PHYSICS OF FLUIDS 19, 036601 共2007兲
An experimental investigation of turbulent flows over a hilly surface
D. Poggia兲
Dipartimento di Idraulica, Trasporti ed Infrastrutture Civili Politecnico di Torino, 10129 Torino, Italy
and Nicholas School of the Environment and Earth Sciences and Department of Civil and Environmental
Engineering, Pratt School of Engineering, Duke University, Durham, North Carolina 27708
G. G. Katulb兲
Nicholas School of the Environment and Earth Sciences and Department of Civil and Environmental
Engineering, Pratt School of Engineering, Duke University, Durham, North Carolina 27708
J. D. Albertsonc兲
Department of Civil and Environmental Engineering, Pratt School of Engineering and Nicholas School
of the Environment and Earth Sciences, Duke University, Durham, North Carolina 27708
L. Ridolfid兲
Dipartimento di Idraulica, Trasporti ed Infrastrutture Civili Politecnico di Torino, 10129 Torino, Italy
共Received 18 August 2006; accepted 9 January 2007; published online 8 March 2007兲
Gentle topographic variations significantly alter the mass and momentum exchange rates between
the land surface and the atmosphere from their flat-world state. This recognition is now motivating
basic studies on how a wavy surface impacts the flow dynamics near the ground for high bulk
Reynolds numbers 共Reh兲. Using detailed flume experiments on a train of gentle hills, we explore the
spatial structure of the mean longitudinal 共u兲 and vertical 共w兲 velocities at high Reh. We show that
classical analytical theories proposed by Jackson and Hunt 共JH75兲 for isolated hills can be extended
to a train of gentle hills if the background velocity is appropriately defined. We also show that these
theories can reproduce the essential 2D structure of the Reynolds stresses. The basic assumptions in
the derivation of the JH75 model are also experimentally investigated. We found that the
linearization of the advective acceleration term and the mixing length proposed in JH75 are
reasonable within the inner layer. We also show that the measured variability in the linearized mean
longitudinal advective acceleration term can explain much of the measured variability in the entire
nonlinear advective acceleration term for the longitudinal mean momentum balance. © 2007
American Institute of Physics. 关DOI: 10.1063/1.2565528兴
I. INTRODUCTION
Turbulent flow over complex surfaces remains an active
research area in modern engineering, geophysical and environmental fluid dynamics. The impact of a wavy surface on
bulk properties of the flow field is of interest in a large number of applications spanning a broad range of spatial scales,
from micromaterials used for drag reduction to drag parametrization of mountains within large-scale weather models,
from designs of optimal roughnesses for heat exchangers to
the evolution and migration of desert sand dunes. The length
and time scales characterizing the flow over wavy surfaces in
such applications can range from microscopic 共␮m, 100−1 s兲
to macroscopic 共km, h兲 thereby resulting in Reynolds numbers that vary from laminar to completely developed turbulent flows 共Reh = 300− 106兲 and from hydraulically smooth to
completely rough flows 共Rek = 0 − 106兲. Here, the two Reynolds number definitions follow standard convention with
Reh = h ubu / ␯, where h is the depth of the boundary layer, ␯ is
the kinematic viscosity, and ubu is the time- and depthaveraged bulk velocity, and Rek = ks u* / ␯, where ks is the
a兲
Electronic mail: davide.poggi@duke.edu
Electronic mail: gaby@duke.edu
c兲
Electronic mail: john.albertson@duke.edu
d兲
Electronic mail: luca.ridolfi@polito.it
b兲
1070-6631/2007/19共3兲/036601/12/$23.00
characteristic roughness of the wavy surface 共representing
both the equivalent sand roughness of the surface and the
global drag induced by the terrain variability on the flow兲
and u* is the friction velocity. While still lacking, any general
theory for flow over wavy surfaces must reproduce the flow
dynamics for the widest possible Rek-Reh configurations. Experimentally, however, much of the data collected over the
past 40 years is clustered within two broad categories: low to
moderate Reh and low Rek 共i.e., hydrodynamically smooth or
transition兲, and very high Reh and Rek. Low to moderate Reh
and low Rek experiments were primarily intended to understand the initial phases of wave generation, coherent structures formation and development,1–9 or mechanisms inducing
passive drag reduction.10,11 The Reh and Rek that characterize
the flow over wavy surfaces interacting with the atmospheric
boundary layer 共ABL兲 are much higher and often necessitate
different experimental and theoretical treatment, the subject
of this investigation.
During the past two decades, interest in atmospheric
boundary layer 共ABL兲 flows over complex terrain significantly increased in micrometeorology and surface
hydrology12 though they did not receive the same level of
attention as flows over uniform flat terrain. However, the
urgent need to link long term CO2 turbulent flux measurements with ecophysiological processes across different
19, 036601-1
© 2007 American Institute of Physics
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036601-2
Poggi et al.
biomes,13,18 often situated on complex terrain, is now elevating the interest in ABL flows over hills.16,17 This interest is
partly driven by the recognition that gentle topographic
variations can lead to significant changes to the climate near
the ground surface, and thereby on exchanges of energy,
mass, and momentum between the land surface and the
atmosphere.16 More significant is the recognition that for the
vertically integrated mean scalar continuity equation, the local balance between turbulent fluxes above the canopy and
integrated ecophysiological sources and sinks within the
canopy can be significantly disrupted by advective terms
originating from topographic variability.14,15,17–20
One possible “fix” to this “imbalance problem” is to directly measure the vertically integrated local advective terms.
Unfortunately, measuring vertically integrated scalar advective terms at a single tower remains difficult in long-term
flux monitoring experiments. Another solution is to model
the advective terms and correct 共or filter兲 flux measurements
when advective terms are predicted to play a significant role
in the mean scalar conservation equation. This modelling
approach may be promising as high resolution digital elevation maps that drive such computations are now available.
The main limitation here is the availability of a simplified
theory that links topographic variations to the mean scalar
advective terms. Clearly, to progress on this latter point, ABL
flows over the “nonflat” terrain must now be confronted with
laboratory and field experiments, detailed numerical simulations, and development of simplified theories that retain clear
analytical tractability.
Interestingly, substantial progress on high Reh flow over
complex terrain did not come initially from laboratory experiments but from analytical theories, the first attributed to
Hunt and co-workers21 共hereafter referred to as the JH75
theory兲. Although these theoretical results are limited in applicability to hills with low slopes 共but a wide range of Rek兲,
they were fundamental to stimulating several field campaigns
as early as 197922–29 共see Mammarella et al.29 for a recent
review兲. The field experiments, primarily conducted for high
Rek and Reh, supported predictions by the linear theory even
for conditions when the theory was not strictly applicable.
Not surprisingly, the spatial resolution of these field experiments were highly restricted given that tower measurements
were often conducted only at the crest of the hill.
To overcome this spatial sampling problem, wind tunnel
experiments have also been conducted.1,30–35 These experiments were successful at resolving the spatial sampling problems that plagued field experiments at the expense of other
limitations. For example, to aerodynamically simulate a real
rough canopy on a hill, the inner layer depth becomes comparable to the roughness height resulting in wind-tunnel experiments having little relevance to inner layer studies of the
ABL.30,36 The inner region is the region of immediate interest here because it exerts the most influence on the environment near the ground surface, and thereby on the exchange
rates of energy, mass, and momentum between the land surface and the atmosphere.
In short, both field and wind tunnel experiments contributed crucial data on ABL flow over a wavy surface for high
Rek and Reh. However, the former suffered from severe spa-
Phys. Fluids 19, 036601 共2007兲
tial resolution issues while the latter did not provide a realistic description of the inner layer dynamics most relevant to
biosphere-atmosphere exchange within the ABL. What is
lacking is a data set with sufficiently thick inner layer depth
and of sufficient spatial resolution to assess closure assumptions 共e.g., eddy diffusivity兲, linear simplifications of advective terms, and background velocity estimation when extending this theory to situations diverging from an isolated hill.
JH75, along with such assumptions, are now forming the
basic foundation for more detailed analytical models of flow
inside tall canopies on hilly terrain.17,20 Hence, this study is
motivated, in part, by the clear need for both simplified models describing flow over hilly terrain, and high spatial resolution experiments that primarily focus on inner layer dynamics under controlled laboratory conditions.
Numerous factors such as thermal stratification, unsteady flow conditions 共including shifts in wind directions兲,
and spatial variability in surface roughness prohibit the immediate extrapolation of these flume experiments to the
ABL. However, resolving the joint effects of all these factors, in addition to topographic variability, is well beyond the
scope of a single study. Hence, the compass of this work is
restricted to the interplay between the topographic variability
and the mean flow field for a train of gentle hills within the
framework of JH75 and the flume experiments described
next. We choose the JH75 framework as a starting point for
our work because it provides the most parsimonious balance
between the resolved physical processes and the “lowdimensionality” in linking topographic variability to mean
flow variability across hilly terrain.
II. EXPERIMENTAL FACILITIES
The experiments were conducted in the OMTIT recirculating channel at the Giorgio Bidone Hydraulics Laboratory,
DITIC Politecnico di Torino. The flume has an 18 m long,
0.90 m wide, and a 1 m deep working section 共Fig. 1兲, and a
recirculating flow rate up to 360 l s−1. At the entrance, the
flow passes through a 1.5 m long diffuser and then into a
honeycomb with a hexagonal cell structure made of reinforced plastic elements 共length-to-cell-size ratio of 6兲. The
channel sides are made of glass to permit optical access.37,38
The topography is constructed using a removable wavy
stainless steel wall composed of four modules, each representing a sinusoidal hill with a shape function given by
f̃共x兲 = H / 2 cos共kX兲, where X is the longitudinal distance, H共
=0.08 m兲 is the hill height, k = ␲ / 共2L兲 and L共=0.8 m兲 is the
hill half length as shown in Fig. 1. This section begins at 4 m
downstream from the channel entrance. The longitudinal 共u兲
and vertical 共w兲 velocity measurements were performed
above the third hill module. To check whether the turbulence
was completely developed, preliminary measurements were
conducted on the second, third, and fourth sections. These
preliminary measurements showed that the u statistics acquired at four locations 共and 10 vertical positions兲 around the
crest of the second and the fourth hills do not significantly
differ from their analogous statistics at the crest of the third
hill 共overall R = 0.95 and m = 1.08, where R is the correlation
coefficient and m is the regression slope兲. During the experi-
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036601-3
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An experimental investigation of turbulent flows
FIG. 1. 共Color online兲 Experimental setup: The test section along with the train of hills 共top right兲. The definition of the displaced coordinate system 共x , z兲 in relation
to the inner layer height 共hi兲 and hill dimensions 共H , L兲
is also shown for reference.
ment planning phase, we conducted model calculations using
first order closure principles and found that the mean u and w
variations above the third hill do not differ from their counterpart above the fourth hill 共overall R = 0.85 and m = 1.18兲.
The velocity measurements were carried out using a
two-component Laser Doppler Anemometry 共LDA兲 operated
in a forward scattering mode. A key advantage of LDA is its
nonintrusive nature and its small averaging volume. Two
separated manual traverse systems were used to position the
LDA apparatus and the receiving optics, allowing a measurement precision on the order of ±0.025 mm. The measurements were performed at ten positions to longitudinally
cover one hill module, and at 0.40 m from the lateral wall in
the spanwise direction. These measurements were performed
along a large number of vertical positions 共⯝35兲 displaced
along a specified coordinate. When modeling or measuring
the flow over complex surfaces, the reference coordinate systems can be externally imposed 共e.g., rectangular Cartesian
systems兲, related to the geometry of the surface 共e.g., terrain
following systems兲, or allowed to adjust according to the
flow dynamics 共e.g., streamline coordinates兲. As discussed in
FB04,17,39 the latter choice is the most preferred for flow
over hills and is adopted here. For our experiment, the hill
shape function is a cosine surface with respect to the rectangular coordinate system 共X and Z兲 resulting in vertical and
longitudinal streamline coordinates, z and x, given by
x = X + H/2 sin共kX兲e−kZ;
共1兲
z = Z − H/2 cos共kX兲e−kZ .
共2兲
The key advantage of a displaced coordinate system is
that it reduces to terrain-following near the ground, and to
the rectangular Cartesian system well above the hill. Hence,
it retains the advantages of both coordinate systems in the
appropriate regions.
We sampled 45 cm 共of the 60 cm water level兲 in the
displaced vertical direction. We concentrated on the vertical
measurement array close to the ground so as to zoom into the
inner layer dynamics. The vertical arrangement of each measurement point was chosen to have an approximate logarithmic vertical distribution. To simplify the acquisition procedure, the inclination of the two velocity components was
kept constant and equal to the slope of the surface. However,
the measurement path from the surface follows the displaced
coordinate system. Numerical postprocessing of the acquired
data was then carried out to readjust the two components of
the velocity with the theoretical coordinates thereby permitting direct comparisons between theory and data.
The water depth was retained at a mean steady state
value of 60 cm throughout the experiment. We are aware that
the ratio between a 60 cm water depth and a 90 cm channel
width does not guarantee that the lateral walls do not influence the statistics of u and w. We carried out a sensitivity
analysis in which the velocity statistics were acquired at several spanwise positions 共ranging from 20 to 40 cm from the
side wall兲 within the inner region and we found no significant difference between them for the first and second moments. Therefore, for the purposes of modelling the mean
flow in the context of JH75, the effects of the lateral walls
were neglected.
One technical challenge to quantifying the flow statistics
within the inner layer was the use of LDA to acquire w near
the ground surface. Tilting the laser beams cannot be employed because such tilting leads to inadequate spatial co-
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036601-4
Phys. Fluids 19, 036601 共2007兲
Poggi et al.
location between the horizontal and vertical velocity. To
avoid such a problem, we used a technique previously developed and tested in this flume.40 A series of ten narrow slits
共0.1 mm wide兲 were made in the stainless steel sheet for
each longitudinal position to permit vertical passage of the
laser. To guarantee the same optical path for every beam, the
lowest part of the sinusoidal surface, which is at 0.15 m
above channel bottom, was filled with water. The presence of
water 共instead of air兲 allows for a decrease in the pressure
difference between the region above and below the thin steel
sheet leading to a decrease in stress and vibrations. To avoid
bubble formation, a low hydraulic discharge of 2 l / s under
the test section was also imposed. Finally, to reduce every
interference between the water below and above the test section, a “pocket” was projected ad hoc and installed below the
steel sheet for each slit. By injecting a fluorescent dye solution 共Red Rhodamine兲 in the fluid under the hill, we confirmed that no interaction between the fluid below and above
the test section occurred. As a final check, we compared the
w measurements with direct predictions from the continuity
equation using the u measurements as input 共described later兲.
The u共x , z兲 and w共x , z兲 measurements were conducted at
Reh = 1.5⫻ 105 共fully developed turbulence兲. The sampling
duration for each run was 300 s, which was shown elsewhere
to be sufficiently long to ensure convergence of the
statistics.41 The sampling frequency for each run ranged from
2500 to 3000 Hz. The analog signals from the processor
were checked by an oscilloscope to verify the Doppler signal
quality at every run. The steadiness of the flow during the
very long experimental times 共about 8 h兲 was verified by
continuously monitoring the flow rate. No artificial seeding
of the channel was employed.
The signal processing was performed by two Dantec
Burst Spectrum Analyzer 共BSA兲 processors. The coincidence
mode was used to obtain reliable measurements of the Reynolds shear stress. To preserve the correlation coefficient between w and u, all data points not exactly temporally coincident were discarded. Further details about the LDA
configuration and signal processing can be found
elsewhere.40
One limitation of this experimental setup pertains to the
surface roughness. To achieve a sufficiently deep inner region, we reduced the roughness height, which necessarily
implies that the viscous effects can play a role in a thin
region close to the ground. For our experiments, the roughness Reynolds number is Rek = u*ks / ␯ = 36, where the measured u* = 0.018 m / s, and ks = 2 mm. The region in which 4
⬍ Rek ⬍ 60 is classically referred to as dynamically slightly
rough.42 However, the presence of a thin viscous region need
not affect the data-model comparison if we restrict the comparisons to the inner region well above the viscous sublayer.
Hence, in the model-data evaluation, we discarded all the
measurements within this viscous region but retained them
when graphically presenting the full data sets. In terms of
impact on the JH75 framework, the presence of a viscous
region necessitates a re-examination of the lower boundary
condition, especially the constant stress assumption, even if
this region is not included in the data-model comparison. We
elaborate on this issue in the results and discussion section.
III. THEORY
According to Jackson and Hunt,21 the domain above a
2D gentle hilly surface can be decomposed into an inner and
an outer region. In the outer region, the perturbations associated with the shear stress gradient induced by the flow over
the hill are of no dynamical significance and the flow can be
treated as inviscid. In the inner region, these perturbations in
the shear stress are comparable with inviscid processes.
Belcher and Hunt48 共hereafter referred to as BH93兲 estimate the inner region depth as the solution of
␥
hi
⯝
,
L
ln共hi/z0兲
共3兲
where z0 is the momentum roughness height 关related to ks
through the mean longitudinal profile as z0 = ks exp共−8.5kv兲兴,
␥ = 2k2v, and kv = 0.4 is the von Karman constant.
In the inner layer and in a thin layer just above 共i.e., the
middle layer as defined in the Appendix兲 the pressure perturbation induced by a sinusoidal surface can be expressed as
p̃共x兲 = − U20H/2k cos共kx兲.
共4兲
Hereafter, tilde denotes solutions for sinusoidal hills.
For analytic tractability, it is convenient to decompose
the mean velocity into an unperturbed or background state
共i.e., assuming the flow is over flat terrain兲 and a perturbation
induced by the hill, given by
u共z,x兲 = Ub共z兲 + ⌬u共x,z兲;
w共z,x兲 = ⌬w共z,x兲;
共5兲
␶共x,z兲 = ␶b共z兲 + ⌬␶共x,z兲.
JH75 suggested that, in the case of gentle hills
共H / L 1兲, 共a兲 the perturbation terms are small compared to
the background terms, and 共b兲 the nonlinear terms can be
neglected. Based on these assumptions, the mean-momentum
equation can be written as
Ub共z兲
⳵ ⌬u共x,z兲
⳵ Ub共z兲
⳵ ⌬p共x兲 ⳵ ⌬␶共x,z兲
+ ⌬w共x,z兲
=−
+
;
⳵x
⳵z
⳵x
⳵z
共6兲
the robustness of these assumptions will be tested in the next
section. Moreover, the above equations imply that the background velocity plays an important role in the estimation of
the perturbation terms. Hence, an unambiguous definition for
Ub becomes critical.
Instead of writing a budget equation for ␶共x , z兲, JH75
introduced another controversial assumption: the eddy viscosity closure with a linear mixing length throughout the
inner layer.47 With this assumption, the equation for the perturbed Reynolds shear stress can be written as a function of
⌬u共x , z兲 using
⌬␶共x,z兲 = 2kvu*z
⳵ ⌬u共x,z兲
,
⳵z
共7兲
where the logarithmic mean velocity profile for the background velocity was used 关i.e., Ub共z兲 = u* / kv log共z / z0兲兴.
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036601-5
Phys. Fluids 19, 036601 共2007兲
An experimental investigation of turbulent flows
While this background velocity assumption may be reasonable for an isolated hill, its validity for more complex terrain
共e.g., a train of hills兲 remains to be investigated.
For the particular case of a flow above a sinusoidal hill,
共x , z兲 is
the solution of ⌬u
⌬u共x,z兲 = −
u*
UI0
kv
冋冉
1 + ln
册
冊
h2i
cos共kx兲 − Re共4B0e−ikx兲 ,
z 0z
共8兲
where UI0 is the dimensionless scaling for the streamwise
velocity in the inner region, UI0 = 共Hk / 2兲共U20 / UI2兲, UI is the
background velocity defined at the inner region height
共Ub共hi兲 = UI兲, B0 = K0共2冑ikLz / hi兲, and K0共.兲 is the modified
Bessel function of the zeroth order. Likewise, the perturbation induced by a sinusoidal hill on the vertical velocity in
real space can be evaluated as
共x,z兲 = − kv␲u*UI0
⌬w
z
sin共kx兲.
hi
共9兲
Following this derivation, it is clear why the JH75 is a
logical starting point to investigate the connections between
topographic variability and mean flow variability. In the
JH75 framework, topographic variability produce pressure
perturbations that induce mean flow perturbations, which in
turn require that the turbulence adjusts to these perturbations
via the turbulent stress gradients. The model simplifications
for linking topographic variability to the pressure perturbations and how the turbulent stresses adjust to the mean flow
perturbations 共i.e., K-theory兲 will be experimentally explored
in greater detail here. We note that these model simplifications often go beyond the JH75 framework and are applied to
almost all closure approaches in Reynolds-Averaged NavierStokes 共RANS兲 approaches. The additional simplification in
JH75 共beyond standard first order closure principles兲 is the
linearization of the advective acceleration term, which we
explore in greater depth as well using the data.
From the dynamics perspective, the JH75 solution reveals several testable hypotheses about the phase relationships between the velocity perturbations and the hill shape
共x , z兲 includes a term in
function. For example, while ⌬u
phase with the hill surface and a term that depends on the
共x , z兲 is always in phase
imaginary and real part of B0, ⌬w
with the pressure gradient and hence out-of-phase with the
hill surface. Moreover, B0 tends to zero exponentially far
above the surface leading to a ⌬u共x , z兲 in-phase with the
surface and out-of-phase with ⳵⌬p共x兲 / ⳵x for large z. Furthermore, the solution for the perturbed shear stress above a
sinusoidal hill is given by
再
冋
⌬␶共x,z兲 = 2u2*UI0 cos kx − Re
共k兲B
ikA
1
1
冑hiik/z
e−ikx
册冎
, 共10兲
where B1 = K1共2冑ikLz / hi兲, and K1共.兲 is the modified Bessel
function of the first kind. Above the surface, similar to
⌬u共x , z兲, ⌬␶共x , z兲 tends 共exponentially兲 to be in-phase with
the surface and out-of-phase with ⳵⌬p共x兲 / ⳵x. But before
testing these hypotheses and simplifications, it is imperative
to address the background velocity formulation for a train of
gentle hills first.
IV. RESULTS
A. The background velocity and the velocity
perturbations
Background velocity: Recall that while the definition of
Ub共z兲 is unambiguous for an isolated hill, being the upstream
mean longitudinal velocity profile, its definition remains ambiguous for complex terrain.39,43 For example, for a train of
ridges, the upstream velocity profile colliding with the nth
hill is also a function of the previous nth− 1 hill. When experimentally studying periodically varying flows, it is plausible to assume that Ub共z兲 can be determined from the mean
velocity averaged over the hill wavelength. On the other
hand, it is theoretically desirable that Ub only be determined
from flow over flat terrain because 共1兲 these profiles are independent of the hill configuration 共except through a surface
roughness兲 and they have been tested in numerous experiments, and 共2兲 the analytical approach in JH75 a priori assumes a logarithmic shape for Ub, known to be the most
robust parametrization of the mean velocity profile for flat
boundary layers. A logical question to explore is whether the
spatially averaged velocity profile, strictly determined from
the data here, is related to the velocity profile determined
from “flat-world” formulations. If the two Ub estimates collapse, then the ambiguity in the definition of Ub is not critical
for our purposes.44,45 This question is explored next in Fig. 2.
Figure 2 shows all ten measured longitudinal velocity
profiles across the hill together with the Ub共z兲 determined
from spatial averaging and its best fit logarithmic velocity
profile 关given by Ul共z兲 = u* / kv log共z / z0兲兴. In Ul共z兲, u*
= 0.018 m s−1 was evaluated using the maximum value of
␶b共z兲, and the roughness length z0 = 0.062 mm was computed
by minimizing the root-mean squared error between Ub共z兲
estimated from spatial averaging and Ul共z兲. The comparison
between Ub共z兲 and Ul共z兲 suggests that the background velocity retains its logarithmic shape.
The ten shear stress profiles measured along the hill are
also presented in Fig. 2 together with the background shear
stress, ␶b共z兲. Within the inner region, it is clear that ␶b共z兲 is
approximately constant in most of the inner region consistent
with the logarithmic background velocity profile assumption.
The combination of the logarithmic Ub共z兲 and constant ␶b共z兲
already hints that the eddy viscosity model, a major assumption of the JH75 theory, may be reasonable in the inner region.
Furthermore, Fig. 2 shows a thin region close to the
ground where ␶b共z兲 decreases toward the origin. This behavior is often used as a robust identification of the viscous
sublayer.46 Here, this layer is limited to a thin region 共z / hi
⬍ 0.2, where hi = 0.08 m is the inner layer depth兲 and only
affects the first two or three sampling points. Recall that the
JH75 solution to the mean longitudinal momentum balance
neglects viscous effects altogether. However, the presence of
a thin viscous subregion does not alter the formulations proposed in JH75. The JH75 model admittedly assumes a
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036601-6
Poggi et al.
Phys. Fluids 19, 036601 共2007兲
FIG. 2. 共Color online兲 The measured
longitudinal mean velocity u共x , z兲 and
turbulent shear stress ␶共x , z兲 across the
hill. Top panel: Color map of the 2D
structure of the measured u共x , z兲 along
with the hill surface and the inner region
boundary
共dotted-dashed兲.
Bottom-left panel: The profiles of the
normalized u at the 10 sections 共s1–
s10兲 marked in the top-left figure 共vertical lines兲. The logarithmic profile
共dotted-dashed line兲 and the spatially
averaged values 共open triangles兲 used
to define Ub are also shown. The velocity is normalized by Uh = Ub共hi兲.
Bottom-right panel: Same as bottomleft but for the shear stress. The normalizing variable is the maximum
horizontally averaged u*.
completely rough surface as a bottom boundary condition,
but this assumption was only invoked to force the turbulent
shear stress to be constant near the wall. In most of the lower
levels of the inner region of this experiment 共but above the
thin viscous sublayer兲, the shear stress appears to be approximately constant with respect to z 共see Fig. 2兲.
The measured 2D spatial structure of the velocity components and Reynolds stress: Having defined the Ub共z兲 and
␶b共z兲 profiles, we show in Fig. 3 the measured 2D spatial
structure of the mean flow perturbations induced by the hill
in the inner region. We focus on three discernible zones
within the inner region of Fig. 3: upwind, summit, and wake.
FIG. 3. The 2D structure of the flow
perturbations around the background
state: Top panel is ⌬u共x , z兲, middle
panel is ⌬w共x , z兲, and bottom panel is
⌬␶共x , z兲. The inner layer depth is also
shown for reference 共dotted-dashed兲.
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036601-7
An experimental investigation of turbulent flows
FIG. 4. 共Color online兲 Profile comparisons between kvz 共solid line兲 and the
effective mixing length l共z兲 estimated from the measured u⬘w⬘ and ⳵u / ⳵z
using the eddy-viscosity model 共circle-line兲 across the hill.
In the upwind zone, ⌬u共x , z兲 progressively increases
with increasing x. Initially, this increase is confined to a zone
very close to the wall 共at x / L = −1.5兲 but extends to the whole
inner layer as the summit of the hill is approached 共x / L
= 0兲. Here, ⌬w共x , z兲 remains slightly negative throughout.
Also, a decrease in ⌬␶共x , z兲 is noted.
Just after the summit, the two noticeable features are a
decrease in ⌬u共x , z兲 compared to its peak value in the upwind region, and a reversal in sign of ⌬␶共x , z兲. Here,
⌬w共x , z兲 remains small in magnitude.
In the so-called wake region, a strong velocity deficit
that extends throughout hi from x / L = 1 to the middle of the
upwind side 共x / L = 2.5兲 is evident. In this region, the magnitude of the Reynolds stress is enhanced, while ⌬w共x , z兲 is
slightly negative. Next, we compare the measured individual
velocity and stress profiles with the JH75 model results and
further explore their assumptions and simplifications.
B. The analytical model
Closure assumption: Before presenting the comparison
between measured and modelled flow statistics, we first analyze the basic assumptions used in the previous sections to
derive the JH75 analytical model. The most controversial
assumption used in JH75 is the eddy viscosity closure with a
linear mixing length throughout the inner layer.47 In Figs. 4,
we show the mixing length determined from measured
u⬘w⬘共x , z兲 and the gradients in u共x , z兲. We found that a linear
mixing length 共=kvz兲 agrees reasonably well with the measurements. Nevertheless, a minor disagreement between the
modelled and the measured mixing length is noticeable on
the lee side of the hill where l = kvz tends to overestimate the
measured mixing length.
Phys. Fluids 19, 036601 共2007兲
FIG. 5. Top panel: Comparisons between the nonlinear and the linearized
advective acceleration terms computed from the velocity measurements
shown in Figs. 3 and 4. Bottom panel: Comparison between the measured
nonlinear advective acceleration term and the measured linearized advective
acceleration assuming w = 0 throughout. The 1:1 line is shown in both
panels.
Another simplification in the JH75 model is the linearization of the advective acceleration terms. We compared the
measured nonlinear advective terms 共u ⳵ u / ⳵x + w ⳵ u / ⳵z兲 with
their linearized counterpart 共Ub ⳵ ⌬u / ⳵x + ⌬w ⳵ Ub / ⳵z兲 in Fig.
5共a兲. It is clear from Fig. 5共a兲 that the linearization proposed
by JH75 is quite reasonable 共R = 0.997 and m = 1.01, where R
is the correlation coefficient and m is the regression slope, as
before兲.
Also, the data permitted us to further explore other simplifying assumptions regarding the advection in the longitudinal momentum balance. We found that the measured
Ub ⳵ ⌬u / ⳵x explains more than 90% of the measured 共nonlinear兲 advective acceleration term as shown in Fig. 5共b兲 共R
= 0.992, m = 1.09兲.
Other assumptions in the JH75 model such as neglecting
⳵⌬p / ⳵z and ⳵u⬘u⬘ / ⳵x can be a source of error in modeling
the velocity statistics. To explore this point further, we conducted a separate experiment on the variation of p at the hill
surface and found that p共x兲 is well approximated by a local
hydrostatic assumption as we moved along the hill 共data not
shown here兲. Furthermore, we conducted preliminary analysis on ⳵u⬘u⬘ / ⳵x and found it to be small 共in a global sense兲
when compared to ⳵ p / ⳵x 共data not shown兲.
Comparison between measured and modelled flow variables: We first present the overall comparisons between measured and modelled u, w, and ␶ in Fig. 6 followed by a
detailed analysis of the spatial structure of the error in ⌬u,
⌬w, and ⌬␶ in Figs. 7–9. From Fig. 6, the best agreement in
terms of regression slope m and correlation coefficient R is
for u 共R = 0.98, m = 1.11兲, followed by ⌬w 共R = 0.84, m
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036601-8
Poggi et al.
Phys. Fluids 19, 036601 共2007兲
FIG. 7. 共Color online兲 Comparison between measured 共open circle兲 and
modelled 共solid line兲 longitudinal velocity perturbation profile across the hill
共s1–s9 as shown in Fig. 2兲.
FIG. 6. An overall comparison between measured and modelled velocity
and shear stress within the inner layer across the hill: Top panel is for u共x , z兲,
middle panel is for w共x , z兲, and bottom panel is for ␶共x , z兲. The modelled
statistics are evaluated using the background and the perturbed expressions
关u is from Eqs. 共A15兲 and Ub, w is from Eq. 共A16兲, and ␶ is from Eq. 共10兲
and ␶b. Ub and ␶b are defined in the text兴. The one-to-one line 共solid兲 is also
shown. Note the small values for w共x , z兲 when compared to u共x , z兲.
= 0.96兲, and the worst agreement is for ⌬␶ 共R = 0.77, m
= 0.74兲. What is clear from this comparison is that the model
biases, for all three variables, are not entirely random.
To explore the 2D structure of the differences between
model and measurements across the hill, we show the profiles of the perturbed flow statistics along the vertically displaced coordinate 共z兲 at ten stations uniformly spaced in the
longitudinal direction 共from s1 at x / L = 0 to s10 at x / L
= 3.64兲 and their 2D spatial structure.
Figure 7 共s1–s10兲 shows the measured 关⌬u共x , z兲兴 and
modelled 关⌬ũ共x , z兲兴 velocity perturbation for the ten vertical
profiles. The comparison between ⌬ũ共x , z兲 and ⌬u共x , z兲
shows that the model is capable of describing the main features of the 2D perturbed longitudinal velocity within the
three zones, at least in terms of signs and order of magnitude
of ⌬u共x , z兲. However, the model clearly underestimates the
extremes.
Figure 8 共s1–s10兲 shows the profile comparison between
⌬w共x , z兲 and ⌬w̃共x , z兲. From Fig. 8 it is clear that the model
predicts an approximate linear profile for ⌬w共x , z兲 throughout, and that ⌬w̃共x , z兲 captures the basic features of ⌬w共x , z兲.
Nevertheless, some disagreement in the phase behavior is
noticeable from Fig. 8. Given the small magnitude of
⌬w共x , z兲 and possible measurement error, we estimated
⌬w共x , z兲 from the measured ⌬u共x , z兲 using the continuity
equation by imposing a zero vertical velocity at the hill surface. The comparison between measured and estimated
⌬w共x , z兲 are also presented in Fig. 8. The difference between
these two estimates is not different from the model-data dif-
FIG. 8. 共Color online兲 Same as Fig. 7 but for the vertical velocity. In addition, estimates of ⌬w共x , z兲 obtained from the measured ⌬u共x , z兲 via the
continuity equation are also shown 共squares兲.
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036601-9
An experimental investigation of turbulent flows
FIG. 9. 共Color online兲 Same as Fig. 5 but for the shear stress. Both modelled
⌬˜␶共x , z兲 and modelled ⌬˜␶共x , z兲 − ⌬˜␶共x , 0兲 共i.e. adjusting the lower boundary
condition to assure a zero-stress in the viscous sublayer兲 are compared to
measured ⌬␶共x , z兲. Note the agreement between modelled ⌬˜␶共x , z兲
− ⌬˜␶共x , 0兲 and measured ⌬␶共x , z兲.
ferences, especially on the upwind side of the hill. The largest difference between the two estimates of ⌬w共x , z兲 and the
model appears to be also on the lee-side of the hill 共Secs. IV
and V兲.
Figure 9 shows the comparison between the profiles of
measured and modelled ⌬␶共x , z兲. From this figure, while
⌬˜␶共x , z兲 is able to reproduce the basic behavior of ⌬␶共x , z兲,
the agreement between the linear model and the data is less
encouraging, especially near the ground. Recall that near the
viscous sublayer, the ⌬␶共x , z兲 is approximately zero, while
the ⌬˜␶共x , z兲 remains finite because the layer is assumed to
reside in a fully developed turbulence zone. If this lower
boundary condition “contamination effect” is minimized by
comparing ⌬˜␶共x , z兲 − ⌬˜␶共x , 0兲 with ⌬␶共x , z兲, then the agreement is significantly improved as shown in Fig. 9.
Phys. Fluids 19, 036601 共2007兲
defining appropriate u* and z0, as a background velocity to
predict the velocity perturbation above gentle terrain. Moreover, Ub, retaining a logarithmic profile, permits us to extend
the applicability of analytical theories previously proposed
for isolated hills 共e.g., JH75兲 to more complex, though
gentle, terrain.
This study is the first to experimentally investigate the
linearization of the advective acceleration terms proposed in
JH75 and found it to be reasonable. However, the often cited
main criticism to the application of analytical theories and
even first order closure models to flows over complex terrain
remains the eddy viscosity models 共or their mixing length
variant兲. Using the data from this experiment, we found that
a linear mixing length describes reasonably well the data
throughout the inner layer. Hence, this comparison suggests
that eddy-viscosity models may be sufficiently robust in
modelling topographically induced perturbations in the mean
flow and turbulent stresses, at least for gentle topography.
When these two findings are taken together, it is clear that
approximations “endogenous” to the flow dynamics in JH75
are reasonable. Note that these conclusions go beyond JH75
and apply to other approaches such as first 共or even higher兲
order closure models in RANS that employ the mixing
length concept. The linearization findings here also go beyond the particulars of the JH75 model. For example, the
linearized approximation can permit explicit tracking of how
individual energetic modes in the topography influence the
mean flow field variability. This finding is particularly timely
given the availability of high resolution digital elevation
maps from remote sensing products.49
Finally, the experiment here permitted us to explore new
simplifications to the mean longitudinal momentum balance
for a train of hills not previously explored. We found that the
measured variability in Ub ⳵ ⌬u / ⳵x captures most of the variability in the nonlinear advective acceleration within the inner layer. The significance of this simplification is that a new
univariate linear PDE may form the basis for future analytical solutions to the 2D structure of the mean flow field. The
generality of this finding remains to be investigated in the
future for terrain variability possessing more than one energetic wave number 共or more complex features兲.
V. SUMMARY AND CONCLUSIONS
APPENDIX: REVIEW OF THE LINEAR ANALYTICAL
MODEL
A new data set was collected above a train of gentle hills
to explore experimentally and theoretically the 2D structure
of the mean velocity. Our choice of a train of sinusoidal hills
here is a logical progression from isolated hills so as to begin
confronting the problem of flow over complex terrain encountered in nature. In extending the analysis from an isolated hill to a train of hills, several issues must be addressed.
The first is that the background velocity is no longer unambiguously defined and its shape cannot be a priori assumed.
Nevertheless, the experiment here suggests that the horizontally averaged velocity is a reasonable and unambiguous interpretation of the background velocity, Ub, given its logarithmic shape. From a prognostic point of view, this result
suggests that a logarithmic velocity profile can be used, after
According to Jackson and Hunt,21 the domain above a
2D gentle hilly surface can be decomposed into an inner and
an outer region. In the outer region, the perturbations associated with the shear stress gradient induced by the flow over
the hill are of no dynamical significance and the flow can be
treated as inviscid. In the inner region, these perturbations in
the shear stress are comparable with inviscid processes.
The analytical derivation commences with scaling arguments reviewed by BH93. BH93 introduced two time scales:
the advection-distortion time scale, TD = ␥L / U共z兲, and the
Lagrangian integral time scale, TL = kvz / u*, and ␥ is a similarity constant. TD characterizes the advection and the distortion of turbulent eddies versus the straining motion associated with perturbations induced by the hill on the mean
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036601-10
Phys. Fluids 19, 036601 共2007兲
Poggi et al.
flow. TL characterizes the decorrelation time scale of the
large energy-containing eddies and the time scale at which
the turbulence comes into equilibrium with the surrounding
mean-flow velocity gradient.50,36 Note that while TD decreases with increasing z, TL increases. Hence, the inner region depth is often defined by the height z = hi at which these
two time scales become equal. BH93 estimated this height as
the solution of
␥
hi
⯝
.
L
ln共hi/z0兲
共A1兲
Hunt et al.47 introduced two additional layers: the inner
surface layer and the middle layer. The inner surface layer
corresponds to a thin layer that was added to take into account the surface boundary conditions. The middle layer is
the lowest region of the outer layer and is introduced to
match inner and outer layer dynamics. In the middle layer,
the Reynolds stress gradient, as in the upper part of the outer
region, is small enough to be neglected but the flow is now
rotational. The depth of the middle layer, hm, differs depending on the ratio between L and the ABL height, hBL. For
elongated hills 共L hBL兲, hm may be assumed to be equal to
the hBL.51 Otherwise, for short hills 共L hBL兲, hm is defined
as the solution of
hm = L ln1/2共hm/z0兲.
共A2兲
Having defined the inner and outer layer depths and their
respective time scale arguments, we proceed to the next scaling arguments; the pressure perturbations at the interface between the middle layer and the upper part of the outer layer,
pm共x兲 = p共x , hm兲. The pressure perturbation induced by the
surface can be expressed as a representative magnitude, po,
and a dimensionless longitudinal function, ␴共x兲, using
pm共x兲 = po␴共x兲,
共A3兲
where ␴共x兲 is a function of the hill shape and po is the forcing due to the flow field in the outer region given by po
= U2b共hm兲 = U20, where Ub共z兲 is the background velocity 共i.e.,
the velocity before the hill is encountered兲. Here U0 is both
the characteristic velocity in the outer layer and the velocity
scale for the pressure perturbation. Again, our interest in the
regions above the inner layer are presented because these
regions are necessary to formulate upper boundary conditions for the inner layer.
Because the middle layer is thin compared to L 共i.e.,
hm / L 1兲, the pressure perturbations in both the middle and
inner layers can be considered constant21,47 with respect to z
and given by pm共x兲 = po␴共x兲. For a two-dimensional hill, ␴共x兲
can be computed from the hill shape function,52 f共x兲. In the
case of sinusoidal hills, where the hill shape function is described by f̃共x兲 = H / 2 cos共kx兲, the terms ␴共x兲 and p共x兲 become
˜␴共x兲 = − H/2k cos共kx兲,
共A4兲
p̃共x兲 = po˜␴共x兲 = − U20H/2k cos共kx兲.
共A5兲
To derive the solution for the mean flow, we consider the
stationary mean longitudinal momentum equation for a 2D
turbulent flow along with the continuity equation, given by
u共x,z兲
⳵ u共x,z兲
⳵ u共x,z兲
⳵ p共x兲 ⳵ ␶共x,z兲
+ w共x,z兲
=−
+
;
⳵x
⳵z
⳵x
⳵z
⳵ u共x,z兲 ⳵ w共x,z兲
+
= 0.
⳵x
⳵z
共A6兲
As stated earlier , it is convenient to decompose the velocity into an unperturbed or background state 共i.e., assuming
the flow is over flat terrain兲 and a perturbation induced by the
hill, given by
u共z,x兲 = Ub共z兲 + ⌬u共x,z兲;
w共z,x兲 = ⌬w共z,x兲;
p共x,z兲 = pb共z兲 + ⌬p共x,z兲;
共A7兲
␶共x,z兲 = ␶b共z兲 + ⌬␶共x,z兲.
When this decomposition is applied to gentle hills
共H / L 1兲 and assuming that the hill perturbations are small
compared to the background terms, JH75 derived the linearized mean-momentum and continuity equations as
Ub共z兲
⳵ ⌬u共x,z兲
⳵ Ub共z兲
⳵ ⌬p共x兲 ⳵ ⌬␶共x,z兲
+ ⌬w共x,z兲
=−
+
;
⳵x
⳵z
⳵x
⳵z
⳵ ⌬u共x,z兲 ⳵ ⌬w共x,z兲
+
= 0.
⳵x
⳵z
共A8兲
The perturbed quantities are then expanded as a power
series in a asymptotically small dimensionless parameter ␦
given by
⌬u共x,z兲 = − po/Ub共h兲共u关0兴 + ␦u关1兴 + ¯ 兲;
⌬w共z,x兲 = − po/Ub共h兲共␦w关1兴 + ␦2w关2兴 + ¯ 兲;
⌬p共x,z兲 = po共␴关0兴共x兲 + ␦2␴关2兴共z,x兲 + ¯ 兲;
共A9兲
⌬␶共x,z兲 = − 2po/U2b共h兲共␶关1兴共x,z兲 + ␦␶关2兴共x,z兲兲,
where all the terms of the power series are dimensionless.
Note that both Ub共h兲 and ␦ have to be separately specified
for each layer depending on the characteristic length and
velocity scales. Once the appropriate length and velocity
scales are determined for each layer, the linear approach can
be employed in the solution. Only terms belonging to the
first order expansion, linear in ␦, are retained. According to
Hunt et al.,47 the asymptotically small dimensionless parameter in Eq. 共A9兲 for the inner layer, ␦, is determined from z0
and hi using ␦ = ln−1共hi / z0兲. Furthermore, the characteristic
velocity Ub共h兲 can be approximated using the background
velocity defined at the inner region height 关Ub共hi兲 = UI兴. Using these two assumptions, the perturbed velocity quantities
共for any 2D hill shape function兲 reduce to
⌬u共x,z兲 = − U20/UI共u关0兴 + ln−1共hi/z0兲u关1兴兲;
共A10兲
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036601-11
Phys. Fluids 19, 036601 共2007兲
An experimental investigation of turbulent flows
⌬w共x,z兲 = − U20/UI共ln−1共hi/z0兲w关1兴兲;
共A11兲
⌬p共x兲 = po␴关0兴共x兲.
共A12兲
The above equations imply that the background velocity
plays an important role in the estimation of the perturbation
terms. Hence, an unambiguous definition for Ub becomes
critical. Instead of writing the equation for ␶共x , z兲 derived
from 共A9兲, we introduce another controversial assumption
used in JH75—the eddy viscosity closure with a linear mixing length throughout the inner layer.47 We note again that
the JH75 formulation assumes that the turbulent viscosity is
much larger than the molecular viscosity within the inner
layer so that the total stress term is dominated by the turbulent stress. With this assumption, the equation for the perturbed Reynolds shear stress can be written as a function of
⌬u共x , z兲 using
⌬␶共x,z兲 = 2kvu*z
⳵ ⌬u共x,z兲
,
⳵z
共A13兲
where the logarithmic mean velocity profile for the background velocity was used. Upon replacing the previous three
equations in the linearized momentum budget and continuity
equations,47 all the unknown terms can be evaluated, albeit
in the Fourier domain, using
␴关0兴共k兲;
û关0兴共k,z兲 = ˆ
冋
␴关0兴共k兲 1 − ln
û关1兴共k,z兲 = ˆ
册
z
− 4B0 ;
hi
共A14兲
z
␴关0兴共k兲ikLk2v ,
ŵ关1兴共k,z兲 = 2ˆ
hi
␴共k兲 is the Fourier transform of ␴共x兲. For the particuwhere ˆ
lar case of a flow above a sinusoidal hill, the solution of
共x , z兲 is
⌬u
⌬u共x,z兲 = −
u*
UI0
kv
冋冉
1 + ln
册
冊
h2i
cos共kx兲 − Re共4B0e−ikx兲 ,
z 0z
共A15兲
where UI0 is the dimensionless scaling for the streamwise
velocity in the inner region, UI0 = 共Hk / 2兲共U20 / UI2兲. Likewise,
the perturbation induced by a sinusoidal hill on the vertical
velocity in real space can be evaluated as
共x,z兲 = −
⌬w
=− kv␲u*UI0
1
冋
册
2HU20k2Lk2v
z
Re i
e−ikx
hi ln共hi/z0兲
2UI
z
sin共kx兲.
hi
共A16兲
共A17兲
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