Definition of the problem and governing equations
Following Kaimal and Finnigan (1994), we define hill as a topographical feature with a characteristic length of less than 10 km, whereas larger features are consid-ered mountains. A hill slope is considconsid-ered low when it never exceeds 100, leading to heights on the order of 1 km or less. This value is the maximum value for which observations and wind tunnel experiments indicate that no separation occurs, even under strong wind condi-tions (Kaimal and Finnigan 1994).
We consider an isolated two-dimensional low-sloped hill in the middle of an otherwise flat terrain of constant or slowly varying surface properties. Suppose the exis-tence of a neutrally stratified atmosphere and a period of the day during which the flow can be considered statistically stationary. Figure 1 illustrates the idealized flow. The vertical coordinate z is defined as the height above the local terrain, and the horizontal coordinate is x. The main geometrical parameters of the hill are its height h, its surface curvature radius Rh, and its hori-zontal length scale Lh. The hill's surface radius of cur-vature is defined as [1 + (dhldx)2]3'21id 2hIdx2I. It can physically be interpreted as the radius of a circle tan-gent to a given curve with the same local rate of varia-tion of its direcvaria-tion along the curve. Figure 1 shows the local radius of curvature of the hilltop (HT). This
nition implies.that Rh is less than 0 at the HT. The
length scale is defined as the horizontal distance from the HT to the half-height point, and the location up-wind of the hilltop at which the velocity profile is not perturbed by the presence of the hill is the reference site (RS).
The vertical profile of the mean horizontal wind at RS la(z) is considered to be essentially logarithmic and is given by Eq. (1). Over the hill, the mean velocity is given by
u(x, z) =170(z) + Au(x, z), (2) where the speedup A1U is caused by topographic and surface property variations. The speedup is positive at HT because the flow is accelerated to satisfy the con-tinuity equation and is negative somewhere at the up-wind slope because of hill curvature effects. By dividing the speedup by the RS velocity, the relative speedup AS can be defined as
AS(x, z) = Au(x, z)/u70(x, z) = 7u(x, z)/iu0(x, z) - 1. (3) Over the last decades many works have been focused on determining the vertical profiles of Aul and AS. Ex-cept for Lemelin et al. (1988), most ol the results are valid only for the HT, although it has been recognized that results applicable for the upwind and downwind slopes would be exceedingly valuable.
a. Governing equations for the ABL
Finnigan (1983) recommends the use of streamline coordinates to treat the ABL equations and describes its advantages in the study of flow over hills. Following Finnigan (1983), the time-averaged governing equa-tions for 2D stationary flow in streamline coordinates, under the Boussinesq approximation, can be written as
--2 1 a aU,2 __ a- + +2-W2 La pO ax ax az La R AT -gx-+ Vx and (4) To 172 1 a) aw2 aW u,2 2 W R_ ax+ R
+2
La RPo
az
azax
RL,and the viscous terms defined as
a82-1 a2T
2
adu
I
au-u
x
aX2
az La dax R az R 2 and (8)a2j7
1al
a
t1al
Vz =V_ + _ + +_ _ + _ (9)
1 axaz La, ax ax (R
RLj
In the preceding set of equations, x represents the direction parallel to the streamlines and -u and u' are the mean and turbulent velocities in the x direction, respectively. The direction normal to the streamlines is represented by z, and the corresponding turbulent ve-locity is w'. The thermodynamic mean pressure is de-noted by p and the mean temperature is T. Respec-tively, To and po are the reference temperature and density of the environment, which is considered to be hydrostatic with the air assumed to be an ideal gas. The difference between the mean temperature and the en-vironmental temperature is denoted by AT. The com-ponents of gravity in the x and z directions are denoted by g. and gz, respectively, and the dynamic viscosity is
v. In Eq. (7), fl represents the mean component of vorticity in the direction normal to the plane of the flow. Finnigan (1983) points out that fl is an invariant of the transformation, and so it can be written in Car-tesian coordinates as
au- aiw
= =
ax'
(10)where w is the z component of the flow velocity, which is zero in streamline coordinates. In Eqs. (4) and (5), La can be interpreted as a length scale for the acceleration term in the x direction, whereas R can be interpreted as the local radius of curvature of the streamlines in 2D flows. Finnigan et al. (1990) state, "both La and R are signed quantities, being positive if their local centers of curvature lie in the positive z and x directions, respec-tively." For real hills, this means that R is less than 0 in the vicinity of the HT and R is greater than 0 in the remaining parts of both slopes. In the set of Eqs. (4)-(10), the mass conservation equation is not included because it is automatically satisfied by the transforma-tion. Finnigan (1992) shows that it must be substituted by the geometrical identity
a 1 a 1 1 1
ax
(L.)
az(R
L 2 R 2 (11)AT
- gz - + z
To
with the characteristic lengths La and R given by
1 1 au
-==- and
La u ax
1 =1 au\
R a V az)
(5) b. The intermediate variable technique
In this section, an overview of the method used here to simplify Eqs. (4)-(10) is presented. The method em-ployed here has its roots on the technique used by (6) Prandtl (1904) to simplify the equations of the flow over a flat plate. In a modern version (e.g., Schetz 1993) the method consists in, first, making the governing (7) equations of the problem nondimensional, which must be done in such a way that every term of the equations
PELLEGRINI AND BODSTEIN
is rewritten as a combination of order-1 variables mul-tiplied by a nondimensional parameter. Depending on the nature of the problem, important phenomena may occur very close to the boundaries of the flow, and so regions in which the nondimensional normal coordinate is nearly zero must be investigated. The next step is to "stretch" the nondimensional normal coordinate through a variable transformation. This is achieved by dividing the normal (nondimensional) coordinate by a (nondimensional) small parameter that is obtained dur-ing the preceddur-ing step. In Prandtl's (1904) work, the procedure was used to magnify mathematically the thin region of the flow field in which friction forces are im-portant. At this point, all terms of the equations are a combination of order-1 variables and nondimensional terms, and it is easy, therefore, to compare them and eventually to neglect the higher-order ones. After this simplification, the resulting equations can be trans-formed back to dimensional form and solved, if they are simple enough. It is evident that the choice of the pa-rameters used to render the equations nondimensional and to stretch the vertical coordinate is crucial.
A number of extensions to the method depicted above appeared in the study of different engineering problems. One was the IVT (Kaplun 1967; Lagerstron and Casten 1972; Mellor 1972; Roberts 1984). In IVT, the governing equations are made nondimensional and are stretched, as in Prandtl's method. The stretching parameter is then allowed to vary continuously be-tween certain limits to magnify different regions of the problem's domain. As a consequence, different regions of the flow field are defined, each governed by certain terms of the original equations, to first order of ap-proximation. Again, the choice of the parameters is a crucial step.
c. Order-of-magnitude analysis
We now apply the IVT to Eqs. (4) and (5). Readers not interested in the mathematical details may skip the next pages and go directly to section 2c(3).
The notation adopted here follows Tennekes and Lumley (1972). If the error involved is less than 30%, the symbol "_" is used. Coarser approximations are represented by "-." This symbol is also used whenever two or more terms of an equation are compared. After the dominant terms have been established, the simpler notation "=" is used in the resulting equation, bearing in mind that the error can be made as small as necessary in some appropriate asymptotic limit. Whenever M is much greater (at least one order of magnitude larger) than N, the notation M > N is used.
Equations (4) and (5) are made nondimensional by the following variable transformations: X = xIL, Z = zIL, U = uilUg, P = pl(pOU2), and flad = LLlUg. Here,
L is the horizontal length scale of the problem under
study and Ug is the geostrophic wind speed. Although ABL problems usually do not consider Coriolis force
effects, the geostrophic velocity is used because it rep-resents an upper limit of the ABL velocity in most situ-ations. The turbulence terms in Eqs. (3), (4), and (5) can be made nondimensional with the friction velocity
u*. After dropping the overbars representing the time
average for simplicity, the result is
U2 Lad aP 2 (au'2 auw' U'2 -W2 - --- _ + -ax *\ ax az Lad - 2 U W GrG + 8RVxd and U2 R ad (12) aP 2(aW'2 au'W' U'2 - W- 2 az * az ax Rad - 2-) - eRGr, + (13) where u* 1 e* = U and 8 R = R ug
~~Re
(14a,b)are the small parameters. They are indeed small, be-cause u* < Ug and Re > 1 in the ABL. In Eqs. (14a,b),
Re =UgL/v is the Reynolds number and Gr, = (gxL3I v2)(ATITo) and Gr, = (g,L3/v2)(ATIT0) are the Grashof
numbers in the x and z directions, respectively. The variables Lad and Rad are defined as
1 1 au
-=--
~and
Lad La UaZ
1 1 (d aU)
R ad U QaZ The nondimensional viscous terms are
d._ (a2U a2U 2
aU
1au
Uax
+ aZ2 Lad aX Rad aZ Rd2) (15) (16) and (17) d 2U 1 au+ a / U )+ U vzd= aXaZ L ad aX ax tR ad R aLad (18) To clarify how the IVT works, we give a short ex-ample at this point. As Jackson and Hunt (1975), Hunt et al. (1988a), and Kaimal and Finnigan (1994) do for other purposes, suppose the existence of a region in which the mean flow advection and the cross-stream divergence of the shearing stresses balance each other in the ABL. Substitution of Eq. (15) into Eq. (12) yieldsuau/az
-- e2(aU'W'/aZ). Recalling that all variables, except possibly Z, must be of order 1, the previous relation suggests that Z - E2* Because Z zIL, this implies that the balance between inertia and turbulence occurs in a region in which z - L2* = L(u 2/U2).903 JUNE 2005
The preceding analysis shows the existence of a re-gion in which advection and shearing stresses balance could be assessed by substituting a stretched variable Z* = Z/42 of order 1 in Eq. (12). This is actually what Prandtl's method does, but what distinguishes it from the IVT is that, in the latter, the small parameter in the denominator is allowed to vary. Therefore, we define
Z z
£
g ' (19)
with variable 8. Setting 8 - £2* in Eq. (19) and assuming
that after the stretching Z* -1 implies that Z - e. Substituting this conclusion into Eq. (12) readily gives
uaU/aZ - (2*(aU'W'/aZ), as before. From Eq. (19) and Z* - 1, it follows that z - sL, meaning that advection and shearing stresses balance in a region defined by z
-L£ 2 = L (U21UQ)
The preceding example illustrates essentially how the IVT can be used to associate flow regions with the dom-inant terms in an equation. Varying the value of £
causes other terms in Eq. (12) to become of the same order in other regions. For example, requiring that 8 -BR/4* implies that *( U'W'/OZ) S £R02u/az, which
means that shearing stresses and viscous effects are of
the same order in a region of the ABL defined by z -L(u4/U2)/Re. The process of varying s can be used sys-tematically to search for values that make the terms containing Z in Eq. (12) change their order of magni-tude-the range for this variation being 0 < s < 1. Equation (19) shows that as e -* 0, the surface is ap-proached, and as s -> 1, the stretching effect disappears. The stretching effect is, therefore, inversely propor-tional to the value of 8. One interesting characteristic of
the IVT is the fact that the simplified equations that are obtained often give relatively good results for regions much larger than that established in the analysis. This general behavior is shared by other asymptotic theo-ries-for example, the linear analyses carried out by Jackson and Hunt (1975). According to Wood (2000), "All ... observational campaigns broadly supported the predictions of the theory .. ., even in cases when the theory is not strictly applicable." The IVT can be applied to a broader class of phenomena, known as boundary layer problems, characterized by a distinctive mathematical behavior near the physical boundaries.
We now return to the problem. Equation (19) is sub-stituted into Eqs. (12) and (13), and the viscous terms and Rad are written out. The result is
-2GrX
82U I a2U 2/ 2 (
+Rax2 82 aZ*2 Lad ax E2U £
aU 1 a
az*
/
c2U
1
aP
21W' 2 au'W' lu' 2 -_w 2 d aud U _w~~ ~3§Z*+
-- (eU)
- 2
U'W]
8Gr,£
az*
az*
ax
fla£)
L aar1
a2u 1 au I a S Qad +aU ) 1 1 (SfQad +aU )I
La t a t aThe analysis of the preceding equations may be sim-plified with a change in notation. The advective term on the left-hand side of Eq. (20) may be denoted A,. On the right-hand side, we denote the pressure term by PA, the turbulence terms (second and third) by Tx, and Tx2,
the curvature terms (fourth and fifth) by Q1l and C.2, the buoyancy term (sixth) by B., and the viscous terms (the remaining) by Vx1, . . ., V.5. Analogous definitions can be used for Eq. (21). After multiplying these equa-tions through by 82 and 8, respectively, the result is
8 Ax = P - T + Tx2- -Cx2) - £282B + S8(e 2 VXI + V,2 - £ 2 VX,3 - VX + VX5) and (22) A, = -Pz - S*(Tj 1 + ET,2 - C,1 sC,2) - s2B, + 8R(-VzI + VZ2 + VZ3 + Vz4). (23)
Equations (22) and (23) can now be stretched and the leading-order terms can be sorted out to determine the approximate governing equations for the various flow regions. This is done in sections 2c(1) and 2c(2), which follow. There, the equations are presented in the sim-plified notation introduced above and their regions of validity are presented in small parameter form. In sec-tion 2c(3), equasec-tions and regions are returned to their dimensional form, and typical values for the heights bounding those regions are analyzed.
1) ANALYSIS OF THE x MOMENTUM EQUATION To carry out the analysis, it is necessary to establish a relation between £* and 8R. In the literature, it is usual to assume that < < 82*. In mathematical terms,
this relation is considered a working hypothesis in the context of asymptotic theories, and it is based on the
U2 'aP 2FaU2 1 aU'W' U'2 - W'2 1 UIWI i'd
au'\1
Lad
L
aX £Z* Lad -2- U L8+azu
(nad - ( £+ aU J and (20)
PELLEGRINI AND BODSTEIN
fact that as Re increases, 8
R decreases and £, increases.
In physical terms, this assumption may be supported by comparison with field data. Typical values of the pa-rameters involved in the definitions of E* and sR are Ug -lOms-1 ,LL-1000m,andv -1.5 x 10-5m2s-1 .For
11*, a typical value of 1.0 m s-1 is assumed, although Holton (1992) suggests a lower value. The reason is that the suggested value (u* - 0.3 m s-1) holds for flat
sur-faces and u* is expected to increase near the HT. Substituting these values into Eqs. (14a,b) results in N/ R
-
4 X 10-5 and42
-1 x 10-2, which shows that R < 82* indeed. Allowing £ to vary in Eq. (22) produces the results below.(i) Advective region (region V)
If one assumes that 2* < e ' 1 in Eq. (22), the largest turbulence and curvature terms, 8*T,X2and 48*SC,2, can be neglected in comparison with the advective and pressure terms, 82AX2 and 2Px. Because /R < 62*
\l-eR < s ' 1 and the largest viscous terms, ERVX2,
ERVY4, and ERVX5, can also be neglected in comparison with the advective and pressure terms. Therefore, Eq. (22), correct to order 8, simplifies to
A, = - - £B (24)
To be rigorous, the buoyancy term BX = Gr -(g.L 3/
i)(ATIT0) in Eq. (24) can only be determined if AT is
known from the coupled solution of the energy equa-tion, which depends on the static stability of the atmo-sphere. In this analysis, B. is seen as an external forcing term, known beforehand. Keeping B. in Eq. (22) means that it is of the same order of magnitude as A, and P]. This assumption implies that B. - 1-4. If BX ' 1I,R the buoyancy term can be neglected and Eq. (22) sim-plifies to A, = -P,. If 1/E2 < Bx, the buoyancy term becomes the largest, and, therefore, it dominates Eq. (24). In these highly nonneutral cases, however, the Boussinesq approximation breaks down and Eq. (4) is no longer valid. Thus, no conclusions can be obtained from the analysis of this case, unless the energy equa-tion is considered. Figure 2 illustrates the behavior of the terms of Eq. (22) in region V and in the other regions that are defined below.
(ii) Turbulent advective region (region TV)
Now suppose that £ - £2* in Eq. (22). In this case, the largest turbulent and curvature terms are of the same order as the advective and pressure terms. Substituting
2 N _
8 £* into R < 8£ yields eR < 82 .Thus, the largest viscous terms can be neglected in comparison with the advective and pressure terms. Equation (22), correct to order e, simplifies to
AX =PX - T 2 + CX2- RBX. (25)
As in region V, keeping B,, in Eq. (25) means that AX
-J -4 RBB, implying that B. -1
~
RIf B 1/ R theE Region
I
Bx decreases Bx increases CR/EC ---Surface I FIG. 2. Schematic diagram that illustrates the behavior of the terms of Eq. (22) in the five regions. Horizontal scale indicates dominant term in Eq. (22). Vertical scale is arbitrary.buoyancy term can be neglected in Eq. (25), which sim-plifies to AX = -PX - T,2 + Cr2. If 1/R < B,, the buoyancy term dominates Eq. (25) and the energy equation has to be considered.
(iii) Fully turbulent region (region III)
If one assumes that SR/I-* < 8 <
42*,
the largest tur-bulent and curvature terms dominate Eq. (22), and the condition\
< '24 is implicitly satisfied. Thus, Eq.(22), correct to order 8, reduces to
0 = 4(-TX2 + C 2) -RBx. R (26)
This time, keeping B, in Eq. (26) implies that E2 T(2
8RB4B, which, in turn, implies that B. - 4/8R,
be-cause TX2 -1. Substituting 8R!I E 4 8 £ into this expression yields 1/e < B, ' 8*/4. If BX - 1/E2, then
the buoyancy term can be neglected, and Eqs. (24)-(26) simplify to0 -TX2 + Cx2. If
4*/
_ B the buoyancyterm dominates Eq. (26) and the energy equation has to be taken into account.
(iv) Turbulent viscous region (region II)
Suppose now that 8 - ERIE* .In this case, the largest turbulent and curvature terms are on, the same order of the largest viscous terms and Eq. (22), correct to order P, simplifies to
0 T.2 + Cx2 -(e/3R)BX + Vx2- Vr4+ Vs5. (27)
Keeping B. implies that B. - */
.
If BX < */R, then the buoyancy term can be neglected and Eq. (27) simplifies to 0 = -TX2 + C,2 + VX2 - VX4 + Vxs5 If4*/
<E BR , the buoyancy term dominates Eq. (27) and the energy equation has to be considered.(v) Viscous region (region I)
Suppose that 8 < ER/I2 in Eq. (22). Now, the largest
viscous terms dominate, and Eq. (22), correct to order s, can be written as
905 JIJNE 2005
0 =-8 2RBX + V,2 -V4 + Vx5.
Keeping BX implies that s4 /83 < B_ If B_ '_E4
the buoyancy term can be neglected and Eq. (28) plifies to 0 = Y12 - VX4 + Vx5. If
4*14R
< M < B., M being a bounded arbitrary value, the buoyancy idominates and the energy equation has to be coi ered.
2) ANALYSIS OF THE Z MOMENTUM EQUATION
For all possible values of E in Eq. (23), with 0 ' 1, the largest turbulent, curvature, and viscous t can be neglected in comparison with the advective pressure terms. In this case, Eq. (23), correct to o
4
with respect to the turbulent and curvature ti and to order eR with respect to the viscous terms, plifies toAZ = -PZ - ££RBZ.
Keeping B, in Eq. (29) means that AZ - PZ - ££ which implies that B, - 1/eE2. Because the ABL not been divided into regions in this case, the use o condition 0 - e - 1 into BZ - 1/e842 only impos
restriction on B, valid for the ABL as a whole. A r useful result can be obtained by rewriting s in tern
BZ according to l 1/BzR. In this case, the magni of BZ determines the region for which the buoy term has to be included in Eq. (29). If BZ < 1/8ER buoyancy term can be neglected within the regior fined by 8 < IlBz 2 and Eq. (29) reduces to Az
-In the region £ > 1/BZ, the buoyancy term domir Eq. (29) and the energy equation has to be consid( 3) SIMPLIFIED EQUATIONS IN DIMENSIONAL FC
It is useful to rewrite the nondimensional equa above in dimensional form so that they are read, general meteorological use in numerical schemes analytical solutions. Their regions of validity must be rewritten in physical terms. Substituting the de tions of the nondimensional variables and small pa eters where appropriate and recalling that Z* - 1 equations in dimensional form, written in strean coordinates for the regions identified above, becon described below.
(i) Advective region (region V)
In this region, u2 La
1 ap AT
PO aXx
TO,
(30)valid for (U*/Ug)2 < zIL - 1 and Grx/Re2 -1. Using the typical values of section 2c(1), the first condition leads to 10 m < z ' 1000 m.
(28) (ii) Turbulent advective region (region IV)
$/83 In this region,
*~ R,
--sim-
-U2
1 a)5 au'w' u,'w' ATwith L, - = - Po ax dz + 2 8z R - -gx T- (31) valid for zIL - (U*IUg)2and Gr./Re2 -1. Typical values yield z - 10 m.
(iii) Fully turbulent region (region III)
In this region,
au'w' u'w' AT
0 = -- +2 -g,o
az
R TO , (32)erms valid for (UgIu*)2/Re < zIL < (u /U8)2 and 1 << Gr,/
sim- Re2 < Re(u*IUg)4. Typical values yield 0.2 mm < z < 10 m.
(29) (iv) Turbulent viscous region (region II) 2RBz, In this region,
has au'w7 utW7 AT
f the 0=- +2 g,
-az R TO
a2i 1 au
+ vaz2
R az
-A),
(33)Rji
valid for zIL - (U)/u*)2/Re and Gr,/Re2 - Re(u*/Ug)4. Typical values yield z - 0.2 mm.
(v) Viscous region (region I)
In this region,
AT 7a2 7 1 a7i i- I 0 =
o azz R aZ R, (34)
valid for zIL < (UgIu*)2/Re and Re(u*/Ug)4 < Gr,/Re2. Typical values yield z < 0.2 mm.
(vi) Entire ABL
For the entire ABL, - 2
R
1 a) AT Po az bzTo '
(35) valid for 0 - z/L c 1 and zIL - Re2/Grz. Equation (35) without the buoyancy term reduces to Euler's equation in the direction normal to the streamlines.
d. Comparison with the regions found in previous studies
Here, we compare the regions found in previous studies with the results obtained in our study. Hunt et al. (1988a) proposed splitting the ABL over hills into two regions, each subsequently divided into two layers. Based on the predominant forces that define them, we
PELLEGRINI AND BODSTEIN
establish the following correspondence: Hunt et al.'s upper and middle layers (their outer region) corre-spond to our region V; Hunt et al.'s shear stress layer corresponds to our region IV, and Hunt et al.'s inner surface layer corresponds to our regions I, II, and III. 3. Solution for the fully turbulent region under
neutral atmosphere
We now focus our attention on a solution for the equation for the fully turbulent region. We consider only the neutral atmosphere case, for which Eq. (32) reads
au'w' ulw'
= 2 - (36)
az R (6
To solve Eq. (36) we suppose that region III is close enough to the surface that the streamlines can be as-sumed to have the same radius of curvature as the hill's surface, that is, R(x, z) R(x, 0) = Rh(x). This
suppo-sition is equivalent to assuming that the flow is parallel to the surface in this region, which is supported by the experimental results of Salmon et al. (1988). Equation (36) can then be rewritten as aqi/az - p(x)$ = 0, where f(x, z) = -u'w' and p(x) = 2/Rh. The solution to this equation is
+i(x, z) = C(x) exp[2z/Rh(x)], (37) where Cl(x) is the integration constant. To determine the value of Cl(x), one boundary condition for the fully turbulent region is necessary. In IVT (as in other types of asymptotic analysis) this condition may come from matching Eq. (37) to the solution of the layers imme-diately below and above, that is, regions II or IV. None of these conditions are available here because no simple analytical solution for regions II or IV could be found. Therefore, an empirical condition must be sought. If we define ( = zIRh, we rewrite Eq. (37) as q(x, z) = C,(x) exp(26).
Field results show that over flat terrain, the momen-tum flux (u'w') does not vary appreciably with z next to the surface, and so f = u'w'(x, z) = u2 for small z. Assuming this behavior to be also valid for low hills and extending the region of validity of Eq. (37) to include the region close to the ground yields u2 = C, exp(260), where 60 is the value of ( on the lower boundary. There-fore, Cl = u2* exp(-2 0), and consequently +(x, z) = u2
exp(-2(0) exp(26). Because 6 = zlRh, we set (o = Zo/Rho in this equation, with Rho being a parameter associated to the radius of curvature defined below. Equation (37) then becomes
{>(x, z) = u* exp(-2zO/Rho) exp(2z/Rh)- (38) The choice of (0 = zolRho is important and deserves special consideration. To clarify the reasoning involved, the reader is briefly reminded of one possible deriva-tion of the log law, Eq. (1) (e.g., Stull 1997). If one
assumes that the mixing length theory holds, the mo-mentum flux in the surface layer can be written as -u'w'(x, z) = (Kzau/az)2. Supposing the flux to be ap-proximately constant in the region considered, we can write u2 = (Kzau/az)2.Taking the square root, separat-ing variables, and integratseparat-ing from zo to z with -u(x, zo) = 0 finally yields Eq. (1). Because in this integration the initial approximation -u'w'(x, z) = (Kzau/az)2 does not
hold at z = 0, then zo 0 0. The roughness length zo is interpreted as the height at which 7F vanishes.
By analogy to the fact that zo 0 0, our analysis sug-gests that 6O0 zolRh because Eq. (37) also does not hold all the way down to z = zo, and 0 = zolRho is then the logical choice. This expression defines Rho. There-fore, in the same way as zo O ho, where ho is the real height of the roughness elements, Rho is expected to be different from Rh. The parameter Rho, hereinafter re-ferred to as the effective radius of curvature, can be physically interpreted as the radius of curvature that the hill should have for the momentum flux, extrapo-lated downward close to ground, to be algebraically equal to its flat terrain value u2.
Adopting a turbulence parameterization allows Eq. (38) to be integrated to yield the velocity profile. As-suming that turbulence is appropriately modeled by the mixing length theory (in streamline coordinates), with the mixing length given by im = KZ (K being the von
K6rman constant), tp may be written as tp = -u'w' = (Kzau/az)2. Substituting this expression into Eq. (38) and separating variables yields
a-i U* exp(z/Rh)
-= - exp( Zf/Rho)
az
K (39)The use of a mixing length parameterization implies that turbulence is in local equilibrium in the region. This can be justified by observing that this hypothesis is known to be valid close to the surface over flat terrain and that we may assume no qualitative change to occur over low-sloped hills. The assumption can ultimately be verified a posteriori by comparison with field data.
Equation (39) indicates that the velocity profile de-pends on both Rho and Rh. Because Rh is not easily modeled and is generally not measured in field cam-paigns, we avoid the explicit dependence of Eq. (39) on it, replacing Rh by Rho and assuming that the difference can be absorbed into u*. Hence, we rewrite Eq. (39) as
a- u* exp(z/RhO)
-
az
=- exp(- zo/RhO) zK (40)
This procedure provides an approximate model for Eq. (39) and yields a practical solution to the problem, because Rho can be estimated from field measurements, as shown in the next section. The use of such an ap-proximation prevents an estimate of the numerical value of Rho from being obtained a priori, despite the physical interpretation of Rho.
Integration of Eq. (39) can now be performed be-907 JUNE 2005
tween zO and z, assuming that u7(x, z,,) = 0 as in the flat-terrain case. The result is
u* exp(-zo/R,1 0)
Ua(x, Z) = K [Ei(z/R/,O) - Ei(zo/RhO)],
(41) for z - zo. The value of zo in Eq. (41) is assumed to be the same as that over flat terrain. The function Ei(x, z) is the exponential integral function, which has well-known properties (Abramowitz and Stegun 1970). If one expresses Ei(x, z) as an infinite power series (Abramowitz and Stegun 1970), Eq. (41) can be rewrit-ten in terms of the logarithmic function as
u* exp(zolRhO)
-U(x, Z) = K
+ (Z/R,1,0) - (ZO/Rho) ]
Zn n=l n * n!
(42) Equation (42) [or Eq. (41)] provides a new law for the ABL flow over low hills under neutral atmosphere, and it is hereinafter referred to as the modified
loga-rithmic law. These equations have no restrictions
re-garding their application to points in x along the slopes of the hill, provided there is no flow separation. Con-sidering that Rho Rh, (as shown below) and that the streamline coordinates reduce to ordinary Cartesian co-ordinates when Rh - -o (the flat-terrain case), Eq. (42) reduces to the logarithmic law, Eq. (1), when R,, -4 O. Equations (41) and (42) can also be extended to include hills covered with tall vegetation simply by displacing the origin, as in the flat-terrain case. Hence, it is as-sumed now that z denotes the displaced height instead of the height above the ground.
Equation (40) can be rearranged in flux-gradient form. Defining the nondimensional velocity gradient as
fn
= (KZIu,)(.oiIaz) results in0,n = exp[(z - zo)/R,o]. (43)
a. Speedup and relative speedup
Equation (41) can be used to write speedup and rela-tive speedup equations. With the recollection that the streamline coordinates reduce to Cartesian coordinates far from the hill, substitution of Eqs. (l) and (41) into Eqs. (2) and (3) yields
u* exp(-zO/RhO)
AU7(X, Z) = K [Ei(z/Rh1O) - Ei(zo/R/,O)]
U*o
-- ln(z/zo0 ) and
u* exp(-ZO/RhO) [Ei(z/Rh,o) - Ei(zo/Rj,
AS(x, z) u*,0 I ln(z/zo0)
- 1,
(44
where the friction velocity and the roughness length at RS are denoted by u*o and zo0, respectively. In power series form, Eqs. (44) and (45) become
U* exp(-ZO/RhO)
AuK =
K
X [In Z + I nZRo -(oR) n!
U*o z -- ln- and K ZOO u* exp(-zo/Rho) u,,,1n(z/zoo) (46) (47) Equations (44) and (46) only hold where both the logarithmic law and the modified logarithmic law hold. In mathematical terms, they are defined for z-zo,, with zo0 = max(zo, zoo). The same is true for Eqs. (45)
and (47), which are defined for z - z, and zo. :t zoo. Equations (46) and (47) show that both A-a and AS -4 0
when Rh -> °-*
b. Heights of maximum speedup and relative speedup
Readers who are not interested in the mathematical analysis that follows may skip most of this section and go directly to the relevant results, Eqs. (53)-(55). The heights at which the speedup and the relative speedup are maximum, l and 1i, respectively, can be calculated by setting aAK/az = 0 and 3AS/az = 0. By assuming that these points of maximum fall into the fully turbulent region, we can apply this procedure to Eqs. (44) and (45). By noting that aAWi/az = ad/daz -aolaz and using the results of Eqs. (38) and (1), aA1a/az becomes
aAd Kr U*o -1
- = - - exp[(z - zo)/Rhol - 1 ?
az
KZ L. U,*0 IJ (48)Setting Eq. (48) equal to zero and recalling that u* > 0 (no reverse flow), u,,o > 0, and z - z, >0 results in u* exp[(zcrit - zo)/Rho] = u*,, which has the unique solution
7u*o
Zcrit = Rho In l + z)
U* )
for Zcr,j > 0. This solution implies that u*,0> u* if Rho >
,o)1 0 and that u*0 < u* if Rho < 0. To decide whether Zcrit calculated through Eq. (49) is a maximum or a mini-mum, the expression u* exp[(z,i,- zo)/Rhol = u.0is
(45) substituted in Eq. (48). It follows that
(49) (ZlRhO)" - (ZOIRhO)'
- [.n Z + 1.
-PELLEGRINI AND BODSTEIN
aAii/az > 0 * u*e(z zo)/R1,O > u*0 => e(z-zORh,, > e(zc,jt-zo)/RhO =A Au increasing aAu/az < 0 => u*e(z Z))/Rh() < U *0=> e(Z zo)/Rho
< e (z.jt-zo)/Rh,O > Au7 decreasing. (50) For Rho > 0, the solutions for the above expressions are
Z > Zcrit => AU increasing f >
z < Zcrit => AU decreasing f
which implies that u*0 > u* and that Zcrit is the absolute minimum. For Rho < 0
Z < Zcrit => AU increasing f Z > Zcrit => AU decreasingJ
which implies that u*0< u* and that Zcrjt is the absolute maximum. In most cases, we are interested in I at the HT, which corresponds to Eq. (52). Substituting for Zcrit with I in Eq. (49) yields
/u*O
I= Rho In ( + z0. (53)
An expression for the maximum speedup, Ai7max, can be obtained by the substitution of Eq. (53) into Eqs. (44) or (46).
The same analysis is repeated to obtain ls from the AS profile, and the complete calculations can be found in Pellegrini (2001). The result is
Is = zo (54)
if zo0 belongs to the domain of AS or limz,z,(z) from the right if it does not. This expression represents an absolute maximum of AS when Rho < 0 and an absolute minimum when Rho > 0, similar to Eq. (53). The ana-lytical result expressed by Eq. (54) has only been ob-tained in the literature from extrapolation of field mea-surements down to the ground (Taylor and Lee 1984; Mickle et al. 1988). Substitution of Eq. (54) into Eq. (47) to obtain ASmax yields
Ul*
ASmax = -- 1. (55)
u*O
Again, Eq. (55) represents a maximum of AS when Rh < 0 and a minimum when R,, > 0.
c. Heights of maximum speedup derived from velocity profiles available in the literature
The procedure to determine I used in the preceding section can also be applied to other velocity profiles available in the literature, such as the profiles of Taylor and Lee (1984), Lemelin et al. (1988), and Finnigan (1992). Taylor and Lee (1984) proposed a simple theory to calculate ASmax and AS(z) in flows over hills, which
can be used to derive an expression for A-a(z) from Eq. (3). Analysis of the expression for AS(z) yields
I+ ln(l+) = C2K2L,h+, (56) where I' lIzo, L' LhIzo, C2 -1IAK
2
and A is a constant that takes different values according to the type of hill considered. Equation (56) is identical to Jackson and Hunt's (1975) equation, l+ In(l+) = 2K2L ,L except that the constant 2 is substituted here by C2, which varies from 1.6 to 2.2 (with K = 0.39), depending
on the value adopted for A. As a consequence, assum-ing the validity of Taylor and Lee's (1984) velocity pro-file is the same as assuming the validity of Jackson and Hunt's (1975) expression for 1.
Lemelin et al. (1988) also proposed an expression for the profile of AS(z). Analysis of their expression pro-duces
(57) where C3 = 1/(2DK2) with the constant D taking on different values according to the type of hill considered. Equation (57) is also similar to Jackson and Hunt's
(1975) equation.
The analysis of Finnigan's (1992) equation for I fur-nishes
(Lh/R) ln(l ) = CK, (58) where C4is a constant to be determined by comparison
with observational data.
d. Speedup ratios
In the present analysis, the relative speedup at any height can be calculated from Eq. (47). However, in some practical applications, it is not necessary to have the entire vertical velocity profile available. Knowledge of the relative speedup at certain standard heights above the ground is often sufficient and can be done by combining Eqs. (47) and (55).
Equation (47) shows that, for a generic height a,
AS(x, a) + 1 = (u*lu*,)f(Rho, zo, zoo), where
f
is a known function. With recollection that AS and (u*Iu*o) are nondimensional parameters, dimensional analysis of the remaining parameters shows that AS(x, a) + 1 =(u*Iu*o)F(Rholzo, RhOjzoo), with F being an unknown function. In addition, we can derive from Eq. (55) that
ASmax + 1 = (u*/u*o). Dividing these two results yields
a ratio
(Rho
Rho - AS(x, a) + 1Ra = G(Zo _ 'Zoo ASax + 1
(59) at height a, where G is an unknown function. From Eq. (59), AS(x, a) can be calculated if ASmax and the depen-dence of Ra on the hill's geometry are known. The value of ASmax can be calculated from Eq. (55) and the dependence of the function Ra on the hill's geometry
JUNE 2005 909
needs to be inferred through comparison with field data, which is presented in the next section.
4. Comparison with observational data
According to Wood (2000), the experimental studies carried out on Askervein Hill (Taylor and Teunissen 1987), Black Mountain (Bradley 1980), and Cooper's Ridge (Coppin et al. 1994) were the only ones capable of providing the near-neutral data for wind speed over isolated low-sloped hills needed for comparison. Other recent studies, such as Walmsley and Taylor (1996) and Founda et al. (1997), provide support to this conclusion. Only one experiment has been conducted under neutral atmosphere since 1996 (Reid 2003), but the density of measurements near the surface is not high enough for the purposes of this paper.
At Askervein Hill and Black Mountain, vertical pro-files of mean wind velocity were measured simulta-neously at RS and HT up to considerable heights. At Askervein, vertical velocity has been measured up to heights of 50 m. Askervein is an essentially elliptical hill, uniformly covered with low vegetation, where the minor axis is 1 km long, the major axis is 2 km long, h = 116 m, and z0 is approximately equal to 0.03 m. At Black Mountain, mean velocities were measured up to heights of 89 m above the displaced origin at the HT and 18 m at the RS. Black Mountain is a slightly asym-metric hill, with h = 170 m and L,, approximately equal to 275 m (in the prevailing wind direction). Its surface is covered with a close tree canopy, with average height of 10 m and zo of about 1.0 m. As mentioned in con-nection with Eq. (39), measurements of the radius of curvature were not made over any of the hills.
At Cooper's Ridge, measurements covered a range of stability classes, including the neutral class, and the hill's curvature was estimated (Coppin et al. 1994). Be-cause the original data are not available, they are not used here for comparison.
a. The modified logarithmic law
Equation (41) depends on four parameters that are not known a priori: z0, K, u*, and Rho. They must be estimated before the modified log law can be used. In atmospheric flow over a flat terrain, the von Karman parameter K is first assigned a value, generally 0.4.
Then, if Eq. (1) is to be valid, field data plotted in semilogarithmic form must follow a straight line. Mea-surement of the linear and angular coefficients yields the values of z, and u*, respectively, for each velocity profile. Application of this procedure to the data at RS determines z(o and u*0. In the ABL flow over hills, this same process can be used if the value of Rho is available beforehand. Because it is not, a different method is then employed. Following the procedure for the flat-terrain case, we define
~
=ln(z) + , and ° °n=1 n n! o - ln(z,) + (zdIRhOY" n=1 (60) (61) With Eqs. (60) and (61), Eq. (42) can be rewritten asu* exp(-ZO/RhO) Tt = K (C - WO) or K -=-exp(z1/R,h0)u + *o. U* (62) In Eqs. (60) and (61), the parameter 11Rho in the argu-ment of the logarithmic functions and the Euler con-stant of the series expansions are omitted because they cancel out.
As in the flat-terrain case, plotting the field data in the variables 4 versus 1u produces a straight line with linear coefficient B = 40 and angular coefficient m = [K exp(zOIRhO)]/u*. Once B has been obtained, 0 becomes known and zo can be calculated from Eq. (61), whereas the determination of m allows u* to be calculated from the previous relation. Both calculations, however, re-quire Rho to be known. Because this condition is not the case, an initial value for Rho is guessed and an iterative procedure is implemented. First, the initial guess of Rho and z00are substituted into Eqs. (60) and (61) so that C
(for each height) and Co can be calculated. The results are then plotted in the form of ; versus -u and a straight line is best-fitted to them, which determines B and m and allows zo and u* to be calculated. Given the as-sumption that the hill is uniformly covered with veg-etation, calculated values of z0 and z00should be equal and a straight line should fit the observed data well. If both of these conditions are satisfied, the value as-sumed for Rho is considered to be correct. If one of the two conditions is not satisfied, a new value of Rho is assumed and the whole procedure is repeated. When both conditions are satisfied, the iteration stops and the last value of Rho is considered to be correct. In the analysis that follows, the von Karman parameter is as-sumed to depend on Re* = u*zoVv, (v being the kine-matic viscosity), according to Frenzen and Voguel (1995), who suggest average values of K = 0.39 for 0.007 < z0 < 0.087 m and K = 0.37 for z0o 1.0 m for most
atmospheric flows. These values are used in our analy-sis.
It can be assumed that Rho depends on the geometry of the hill (through R,,), on viscous and pressure forces, and on z0. Based on that assumption, dimensional analysis shows that Rholh = g,(Rhlh, Re*, Ro*), where Ro* = UJfzo is the roughness Rossby number and
f
is the the Coriolis factor. Because detailed hill profile shapes are not available to determine R,,, an approxi-mate relation between Rh and the global geometry of the hill is used, based on the calculation of thecurva-PELLEGRINI AND BODSTEIN
ture radius of two analytical functions (the bell and the Gaussian shapes) that approximate real hill shapes. Both calculations depend on L /2h as a scaling param-eter, suggesting that Rh cc L /2h and, thus, that Rhlh cc Lhlh and Rholh = g2(Lhlh, Re*, Ro*).
The dependence of Rholh on Lhlh is shown in Fig. 3 for the Askervein data, where different values of Lhlh correspond to different wind directions. The depen-dence of Rhlh on Lhlh for the HT of Black Mountain is not investigated because all of the measurements were
obtained for the same incident wind direction, which gives Rhlh = -0.35 for Lhlh = 1.62. The straight line fitted to the data has a slope that is consistently high, confirming the suspected dependence between Rholh and Lhlh, despite the relatively large data scatter (shown by the correlation coefficient, R2). No strong dependence of Rholh with Re* or Ro* was detected in our study. Therefore,
Rho/h g3(Lh/h). (63)
In the analysis above, the Askervein experimental data acquired on days 25 September 1982, 26 and 30 September 1983, and 5 October 1983 were not consid-ered to be adequate for the purpose of this work and were eventually disregarded. On 25 September 1982 and 30 September 1983 (hereinafter referred to as the 3D days), the wind approached the hill from a direction almost parallel to the major axis, suggesting the occur-rence of nonnegligible 3D effects. On 5 October 1983 (hereinafter referred to as the wake day), the wind blew from the direction of some nearby farm buildings and showed nonlogarithmic profile at RS. The data for 26 September 1982 cannot be well fitted by the modified log law, probably because the first two measurement levels (1 and 5 m) are missing. A thorough investigation of Fig. 3 to assess the effects of the 3D and wake days leads to the conclusion that 25 September 1982 and 30 September 1983 are indeed strongly affected by 3D ef-fects, whereas 5 October 1983 is weakly (but observ-ably) influenced by wake effects. Therefore, the 3D and wake days are disregarded in the discussion that fol-lows. The data for 26 September 1982 are permanently neglected.
Additional analysis of Fig. 3 shows that Rho is differ-ent from the estimated average value of Rh, as sus-pected in section 3. With the values of Rho calculated above, the modified log law can be evaluated for all available observed profiles. Figure 4 shows the experi-mental data of Askervein and Black Mountain plotted in ; versus ii form. The experimental points are average values, weighted by their respective measuring periods. The straight line corresponds to the modified logarith-mic law and shows good agreement with the observa-tions. The influence of the wake day, excluded from the 1983 Askervein data, is shown below.
At Askervein, the results show the expected behav-ior up to heights around 15 m. At Black Mountain, the observed profiles agree well with the theory from 30 to
0.15 0.10 -0.05 -0.00 0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 L h/h
FIG. 3. Nondimensional effective radius of curvature for Askervein data with wake and 3D days excluded.
89 m. The result for Black Mountain is predictable in some sense. Kaimal and Finnigan (1994) observed that the log law is not expected to be valid at the roughness sublayer (the layer going from the ground to 3h0, where ho is the average height of the roughness elements). At
Black Mountain, 3ho = 30 m. At Askervein, however, 3ho is approximately equal to 1 m and the effects of the roughness sublayer are not detected.
b. The relative speedup
Figures 5-8 show the comparison of Eq. (47) with the observational data. The experimental points are aver-age values, weighted by their respective measuring pe-riods. The wake day is not included in Fig. 6 and its effects are shown in Fig. 7, which supports the indica-tion that this point should be disregarded for the rela-tive speedup. In general, the theory agrees very well with observation above the roughness sublayer. Equa-tion (47) is observed to be sensitive to the values of Rho,
u*, and u*,, but its sensitivity to z0 and z00 is negligible. Figures 5 and 6 also show the comparison between Eq. (47) and the equations proposed by Taylor and Lee (1984), Weng et al. (2000), and Lemelin et al. (1988). Overall, Eq. (47) shows the best agreement with obser-vation, whereas Figs. 5 and 6 indicate that ASmax is different from AS(z = 10 m), as assumed by Taylor and Lee (1984) and Weng et al. (2000).
For the Askervein data, Figs. 5 and 6 confirm that I5
- z0, as predicted by Eq. (54). At Black Mountain, on
the other hand, Fig. 8 shows the maximum clearly lo-cated in the jet, far from the ground. This result is ex-pected because, as mentioned before, Eq. (44) is ob-tained from Eq. (54), which is valid for region III, as-suming it extends down to z = z0. At Askervein, where the roughness sublayer is thin, AS does not vary much in this region and, therefore, the approximation is good. At Black Mountain, however, the roughness sublayer is large, AS varies considerably, and Eqs. (54) and (55) are no longer valid in that region. A solution for region I is likely to correct for this effect. As a consequence, no predictions for AS ax are compared with the observa-tions at Black Mountain.
y= 0.0175x- 0.oo07 R2 = 0.5955 o 0 °0 / 0 _ 911 JUNE 2005
4.0 3.0 2.0 1.0 0.0 4.0 3.0 2.0 1.0 o,n I Askervein 1982 I I 5 7 9 11 Velocity at RS (mls) 6 8 10 Velocity at RS (m/s) 4.0 3.0 2.0 1.0 0.0 12 1.0 0.8 0.6 0.4 0.2 0.0 I n 0.8 0.6 0.4 0.2 -0.0 --0.2 -I I I I 11 12 13 14 15 Velocity at HT (m/s) I~~~~ 10 12 14 16 Velocity at HT (mrs) Ir 3.8 3.6 3.4 3.2 3.0 18 0 2 4 6 8 9 10 Velocity at RS (rn/s) Velocity at HT (m/s) 11 FIG. 4. Vertical velocity profiles for Askervein and Black Mountain data. Filled circles are the observations, and the solid line is the best fit. The wake day is excluded from the Askervein 1983 data.
Predicted ASmax values are compared indirectly with the Askervein data because no velocity measurements were made at z,. From Figs. 5 and 6 we see that the observed values of the velocity measurements close to the ground agree well with the theoretical curve. This fact allows us to infer that the predicted value for ASmax also agrees well with the observed value. The same reasoning indicates that ASmax is roughly 90% larger than AS(z = 10 m) for the data in Fig. 5 and is 70% larger for the data in Fig. 6.
Figure 9 shows the dependence of the calculated
val-ues of ASmax on the incident wind direction
4.
The error bars are set as 10% of ASmax. The predictions of Taylor and Lee (1984) are also plotted for comparison and are seen to be substantially smaller than the values ob-tained with Eq. (55). The dependence of L,, on 4) is also represented in Fig. 9, and the expected negative corre-lation with ASma, is observed. Polynomial best-fit curves are fitted to the data, and some Lh values are interpo-lated from Taylor and Lee (1984). Both the 3D and wake days are included (extreme points of the graphs) because they aid in the polynomial best-fit procedure. Askervein 1983 Black Mountain I l Il I I-U .4 v.v. PELLEGRINI AND BODSTEIN
* * Askervein 1983 - Theory - - Taylor and Lee (1984) - - Weng et aL (2000) - -- Lemelin et al. (1988) 'I \
0.2
0.4
0.6
0.8
1.0
Relafive speed-up
FIG. S. Vertical relative speedup profile for Askervein 1982 data and various theoretical results.
c. The height of maximum speedup
The most well-known expressions found in the litera-ture during the last decades to calculate the maximum of the function Au7 come from the works of Jackson and Hunt (1975), Jensen et al. (1984), Claussen (1988), and Beljaars and Taylor (1989). Walmsley and Taylor (1996) present a good review of these expressions. In a recent paper, Pellegrini and Bodstein (2000) propose a new constant for Jensen's equation based on a best fit to available field data. The result is
l+ In2(1+) = 2.41K2L+. (64) Pellegrini and Bodstein (2000) conclude that Eq. (64) describes the experimental data slightly better than the equations obtained in the references above. The work in this paper proposes a new expression to calculate I [Eq. (53)], and only this equation is compared with Eq. (64) in Fig. 10, which shows predictions for l for the Askervein Hill data. All of the days are included so that the influence of 3D and wake days can be accessed. It is clear that Eq. (53) provides the best description of the observed values. Equation (64) tends to overestimate field results and shows large dispersion and poor pre-diction for the 3D and wake days (appearing far to the left and above the line). The average difference be-tween theoretical and observed values is 44% and the standard deviation of these differences is 117%,
con-0.2
0.4
0.6
0.8
1.0
1.2
Relative speed-up
FIG. 6. Vertical relative speedup profile for Askervein 1983 data (wake day excluded) and various theoretical results. sidering all of the days. Equation (53), on the other hand, is considerably more accurate and shows less dis-persion. The average difference is close to -13% and the standard deviation is close to 20%, no matter which days are considered. A more thorough analysis of this equation is presented in Pellegrini and Bodstein (2004). For Black Mountain, I is observed to be located al-ways at a height of about 30 m. An average value for I is obtained through Eq. (53), using average values of u, and u,0calculated in section 4a. The result is I = 41.8 m,
overestimating the observed value by 39%. The result obtained from Eq. (64) produces I = 14.0 m, underes-timating the observed value by 53%.
d. The speedup ratios
The experimental data of Askervein are now used to obtain the dependence of Ra on R1,0/zo in Eq. (59). The dependence of Ra on z00cannot be explicitly detected because zo = z00. Because the measurement levels were different during the 1982 and 1983 campaigns, these years are considered separately. For the 1982 data, only seven velocity profiles are available, and these results are disregarded. Because the measurements for the 1983 campaign closest to the standard levels of 2 and 10 m are 3 and 8 m, respectively, the ratios R3and R8are calculated instead. The results of Black Mountain are disregarded because ASmax cannot be calculated (sec-tion 4c).
