An experimental investigation of turbulent ﬂows over a ... · inside tall canopies on hilly terrain.17,20 Hence, this study is motivated, in part, by the clear need for both simpliﬁed

An experimental investigation of turbulent flows over a hilly surfaceD. Poggia�

Dipartimento di Idraulica, Trasporti ed Infrastrutture Civili Politecnico di Torino, 10129 Torino, Italyand Nicholas School of the Environment and Earth Sciences and Department of Civil and EnvironmentalEngineering, Pratt School of Engineering, Duke University, Durham, North Carolina 27708

G. G. Katulb�

Nicholas School of the Environment and Earth Sciences and Department of Civil and EnvironmentalEngineering, Pratt School of Engineering, Duke University, Durham, North Carolina 27708

J. D. Albertsonc�

Department of Civil and Environmental Engineering, Pratt School of Engineering and Nicholas Schoolof the Environment and Earth Sciences, Duke University, Durham, North Carolina 27708

L. Ridolfid�

Dipartimento di Idraulica, Trasporti ed Infrastrutture Civili Politecnico di Torino, 10129 Torino, Italy

�Received 18 August 2006; accepted 9 January 2007; published online 8 March 2007�

Gentle topographic variations significantly alter the mass and momentum exchange rates betweenthe land surface and the atmosphere from their flat-world state. This recognition is now motivatingbasic studies on how a wavy surface impacts the flow dynamics near the ground for high bulkReynolds numbers �Reh�. Using detailed flume experiments on a train of gentle hills, we explore thespatial structure of the mean longitudinal �u� and vertical �w� velocities at high Reh. We show thatclassical analytical theories proposed by Jackson and Hunt �JH75� for isolated hills can be extendedto a train of gentle hills if the background velocity is appropriately defined. We also show that thesetheories can reproduce the essential 2D structure of the Reynolds stresses. The basic assumptions inthe derivation of the JH75 model are also experimentally investigated. We found that thelinearization of the advective acceleration term and the mixing length proposed in JH75 arereasonable within the inner layer. We also show that the measured variability in the linearized meanlongitudinal advective acceleration term can explain much of the measured variability in the entirenonlinear advective acceleration term for the longitudinal mean momentum balance. © 2007American Institute of Physics. �DOI: 10.1063/1.2565528�

I. INTRODUCTION

Turbulent flow over complex surfaces remains an activeresearch area in modern engineering, geophysical and envi-ronmental fluid dynamics. The impact of a wavy surface onbulk properties of the flow field is of interest in a large num-ber of applications spanning a broad range of spatial scales,from micromaterials used for drag reduction to drag param-etrization of mountains within large-scale weather models,from designs of optimal roughnesses for heat exchangers tothe evolution and migration of desert sand dunes. The lengthand time scales characterizing the flow over wavy surfaces insuch applications can range from microscopic ��m, 100−1 s�to macroscopic �km, h� thereby resulting in Reynolds num-bers that vary from laminar to completely developed turbu-lent flows �Reh=300−106� and from hydraulically smooth tocompletely rough flows �Rek=0−106�. Here, the two Rey-nolds number definitions follow standard convention withReh=h ubu/�, where h is the depth of the boundary layer, � isthe kinematic viscosity, and ubu is the time- and depth-averaged bulk velocity, and Rek=ks u* /�, where ks is the

characteristic roughness of the wavy surface �representingboth the equivalent sand roughness of the surface and theglobal drag induced by the terrain variability on the flow�and u* is the friction velocity. While still lacking, any generaltheory for flow over wavy surfaces must reproduce the flowdynamics for the widest possible Rek-Reh configurations. Ex-perimentally, however, much of the data collected over thepast 40 years is clustered within two broad categories: low tomoderate Reh and low Rek �i.e., hydrodynamically smooth ortransition�, and very high Reh and Rek. Low to moderate Reh

and low Rek experiments were primarily intended to under-stand the initial phases of wave generation, coherent struc-tures formation and development,1–9 or mechanisms inducingpassive drag reduction.10,11 The Reh and Rek that characterizethe flow over wavy surfaces interacting with the atmosphericboundary layer �ABL� are much higher and often necessitatedifferent experimental and theoretical treatment, the subjectof this investigation.

During the past two decades, interest in atmosphericboundary layer �ABL� flows over complex terrain signifi-cantly increased in micrometeorology and surfacehydrology12 though they did not receive the same level ofattention as flows over uniform flat terrain. However, theurgent need to link long term CO2 turbulent flux measure-ments with ecophysiological processes across different

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biomes,13,18 often situated on complex terrain, is now elevat-ing the interest in ABL flows over hills.16,17 This interest ispartly driven by the recognition that gentle topographicvariations can lead to significant changes to the climate nearthe ground surface, and thereby on exchanges of energy,mass, and momentum between the land surface and theatmosphere.16 More significant is the recognition that for thevertically integrated mean scalar continuity equation, the lo-cal balance between turbulent fluxes above the canopy andintegrated ecophysiological sources and sinks within thecanopy can be significantly disrupted by advective termsoriginating from topographic variability.14,15,17–20

One possible “fix” to this “imbalance problem” is to di-rectly measure the vertically integrated local advective terms.Unfortunately, measuring vertically integrated scalar advec-tive terms at a single tower remains difficult in long-termflux monitoring experiments. Another solution is to modelthe advective terms and correct �or filter� flux measurementswhen advective terms are predicted to play a significant rolein the mean scalar conservation equation. This modellingapproach may be promising as high resolution digital eleva-tion maps that drive such computations are now available.The main limitation here is the availability of a simplifiedtheory that links topographic variations to the mean scalaradvective terms. Clearly, to progress on this latter point, ABLflows over the “nonflat” terrain must now be confronted withlaboratory and field experiments, detailed numerical simula-tions, and development of simplified theories that retain clearanalytical tractability.

Interestingly, substantial progress on high Reh flow overcomplex terrain did not come initially from laboratory ex-periments but from analytical theories, the first attributed toHunt and co-workers21 �hereafter referred to as the JH75theory�. Although these theoretical results are limited in ap-plicability to hills with low slopes �but a wide range of Rek�,they were fundamental to stimulating several field campaignsas early as 197922–29 �see Mammarella et al.29 for a recentreview�. The field experiments, primarily conducted for highRek and Reh, supported predictions by the linear theory evenfor conditions when the theory was not strictly applicable.Not surprisingly, the spatial resolution of these field experi-ments were highly restricted given that tower measurementswere often conducted only at the crest of the hill.

To overcome this spatial sampling problem, wind tunnelexperiments have also been conducted.1,30–35 These experi-ments were successful at resolving the spatial sampling prob-lems that plagued field experiments at the expense of otherlimitations. For example, to aerodynamically simulate a realrough canopy on a hill, the inner layer depth becomes com-parable to the roughness height resulting in wind-tunnel ex-periments having little relevance to inner layer studies of theABL.30,36 The inner region is the region of immediate inter-est here because it exerts the most influence on the environ-ment near the ground surface, and thereby on the exchangerates of energy, mass, and momentum between the land sur-face and the atmosphere.

In short, both field and wind tunnel experiments contrib-uted crucial data on ABL flow over a wavy surface for highRek and Reh. However, the former suffered from severe spa-

tial resolution issues while the latter did not provide a real-istic description of the inner layer dynamics most relevant tobiosphere-atmosphere exchange within the ABL. What islacking is a data set with sufficiently thick inner layer depthand of sufficient spatial resolution to assess closure assump-tions �e.g., eddy diffusivity�, linear simplifications of advec-tive terms, and background velocity estimation when extend-ing this theory to situations diverging from an isolated hill.JH75, along with such assumptions, are now forming thebasic foundation for more detailed analytical models of flowinside tall canopies on hilly terrain.17,20 Hence, this study ismotivated, in part, by the clear need for both simplified mod-els describing flow over hilly terrain, and high spatial reso-lution experiments that primarily focus on inner layer dy-namics under controlled laboratory conditions.

Numerous factors such as thermal stratification, un-steady flow conditions �including shifts in wind directions�,and spatial variability in surface roughness prohibit the im-mediate extrapolation of these flume experiments to theABL. However, resolving the joint effects of all these fac-tors, in addition to topographic variability, is well beyond thescope of a single study. Hence, the compass of this work isrestricted to the interplay between the topographic variabilityand the mean flow field for a train of gentle hills within theframework of JH75 and the flume experiments describednext. We choose the JH75 framework as a starting point forour work because it provides the most parsimonious balancebetween the resolved physical processes and the “low-dimensionality” in linking topographic variability to meanflow variability across hilly terrain.

II. EXPERIMENTAL FACILITIES

The experiments were conducted in the OMTIT recircu-lating 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 �Fig. 1�, and arecirculating flow rate up to 360 l s−1. At the entrance, theflow passes through a 1.5 m long diffuser and then into ahoneycomb with a hexagonal cell structure made of rein-forced plastic elements �length-to-cell-size ratio of 6�. Thechannel sides are made of glass to permit optical access.37,38

The topography is constructed using a removable wavystainless steel wall composed of four modules, each repre-senting a sinusoidal hill with a shape function given by

f�x�=H /2 cos�kX�, where X is the longitudinal distance, H�=0.08 m� is the hill height, k=� / �2L� and L�=0.8 m� is thehill half length as shown in Fig. 1. This section begins at 4 mdownstream from the channel entrance. The longitudinal �u�and vertical �w� velocity measurements were performedabove the third hill module. To check whether the turbulencewas completely developed, preliminary measurements wereconducted on the second, third, and fourth sections. Thesepreliminary measurements showed that the u statistics ac-quired at four locations �and 10 vertical positions� around thecrest of the second and the fourth hills do not significantlydiffer from their analogous statistics at the crest of the thirdhill �overall R=0.95 and m=1.08, where R is the correlationcoefficient and m is the regression slope�. During the experi-

036601-2 Poggi et al. Phys. Fluids 19, 036601 �2007�


ment planning phase, we conducted model calculations usingfirst order closure principles and found that the mean u and wvariations above the third hill do not differ from their coun-terpart above the fourth hill �overall R=0.85 and m=1.18�.

The velocity measurements were carried out using atwo-component Laser Doppler Anemometry �LDA� operatedin a forward scattering mode. A key advantage of LDA is itsnonintrusive nature and its small averaging volume. Twoseparated manual traverse systems were used to position theLDA apparatus and the receiving optics, allowing a measure-ment precision on the order of ±0.025 mm. The measure-ments were performed at ten positions to longitudinallycover one hill module, and at 0.40 m from the lateral wall inthe spanwise direction. These measurements were performedalong a large number of vertical positions ��35� displacedalong a specified coordinate. When modeling or measuringthe flow over complex surfaces, the reference coordinate sys-tems can be externally imposed �e.g., rectangular Cartesiansystems�, related to the geometry of the surface �e.g., terrainfollowing systems�, or allowed to adjust according to theflow dynamics �e.g., streamline coordinates�. As discussed inFB04,17,39 the latter choice is the most preferred for flowover hills and is adopted here. For our experiment, the hillshape function is a cosine surface with respect to the rectan-gular coordinate system �X and Z� resulting in vertical andlongitudinal streamline coordinates, z and x, given by

x = X + H/2 sin�kX�e−kZ; �1�

z = Z − H/2 cos�kX�e−kZ. �2�

The key advantage of a displaced coordinate system isthat 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 theappropriate regions.

We sampled 45 cm �of the 60 cm water level� in thedisplaced vertical direction. We concentrated on the verticalmeasurement array close to the ground so as to zoom into theinner layer dynamics. The vertical arrangement of each mea-surement point was chosen to have an approximate logarith-mic vertical distribution. To simplify the acquisition proce-dure, the inclination of the two velocity components waskept constant and equal to the slope of the surface. However,the measurement path from the surface follows the displacedcoordinate system. Numerical postprocessing of the acquireddata was then carried out to readjust the two components ofthe velocity with the theoretical coordinates thereby permit-ting direct comparisons between theory and data.

The water depth was retained at a mean steady statevalue of 60 cm throughout the experiment. We are aware thatthe ratio between a 60 cm water depth and a 90 cm channelwidth does not guarantee that the lateral walls do not influ-ence the statistics of u and w. We carried out a sensitivityanalysis in which the velocity statistics were acquired at sev-eral spanwise positions �ranging from 20 to 40 cm from theside wall� within the inner region and we found no signifi-cant difference between them for the first and second mo-ments. Therefore, for the purposes of modelling the meanflow in the context of JH75, the effects of the lateral wallswere neglected.

One technical challenge to quantifying the flow statisticswithin the inner layer was the use of LDA to acquire w nearthe ground surface. Tilting the laser beams cannot be em-ployed because such tilting leads to inadequate spatial co-

FIG. 1. �Color online� Experimental setup: The test sec-tion along with the train of hills �top right�. The defini-tion of the displaced coordinate system �x ,z� in relationto the inner layer height �hi� and hill dimensions �H ,L�is also shown for reference.

036601-3 An experimental investigation of turbulent flows Phys. Fluids 19, 036601 �2007�


location between the horizontal and vertical velocity. Toavoid such a problem, we used a technique previously devel-oped and tested in this flume.40 A series of ten narrow slits�0.1 mm wide� were made in the stainless steel sheet foreach longitudinal position to permit vertical passage of thelaser. To guarantee the same optical path for every beam, thelowest part of the sinusoidal surface, which is at 0.15 mabove channel bottom, was filled with water. The presence ofwater �instead of air� allows for a decrease in the pressuredifference between the region above and below the thin steelsheet leading to a decrease in stress and vibrations. To avoidbubble formation, a low hydraulic discharge of 2 l / s underthe test section was also imposed. Finally, to reduce everyinterference between the water below and above the test sec-tion, a “pocket” was projected ad hoc and installed below thesteel sheet for each slit. By injecting a fluorescent dye solu-tion �Red Rhodamine� in the fluid under the hill, we con-firmed that no interaction between the fluid below and abovethe test section occurred. As a final check, we compared thew measurements with direct predictions from the continuityequation using the u measurements as input �described later�.

The u�x ,z� and w�x ,z� measurements were conducted atReh=1.5�105 �fully developed turbulence�. The samplingduration for each run was 300 s, which was shown elsewhereto be sufficiently long to ensure convergence of thestatistics.41 The sampling frequency for each run ranged from2500 to 3000 Hz. The analog signals from the processorwere checked by an oscilloscope to verify the Doppler signalquality at every run. The steadiness of the flow during thevery long experimental times �about 8 h� was verified bycontinuously monitoring the flow rate. No artificial seedingof the channel was employed.

The signal processing was performed by two DantecBurst Spectrum Analyzer �BSA� processors. The coincidencemode was used to obtain reliable measurements of the Rey-nolds shear stress. To preserve the correlation coefficient be-tween w and u, all data points not exactly temporally coin-cident were discarded. Further details about the LDAconfiguration and signal processing can be foundelsewhere.40

One limitation of this experimental setup pertains to thesurface roughness. To achieve a sufficiently deep inner re-gion, we reduced the roughness height, which necessarilyimplies that the viscous effects can play a role in a thinregion close to the ground. For our experiments, the rough-ness Reynolds number is Rek=u*ks /�=36, where the mea-sured u*=0.018 m/s, and ks=2 mm. The region in which 4�Rek�60 is classically referred to as dynamically slightlyrough.42 However, the presence of a thin viscous region neednot affect the data-model comparison if we restrict the com-parisons to the inner region well above the viscous sublayer.Hence, in the model-data evaluation, we discarded all themeasurements within this viscous region but retained themwhen graphically presenting the full data sets. In terms ofimpact on the JH75 framework, the presence of a viscousregion necessitates a re-examination of the lower boundarycondition, especially the constant stress assumption, even ifthis region is not included in the data-model comparison. Weelaborate on this issue in the results and discussion section.

III. THEORY

According to Jackson and Hunt,21 the domain above a2D gentle hilly surface can be decomposed into an inner andan outer region. In the outer region, the perturbations asso-ciated with the shear stress gradient induced by the flow overthe hill are of no dynamical significance and the flow can betreated as inviscid. In the inner region, these perturbations inthe shear stress are comparable with inviscid processes.

Belcher and Hunt48 �hereafter referred to as BH93� esti-mate the inner region depth as the solution of

hi

L�

�

ln�hi/z0�, �3�

where z0 is the momentum roughness height �related to ks

through the mean longitudinal profile as z0=ks exp�−8.5kv��,�=2kv

2, and kv=0.4 is the von Karman constant.In the inner layer and in a thin layer just above �i.e., the

middle layer as defined in the Appendix� the pressure pertur-bation induced by a sinusoidal surface can be expressed as

p�x� = − U02H/2k cos�kx� . �4�

Hereafter, tilde denotes solutions for sinusoidal hills.For analytic tractability, it is convenient to decompose

the mean velocity into an unperturbed or background state�i.e., assuming the flow is over flat terrain� and a perturbationinduced by the hill, given by

u�z,x� = Ub�z� + �u�x,z�;

w�z,x� = �w�z,x�; �5�

��x,z� = �b�z� + ��x,z� .

JH75 suggested that, in the case of gentle hills�H /L�1�, �a� the perturbation terms are small compared tothe background terms, and �b� the nonlinear terms can beneglected. Based on these assumptions, the mean-momentumequation can be written as

Ub�z��u�x,z�

�x+ �w�x,z�

�Ub�z��z

= −��p�x�

�x+

��x,z��z

;

�6�

the robustness of these assumptions will be tested in the nextsection. Moreover, the above equations imply that the back-ground velocity plays an important role in the estimation ofthe perturbation terms. Hence, an unambiguous definition forUb becomes critical.

Instead of writing a budget equation for ��x ,z�, JH75introduced another controversial assumption: the eddy vis-cosity closure with a linear mixing length throughout theinner layer.47 With this assumption, the equation for the per-turbed Reynolds shear stress can be written as a function of�u�x ,z� using

��x,z� = 2kvu*z��u�x,z�

�z, �7�

where the logarithmic mean velocity profile for the back-ground velocity was used �i.e., Ub�z�=u* /kv log�z /z0��.



While this background velocity assumption may be reason-able for an isolated hill, its validity for more complex terrain�e.g., a train of hills� remains to be investigated.

For the particular case of a flow above a sinusoidal hill,the solution of �u�x ,z� is

�u�x,z� = −u*

kvUI0��1 + ln

hi2

z0z�cos�kx� − Re�4B0e−ikx� ,

�8�

where UI0 is the dimensionless scaling for the streamwisevelocity in the inner region, UI0= �Hk /2��U0

2 /UI2�, UI is the

background velocity defined at the inner region height�Ub�hi�=UI�, B0=K0�2ikLz /hi�, and K0�.� is the modifiedBessel function of the zeroth order. Likewise, the perturba-tion induced by a sinusoidal hill on the vertical velocity inreal space can be evaluated as

�w�x,z� = − kv�u*UI0z

hisin�kx� . �9�

Following this derivation, it is clear why the JH75 is alogical starting point to investigate the connections betweentopographic variability and mean flow variability. In theJH75 framework, topographic variability produce pressureperturbations that induce mean flow perturbations, which inturn require that the turbulence adjusts to these perturbationsvia the turbulent stress gradients. The model simplificationsfor linking topographic variability to the pressure perturba-tions and how the turbulent stresses adjust to the mean flowperturbations �i.e., K-theory� will be experimentally exploredin greater detail here. We note that these model simplifica-tions often go beyond the JH75 framework and are applied toalmost all closure approaches in Reynolds-Averaged Navier-Stokes �RANS� approaches. The additional simplification inJH75 �beyond standard first order closure principles� is thelinearization of the advective acceleration term, which weexplore in greater depth as well using the data.

From the dynamics perspective, the JH75 solution re-veals several testable hypotheses about the phase relation-ships between the velocity perturbations and the hill shapefunction. For example, while �u�x ,z� includes a term inphase with the hill surface and a term that depends on theimaginary and real part of B0, �w�x ,z� is always in phasewith the pressure gradient and hence out-of-phase with thehill surface. Moreover, B0 tends to zero exponentially farabove the surface leading to a �u�x ,z� in-phase with thesurface and out-of-phase with ��p�x� /�x for large z. Further-more, the solution for the perturbed shear stress above asinusoidal hill is given by

��x,z� = 2u*2UI0�cos kx − Re� ikA˜1�k�B1

hiik/ze−ikx� , �10�

where B1=K1�2ikLz /hi�, and K1�.� is the modified Besselfunction of the first kind. Above the surface, similar to�u�x ,z�, ��x ,z� tends �exponentially� to be in-phase withthe surface and out-of-phase with ��p�x� /�x. But beforetesting these hypotheses and simplifications, it is imperative

to address the background velocity formulation for a train ofgentle hills first.

IV. RESULTS

A. The background velocity and the velocityperturbations

Background velocity: Recall that while the definition ofUb�z� is unambiguous for an isolated hill, being the upstreammean longitudinal velocity profile, its definition remains am-biguous for complex terrain.39,43 For example, for a train ofridges, the upstream velocity profile colliding with the nthhill is also a function of the previous nth−1 hill. When ex-perimentally studying periodically varying flows, it is plau-sible to assume that Ub�z� can be determined from the meanvelocity averaged over the hill wavelength. On the otherhand, it is theoretically desirable that Ub only be determinedfrom flow over flat terrain because �1� these profiles are in-dependent of the hill configuration �except through a surfaceroughness� and they have been tested in numerous experi-ments, and �2� the analytical approach in JH75 a priori as-sumes a logarithmic shape for Ub, known to be the mostrobust parametrization of the mean velocity profile for flatboundary layers. A logical question to explore is whether thespatially averaged velocity profile, strictly determined fromthe data here, is related to the velocity profile determinedfrom “flat-world” formulations. If the two Ub estimates col-lapse, then the ambiguity in the definition of Ub is not criticalfor our purposes.44,45 This question is explored next in Fig. 2.

Figure 2 shows all ten measured longitudinal velocityprofiles across the hill together with the Ub�z� determinedfrom spatial averaging and its best fit logarithmic velocityprofile �given by Ul�z�=u* /kv log�z /z0��. In Ul�z�, u*

=0.018 m s−1 was evaluated using the maximum value of�b�z�, and the roughness length z0=0.062 mm was computedby minimizing the root-mean squared error between Ub�z�estimated from spatial averaging and Ul�z�. The comparisonbetween Ub�z� and Ul�z� suggests that the background veloc-ity retains its logarithmic shape.

The ten shear stress profiles measured along the hill arealso presented in Fig. 2 together with the background shearstress, �b�z�. Within the inner region, it is clear that �b�z� isapproximately constant in most of the inner region consistentwith the logarithmic background velocity profile assumption.The combination of the logarithmic Ub�z� and constant �b�z�already hints that the eddy viscosity model, a major assump-tion of the JH75 theory, may be reasonable in the inner re-gion.

Furthermore, Fig. 2 shows a thin region close to theground where �b�z� decreases toward the origin. This behav-ior is often used as a robust identification of the viscoussublayer.46 Here, this layer is limited to a thin region �z /hi

�0.2, where hi=0.08 m is the inner layer depth� and onlyaffects the first two or three sampling points. Recall that theJH75 solution to the mean longitudinal momentum balanceneglects viscous effects altogether. However, the presence ofa thin viscous subregion does not alter the formulations pro-posed in JH75. The JH75 model admittedly assumes a



completely rough surface as a bottom boundary condition,but this assumption was only invoked to force the turbulentshear stress to be constant near the wall. In most of the lowerlevels of the inner region of this experiment �but above thethin viscous sublayer�, the shear stress appears to be approxi-mately constant with respect to z �see Fig. 2�.

The measured 2D spatial structure of the velocity com-ponents and Reynolds stress: Having defined the Ub�z� and�b�z� profiles, we show in Fig. 3 the measured 2D spatialstructure of the mean flow perturbations induced by the hillin the inner region. We focus on three discernible zoneswithin the inner region of Fig. 3: upwind, summit, and wake.

FIG. 2. �Color online� The measuredlongitudinal mean velocity u�x ,z� andturbulent shear stress ��x ,z� across thehill. Top panel: Color map of the 2Dstructure of the measured u�x ,z� alongwith the hill surface and the inner re-gion boundary �dotted-dashed�.Bottom-left panel: The profiles of thenormalized u at the 10 sections �s1–s10� marked in the top-left figure �ver-tical lines�. The logarithmic profile�dotted-dashed line� and the spatiallyaveraged values �open triangles� usedto define Ub are also shown. The ve-locity is normalized by Uh=Ub�hi�.Bottom-right panel: Same as bottom-left but for the shear stress. The nor-malizing variable is the maximumhorizontally averaged u*.

FIG. 3. The 2D structure of the flowperturbations around the backgroundstate: Top panel is �u�x ,z�, middlepanel is �w�x ,z�, and bottom panel is��x ,z�. The inner layer depth is alsoshown for reference �dotted-dashed�.



In the upwind zone, �u�x ,z� progressively increaseswith increasing x. Initially, this increase is confined to a zonevery close to the wall �at x /L=−1.5� but extends to the wholeinner layer as the summit of the hill is approached �x /L=0�. Here, �w�x ,z� remains slightly negative throughout.Also, a decrease in ��x ,z� is noted.

Just after the summit, the two noticeable features are adecrease in �u�x ,z� compared to its peak value in the up-wind region, and a reversal in sign of ��x ,z�. Here,�w�x ,z� remains small in magnitude.

In the so-called wake region, a strong velocity deficitthat extends throughout hi from x /L=1 to the middle of theupwind side �x /L=2.5� is evident. In this region, the magni-tude of the Reynolds stress is enhanced, while �w�x ,z� isslightly negative. Next, we compare the measured individualvelocity and stress profiles with the JH75 model results andfurther explore their assumptions and simplifications.

B. The analytical model

Closure assumption: Before presenting the comparisonbetween measured and modelled flow statistics, we first ana-lyze the basic assumptions used in the previous sections toderive the JH75 analytical model. The most controversialassumption used in JH75 is the eddy viscosity closure with alinear mixing length throughout the inner layer.47 In Figs. 4,we show the mixing length determined from measuredu�w��x ,z� and the gradients in u�x ,z�. We found that a linearmixing length �=kvz� agrees reasonably well with the mea-surements. Nevertheless, a minor disagreement between themodelled and the measured mixing length is noticeable onthe lee side of the hill where l=kvz tends to overestimate themeasured mixing length.

Another simplification in the JH75 model is the linear-ization of the advective acceleration terms. We compared themeasured nonlinear advective terms �u�u /�x+w�u /�z� withtheir linearized counterpart �Ub��u /�x+�w�Ub /�z� in Fig.5�a�. It is clear from Fig. 5�a� that the linearization proposedby JH75 is quite reasonable �R=0.997 and m=1.01, where Ris the correlation coefficient and m is the regression slope, asbefore�.

Also, the data permitted us to further explore other sim-plifying assumptions regarding the advection in the longitu-dinal momentum balance. We found that the measuredUb��u /�x explains more than 90% of the measured �nonlin-ear� advective acceleration term as shown in Fig. 5�b� �R=0.992, m=1.09�.

Other assumptions in the JH75 model such as neglecting��p /�z and �u�u� /�x can be a source of error in modelingthe velocity statistics. To explore this point further, we con-ducted a separate experiment on the variation of p at the hillsurface and found that p�x� is well approximated by a localhydrostatic assumption as we moved along the hill �data notshown here�. Furthermore, we conducted preliminary analy-sis on �u�u� /�x and found it to be small �in a global sense�when compared to �p /�x �data not shown�.

Comparison between measured and modelled flow vari-ables: We first present the overall comparisons between mea-sured and modelled u, w, and � in Fig. 6 followed by adetailed analysis of the spatial structure of the error in �u,�w, and �� in Figs. 7–9. From Fig. 6, the best agreement interms of regression slope m and correlation coefficient R isfor u �R=0.98, m=1.11�, followed by �w �R=0.84, m

FIG. 4. �Color online� Profile comparisons between kvz �solid line� and theeffective mixing length l�z� estimated from the measured u�w� and �u /�zusing the eddy-viscosity model �circle-line� across the hill.

FIG. 5. Top panel: Comparisons between the nonlinear and the linearizedadvective acceleration terms computed from the velocity measurementsshown in Figs. 3 and 4. Bottom panel: Comparison between the measurednonlinear advective acceleration term and the measured linearized advectiveacceleration assuming w=0 throughout. The 1:1 line is shown in bothpanels.



=0.96�, and the worst agreement is for �� R=0.77, m=0.74�. What is clear from this comparison is that the modelbiases, for all three variables, are not entirely random.

To explore the 2D structure of the differences betweenmodel and measurements across the hill, we show the pro-files of the perturbed flow statistics along the vertically dis-placed coordinate �z� at ten stations uniformly spaced in thelongitudinal direction �from s1 at x /L=0 to s10 at x /L=3.64� and their 2D spatial structure.

Figure 7 �s1–s10� shows the measured ��u�x ,z�� andmodelled ��u�x ,z�� velocity perturbation for the ten verticalprofiles. The comparison between �u�x ,z� and �u�x ,z�shows that the model is capable of describing the main fea-tures of the 2D perturbed longitudinal velocity within thethree zones, at least in terms of signs and order of magnitudeof �u�x ,z�. However, the model clearly underestimates theextremes.

Figure 8 �s1–s10� shows the profile comparison between�w�x ,z� and �w�x ,z�. From Fig. 8 it is clear that the modelpredicts an approximate linear profile for �w�x ,z� through-out, and that �w�x ,z� captures the basic features of �w�x ,z�.

Nevertheless, some disagreement in the phase behavior isnoticeable from Fig. 8. Given the small magnitude of�w�x ,z� and possible measurement error, we estimated�w�x ,z� from the measured �u�x ,z� using the continuityequation by imposing a zero vertical velocity at the hill sur-face. The comparison between measured and estimated�w�x ,z� are also presented in Fig. 8. The difference betweenthese two estimates is not different from the model-data dif-

FIG. 6. An overall comparison between measured and modelled velocityand shear stress within the inner layer across the hill: Top panel is for u�x ,z�,middle panel is for w�x ,z�, and bottom panel is for ��x ,z�. The modelledstatistics are evaluated using the background and the perturbed expressions�u is from Eqs. �A15� and Ub, w is from Eq. �A16�, and � is from Eq. �10�and �b. Ub and �b are defined in the text�. The one-to-one line �solid� is alsoshown. Note the small values for w�x ,z� when compared to u�x ,z�.

FIG. 7. �Color online� Comparison between measured �open circle� andmodelled �solid line� longitudinal velocity perturbation profile across the hill�s1–s9 as shown in Fig. 2�.

FIG. 8. �Color online� Same as Fig. 7 but for the vertical velocity. In addi-tion, estimates of �w�x ,z� obtained from the measured �u�x ,z� via thecontinuity equation are also shown �squares�.



ferences, especially on the upwind side of the hill. The larg-est difference between the two estimates of �w�x ,z� and themodel appears to be also on the lee-side of the hill �Secs. IVand V�.

Figure 9 shows the comparison between the profiles ofmeasured and modelled ��x ,z�. From this figure, while��x ,z� is able to reproduce the basic behavior of ��x ,z�,the agreement between the linear model and the data is lessencouraging, especially near the ground. Recall that near theviscous sublayer, the ��x ,z� is approximately zero, whilethe ��x ,z� remains finite because the layer is assumed toreside in a fully developed turbulence zone. If this lowerboundary condition “contamination effect” is minimized bycomparing ��x ,z�−��x ,0� with ��x ,z�, then the agree-ment is significantly improved as shown in Fig. 9.

V. SUMMARY AND CONCLUSIONS

A new data set was collected above a train of gentle hillsto explore experimentally and theoretically the 2D structureof the mean velocity. Our choice of a train of sinusoidal hillshere is a logical progression from isolated hills so as to beginconfronting the problem of flow over complex terrain en-countered in nature. In extending the analysis from an iso-lated hill to a train of hills, several issues must be addressed.The first is that the background velocity is no longer unam-biguously defined and its shape cannot be a priori assumed.Nevertheless, the experiment here suggests that the horizon-tally averaged velocity is a reasonable and unambiguous in-terpretation of the background velocity, Ub, given its loga-rithmic shape. From a prognostic point of view, this resultsuggests that a logarithmic velocity profile can be used, after

defining appropriate u* and z0, as a background velocity topredict the velocity perturbation above gentle terrain. More-over, Ub, retaining a logarithmic profile, permits us to extendthe applicability of analytical theories previously proposedfor isolated hills �e.g., JH75� to more complex, thoughgentle, terrain.

This study is the first to experimentally investigate thelinearization of the advective acceleration terms proposed inJH75 and found it to be reasonable. However, the often citedmain criticism to the application of analytical theories andeven first order closure models to flows over complex terrainremains the eddy viscosity models �or their mixing lengthvariant�. Using the data from this experiment, we found thata linear mixing length describes reasonably well the datathroughout the inner layer. Hence, this comparison suggeststhat eddy-viscosity models may be sufficiently robust inmodelling topographically induced perturbations in the meanflow and turbulent stresses, at least for gentle topography.When these two findings are taken together, it is clear thatapproximations “endogenous” to the flow dynamics in JH75are reasonable. Note that these conclusions go beyond JH75and apply to other approaches such as first �or even higher�order closure models in RANS that employ the mixinglength concept. The linearization findings here also go be-yond the particulars of the JH75 model. For example, thelinearized approximation can permit explicit tracking of howindividual energetic modes in the topography influence themean flow field variability. This finding is particularly timelygiven the availability of high resolution digital elevationmaps from remote sensing products.49

Finally, the experiment here permitted us to explore newsimplifications to the mean longitudinal momentum balancefor a train of hills not previously explored. We found that themeasured variability in Ub��u /�x captures most of the vari-ability in the nonlinear advective acceleration within the in-ner layer. The significance of this simplification is that a newunivariate linear PDE may form the basis for future analyti-cal solutions to the 2D structure of the mean flow field. Thegenerality of this finding remains to be investigated in thefuture for terrain variability possessing more than one ener-getic wave number �or more complex features�.

APPENDIX: REVIEW OF THE LINEAR ANALYTICALMODEL

According to Jackson and Hunt,21 the domain above a2D gentle hilly surface can be decomposed into an inner andan outer region. In the outer region, the perturbations asso-ciated with the shear stress gradient induced by the flow overthe hill are of no dynamical significance and the flow can betreated as inviscid. In the inner region, these perturbations inthe shear stress are comparable with inviscid processes.

The analytical derivation commences with scaling argu-ments reviewed by BH93. BH93 introduced two time scales:the advection-distortion time scale, TD=�L /U�z�, and theLagrangian integral time scale, TL=kvz /u*, and � is a simi-larity constant. TD characterizes the advection and the dis-tortion of turbulent eddies versus the straining motion asso-ciated with perturbations induced by the hill on the mean

FIG. 9. �Color online� Same as Fig. 5 but for the shear stress. Both modelled��x ,z� and modelled ��x ,z�−��x ,0� �i.e. adjusting the lower boundarycondition to assure a zero-stress in the viscous sublayer� are compared tomeasured ��x ,z�. Note the agreement between modelled ��x ,z�−��x ,0� and measured ��x ,z�.



flow. TL characterizes the decorrelation time scale of thelarge energy-containing eddies and the time scale at whichthe turbulence comes into equilibrium with the surroundingmean-flow velocity gradient.50,36 Note that while TD de-creases with increasing z, TL increases. Hence, the inner re-gion depth is often defined by the height z=hi at which thesetwo time scales become equal. BH93 estimated this height asthe solution of

hi

L�

�

ln�hi/z0�. �A1�

Hunt et al.47 introduced two additional layers: the innersurface layer and the middle layer. The inner surface layercorresponds to a thin layer that was added to take into ac-count the surface boundary conditions. The middle layer isthe lowest region of the outer layer and is introduced tomatch inner and outer layer dynamics. In the middle layer,the Reynolds stress gradient, as in the upper part of the outerregion, is small enough to be neglected but the flow is nowrotational. The depth of the middle layer, hm, differs depend-ing on the ratio between L and the ABL height, hBL. Forelongated hills �L�hBL�, hm may be assumed to be equal tothe hBL.51 Otherwise, for short hills �L�hBL�, hm is definedas the solution of

hm = L ln1/2�hm/z0� . �A2�

Having defined the inner and outer layer depths and theirrespective time scale arguments, we proceed to the next scal-ing arguments; the pressure perturbations at the interface be-tween the middle layer and the upper part of the outer layer,pm�x�= p�x ,hm�. The pressure perturbation induced by thesurface can be expressed as a representative magnitude, po,and a dimensionless longitudinal function, �x�, using

pm�x� = po�x� , �A3�

where �x� is a function of the hill shape and po is the forc-ing due to the flow field in the outer region given by po

=Ub2�hm�=U0

2, where Ub�z� is the background velocity �i.e.,the velocity before the hill is encountered�. Here U0 is boththe characteristic velocity in the outer layer and the velocityscale for the pressure perturbation. Again, our interest in theregions above the inner layer are presented because theseregions are necessary to formulate upper boundary condi-tions for the inner layer.

Because the middle layer is thin compared to L �i.e.,hm /L�1�, the pressure perturbations in both the middle andinner layers can be considered constant21,47 with respect to zand given by pm�x�= po�x�. For a two-dimensional hill, �x�can be computed from the hill shape function,52 f�x�. In thecase of sinusoidal hills, where the hill shape function is de-

scribed by f�x�=H /2 cos�kx�, the terms �x� and p�x� be-come

�x� = − H/2k cos�kx� , �A4�

p�x� = po�x� = − U02H/2k cos�kx� . �A5�

To derive the solution for the mean flow, we consider thestationary mean longitudinal momentum equation for a 2Dturbulent flow along with the continuity equation, given by

u�x,z��u�x,z�

�x+ w�x,z�

�u�x,z��z

= −�p�x�

�x+

��x,z��z

;

�u�x,z��x

+�w�x,z�

�z= 0. �A6�

As stated earlier , it is convenient to decompose the ve-locity into an unperturbed or background state �i.e., assumingthe flow is over flat terrain� and a perturbation induced by thehill, given by

u�z,x� = Ub�z� + �u�x,z�;

w�z,x� = �w�z,x�;�A7�

p�x,z� = pb�z� + �p�x,z�;

��x,z� = �b�z� + ��x,z� .

When this decomposition is applied to gentle hills�H /L�1� and assuming that the hill perturbations are smallcompared to the background terms, JH75 derived the linear-ized mean-momentum and continuity equations as

Ub�z��u�x,z�

�x+ �w�x,z�

�Ub�z��z

= −��p�x�

�x+

��x,z��z

;

��u�x,z��x

+��w�x,z�

�z= 0. �A8�

The perturbed quantities are then expanded as a powerseries in a asymptotically small dimensionless parameter given by

�u�x,z� = − po/Ub�h��u�0� + u�1� + ¯ �;

�w�z,x� = − po/Ub�h��w�1� + 2w�2� + ¯ �;�A9�

�p�x,z� = po��0��x� + 2�2��z,x� + ¯ �;

��x,z� = − 2po/Ub2�h��1��x,z� + ��2��x,z�� ,

where all the terms of the power series are dimensionless.Note that both Ub�h� and have to be separately specifiedfor each layer depending on the characteristic length andvelocity scales. Once the appropriate length and velocityscales are determined for each layer, the linear approach canbe employed in the solution. Only terms belonging to thefirst order expansion, linear in , are retained. According toHunt et al.,47 the asymptotically small dimensionless param-eter in Eq. �A9� for the inner layer, , is determined from z0

and hi using =ln−1�hi /z0�. Furthermore, the characteristicvelocity Ub�h� can be approximated using the backgroundvelocity defined at the inner region height �Ub�hi�=UI�. Us-ing these two assumptions, the perturbed velocity quantities�for any 2D hill shape function� reduce to

�u�x,z� = − U02/UI�u�0� + ln−1�hi/z0�u�1��; �A10�



�w�x,z� = − U02/UI�ln−1�hi/z0�w�1��; �A11�

�p�x� = po�0��x� . �A12�

The above equations imply that the background velocityplays an important role in the estimation of the perturbationterms. Hence, an unambiguous definition for Ub becomescritical. Instead of writing the equation for ��x ,z� derivedfrom �A9�, we introduce another controversial assumptionused in JH75—the eddy viscosity closure with a linear mix-ing length throughout the inner layer.47 We note again thatthe JH75 formulation assumes that the turbulent viscosity ismuch larger than the molecular viscosity within the innerlayer so that the total stress term is dominated by the turbu-lent stress. With this assumption, the equation for the per-turbed Reynolds shear stress can be written as a function of�u�x ,z� using

��x,z� = 2kvu*z��u�x,z�

�z, �A13�

where the logarithmic mean velocity profile for the back-ground velocity was used. Upon replacing the previous threeequations in the linearized momentum budget and continuityequations,47 all the unknown terms can be evaluated, albeitin the Fourier domain, using

u�0��k,z� = �0��k�;

u�1��k,z� = �0��k��1 − lnz

hi− 4B0; �A14�

w�1��k,z� = 2�0��k�ikLkv2 z

hi,

where �k� is the Fourier transform of �x�. For the particu-lar case of a flow above a sinusoidal hill, the solution of�u�x ,z� is

�u�x,z� = −u*

kvUI0��1 + ln

hi2

z0z�cos�kx� − Re�4B0e−ikx� ,

�A15�

where UI0 is the dimensionless scaling for the streamwisevelocity in the inner region, UI0= �Hk /2��U0

2 /UI2�. Likewise,

the perturbation induced by a sinusoidal hill on the verticalvelocity in real space can be evaluated as

�w�x,z� = −2HU0

2k2Lkv2

2UIRe�i

z

hi ln�hi/z0�e−ikx �A16�

=− kv�u*UI0z

hisin�kx� . �A17�

1W. M. Gong, P. A. Taylor, and A. Dornbrack, “Turbulent boundary-layerflow over fixed aerodynamically rough two-dimensional sinusoidalwaves,” J. Fluid Mech. 312, 1 �1996�.

2J. D. Hudson, L. Dykhno, and T. J. Hanratty, “Turbulence production inflow over a wavy wall,” Exp. Fluids 20, 257 �1996�.

3P. Cherukat, Y. Na, T. J. Hanratty, and J. B. McLaughlin, “Direct numeri-cal simulation of a fully developed turbulent flow over a wavy wall,”Theor. Comput. Fluid Dyn. 11, 109 �1998�.

4D. S. Henn and R. I. Sykes, “Large-eddy simulation of flow over wavysurfaces,” J. Fluid Mech. 383, 75 �1999�.

5A. Gunther and P. R. von Rohr, “Structure of the temperature field in aflow over heated waves,” Exp. Fluids 33, 920 �2002�.

6Y. S. Chang and A. Scotti, “Entrainment and suspension of sediments intoa turbulent flow over ripples,” J. Turbul. 4, 019 �2003�.

7A. Gunther and P. R. von Rohr, “Large-scale structures in a developedflow over a wavy wall,” J. Fluid Mech. 478, 257 �2003�.

8Y. H. Tseng and J. H. Ferziger, “Large-eddy simulation of turbulent wavyboundary flow-illustration of vortex dynamics,” J. Turbul. 5, 034 �2004�.

9C. Marchioli, V. Armenio, M. V. Salvetti, and A. Soldati, “Mechanisms fordeposition and resuspension of heavy particles in turbulent flow over wavyinterfaces,” Phys. Fluids 18, 025102 �2006�.

10D. W. Bechert, M. Bruse, W. Hage, J. G. T. VanderHoeven, and G. Hoppe,“Experiments on drag-reducing surfaces and their optimization with anadjustable geometry,” J. Fluid Mech. 338, 59 �1997�.

11D. W. Bechert, M. Bruse, and W. Hage, “Experiments with three-dimensional riblets as an idealized model of shark skin,” Exp. Fluids 28,403 �2000�.

12J. C. Kaimal and J. J. Finnigan, Atmospheric Boundary Layer Flows:Their Structure and Measurement �Oxford University Press, New York,1994�.

13D. Baldocchi, J. Finnigan, K. Wilson, K. T. Paw U, and E. Falge, “Onmeasuring net ecosystem carbon exchange over tall vegetation on complexterrain,” Boundary-Layer Meteorol. 96, 257 �2000�.

14M. Aubinet, B. Heinesch, and M. Yarneaux, “Horizontal and vertical CO2

advection in a sloping forest,” Boundary-Layer Meteorol. 108, 397�2003�.

15C. Feigenwinter, C. Bernhofer, and R. Vogt, “The influence of advectionon short term CO2 budget in and above a forest canopy,” Boundary-LayerMeteorol. 113, 201 �2004�.

16M. R. Raupach and J. J. Finnigan, “The influence of topography on me-teorological variables and surface-atmosphere interactions,” J. Hydrol.190, 182 �1997�.

17J. J. Finnigan and S. E. Belcher, “Flow over a hill covered with a plantcanopy,” Q. J. R. Meteorol. Soc. 130, 1 �2004�.

18M. Aubinet, P. Berbigier, C. H. Bernhofer, A. Cescatti, C. Feigenwinter,A. Granier, H. Grunwald, K. Havrankova, B. Heinesch, B. Longdoz, B.Marcolla, L. Montagnani, and P. Sedlak, “Comparing CO2 storage fluxesand advection at night at different carboeuroflux sites,” Boundary-LayerMeteorol. 116, 63 �2005�.

19R. M. Staebler and D. Fitzjarrald, “Observing subcanopy CO2 advection,”Agric. Forest Meteorol. 122, 139 �2004�.

20G. G. Katul, J. J. Finnigan, D. Poggi, R. Leuning, and S. Belcher, “Theinfluence of hilly terrain on canopy-atmosphere carbon dioxide exchange,”Boundary-Layer Meteorol. 118, 189 �2006�.

21P. S. Jackson and J. C. R. Hunt, “Turbulent wind flow over a low hill,” Q.J. R. Meteorol. Soc. 101, 929 �1975�.

22P. J. Mason and R. I. Sykes, “Flow over an isolated hill of moderateslope,” Q. J. R. Meteorol. Soc. 105, 383 �1979�.

23E. F. Bradley, “Experimental-study of the profiles of wind-speed, shearingstress and turbulence at the crest of a large hill,” Q. J. R. Meteorol. Soc.106, 101 �1980�.

24P. J. Mason and J. C. King, “Measurements and predictions of flow andturbulence over an isolated hill of moderate slope,” Q. J. R. Meteorol. Soc.111, 617 �1985�.

25P. A. Taylor and H. W. Teunissen, “The Askervein hill project: Overviewand background data,” Boundary-Layer Meteorol. 39, 15 �1987�.

26J. R. Salmon, A. J. Bowen, A. M. Hoff, R. Johnson, R. E. Mickle, P. A.Taylor, G. Tetzlaff, and J. L. Walmsley, “The Askervein hill project - meanwind variations at fixed heights above ground,” Boundary-Layer Meteorol.43, 247 �1988�.

27J. R. Salmon, H. W. Teunissen, R. E. Mickle, and P. A. Taylor, “Thekettles hill project: Field observations, wind-tunnel simulations andnumerical-model predictions for flow over a low hill,” Boundary-LayerMeteorol. 43, 309 �1988�.

28J. L. Walmsley and P. A. Taylor, “Boundary-layer flow over topography:Impacts of the Askervein study,” Boundary-Layer Meteorol. 78, 291�1996�.

29I. Mammarella, F. Tampieri, M. Tagliazucca, and M. Nardino, “Turbulenceperturbations in the neutrally stratified surface layer due to the interactionof a katabatic flow with a steep ridge,” Env. Fluid Mech. 5 �3�, 227–246�2005�.

30J. J. Finnigan, M. R. Raupach, E. F. Bradley, and G. K. Aldis, “A wind-



tunnel study of turbulent-flow over a 2-dimensional ridge,” Boundary-Layer Meteorol. 50, 277 �1990�.

31W. M. Gong and A. Ibbetson, “A wind-tunnel study of turbulent-flow overmodel hills,” Boundary-Layer Meteorol. 49, 113 �1989�.

32M. Athanassiadou and I. P. Castro, “Neutral flow over a series of roughhills: A laboratory experiment,” Boundary-Layer Meteorol. 101, 1 �2001�.

33A. R. Brown, J. M. Hobson, and N. Wood, “Large-eddy simulation ofneutral turbulent flow over rough sinusoidal ridges,” Boundary-LayerMeteorol. 98, 411 �2001�.

34A. N. Ross, S. Arnold, S. B. Vosper, S. D. Mobbs, N. Dixon, and A. G.Robins, “A comparison of wind tunnel experiments and numerical simu-lations of neutral and stratified flow over a hill,” Boundary-Layer Meteo-rol. 113, 427 �2004�.

35F. Tampieri, I. Mammarella, and A. Maurizi, “Turbulence in complexterrain,” Boundary-Layer Meteorol. 109, 85 �2003�.

36R. E. Britter, J. C. R. Hunt, and K. J. Richards, “Air flow over a two-dimensional hill: Studies of velocity speed-up, roughness effects and tur-bulence,” Q. J. R. Meteorol. Soc. 107, 91 �1981�.

37D. Poggi, G. G. Katul, and J. Albertson, “Moment transfer and turbulentkinetic energy budgets within a dense model canopy,” Boundary-LayerMeteorol. 111-3, 589 �2004�.

38D. Poggi, A. Porporato, L. Ridolfi, G. G. Katul, and J. Albertson, “Theeffect of vegetation density on canopy sublayer turbulence,” Boundary-Layer Meteorol. 111-3, 565 �2004�.

39D. Poggi and G. G. Katul, “The ejection-sweep cycle over bare and for-ested gentle hills: A laboratory experiment,”Boundary-Layer Meteorol.122, 3 �2007�.

40D. Poggi, A. Porporato, and L. Ridolfi, “An experimental contribution to

near-wall measurements by means of a special laser Doppler anemometrytechnique,” Exp. Fluids 32, 366 �2002�.

41D. Poggi, A. Porporato, and L. Ridolfi, “Analysis of the small-scale struc-ture of turbulence on smooth and rough walls,” Phys. Fluids 15, 35�2003�.

42A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics: Mechanicsof Turbulence �MIT Press, Cambridge, 1971�, Vol. 1.

43K. W. Ayotte, “Optimization of upstream profiles in modelled flow overcomplex terrain,” Boundary-Layer Meteorol. 83, 285 �1997�.

44S. Besio, A. Mazzino, and C. F. Ratto, “Local log-law of the wall: Nu-merical evidences and reasons,” Phys. Lett. A 275, 152 �2000�.

45S. Besio, A. Mazzino, and C. F. Ratto, “Local law-of-the-wall in complextopography: A confirmation from wind tunnel experiments,” Phys. Lett. A282, 325 �2001�.

46J. O. Hinze, Turbulence �McGraw-Hill, New York, 1959�.47J. C. R. Hunt, S. Leibovich, and K. J. Richards, “Turbulent shear flows

over low hills,” Q. J. R. Meteorol. Soc. 114, 1435 �1988�.48S. E. Belcher and J. C. R. Hunt, “Turbulent shear-flow over slowly moving

waves,” J. Fluid Mech. 251, 109 �1993�.49M. Lefsky, W. B. Cohen, G. Parker, and D. Harding, “Lidar remote sens-

ing for ecosystem studies,” BioScience 52, 19 �2002�.50H. Tennekes and J. L. Lumley, A First Course in Turbulence �MIT Press,

Cambridge, 1972�, Vol. I.51S. E. Belcher, D. P. Xu, and J. C. R. Hunt, “The response of a turbulent

boundary-layer to arbitrarily distributed 2-dimensional roughnesschanges,” Q. J. R. Meteorol. Soc. 116, 611 �1990�.

52S. E. Belcher and J. C. R. Hunt, “Turbulent flow over hills and waves,”Annu. Rev. Fluid Mech. 30, 507 �1998�.



Documents

An experimental investigation of turbulent ﬂows over a ... · inside tall canopies on hilly terrain.17,20 Hence, this study is motivated, in part, by the clear need for both simpliﬁed