Roughness length, z0 and friction velocity, u* are the defining parameters of wind log profile that must be matched in wind tunnel simulation. To fully understand the role of these parameters, the basics and review from the primitive equations and its relation to the logarithmic profile obtained for wind tunnel conditions were discussed. The problem of roughness, although well known, still needs to be addressed more rigorously especially when determining values of z0 and u* from wind tunnel data and their relation to the roughness element geometry. A review of classic literature and new published material were carried out, focusing on the applicability to wind tunnel modelling

Beeck, J. V.

Buckingham, S.

Conan, B.

Cuerva Tejero, Alvaro

Pérez Grande, María Isabel

Yeow, Tee Seong

English

Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid

   
European Academy of Wind Energy  Norwegian University of Science and Technology 
teeseong.yeow@upm.es 
Modelling of atmospheric boundary layer: Generation of shear 
profile in wind tunnel 
Yeow, T.S.1), Cuerva, A.2) Pérez, J.3) Conan, B.4), Buckingham, S.5) Beeck, J.V.6) 
 
1) 2) 3) Universidad Politecnica de Madrid, ETSI Aeronauticos, Instituto Universitario  
Ignacio Da Riva, IDR-UPM, Spain,   
4) 5) 6) Von Karman Institute, Belgium 
ABSTRACT 
Roughness length, z0 and friction velocity, u* are the defining parameters of wind log profile 
that must be matched in wind tunnel simulation. To fully understand the role of these 
parameters, the basics and review from the primitive equations and its relation to the 
logarithmic profile obtained for wind tunnel conditions were discussed. The problem of 
roughness, although well known, still needs to be addressed more rigorously especially when 
determining values of z0 and u* from wind tunnel data and their relation to the roughness 
element geometry. A review of classic literature and new published material were carried out, 
focusing on the applicability to wind tunnel modelling.  
 
KEYWORDS 
log profile, roughness length, roughness height, friction velocity  
1 INTRODUCTION 
1.1 Logarithmic Profile 
In describing the atmosphere, the set of seven equations configured by the Navier-
Stokes ones plus continuity, energy, state equation and conservation of humidity is the most 
rigorous model known. The full set of equations, while giving the best solution, is practically 
impossible to solve due to non-linearity and large number of complex initial and boundary 
conditions that need to be considered. Starting from these equations, performing the 
ensemble averaging and applying the corresponding averaging rules to obtain the RANS 
equations, and confining the problem to wind tunnel conditions (channel flow hypothesis, see 
Schlichting and Gersten, 2000 and Wyngaard, 2010), therefore adopting a few assumptions; 
isentropic flow, dry air considered and incompressibility (density,  = constant); the problem 
is reduced to a much simpler approximation though still leaves much to be solved. The 
classic logarithmic profile equation is attained in the case of a flow on a smooth, flat surface, 
   
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for the mean value of the longitudinal component of the wind speed, U (see expression (0.1)
), in terms of the well known non-dimensional “+” variables, after using as characteristic 
length, l+ = /u*, being  the kinematic viscosity, and as characteristic velocity, the friction 
velocity, u* (u* is a reference velocity applied to the motion near the surface where the shear 
stress is not a function of the distance from the wall and defined as *u   , where  is 
the Reynolds stress and  is the density of the fluid). 
*
*( ) ln( )
uU z z Bu           (0.1) 
z+ is the non-dimensional distance from the wall,  is the von Karman constant (taken as 0.4) 
and B is an integration constant. In the considered case of smooth wall, B is usually taken to 
be 5.1 though experimental values give a range 5.0 – 5.5 (Raupach, 1991). In the figure 1, 
the theoretical result (0.1) (log law in the figure) is included along with experimental data from 
different authors, the viscous region (viscous sub-layer in the figure, where U = u*z+) and the 
buffer region (buffer layer in the figure, adapted from Garratt, 1992). For a deeper 
explanation on the viscous and buffer regions, see Jimenez (2004) and Durbin & Pettersson 
Reif (2001). 
 
1.2 Rough Wall Boundary Layer 
For rough walls, the logarithmic law can be written as  
*
*( ) ln( ) ( )h
uU z z B h u            (0.2) 
where the constant B is replaced by a function Bh  that depends on the non-dimensional 
roughness geometry. A basic dependency on the roughness geometry for Bh can be defined 
in terms of the non-dimensional equivalent sand grain roughness height h+ = h/l+ (being h the 
dimensional equivalent sand grain roughness height, see Durbin & Peterson-Reif, 2001). The 
equation above can also be written as  
*( ) ln( ) BuU z z            (0.3) 
where B  is a function of roughness height in the form of  
1( ) ln( ) ( )hh h B h             (0.4) 
The logarithmic profile for rough wall can now be rewritten alternatively as  
*
0
( ) lnu zU z
z
    
         (0.5) 
   
European Academy of Wind Energy  Norwegian University of Science and Technology 
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with z0 the hydrodynamic roughness length (also known as aerodynamic roughness length or 
simply roughness length), which is defined from (0.3) and (0.5) as: 
0 exp( B)z h             (0.6) 
 
For smooth walls, expressions (0.1) and (0.5) give the minimum value of hydrodynamic 
roughness length, z0,min = exp(–B)u*  0.14u*. For fully rough surface values B = 8.5 and 
z0 = 0.033h are obtained (see Ligrani & Moffat, 1986). This practice of defining roughness 
influence was first introduced by Schlicting (1936). The effect of roughness on the logarithmic 
profile is a matter of intense study since the pioneering work of Schlicting (1936). As an 
example, in figure 2, the measurements obtained by Schultz and Flack (2007) are shown. It 
can be observed that the higher the roughness height, h+ the lower the wind speed in the log 
region. This is a common result but it is not universal since some types of roughness 
geometry can lead to flow acceleration (see Jimenez, 2004).  
U+
0
100
Viscous 
sub-layer
2
4
6
8
18
16
14
12
10
101 102 103
z+
Buffer layer Log law
 
Figure 1: Velocity distribution above a smooth surface, from laminar flow to fully 
turbulent flow (viscous sub-layer to log law regions). Experimental data 
represented by symbols. The curve at low Re number represents a linear velocity 
profile while at higher Re. represents the logarithmic law. (Schlichting, 2000, 
Garratt, 1992). Observe that z+ = zu*/ can be interpreted as a Reynolds number 
based on the friction velocity and the height on the wall. 
 
   
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h  = 2.3
20
25
15
10
5
0
10 0 
U+=z+
3.2
9.2
16
23
26
Log-law
10 1 102 103 10 4 
U + 
z+  
Figure 2: Mean velocity profile for different roughness heights compared to 
logarithmic profile of McKeon et al. (2004). (Schultz and Flack, 2007) 
 
The relation between the hydrodynamic roughness length, z0, and the roughness geometry 
has been an issue of main concern up to know. In the figure 3, the so called roughness 
function, U/u* (see Jimenez, 2004), which is related to the hydrodynamic roughness length 
by (see Raupach et al. 1991)  
0
* *
exp Uz B
u u
               (0.7) 
is presented for different values of non-dimensional roughness height in lab and the free 
atmosphere. 
 
   
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30
20
10
0
100
Roughness
G
Atmosphere
101 101 103 104 105 106
Laboratory
D
FC
K
BH
AME
I
wire screen
square bar
sand grain
natural vegetation
h+
U/u*
J
 
Figure 3: Roughness function, U/u*, as a function of the roughness Reynolds 
number, h+=hu*/ for different types of roughness elements in the wind tunnel and 
in the atmosphere (from Raupach et al., 1991). Codes for natural vegetation are 
described in the reference. 
 
Studies of different roughness elements by Lettau (1969), Wooding (1973) and Raupach et 
al. (1991), amongst others, established a relation between the hydrodynamic roughness 
length, z0, roughness height, h, and roughness density, , which is the total projected frontal 
roughness area per unit area on the wind tunnel floor corresponding to a single roughness 
element of a homogenous roughness layout, in the form 
0 / 0.5z h            (0.8) 
The relation (0.8) is widely used to define the density, , and height, h, of homogeneously 
distributed roughness elements to match a given value of hydrodynamic roughness length, 
z0. (Dyrbye and Hansen, 1996) A scheme on the definition of roughness density, , is 
presented in figure 4. 
   
European Academy of Wind Energy  Norwegian University of Science and Technology 
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SP
SG
P
G
S
S
 
dC
hC
 
Figure 4: Schematic for determining the roughness density,  with SP being the 
projected frontal area to the averaged wind speed, and SG being the unit area on 
the wind tunnel floor corresponding to a single roughness element.  
 
Therefore, with this method, a simple estimation of roughness element needed to generate 
the required value of z0 can be obtained. Here there is a maximum value of max where the 
equation holds which has been studied to be 0.15. Beyond max, z0/h decreases with further 
increase of , which is attributed to the mutual sheltering of roughness elements (Wooding, 
1973). However, changes of the value of max depending on the roughness geometry, 
suggest a need to study different aspect ratios or additional characteristic lengths (Raupach 
et al. 1991), see figure 5. Lettau (1969) who first proposed equation (0.8), also remarked that 
a sufficient extent of roughness in the upwind direction is needed for the equation (0.8) to be 
valid. This was confirmed by Counehan (1971) who suggested a minimum length of 1000h 
needed to achieve an almost equilibrium boundary layer in the wind tunnel. 
 
   
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10-2
0.01 0.1
10-3
10-1
-200%
+200%

z0/h Cylinders (realistic d values)
Cylinders (d = 0)
Cubes
Spheres
VKI (measured)
3D Elements
0 0.5z
h

max
 
Figure 5: Normalized roughness length z0/h as a function of roughness density,  
for three dimensional elements. The empirical correlation z0/h = 0.5  is shown 
by the dashed line. Measured values of z0/h from VKI initial wind tunnel tests 
(Buckingham, 2010) are included along with the uncertainty levels due to different 
methods of determining z0 from Iyengar & Farell (2001). 
 
Some authors have also paid attention to the limits related to the geometrical roughness 
height, h, for instance Jimenez (2004) remarks that the ratio of the total boundary layer 
height, , to the roughness element height, h, must satisfy /h > 80 if one wants to apply the 
previous considerations. For larger heights, the flow must be better analysed as flow over 
obstacles. 
2 FRICTION VELOCITY AND ROUGHNESS LENGTH DETERMINATION 
Some studies have been devoted to compare the different methods to estimate the 
friction velocity, u* and hydrodynamic roughness length, z0 in wind tunnel flows (see for 
instance, Iyengar & Farell, 2001). These authors compared indirect methods of obtaining z0, 
u* and d, zero-displacement height, (Schlichting, 2000) (using Hama’s law fits and log-power 
law fits) with values obtained from direct measurements of u* (from Reynolds shear stress 
and balance measurements). Estimates by indirect methods, used successfully in smooth 
wall flow studies, can give differences of up to 200% in rough wall cases due to the 
possibility of having several sets of values z0, u* and d giving comparable fits to the same 
velocity profile (Iyengar & Farell, 2001). Direct measurements give errors of up to 15 % which 
is mainly due to the X-wire probe errors in measuring turbulent flows. The predicted value of 
z0/h from the roughness elements (33mm cups with base to top diameter variation of 40mm 
   
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to 30mm) used in the initial wind tunnel study in VKI is shown in the figure 5 along with 
possible value range of z0/h according to the indirect method uncertainty factor of 2 
documented in (Iyengar & Farell, 2001). 
 
2.1 Scaling Factor Effects 
After deciding the model scale, S, at the beginning and considering an objective value 
for z0(real), the value of the required hydrodynamic roughness length in the wind tunnel, z0(WT) 
is determined. As a preliminary attempt, the relation (0.8) can be used for determining the 
geometry of the roughness element and its density to be used in the wind tunnel. By taking  
= SP/SG as shown in figure 5, a range of possible roughness element dimensions can be 
calculated (see figure 6). 
 
0
10
10
n = 5
n = 20
20 30 50 60 80 90 10040 70
20
30
40
50
60
70
80
90
100
n = 12
R
ou
gh
ne
ss
 e
le
m
en
t w
id
th
, d
C
(m
m
)
Roughness element height, hC (mm)
max= 0.15
 
Figure 6: Relation roughness element dimensions, dC and hC, and roughness 
density  for a chosen model scale, S, and hydrodynamic roughness length, z0(real) 
(in the figure, S=500 and z0(real)= 50mm, therefore z0(WT) = z0(real)/S = 0.1mm). A 
homogenous distribution of roughness elements in wind tunnel has been 
assumed, lines plotted are from the different number of elements in a row, n 
perpendicular to the mean flow speed. The limit  = 0.15 is shown with the right 
side of this line for  << max.  
 
Considering the Alaiz site, the large scaling factor proposed for wind tunnel modelling posed 
a new challenge as most wind tunnel modelling have been done to a scale of S <500. 
Considered the minimum value reproducible in the wind tunnel z0,min = 0.14u* a minimum 
   
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real value for the hydrodynamic roughness length can be estimated as z0,min(real) = 0.14Su*. If 
the free upflow to the Alaiz site is assumed to be between Type II and Type III terrain type 
according to the Eurocode classification, which corresponds to a z0(real) range of 0.05-0.3m, 
choosing the model scale to be S 5000, the z0(WT) needed to be reproduced in the wind 
tunnel can be calculated (see table 1). 
 
Table 1: z0 of real terrain and wind tunnel model with a model scale, S= 5000. 
 Terrain Model 
Type II: z0 (m) 0.05 1 10–5 
Type III: z0 (m) 0.3 6 10–5 
 
For flat and smooth wall boundary layers, it is quite well established that the viscous region 
(the region close to the wall where the log profile is no applicable) extends to non-
dimensional values of height, z+  [0, 40]. In case of using large scale models, such as the 
one for Alaiz (S 1:5000) this range of non-dimensional heights could correspond to values 
comparable to the hub height in the real case. 
 
Roughness and non-homogeneity of the terrain lead to variations of this interval of height for 
which the log profile is not applicable (see figure 2). Taking as valid the estimation for smooth 
and flat wall, and considering typical values of kinematic viscosity and friction velocity ( = 
1.510–5 m2/s, u* = 0.5 m/s) the viscous limitation in the case of Alaiz might range in the 
interval z  [0, 1.2mm], which corresponds to an interval in the real terrain z  [0, 6.5m]. So 
no-conclusions about the log profile from the wind tunnel model could be obtained for height 
values less than 6.5m in the reality. Hence, a rough estimation on the minimum analyzable 
height is zmin,(real) = 40Su*. 
 
Table 2: Wind turbine height and distance from model with scaling. 
 Wind Turbine Distance from 
model surface 
Height: (m) 70 0.014 
Height: (m) 100 0.02 
 
The distances shown in the table above indicate that measurements taken on a model with 
scale S = 1:5000 are well into the logarithmic region of the wind profile (non viscous region). 
   
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In this case the limitation would be associated to the precision of the traverse system used in 
the case of a hot-wire system. 
3 CONCLUSIONS 
This paper presents a preliminary study to conduct a modelling of complex terrain in wind 
tunnels with large scales (5000). The analysis is focussed on the generation of a required 
log profile for the mean wind speed. The relation between the hydrodynamic roughness 
length and the roughness geometry has been analysed particularly for the case of large 
scale models. Two issues are outlined, first the fact that a large scale factor might provoke 
that viscous sub-layer on the model extends to heights that in the real field correspond to 
distances from the ground comparable to the hub height, being the minimum analyzable 
height zmin,(real) = 40Su*. Secondly, there is a limit for the generation of the hydrodynamic 
roughness height in the wind tunnel which establishes a minimum reproducible value for the 
real value z0(real) = 0.14Su*. 
 
BIBLIOGRAPHY 
[1] Buckingham,S. 2010. Wind park siting in complex terrains assessed by wind tunnel 
simulations. Master’s thesis, VKI, Brussels, Belgium 
[2] Counehan, J. 1971. Wind tunnel determination of the roughness length as a function of 
the fetch and the roughness density of three-dimensional roughness elements. 
Atmospheric Environment (1967) 5, 637-642. 
[3] Drybye, C., Hansen, S.O. 1996. Wind Loads on Structures. Wiley, West Sussex, 229 
pp. 
[4] Durbin, P. A., Pettersson Reif, B. A. 2001. Statistical Theory and Modelling for 
Turbulent Flows. Wiley, West Sussex, 302 pp.  
[5] Garratt, J. R. 1992. The Atmospheric Boundary Layer. Cambridge Atmospheric and 
Space Science Series, New York, 336 pp. 
[6] Iyengar, A.K.S., Farell, C. 2001. Experimental issues in atmospheric boundary layer 
simulations: roughness length and integral length scale determination. Journal of Wind 
Engineering and Industrial Aerodynamics 89, 1059-1080.  
[7] Jimenez, J. 2004. Turbulent flows over rough walls. Annual Review of Fluid Mechanics 
36, 173-196.  
[8] Lettau, H. 1969. Note on aerodynamic roughness-parameter estimation on the basis of 
roughness-element description. Journal of Applied Meteorology 8, 828-832. 
   
European Academy of Wind Energy  Norwegian University of Science and Technology 
teeseong.yeow@upm.es 
[9] Ligrani, P.M., Moffat, R.J. 1986. Structure of Transitionally Rough and Fully Rough 
Turbulent Boundary-Layers. Journal of Fluid Mechanics 162, 69-98.  
[10] Raupach, M.R., Antonia, R.A., Rajagopalan, S. 1991. Rough-wall turbulent boundary 
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[11] Schlichting H. 1936. Experimentelle Untersuchungen zum Rauhigkeits problem. Ing-
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[12] Schlichting H. 2000. Boundary Layer Theory, 8th ed. McGraw-Hill, New York, 801 pp. 
[13] Schultz, M.P., Flack, K.A. 2007. The rough-wall turbulent boundary layer from the 
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[14] Wooding, R., Bradley, E., Marshall, J. 1973. Drag due to regular arrays of roughness 
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[15] Wyngaard, J.C. 2010. Turbulence in the Atmosphere. Cambridge University Press, 
New York, 406 pp. 
ACKNOWLEDGEMENT 
The work has been carried out with funding from the EU FP7-PEOPLE program, under 
WAUDIT Marie-Curie Initial Training Network. 
Special acknowledgements to Boris Conan, Sophie Buckingham and Jeroen van Beeck from 
VKI for carrying out initial experiments in wind tunnel on a model of Alaiz and sharing the 
results. 


Modelling of atmospheric boundary layer: Generation of shear.

Archivo Digital UPM

Drag due to regular arrays of roughness elements of varying geometry.

Experimental issues in atmospheric boundary layer simulations: roughness length and integral length scale determination.

Experimentelle Untersuchungen zum Rauhigkeits problem.

Note on aerodynamic roughness-parameter estimation on the basis of roughness-element description.

Rough-wall turbulent boundary layers.

Statistical Theory and Modelling for Turbulent Flows.

Structure of Transitionally Rough and Fully Rough Turbulent Boundary-Layers.

The Atmospheric Boundary Layer. Cambridge Atmospheric and Space Science Series,

The rough-wall turbulent boundary layer from the hydraulically smooth to the fully rough regime.

Turbulence in the Atmosphere.

Turbulent flows over rough walls.

Wind Loads on Structures.

Wind park siting in complex terrains assessed by wind tunnel simulations. Master’s thesis, VKI,

Wind tunnel determination of the roughness length as a function of the fetch and the roughness density of three-dimensional roughness elements. Atmospheric Environment

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Modelling of atmospheric boundary layer: Generation of shear.

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