RAUPACH1991

Prévia do material em texto

Rough-wall turbulent boundary layers 
M R Raupach 
CSIRO Centre for Environmental Mechanics, GPO Box 821, Canberra, ACT 2601, 
Australia 
R A Antonia and S Rajagopalan 
Department of Mechanical Engineering, University of Newcastle, NSW 2308, Australia 
This review considers theoretical and experimental knowledge of rough-wall turbulent 
boundary layers, drawing from both laboratory and atmospheric data. The former apply 
mainly to the region above the roughness sublayer (in which the roughness has a direct 
dynamical influence) whereas the latter resolve the structure of the roughness sublayer 
in some detail. Topics considered include the drag properties of rough surfaces as 
functions of the roughness geometry, the mean and turbulent velocity fields above the 
roughness sublayer, the properties of the flow close to and within the roughness canopy, 
and the nature of the organized motion in rough-wall boundary layers. Overall, there is 
strong support for the hypothesis of wall similarity: At sufficiently high Reynolds 
numbers, rough-wall and smooth-wall boundary layers have the same turbulence struc­
ture above the roughness (or viscous) sublayer, scaling with height, boundary-layer 
thickness, and friction velocity. 
CONTENTS 
1. Introduction 1 
2. Mean Velocity Above the Roughness Sublayer 2 
2.1 Dimensional considerations and the logarithmic 
profile 2 
2.2 Fully rough flow 4 
2.3 Other scaling possibilities 6 
2.4 Transitional roughness 7 
3. Turbulence Above the Roughness Sublayer 7 
3.1 Wall similarity 7 
3.2 Turbulence velocity scales, length scales, and 
spectra 8 
3.3 The attached-eddy hypothesis 11 
3.4 Observations of velocity variances 11 
4. Flow Close to and Within the Roughness 12 
4.1 Mean velocity 12 
4.2 Basic properties of the turbulence 13 
4.3 Measurement problems 15 
4.4 Second-moment budgets 16 
5. Organized Motion in Rough-Wall Boundary Layers 17 
5.1 Two-point velocity correlation functions 17 
5.2 Manifestations of organized motion 18 
5.3 Inferred structure of the organized motion 20 
6. Conclusions 21 
Appendix: Spatial Averaging of the Flow Equations 22 
References 22 
1. INTRODUCTION 
In basic turbulence research, much attention has been given to 
the structure of the turbulent boundary layer over a smooth 
wall with zero pressure gradient: see, for example, the reviews 
Transmitted by AMR Associate Editor M Gad-el-Hak. 
by Kovasznay (1970), Willmarth (1975), Cantwell (1981), Kline 
(1978), Sreenivasan (1989), and Kline and Robinson (1990). 
By contrast, the corresponding boundary layer over a rough 
wall has received far less attention—a situation which at first 
sight appears justifiable on the grounds that one should try to 
understand wall-bounded flow with the simplest possible 
boundary condition before introducing complexities such as 
roughness, pressure gradients, curvature, and so on. 
However, this comparative neglect may obscure the poten­
tial contribution of rough-wall boundary layer studies to some 
continuing problems of boundary-layer research in general. 
Over either a smooth or a rough wall, the turbulent boundary 
layer consists (in the simplest view) of an outer region where 
the length scale is the boundary-layer thicknessdU/dz 
attenuating within the canopy at a rate depending on the 
roughness density A and other geometrical properties. The 
upper part of the within-canopy U(z) profile is fairly well 
approximated by the empirical "exponential wind profile": 
where the coefficient a tends overall to increase with \, but 
with considerable scatter (see Table 3). 
In the lower part of the canopy, some workers have reported 
"bulges" in the profile of U(z)—see, for example, the data for 
Uriarra Forest in Fig 8a. Such a bulge, if real, implies counter-
gradient momentum transfer in the region where dU/dz /(> 
I 
-
_, 
-
\ VW 
v.\% 
';\\ 
3 
-
1 
-
_ 
. v~^— 
i 
-a ' 
OS 
o 
tu 
oo 
o u. 
o 
UH 
4 
T 1 
a 
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Appl Mech Rev vol 44, no 1, January 1991 Raupach et al: Rough-wall turbulent boundary layers 15 
TABLE 3. Physical and aerodynamic properties of seven canopies in Fig 8" 
Sensors 
Canopy /; A U(h)/ut Mean Turbulence 
WT strips 60 mm 0.23 3.3 T T 
WT wheat 47 mm 0.47 3.6 T T 
WTrods 19 cm 1.00 5.0 X X 
Shaw corn 
225 cm 1.45 3.2 C, F F 
Moga forest 12 m 0.5 2.9 C. S3 S3 
Uriarra forest 16, 20 m Z0 Z5 C SI 
* See Raupach (1988, 1989a) for primary references. WT denotes wind tunnel. Sensors: C, cup anemometer; X, X-wire probe; T, coplanar 
triple-wire probe; SI, single-dimensional (vertical) sonic anemometer; S3, three-dimensional sonic anemometer. For Shaw corn, Wilson corn, 
Moga forest, and Uriarra forest, A taken as LAI/2. [Note: in Raupach (1988, 1989a), values of A and U(h)/ut for "WT wheat" were slightly 
wrong; present values are correct.] 
gusts. This indication can be made precise by quadrant analysis 
(section 5.2). Kurtoses for it and w, not shown here, reveal the 
same trend towards very high intermittency in the canopy 
(Maitani 1979). 
The single-point Eulerian length scales L„ and L„ can be 
estimated from the single-point it and w integral time scales by 
applying Taylor's frozen-turbulence hypothesis: 
L„. = — w(t)w(t + T) dr. (4.2) 
07,. Jo 
and similarly for Lu. Near z = /;, Lu is of order /; and L„ of 
order h/3 (Figs 8g and 8h), so that the turbulence length scales 
are comparable with h. It follows that the gusts inferred from 
the skewness profiles are large structures, coherent over stream-
wise and vertical distances of order /;. The existence of such 
motions can be verified visually by watching "honami," the 
traveling wind waves seen on fields of grass, wheat, or barley 
on windy days (Inoue 1955, Finnigan 1979a, b). 
Figure 8 suggests that the dominant velocity and length 
scales for the turbulence in the canopy are u„, and // (or the 
closely related length scale h - d). These scales provide an 
approximate collapse of turbulence data from experiments in 
which h ranges over a factor of 400 and u^ over a factor of 10 
or more. The scatter in the data indicates the influence on the 
canopy turbulence of other length and velocity scales related to 
canopy morphology, the fluttering of leaves and the waving of 
whole plants, and viscous (Reynolds number) effects which 
influence the drag on individual leaves (Thom 1968, 1971). In 
the field, an additional important complication is buoyancy, 
though its effects are absent from the data in Figure 8 which 
pertain only to thermally neutral or slightly unstable daytime 
conditions. 
4.3. Measurement problems 
We have referred several times to measurement problems in 
the high-intensity turbulence near and within the roughness. 
These have proved troublesome and (at times) confusing, es­
pecially in laboratory situations where X-wire probes have been 
the main turbulence sensors.The most obvious symptom is a 
decrease in the measured shear stress —TTTv just above z = /;, 
seen in most laboratory measurements over rough walls with 
X-wire probes. Examples are the X-wire TTvP profiles measured 
by Antonia and Luxton (1971a, b, 1972), Mulhearn and Fin­
nigan (1978), and Raupach et al (1980) (see Fig 5). Such a 
decrease, if real, would violate momentum conservation in the 
constant-stress layer close to the surface, unless an extra mo­
mentum transfer mechanism exists in the roughness sublayer. 
There has been speculation that such a mechanism could be 
a systematic, time-averaged spatial variation in the mean veloc­
ity field imposed by the horizontal heterogeneity of the canopy, 
leading to a horizontally averaged momentum flux (U"W"). 
Here, angle brackets denote a horizontal plane average and 
double primes a departure of a time-averaged quantity from its 
horizontal average. (Spatial averages are discussed in the Ap­
pendix.) Fluxes of the type {U" W") were identified by Wilson 
and Shaw (1977) for vegetation canopies, and labeled "disper­
sive fluxes." An early estimate by Antonia (1969) indicated that 
this type of momentum flux is unlikely to account for obser­
vations with X-wire probes of apparent stress decreases near 
transverse bar roughness. Later, detailed measurements by 
Mulhearn (1978) (bar roughness), Raupach et al (1980) (cylin­
der roughness), Raupach et al (1986) (model plant canopy), 
and Perry et al (1987) (mesh roughness) demonstrated that the 
magnitude of (U" W") is less than a few percent of;/;, at most. 
This leaves no possible explanation for the apparent stress 
decrease just above the roughness, other than the measurement 
errors of X-wire probes. 
Further evidence that measurement error is the problem is 
provided by the 7JTP data in Fig 8b, which show convincingly 
that no apparent stress decrease is found in field data measured 
with omnidirectional sonic anemometers, and in laboratory 
data obtained with coplanar triple-wire probes, which have far 
better directional response than X-wire probes (Kawall et al 
1983, Legg et al 1984). All of these sensors indicate a layer of 
constant stress —liw within the expected limits of the constant-
stress layer (section 3.2). 
Theoretical and empirical error analyses on X-wire probes 
were made by Tutu and Chevray (1975), Raupach et al (1980), 
Legg et al (1984), and Perry et al (1987). All these studies agree 
that the main problem is the limited velocity-vector acceptance 
angle of ±45° in a conventional X-wire probe, with secondary 
problems being contamination of streamwise and vertical ve­
locity signals by the lateral velocity component, and finite wire 
length (in order of decreasing significance). Recent measure­
ments of 1m have addressed some of these problems, and are 
of better quality than the earlier data. Perry et al (1987) showed 
that, by increasing the acceptance angle from the usual ±45° to 
±60° and/or "flying" the probe in the streamwise direction to 
reduce the turbulence intensity a„/U, acceptable liw measure­
ments can be made with X-wire probes. Acharya and Escudier 
(1987) confirmed the improvement in TJTT' measurements re­
sulting from ±60° X-wire probes. Li and Perry's (1989) meas­
urements of mv over a rough-wall boundary layer, obtained 
with either a ±60° stationary or a ±45° flying X-wire probe, 
were in close agreement with an analytical expression of 77TP 
obtained by integrating the mean streamwise momentum equa­
tion (3.3), using a logarithmic profile law and Coles wake 
function to specify U(z). 
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16 Raupach et al: Rough-wall turbulent boundary layers Appl Mech Rev vol 44, no 1, January 1991 
4.4. Second-moment budgets 
The mechanisms maintaining the turbulence in the rough­
ness sublayer, both above and within the roughness, are partly 
elucidated by the turbulent kinetic energy and shear stress 
budgets. The budgets must be considered in a spatially (in 
practice, horizontally) averaged form because a significant dy­
namical role is played by processes associated with spatial 
heterogeneity at the length scales of individual roughness 
elements. 
Turbulent kinetic energy budget: For a steady flow over a 
horizontal, immobile rough surface at high Reynolds number 
(so that molecular transport terms are negligible), and with 
negligible advection, thermal forcing, and mean pressure 
gradient, the horizontally average turbulent kinetic energy 
budget is 
the rate of working of the mean flow against drag: 
dt 
= 0 = Ps + P,, + T, + T,i + T„ - (e) (4.3a) 
with 
P, = -(liw) 
/ > „ • = -[UiUj 
d(U) 
dz 
dur 
dXj, 
T, = -
dz 2 
(4.3b) 
T" = ~7Z 
d jW"q2" 
2 
dz 
Equation (4.3) is derived by methods outlined in the Appendix. 
The terms denote shear production (Ps), wake production (/>„•), 
turbulent, dispersive and pressure transport (7), Td, Tp, respec­
tively), and dissipation (—(«)). It is convenient to write the 
wake production term in tensor notation, with x, = (x, y, z), 
Ui = (U, V, W), H, = (u, v, w), and the summation convention 
effective. Of these terms, Piy T,, Tp, and (t) are familiar as 
spatial averages of the corresponding terms in the single-point 
turbulent kinetic energy equation (3.4), whereas the less familiar 
terms P„ and Ti arise from spatial heterogeneity at roughness-
element length scales. The dispersive transport term Td is the 
vertica] gradient of a dispersive turbulent kinetic energy flux 
{W"q2")/2, directly analogous to the dispersive momentum 
flux {U" W"). Since the dispersive momentum flux is negligible 
relative to the turbulent momentum flux (section 4.3), it is 
likely that the dispersive turbulent kinetic energy flux is likewise 
negligible, so that Td is negligible relative to T,. 
The wake production term Pw is far ffom negligible (Wilson 
and Shaw 1977). It is the production rate of turbulent kinetic 
energy in the wakes of roughness elements by the interaction 
of local turbulent stresses and time-averaged strains. Like Ps, 
P„ represents a conversion of mean to turbulent kinetic energy, 
but the two terms operate at different scales: Ps creates "shear 
turbulence" with a length scale of order /; within and just above 
the canopy (section 4.2), whereas P,t creates "wake turbulence" 
with a length scale of the order of a typical roughness-element 
wake width. In vegetation canopies, wake turbulence is usually 
much smaller-scale than shear turbulence. It can be shown 
(Raupach and Shaw 1982) that P„. is approximately equal to 
P»- -(U)fx»-{U) 
3{uw] 
dz 
(4.4) 
where /v is the horizontally-averaged total force exerted by the 
elements on the flow, a negative quantity. It is shown in the 
Appendix that/v ~ 3(7w)/dz. 
Figure 9a shows measurements, from Raupach et al (1986), 
of the terms Ps, P„. [using (4.4)] and T, in the "WT strips" 
wind-tunnel model plant canopy (Fig 8 and Table 3). The 
curve D is the residual —Ps — P». - T„ equal to T„ - (c) if Td 
is negligible as argued above. The main features are the peak 
in shear production Px near z = h, the large wake production 
P„ in the upper part of the canopy, and the major role of 
turbulent transport T, in carrying turbulent kinetic energy from 
the regions of strong production near z = h to lower levels in 
the canopy. In the lower part of the canopy, the turbulent 
kinetic energy budget reduces to an approximate balance be­
tween transport and dissipation. The importance of T, is related 
to the dominant role of sweep motions, or gusts, in momentum 
transfer (section 5.2). 
Two aspects of the turbulent kinetic energy budget do not 
emerge from Fig 9a. First, Tp (which could not be measured) 
is probably significant. Maitani and Seo (1985) estimated ~wp 
in a cereal canopy in the field from surface pressure measure­
ments, concluding that wp is downward within the canopy and 
about half of wq2/2 (which is also downward);this suggests that 
Tp is comparable with T, and likewise acts as a gain term in the 
budget deep in the canopy. Second, PK converts not only mean 
kinetic energy but also large-scale (shear) turbulent kinetic 
energy into wake-scale turbulent kinetic energy. This conver­
sion is not evident in (4.3), which is spectrally integrated over 
all turbulence scales. However, since the small-scale wake tur­
bulence is much more quickly dissipated than the larger-scale 
shear turbulence, the effect is that the dissipation rate of the 
shear turbulence within the canopy is much greater than would 
occur for free turbulence with similar velocity and length scales. 
The rapid dissipation rate of the wake turbulence also accounts 
(Raupach and Shaw 1982) for the fact that, in velocity spectra 
measured within canopies, little extra energy is seen at wave-
M1 
Ml 
ill 1 
i!| 
- l \ 
ps 
- P w 
- T t 
i i i r i i i 
- 6 - 4 - 2 0 2 4 6 - 6 - 4 - 2 0 2 4 6 
h a?/2 h 3 (fw") 
' (4.5a) 5 j _ Two-point velocity correlation functions 
with 
Pi = - 
d(U) 
dz 
p, , „ am _, „ dU" 
OXj OX, 
a 
dz 
T'cl = --{W"uw"), 
dz 
T'p = -—(up), 
(4.5b) 
, Idll dvt 
The terms in (4.5), distinguished by primes, correspond in 
name and mnemonic to the terms in (4.3) except for the 
pressure-strain term # ' , which is the main destruction term for 
shear stress. The dispersive transport term 77/ is usually negli­
gible in practice, just as for Td in (4.3). However, in contrast to 
P„, which plays a very important part in (4.3), the wake 
production term P'„ in (4.5) is also usually negligible. 
Figure 9b shows direct measurements of the terms P's and 
77 in the shear stress budget (Raupach et al 1986). As for the 
turbulent kinetic energy budget, shear production (P't) peaks 
strongly near z = //, while turbulent transport (77) is a loss 
near z = h and a gain lower down (noting that, because Tiw 
is negative whereas q2/2 is positive, gain terms are on the 
right of Fig 9a but the left of Fig 9b). The role of transport 
in the shear stress budget is relatively much smaller than in 
the turbulent kinetic energy budget, because the trans-
port term ratio | T',/T, | is of order | (uw2)/(wq2/2) |, 
which is only about 0.2 in the canopy, whereas the two pro­
duction terms are comparable since \P'S/PS\ = | (w2)/(Uw) | 
~ 1.5 near z = /;. 
The main features of Figs 9a and 9b are confirmed by a 
growing number of measurements from both wind-tunnel 
models and field canopies. However, the discussion has been 
restricted to vegetationlike roughness, for observational reasons 
already outlined. The conceptual framework of (4.3), (4.4), and 
(4.5) is valid for any roughness type, but the quantitative 
behavior of the budgets is another matter; although the 
main features of Fig 9 probably carry over at least to three-
dimensional roughness such as sandgrain roughness, separate 
investigation is required for two-dimensional bar roughness, 
either widely-spaced ("k-type") or narrow-cavity ("d-type"). 
5. O R G A N I Z E D M O T I O N IN R O U G H - W A L L 
B O U N D A R Y L A Y E R S 
It is now generally recognized that turbulent flows univer­
sally exhibit various forms of organized motion, sometimes 
A traditional but useful starting point for an examination of 
organized motion is the two-point, time-delayed velocity cor­
relation function 
///(A, v, z, T zK) = — — — , (5.1) 
luf(z)u2(z,t)}"2 
where zR is the height of a reference sensor at (A, y) = (0, 0). 
The correlation function depends explicitly on both the heights 
z and Z/ 0, contour interval = 0.1; ( ) /•],» _ 
— • - _ „ 
T>ih/h 
FIG 11. Vertically separated space-time correlations r„(0, 0, r, 
T; ZR) in and above the "WT wheat" canopy (see Table 3), from 
Raupach et al (1989), with zR = 2h. 
with sonic anemometers offer an unambiguous resolution of 
all three velocity components which is not achievable in labo­
ratory roughness sublayers, whereas laboratory measurements 
(Fig 11) offer higher measurement density and reproducibility 
than the field. 
A striking feature of Figs 11 and 12 is the difference in the 
correlation functions for it, u, w, and 0 . For u and 0 , and to a 
lesser extent for u, the maximum correlation occurs at a time 
delay r which increases as the height separation increases, 
consistent with Fig 10 and the Brown and Thomas (1977) 
result. It follows that the motions dominating the it, u, and 0 
correlations are inclined structures leaning with the shear. For 
vv, however, the maximum correlation occurs with zero time 
delay, so that organized fluctuations in w are aligned vertically, 
both within and above the roughness. The region of strong w 
correlation is also more localized than for u, u, or 0 . As with 
other features of rih these results agree well with smooth-wall 
data: Antonia et al (1988) found that the maximum w corre­
lation over a smooth wall occurs at T = 0 for a wide range of 
both zR and z, again implying a vertical alignment of organized 
w fluctuations. 
In summary, two-point correlation functions confirm wall 
similarity above the roughness sublayer, and yield eddy length 
scales, orientations, and convection velocities both above and 
within the roughness sublayer. However, they are only weak 
d. 8 
A. 
i 
^2 + >4. (5.2) 
Nakagawa and Nezu (1977), using an open-channel water flow 
over glass-bead roughness, made several significant findings: 
(1) Sweep events are more important than ejection events for 
momentum transfer close to a rough wall, with the sweep-to-
ejection ratio(with / + j = 3) are the normalized 
third moments or skewnesses. The constants in (5.3) are 
experimental, derived from measurements throughout the 
smooth-wall and rough-wall flows including the region within 
the roughness, but very similar values emerge from the 
cumulant-discard theory. Later unpublished measurements by 
one of us (RAA) have confirmed that (5.3) also applies in a 
smooth-wall boundary layer, but only if the Reynolds number 
is sufficiently large.1 For vegetation, the early work of Finnigan 
(1979a) (with a small data set) was followed by Shaw et al 
(1983), who applied quadrant analysis to turbulence data from 
a corn canopy, finding ((iiw))4/((uw))2 values of about 2 
near z = /; and higher within the canopy, thus confirming that 
sweeps dominate the momentum transfer close to and within 
field canopies. 
One dramatic visualization of the spatial structure of sweeps 
in a rough-wall boundary layer is the phenomenon of 
"honami," or traveling wind waves in cereal (wheat, barley, 
rice, grass) canopies. For one engaged in research on boundary-
layer turbulence, watching these waves is time well spent. The 
phenomenon of honami was named and first studied by Inoue 
(1955), and has since been investigated in detail by Finnigan 
(1979a, b). He found that the waves are initiated by gust fronts, 
or sweeps, moving across the canopy at convection speeds 
substantially greater than the local mean wind speed. Each gust, 
as it advances, bends over a patch of stalks which undergoes 
damped oscillation (typically for about two cycles) after the 
gust has passed, thus creating the impression of a wave moving 
through the canopy. By studying motion pictures of waving 
canopies and by analyzing short-time, vertically separated, 
space-time correlations, Finnigan found that the streamwise 
separation between gusts was about 5/;-8/z, close to the value 
8/; inferred by several spectral and correlation methods for the 
typical streamwise separation between quasicoherent eddies 
(Raupach et al 1989). 
(c) Ramp-jump structures in signals: Chen and Blackwelder 
(1978) observed correlated ramp-jump (sawtooth) patterns in 
temperature signals throughout a smooth-wall boundary layer, 
which they suggested to be a direct link between the wall region 
and the outer layer. This suggestion should be equally true in 
a rough-wall boundary layer. Indeed, such patterns were first 
observed in the (definitely rough-wall) atmospheric surface 
layer by Taylor (1958) and Priestley (1959), though the tem­
perature structure in many of these early observations was 
largely determined by free convection rather than thermally 
near-neutral shear turbulence. Nevertheless, for moderately 
unstable conditions in the atmosphere, Antonia et al (1979) 
concluded that the observed similarity between laboratory and 
atmospheric ramp-jump temperature structures should be in­
terpreted as the signature of an organized large-scale shear-
driven motion, rather than as a consequence of the buoyancy 
field (which, of course, also produces large-scale organized 
motion). Wyngard (1988) reinforced the dominance of shear 
turbulence close to the surface, even in strongly unstable 
atmospheric boundary layers. 
Many subsequent observations have confirmed that ramp-
jump structures are universally observed in both rough-wall 
and smooth-wall boundary layers, in both the atmosphere (over 
land and sea) and the laboratory, and also for velocity com­
ponents as well as temperature (for example, Antonia and 
Chambers 1978, Antonia et al 1979, Phong-Anant et al 1980, 
Antonia et al 1982). The velocity and temperature signals 
yielding the two-point correlation functions in Figs 10-12 all 
' This raises a significant point in the context of using laboratory data to 
explore new ideas or theories about boundary-layer turbuience: asymptotic sim­
ilarity appears to be reached at moderate Reynolds numbers over a rough wail 
by comparison with a smooth wall. The pipe-flow measurements of Perry and 
Abell (1977) support this. 
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20 Raupach et al: Rough-wall turbulent boundary layers Appl Mech Rev vol 44, no 1, January 1991 
exhibited these structures, to the extent that they substantially 
determine the shape of the correlation functions; Figs 10-12 
therefore indicate that the ramp-jump structures extend into 
the roughness itself. Further unpublished wind-tunnel meas­
urements by us, over a slightly heated gravel roughness, have 
confirmed that temperature ramp-jumps are observed coher­
ently throughout the whole (rough-wall) boundary layer, from 
zsmooth 
wall (Kovasznay et al 1970) and widely spaced bar roughness 
(Antonia 1972). The latter paper showed that the outer-layer 
similarity between these two flows is also evident from profiles 
of mean velocity and a wide range of velocity moments, further 
supporting the wall similarity discussion in section 3.1. 
2.5 
2.0 
1.5 
•z i.o 
0.5 
0.0 
^v- i J N - " ~ >budget, especially the role of 
turbulent transport) tend to be more reminiscent of a mixing 
layer than a boundary layer. 
This proposed mechanism is fairly easy to visualize for 
vegetation and similar roughness where the element (leaf) 
length scales are small compared with h and horizontal 
heterogeneity is relatively unimportant. For laboratory three-
dimensional and two-dimensional roughness with element di­
mensions comparable with /;, individual roughness elements 
can generate strong wakes (for example, streamwise vortices in 
the case of discrete three-dimensional roughness elements) 
which may play a role in the transfer process. However, there 
is almost always a strong vertical shear just above the elements, 
so the effects of individual element wakes may well be consid­
ered as superposed upon some more general process such as 
that just described. 
6. CONCLUSIONS 
We have attempted to place within a single framework two 
bodies of research which have hitherto been largely separate: 
laboratory and theoretical work on rough-wall turbulent bound­
ary layers, and micrometeorological studies in the atmospheric 
surface layer. By combining insights from both fields, a fairly 
complete picture of the rough-wall turbulent boundary layer 
emerges. The hypothesis of wall similarity, that rough-wall and 
smooth-wall boundary layers at sufficiently high Reynolds 
numbers are structurally similar outside the roughness (or 
viscous) sublayer, is well supported by many kinds of observa­
tion. The flow in the roughness sublayer is more difficult to 
measure than that in the overlying boundary layer, not only 
because of spatial heterogeneity but also because of high tur­
bulence intensities, which introduce unacceptable errors with 
many laboratory velocity sensors, including X-wire probes. 
However, careful measurement techniques in the laboratory, 
using flying X-wire probes or coplanar triple-wire probes, have 
eliminated some of these difficulties. For field vegetation, three-
dimensional sonic anemometers provide an unambiguous 
measure of all three velocity components superior to anything 
obtainable with current laboratory sensors. Together, these 
techniques have facilitated the exploration of the main prop­
erties of the roughness sublayer, including its spatial heteroge­
neity, its turbulence structure in terms of velocity moments 
and second-moment budgets, and the organized motion within 
it. To a surprising extent, these properties are common across 
a wide variety of roughness types. 
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22 Raupaoh et al: Rough-wall turbulent boundary layers Appl Mech Rev vol 44, no 1, January 1991 
An important fundamental role for the study of rough-wall 
boundary layers is in tackling the general problem of boundary-
layer turbulence and its dominant forms of organized motion. 
It is clear that conditions at the wall can be drastically altered 
by roughness without changing the main boundary-layer 
structure (outside the roughness or viscous sublayer) in any 
fundamental way. This provides a strong clue about the self-
organizing properties of boundary-layer turbulence, which, 
when properly understood, will offer much to the study of 
turbulence in general. 
A P P E N D I X : S P A T I A L AVERAGING 
O F T H E F L O W E Q U A T I O N S 
Steady flow about a rough surface is characterized by several 
processes which are strongly spatially heterogeneous (even after 
time-averaging) over roughness-element length scales. The most 
important are (1) drag, (2) scalar transfer, (3) wake turbulence 
production and decay, and (4) deviation of mean streamlines 
from unidirectional, leading to "dispersive fluxes" (section 4.3). 
In contrast, most applications are concerned only with spatially 
averaged properties of the roughness. To link these two levels 
of description, it is necessary to spatially average the flow 
equations: As well as applying a time-averaging operator to the 
Navier-Stokes equations to obtain the conventional turbulence 
(Reynolds) equations, a second, spatial-averaging operation is 
applied to obtain equations containing terms which represent 
explicitly the critical processes dependent on heterogeneity. 
Spatial averaging in this way was first introduced as a horizontal 
plane average, for flow in vegetation canopies, by Wilson and 
Shaw (1977) and Raupach and Shaw (1982); later, Finnigan 
(1985) and Raupach et al (1986) discussed the more general 
volume average, of which the horizontal plane average is a 
special case when the averaging volume is a thin horizontal 
slab. 
The volume average of a scalar (or vector component) 4>, 
denoted by angle brackets, is 
W(A', /) = 
J Jv J 
4>(x + r, t) dr, (Al) 
where the averaging volume V excludes the solid roughness 
elements. The decomposition of ) + " (A2) 
with (") = 0. The volume average operator does not commute 
with spatial differentiation, or with temporal averaging if the 
rough surface is moving relative to the fixed coordinate frame, 
as in the case of a waving plant canopy. Instead, it can be 
shown that (Finnigan 1985, Raupach et al 1986): 
ax, 
d4> 
dl 
d() 
rlV, 
aw 
dl 
(prii dS 
V 
U 
(A3) 
dS 
where So is the outer or free part of the bounding surface S of 
V; Si is the part of S coinciding with the rough surface (so that 
S = So + Si); Hi is the unit normal vector pointing away from 
5 into V; and i>, is the velocity of S. For static laboratory or 
field roughness v, = 0, but u, is nonzero over £", for waving 
vegetation canopies (Finnigan 1979a, b, 1985). 
The volume-averaging operator (Al) can be applied to the 
time-averaged mass and momentum conservation equations. 
In tensor notation, with A, = (x, y, z), Ui = (U, V, W), and 
it, = (u, v, w) and the summation convention effective, 
the corresponding volume-averaged equations are 
d{U.) 
dx, 
= 0, (A4) 
dl 
+for (77vT') (see section 4.4). 
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Appl Mech Rev vol 44, no 1, January 1991 Raupach et al: Rough-wall turbulent boundary layers 25
Wygnanski I and Fiedler H E (1970), The two-dimensional mixing region,
J Fillid Mech 41,327-36 L
Wyngaard J C (1988), Convective processes in the lower atmosphere, in FI(Ilr
IIl1d IIWI.IJ!or! ill the !w/lInl! elll'irollll1C1lt: adrallees alld applicatiolls, Stellen
W Land Denmead 0 T, Eds, Springer, Berlin, pp 240-260,
Michael R Raupach received
his BSc degree/imll the Univer­
sitv o(Adelaide in 1970 and his
PJiD'in 1976 limn the Flinders
Universitv o/South Australia,
where he'lwJrked with PI"!?fessor
Peter Schwerd(/eger, A/ier a
two-year post-doctoral position
in the Department ofMeteoro1­
ogy at the University (~(Edin­
burgh, he joined the CSIRO Di­
vision o(Environmental Me­
chanics in Canberra, Australia,
where he is currently a Principal Research Scientist, Dr
Raupach has held visiting positions at Reading and Cam­
bridge Universities, He has published over 50 scientific papers
on the physics qlturbulent/luid/low and the mechanisms (~l
turbulent tran,~/'er processes, Dr Raupach 's current research
interests inc/ude both experimental and theoretical aspects q(
turbulence: the transfer qlscalars and momentum in the lower
atmosphere, especial/y in the vegetation canopy environment;
evaporation; soil erosion by wind; planetary boundary layer
processes; andjlow in inhomogeneousare used interchangeably to denote the overall 
rough surface, assumed to be horizontal on average with flow 
above it. The streamwise, lateral and vertical coordinates are 
(x, y, z), with the plane z = 0 being the substrate surface upon 
which the roughness elements are located (the underlying 
ground surface in the case of vegetation). The mean and 
fluctuating velocity vectors will be denoted (U, V, W) 
and (/(, v, M the fluid density). 
Suppose also that the flow is in the state called "moving 
equilibrium" by Yaglom (1979), in which /ut together 
with the roughness height /; and all additional lengths L, needed 
to completely characterize the roughness. Typically, L, includes 
at least the roughness element dimensions in the x and y 
directions (/v and /,., respectively), and the mean element sepa­
ration distance (D) [defined by D = (A/n)l/2, where /; is the 
number of roughness elements in a horizontal area A]. Other 
lengths may also be relevant in some circumstances. 
Of course, U(z) also depends on z itself. However, care is 
necessary in defining the origin of z for a rough surface, since 
the roughness itself displaces the entire flow upwards. To ac­
count for this, we define the displaced height Z = z - d, where 
c/is the fluid-dynamic height origin or zero-plane displacement, 
dependent on both the flow and the roughness. Thorn (1971) 
proposed, and Jackson (1981) verified theoretically, that d is 
the mean height of momentum absorption by the surface; this 
definition of d is adopted here. It follows that d automatically 
satisfies the constraints 0 -- ' • 
1 1 
10 104 
FIG 1. The relationship between At////, and the roughness 
Reynolds number /;+ = ////,/>'.and hi//y terrain He is
a Fel/mv (}lthe Royal Meteorological Society,
S Rajagopalan obtained his B E
(Mech Eng) ji-om the Univer­
sity ofMadras and his ME
and PhD (Aero Eng) ji-om the
Indian Institute o(Science
(Bangalore), He worked as a
Scientist in the National Aero­
nautical Laboratory (India) and
as a Post-Doctoral Fellow at the
Universities o(Newcastle and
Adelaide. He'was a Lecturer at
the Universitv o( vVol/ongong
be.fiJre movin:f5 /0 the University
(}(Newcastle. His research areas are structure and control (~(
turbulent/lows and instrumentation.
Yaglom A tv! (1977), Comments on wind and temperature nux-prolile relation­
ships, BOlilldan'-Layer Mcteorolll, 89-102,
Yaglom A tv! (1979), Similarity laws I'lr constant-pressure and pressure-gradient
turhulent wall nows, .'11111 ReI' FllIid Mech II, 505-540,
Robert Anthony Antonia re­
ceived his BE (1963), Master o(
Eng Sc (1965) and PhD (1969)
ji-om the Universitv o(Swlnev
'in the Department' q/lv{ecJzaili­
cal Engineering. He was a
CSIRO Post-Doctoral Research
Fellow at ltnperial Col/ege,
Universitv o(London in 1970.
He was {IPP(Jinted to a lecture­
ship at the University olSydney
in 1972. He was appointed to
the Chair qlMechanical Engi­
neering at the University olNewcastle in 1976, His research
interests have included the structure o!'turbulence and the
transport (~(momentum and heat in (i number ()(d([!'erent
shear/lows, including smooth- and rough-wall boundary lay­
ers, He has also taken part injoint cooperative research on the
transport properties ()!'the atmospheric sur/ace layer. He is a
Fel/ow (?!'the Institution (J(Engineers, Australia.
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 06/13/2015 Terms of Use: http://asme.org/termsLaboratory data from survey by 
Bandyopadhyay (1987): (•) wire screen roughness (Hama 1954); 
(•) bar roughness (Moore 1951); (O) bar roughness, D/h = 4.0 
(Perry and Joubert 1963); (•) bar roughness, D/h = 3.6 (Perry et al 
1969); (O) bar roughness, D/h = 3.8 (Bandyopadhyay 1987); 
(x) bar roughness, D/h = 4 (Liu et al 1966); (+) bar roughness, 
D/h = 12 (Liu et al 1966); (A) sandgrain roughness (Colebrook and 
White 1937); (A) sandgrain roughness (Bandyopadhyay 1987); 
( - - ) sandgrain roughness (Prandtl and Schlichting 1934). Atmos­
pheric data from Table 1. 
TABLE 1. Data used in Figs 1 and 3° 
A 
B 
C 
D 
E 
F 
G 
H 
I 
J 
K 
L 
M 
Surface 
Trees (Ml) 
Trees (M2) 
Earlv wheat 
Late wheat 
Pines 
Vinevard: rows 
Vinevard: rows 
Forest (Bergen) 
Forest (Martin) 
Forest (Oliver) 
Forest (Kondo) 
Forest (Kondo) 
Forest (Landsbe 
WT Strips 
WT Wheat 
ilong wind 
icross wind 
"g and Jarvis) 
h(m) 
9 
9 
0.4 
1.0 
13 
0.9 
0.4 
10 
22 
15.5 
4.5 
23 
11.5 
0.060 
0.047 
A 
0.06 
0.21 
0.10 
0.25 
2.3 
0.04 
0.22 
2.8 
3.1 
4.3 
0.8 
1.7 
9.6 
0.23 
0.47 
z0(m) 
0.45 
0.8 
0.015 
0.05 
0.4 
0.023 
0.12 
0.50 
0.66 
0.93 
0.45 
1.15 
0.35 
0.0087 
0.0040 
d(m) 
7.6 
19.8 
11.8 
3.0 
19.1 
9.7 
0.043 
0.035 
10"5//+ 
3.0 
3.0 
0.13 
0.33 
4.3 
0.30 
0.13 
3.3 
7.3 
5.2 
1.5 
7.7 
3.8 
0.040 
0.031 
A/7///, 
29.0 
30.5 
20.5 
23.5 
28.7 
21.6 
25.7 
29.3 
30.0 
30.9 
29.0 
31.4 
28.4 
20.9 
19.0 
[/(/;)/;/, 
3.8 
2.9 
3.3 
2.9 
3.1 
3.4 
3.3 
3.6 
"Surfaces A-G from Garratt (1977), and H-M from Jarvis et al (1976); see these papers for primary sources. In calculation of h+ and 
AC'///, [using (2.12)], assumed values were //, = 0.5 m s~', v — 0.15 x 10~5 m2 s~', C0 = 5, K = 0.4. For surfaces A-G, A taken as A, in 
Garratt (1977); for surfaces H-M, A taken as LAI/2. 
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4 Raupach et al: Rough-wall turbulent boundary layers Appl Mech Rev vol 44, no 1, January 1991 
where 70. At intermediate values of /;+, the flow is 
called transitional. In the fully rough state, the data for sand 
roughness show that cm = 8.5, giving z0 ~ h/30 from (2.10). 
It is also useful to define a Reynolds number based on z0 by 
writing (consistent with previous notation) z0+ = z0ujv. (This 
is sometimes called the roughness Reynolds number, but we 
reserve that term for //+.) From (2.11), the minimum value of 
Z(>+ is 0.14, on a smooth wall. The relation between the rough­
ness function AU/u* and the roughness length z0 is most easily 
expressed in terms of z0+: 
AU 
u* 
= Co + K 'In z0+ (2.12) 
which permits simple conversion between the engineering and 
meteorological measures of roughness. 
Before leaving the dimensional analysis, it is necessary to 
consider the zero-plane displacement d, which is required to 
fix the origin of Z in (2.4)-(2.8). From the definition of d as 
the mean level of momentum absorption by the (rough) surface, 
it follows that d is a fluid-dynamic property of the surface which 
obeys dimensional constraints similar to those on z0. Hence, 
when 5 » ("/"*, /'. L,), the normalized displacement d/h is a 
function only of the surface properties //+ and a,, independent 
of/;+ as /;+ -^ oo, like zu/h from (2.9) and (2.10). 
2.2. Fully rough flow 
At Reynolds numbers large enough for the flow to obey Rey­
nolds number similarity, the problem of determining the mean 
velocity profile in the logarithmic region devolves to finding 
the functional dependence of z0//z (or AU/u*) and d/h upon 
the roughness geometry as specified by a,. The question of 
whether there are kinds of roughness which do not achieve a 
fully rough state (even at very high Reynolds numbers) is 
considered in the next section. 
The earliest approach to the problem of characterizing z0/h 
or AU/u.t was to define roughness by analogy with particular, 
well-studied forms such as the sand roughness of Nikuradse 
(1933), for which ev, = 8.5 and z0 ~ h/30. It is still common in 
engineering to define roughness in terms of the "equivalent 
sandgrain roughness height" hs = 32.6z0 introduced by Schlicht-
ing (1936). In micrometeorology, surveys of early data by 
Tanner and Pelton (1960) and Stanhill (1969) gave z0/h = 0.13 
(tv. = 5.10) and d/h = 0.64 for field crops and grass canopies, 
which have proved to be good rules-of-thumb in many cases 
and are still in widespread use. For forests, measurements 
reviewed by Jarvis et al (1976) suggested the rather different 
typical values z0/h ~ 0.06, d/h ~ 0.8. 
The large differences between sandgrain, crop, and forest 
values of z0/h and d/h reinforces the need for understanding 
the influence of geometry. To do this, it is necessary to identify 
and study experimentally the particular aspect ratios a, which 
dominate the behavior of the roughness as a momentum ab­
sorber. The main ones studied hitherto are the element aspect 
ratios ax = lx/h, Xmax (Wooding et al 1973). However, the 
function [z«//i](X) and the location of Xmas depend on the type 
of roughness, indicating that other aspect ratios besides X are 
required for a complete specification. 
At low roughnessdensities (X 
(b) 
FIG 2. Normalized roughness length z0/h as a function 
of roughness density X. (a) For three-dimensional rough surfaces 
with elements of several shapes: cylinders (with realistic values of 
the zero-plane displacement d and also with the assumption d = 0) 
(Raupach et al 1980); cubes (O'Loughlin 1965, Koloseus and 
Davidian 1966); and spheres (Koloseus and Davidian 1966). 
The solid line is the approximate best fit to cube data, (b) For two-
dimensional rough surfaces: the heavy solid line, square bar data 
from Koloseus and Davidian (1966); light dashed line, prediction 
(2.13) (Dvorak 1969); light solid lines, prediction (2.14) (Kader 
1977). 
eluded from data on flow over bushel baskets on a frozen lake 
(Kutzbach 1961) that z0/h = 0.5X; however, he imposed no 
restriction analogous to X 
0.2, permanent separated vortices occupy the entire cavity 
between adjacent square bars, whereas for Xbut // — d (which they called the error-in-origin). 
This leads to a revised version of (2.10) in which h is replaced 
by /; — d, so that (as h+ —* oo) 
z0/(h-d) = D,, 
AC/V = D2 + K~ 'ln(/; - d)+, 
(2.15) 
where A and D2 are constants related by A = exp[K(£>2 -
Co)], and dependent on the nature of the roughness elements. 
Perry et al verified experimentally that (2.15) does indeed 
S3 0 0 8 -
(a) 
0.25 
09 
0 6 
05 5 
0 4 
0 3 
0.5 
1 
-
1 
1 1 1 
Zm"/h / / / ' 
^ / ^ / ^ / / 0 3 / 
^^^-^S^>-Ox'0'2 
. . 1 .. 1 1 ... 
1 1 ^ 
— 
1 1 
(b) 
FIG 4. Predictions from a second-order closure model (Shaw and Pereira 1982) of (a) normalized roughness length z0/h and (b) zero-
plane displacement d/h, for field vegetation. Upper abscissa is plant area index (PAI), related to roughness density A by A = PAI/2 if leaf and 
stem orientations are isotropic. Plant area assumed to have a triangular distribution with height, peaking at ?,„»//;; plant element drag 
coefficient Cd = 0.2, including influence of turbulence and interelement shelter as in (A9). 
Downloaded From: http://appliedmechanicsreviews.asmedigitalcollection.asme.org/ on 06/13/2015 Terms of Use: http://asme.org/terms
Appl Mech Rev vol 44, no 1, January 1991 Raupaoh et al: Rough-wall turbulent boundary layers 7 
describe "d-type" narrow-cavity bar roughness and is also ap­
plicable to "k-type" sandgrain roughness, with constants Z)2 = 
-0.4, D, =0.12. Thorn (1971) independently proposed (2.15) 
for vegetation canopies, finding from experiments on a model 
crop and a bean canopy that D, = 0.36. 
The influence of roughness geometry appears in (2.15) in 
two ways, the first being the dependence of D, and D2 upon 
the nature of the roughness elements, illustrated by the differ­
ence between the D, values of Thom for vegetation and Perry 
et al for laboratory roughness. Second, the normalized displace­
ment d/h is itself a function of the aspect ratios o-, (and also //+, 
unless /;+ —» oo). Hence, to determine AU/ut or z0/h for an 
arbitrary rough surface using (2.15), one must first determine 
d/h, which is equally difficult in practice. 
There are also difficulties with the division of roughness into 
"k-type" and "d-type" classes. First, the division implies that 
AU/u* is determined for each class by a single length scale, but 
the evidence reviewed in the previous section shows that this is 
never the case; a consideration of other aspect ratios, including 
X, 70 (fully 
rough). In dynamically smooth flow the surface shear stress is 
entirely viscous whereas in fully rough flow the stress is domi­
nated by form drag on the roughness elements. In the transi­
tional range, both mechanisms are significant. The above 
Reynolds number values come from studies of sand-grain 
roughness by Nikuradse (1933), but different values are ob­
tained for different kinds of roughness as seen in Fig 1. For 
instance, /z+(upper) and //+(lower), the upper and lower h+ 
limits of the transition range, depend on X (Dvorak 1969). 
Bandyopadhyay (1987) showed experimentally that 
/;+(upper) and /;+(lower) decrease as the aspect ratio ay in­
creases, and that curves of AU/u^ against /;+ for different 
surfaces become similar when normalized by //+(upper) and 
the value of AU/ut at //+(upper). This was verified by Ligrani 
and Moffat (1986). Bandyopadhyay (1987) also correlated 
//+(upper) and /;+(lower) with the Reynolds number associated 
with the onset of, and development of irregularities in, the 
vortex street shed from an isolated roughness element em­
bedded in a laminar boundary layer. 
For vegetation, viscous drag can still be important despite 
large values of /;+ (Table 1), because the drag-inducing rough­
ness elements have Reynolds numbers Ul/v orders of magni­
tude smaller than h+ (where / is an element dimension such as 
a pine needle diameter or a leaf width, and U is the ambient 
velocity about the element). For pine needles Ul/v is around 
30-200; for wheat leaves, around 500-2000. Thom (1968) 
estimated the ratio of form to viscous drag on a typical bean 
leaf as 3:1. Thus, viscosity provides significant drag in many 
canopies. 
3. TURBULENCE ABOVE THE ROUGHNESS 
SUBLAYER 
Because dimensional arguments establish many properties of 
the mean velocity field in a rough-wall boundary layer, it is 
worth examining the extent to which dimensional reasoning 
also determines properties of the turbulence, especially velocity 
variances and turbulence length scales. It turns out that a more 
physically based form of dimensional analysis is needed to 
understand the turbulence statistics. This section examines 
three complementary hypotheses about length and velocity 
scales for the turbulence above the roughness (or viscous) 
sublayer: the wall-similarity, equilibrium-layer, and attached-
eddy hypotheses. Together, they lead to a set of predictions for 
turbulence length scales and velocity variances which are com­
parable with the logarithmic profile law for the mean velocity. 
For this analysis, a sufficiently general turbulence statistic for 
consideration is the two-point velocity covariance 
H,-(Z)!(f;(Z + r) = iilR.AZ, r; 5, h, L„ v/ut), (3.1) 
in which the displaced height Z = z - cl and the separation r 
are primary arguments, the outer and surface length scales are 
secondary arguments, and velocityscaling with u.t is assumed. 
However, similar dimensional arguments apply to higher ve­
locity moments as well. 
3.1. Wal l similarity 
The first hypothesis is one of flow similarity over different 
surface types: 
Outside the roughness (or viscous) sublayer, the turbulent 
motions in a boundary layer at high Reynolds number are 
independent of the wall roughness and the viscosity, except 
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8 Raupach et al: Rough-wall turbulent boundary layers Appl Mech Rev vol 44, no 1, January 1991 
for the role of the wall in setting the velocity scale u^, the 
height Z = z — d and the boundary-layer thickness S. 
This "wall similarity" hypothesis (our label) is an extension of 
the usual postulate of Reynolds number similarity, which 
Townsend (1976, p 53) expresses thus: "while geometrically 
similar flows are expected to be dynamically similar if their 
Reynolds numbers are the same, their structures are also very 
nearly similar for all Reynolds numbers which are large enough 
to allow (fully) turbulent flow." Provided that the Reynolds 
number is sufficiently high, Reynolds number similarity implies 
that, outside the viscous sublayer, the viscous length scale v/u:f 
has no influence in (3.1); the wall similarity hypothesis makes 
the further claim that, outside the roughness sublayer, the 
roughness length scales h and L, are also irrelevant. It appears 
that Perry and Abell (1977) were the first to advance this 
hypothesis in essentially the form stated above; they called it 
the "Townsend hypothesis," since it is implicit in the similarity 
arguments of Townsend (1961, 1976). They supported the 
hypothesis with an analysis of scaling laws for velocity spectra 
in several overlapping spectral ranges, an idea extended later 
by Perry et al (1986, 1987) (see section 3.2). 
For the velocity covariance, the wall similarity hypothesis is 
Ui(Z)iij(Z + r) = ulRij(Z, r; 5) (3.2) 
for large Reynolds numbers and for both Z and Z + r above 
the roughness sublayer. An a priori motivation for the hypoth­
esis (not a derivation) is that (3.2) is a dimensional statement 
analogous to the outer-layer law (2.1) for the mean velocity 
U(Z). The foregoing discussion of the mean velocity profile 
shows that both dU/dZ and U(Z) itself, apart from a height-
independent but roughness-dependent translational velocity, 
are independent of surface length scales (h, Lh v/u^) in the 
outer layer, including the overlap region with the inner layer. 
Therefore, wall similarity holds for relative mean motion at all 
heights above the roughness sublayer. Since the turbulence 
maintains and is maintained by the mean velocity profile, it is 
unlikely that surface length scales which are irrelevant for the 
mean velocity profile are important for the dominant turbulent 
motions. 
There is strong experimental support for the wall similarity 
hypothesis, of at least three kinds. Two of these (stress-to-shear 
relationships and measurements of single-point velocity mo­
ments) are reviewed now, while a third (two-point velocity 
covariance measurements) is considered in section 5.1. 
Stress-to-shear relationships: Strong, though indirect, evidence 
that the turbulence structure is essentially independent of the 
nature of the wall is provided by the universal value of the von 
Karman constant K (the ratio of the turbulent velocity scale u^ 
to the normalized mean shear ZdU/dZ in the inertial sublayer). 
It is found that K is independent of wall roughness to within 
experimental accuracy in both the laboratory and in the at­
mospheric boundary layer [see Yaglom (1977) for a review of 
atmospheric measurements]. Townsend (1976) pointed out the 
support that this fact provides for Reynolds number similarity, 
since the data span a Reynolds number range from 104 to 108. 
The support for wall similarity is equally striking, since the data 
also span surface types from smooth walls to natural vegetation. 
Single-point measurements of velocity moments: A conse­
quence of (3.2) is that, provided that the Reynolds number is 
sufficiently large, vertical profiles of single-point velocity mo­
ments («2, v2, vt'2, TTvT', and higher moments) should collapse to 
common curves independent of wall roughness, when normal­
ized with ut and 
(Fig 5a) and the standard deviationsAppl Mech Rev vol 44, no 1, January 1991 Raupaoh et al: Rough-wall turbulent boundary layers 9 
a v / U * 
z -d 
6 
1-2 
1-0 
0-8 
0-6 
0 4 
0-2 
o A 
O
 O
 
ED
 
" E 
• F 
- ^ 
• s ^ * , 
fv. 
^' 
1 
e 
^* 
° 
0 
o 
9 - * ' 
-
z -d 
6 
-1-2 -1-0 "0-8 - 0 6 -0-4 -0-2 
u w / u * 
(a) 
1-2 
1-0 
0-8 
0-6 
0-4 
0-2 
0 
o ' 
o 
- o 
o 
, 
' ' ' I 
\ 
°%.w 
/dz = 0 as Z/5 = (z — d)/8 —» 0 (but without 
going below the top of the roughness at z = //). Hence there is 
a "constant-stress" layer near the surface in which liw(z) ~ 
—u\. In practice, the constant-stress layer is roughly the region 
h — d mt{k), $uu(/c), 
4>p, whereas in range AB, the u and u spectra are 
proportional to k~x. Ranges A and AB do not exist in the w 
spectrum since the inactive eddies have negligible vertical 
motion. 
Qualified support for the spectral scaling hypothesis (3.7) is 
provided by spectral measurements in both laboratory bound­
ary layers and the atmospheric surface layer. Comprehensive 
laboratory spectral measurements were obtained by Perry et al 
(1987) over smooth and mesh-roughened walls. Figures 6 and 
7 show their it spectra over both wall types, which collapse in 
the outer (A, AB) and inner (AB, B, BC) spectral ranges when 
normalized with outer-layer and inner-layer scales, respectively. 
The Reynolds numberbroad spectra with 
extensive Ar~' and k~*/3 behaviors in spectral ranges AB and 
BC, respectively, a typical limitation in the laboratory. A further 
problem is the spread of convection velocities at low wavenum-
bers, to which Perry et al attributed the less than satisfactory 
k6 
10- ! 10-' 10" 10' 
I ! HUM—TTTTTini—TTT77n§ 
(b) Outer-flow scalin; 
„(kz) 
FIG 6. it spectra for varying values of Z/& in a smooth-wall 
boundary layer: (a) inner-layer scaling; (b) outer-layer scaling. From 
Perry et al (1987). 
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Appl Mech Rev vol 44, no 1, January 1991 Raupach et al: Rough-wall turbulent boundary layers 11 
k6 
10-= 10-' 10° 10' 10" 10" 
i—i i nimi—rrrmrn—t I uum—rrrrmii—I I mm 
(b) Outer-flow scaling 
*„„(Kz) +uu(kz> 
FIG 7. (/ spectra for varying values of Z//ut «: Z '„, z„) is the center of a particular eddy and u0 
is its velocity scale. The term "attached" implies that all eddies 
are not only geometrically similar but have the same geomet­
rical relationship with the wall, scaling with the center height 
z„. By ensemble-averaging an eddy field consisting of a random 
superposition of eddies of the form (3.9), imposing a continuity 
condition on f to account for the presence of the wall, and 
requiring TTtt'(z) = u\ = const, Townsend derived the high 
Reynolds-number limit of (3.8) without further specifying.//. 
Perry and Chong (1982) were more specific about/, invok­
ing flow visualizations of hairpin vortices in a smooth-wall 
boundary layer by Head and Bandyopadhyay (1981) to suggest 
a model attached eddy consisting of a "A-vortex" inclined with 
the shear and with legs trailing along the wall. Although this 
structure leads to the spectral scaling laws (3.7), and to (3.8) for 
the velocity variances, it is apparent that many other choices 
for /,' lead to similar results. This suggests that observations of 
spectra and variances alone are insufficient to specify details of 
the eddy structure beyond those implied by the dimensional 
arguments given above. 
3.4. Observations of velocity variances 
One test of the wall similarity, equilibrium-layer, and at­
tached-eddy hypotheses is a comparison of their predictions 
with laboratory and atmospheric data on the velocity variances 
ir = err,, v2 = al, and w2 = al. A small portion of the 
vast quantity of available data is summarized in Table 2 in the 
form of values of aju^, aL,/u.t, anddata. Later, Perry et al (1988) 
reported reasonable agreement between slightly modified forms 
of the third relation in (3.8) and carefully selected data for 
Ow/u*; the data selection ensured that spatial resolution and 
cone angle problems were minimized and that the X-wire 
probes yielded values of Tm> consistent with Clauser-chart or 
Preston-tube values. 
In the atmosphere over grassland sites in flat terrain, 
oju*, and //, 
from (3.3) et seq, the reduced shear implies an enhanced 
turbulent diffusivity K for momentum in the roughness sub­
layer, relative to the inertial-sublayer form K = KIIA[Z — d). An 
approximate form for K is K = KH,.(Z„ — d), independent of 
height for /;