turbulent intensities and velocity spectra for bare and ......turbulent intensities and velocity...

Boundary-Layer Meteorol (2008) 129:25–46DOI 10.1007/s10546-008-9308-8

ORIGINAL PAPER

Turbulent Intensities and Velocity Spectra for Bareand Forested Gentle Hills: Flume Experiments

Davide Poggi · Gabriel G. Katul

Received: 22 October 2007 / Accepted: 21 August 2008 / Published online: 17 September 2008© Springer Science+Business Media B.V. 2008

Abstract To investigate how velocity variances and spectra are modified by thesimultaneous action of topography and canopy, two flume experiments were carried outon a train of gentle cosine hills differing in surface cover. The first experiment was con-ducted above a bare surface while the second experiment was conducted within and above adensely arrayed rod canopy. The velocity variances and spectra from these two experimentswere compared in the middle, inner, and near-surface layers. In the middle layer, and forthe canopy surface, longitudinal and vertical velocity variances (σ 2

u , σ 2w) were in phase with

the hill-induced spatial mean velocity perturbation (�u) around the so-called backgroundstate (taken here as the longitudinal mean at a given height) as predicted by rapid distortiontheory (RDT). However, for the bare surface case, σ 2

u and σ 2w remained out of phase with �u

by about L/2, where L is the hill half-length. In the canopy layer, wake production was asignificant source of turbulent energy for σ 2

w, and its action was to re-align velocity varianceswith �u in those layers, a mechanism completely absent for the bare surface case. Such alower ‘boundary condition’ resulted in longitudinal variations of σ 2

w to be nearly in phasewith �u above the canopy surface. In the inner and middle layers, the spectral distortionsby the hill remained significant for the background state of the bare surface case but not forthe canopy surface case. In particular, in the inner and middle layers of the bare surface case,the effective exponents derived from the locally measured power spectra diverged from theirexpected −5/3 value for inertial subrange scales. These departures spatially correlated withthe hill surface. However, for the canopy surface case, the spectral exponents were near −5/3above the canopy though the minor differences from −5/3 were also correlated with the hillsurface. Inside the canopy, wake production and energy short-circuiting resulted in signifi-cant departures from −5/3. These departures from −5/3 also appeared correlated with the

D. Poggi (B)Dipartimento di Idraulica, Trasporti ed Infrastrutture Civili, Politecnico di Torino, Torino, Italye-mail: [email protected]

G. G. KatulNicholas School of the Environment and Earth Sciences, Duke University, Durham, NC, USA

G. G. KatulDepartment of Civil and Environmental Engineering, Duke University, Durham, NC, USA

123

26 D. Poggi, G. G. Katul

hill surface through the wake production contribution and its alignment with �u. Moreover,scales commensurate with Von Karman street vorticies well described wake production scalesinside the canopy, confirming the important role of the mean flow in producing wakes. Thespectra inside the canopy on the lee side of the hill, where a negative mean flow delineateda recirculation zone, suggested that the wake production scales there were ‘broader’ whencompared to their counterpart outside the recirculation zone. Inside the recirculation zone,there was significantly more energy at higher frequencies when compared to regions outsidethe recirculation zone.

Keywords Canopy turbulence · Flow over gentle hills · Rapid distortion theory ·Turbulent intensities · Turbulent spectra · Spectral short-circuiting · Von Karman streets ·Wake production

1 Introduction

Theories and models on how hills, bare or covered by a canopy, modify the mean flow prop-erties have received significant theoretical and experimental attention over the past threedecades. These theories and models have sufficiently matured to be employed in wide rang-ing applications spanning flows in vegetated canopies (Finnigan et al. 1990; Raupach andFinnigan 1995, 1997; Wilson et al. 1998; Van Gorsel et al. 2003; Finnigan 2004; Finnigan andBelcher 2004; Rotach et al. 2004; Katul et al. 2006), to wind-load design and wind energyharvesting (Morfiadakis et al. 1996; Miller and Davenport 1998; Mann 2000; Riziotis andVoutsinas 2000; Bitsuamlak et al. 2004), to improvements in numerical weather prediction(Belcher and Hunt 1998), to air flow over water waves (Belcher and Hunt 1993; Mastenbroeket al. 1996; Belcher and Hunt 1998; Cohen and Belcher 1999), to wind breaks (Finnigan andBradley 1983; Judd et al. 1996), to predictions of sediment movement and corrolary aelo-lian processes (van Boxel et al. 1999), and to particulate matter and chemical deposition incomplex terrain (Hill et al. 1987; Parker and Kinnersley 2004).

While these theories suggest that the mean flow properties are strongly forced by topo-graphy and can exhibit ‘discontinuities’ such as separation or recirculation regions, it is notclear how the turbulence is affected by such discontinuities in the mean flow states. Earlystudies on flow above gentle hills suggest that such mean-flow discontinuities do not lead toconcomitant discontinuities in some of the turbulence moments, at least in the absence of tallcanopies (Finnigan et al. 1990). However, the presence of a canopy on the hill surface canmodify the mean flow significantly, to produce a recirculation zone inside the canopy even ifthe flow above the canopy does not separate (Finnigan and Belcher 2004; Katul et al. 2006;Poggi and Katul 2007c).

The modulation of turbulent energics and their canonical length scales inside canopies oncomplex terrain remains largely an unexplored topic (Poggi and Katul 2007a) though interestin these properties is proliferating, whether be it for improving large-eddy simulation subgridmodels (Walko et al. 1992; Allen and Brown 2002; Stoll and Porte-Agel 2006; Wan et al.2007) or constructing novel closure schemes in Reynolds-averaged Navier-Stokes (RANS)models (Beljaars et al. 1987; Zeman and Jensen 1987; Frank 1996; Ying and Canuto 1996;Apsley and Castro 1997; Eidsvik and Utnes 1997; Kim et al. 1997; Ying and Canuto 1997;Ross et al. 2004; Bitsuamlak et al. 2006; Katul et al. 2006). One complication stems from thefact that the canopy alters the energy spectrum of turbulence by short-circuiting the energycascade and injecting energy by wake production (Finnigan 2000; Poggi et al. 2004c; Poggiand Katul 2006; Cava and Katul 2008). How the hills modulate these spectral attributes has

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Turbulent Intensities and Velocity Spectra for Bare and Forested Gentle Hills 27

not been methodically studied. Progress on this problem is also hindered by the absence ofdetailed experiments on velocity spectra for flows inside canopies on gentle hills.

Because the interplay between canopy processes and topographic variability is difficultto separate, two flume experiments were carried out. The first experiment considered a trainof gentle hills covered with a bare surface, while the second included a vegetated surfacecomposed of densely arrayed rods but with topopgraphic variations being identical to thefirst experiment. The rod canopy experiment was designed such that topographic variationswere comparable to the canopy height, and the canopy density was sufficiently large sothat within-canopy local advection and interactive pressure effects with the outer layer arenon-negligible. The geometery of the train of hills, the flow rate, and the mean water depthwere all identical in these two experiments. Laser doppler anemometry (LDA) was used tocollect high frequency velocity time series, and the velocity statistics from these two exper-iments were then compared within the various layers above the hill surface. Naturally, otherprocesses such as unsteadiness in the mean flow conditions, density gradients, variabilityin canopy morphology (both horizontal and vertical), steep slopes and multiple modes ofterrain variability do not permit the immediate extrapolation of these experiments to fieldconditions; however, exploring all of these effects simultaneously is well beyond the scopeof a single study. Hence, the compass of this work is restricted to how the longitudinal andvertical velocity variances, as well as their spectra, are modified by the simultaneous actionof the hill and the canopy.

2 Experimental Set-up

Much of the experimental set-up is described elsewhere (Poggi and Katul 2007a,b,c; Poggiet al. 2007); however, an overview of the salient features are provided below for complete-ness. The two experiments were conducted in a recirculating channel that is 18 m long,0.90 m wide, and 1 m deep with sides made of glass to permit optical access (Fig. 1). Thewater depth (hw) was maintained constant at 0.6 m, and the steadiness of the recirculatingflow rate (Qr = 120 l s−1) was verified by continuous monitoring. As earlier noted, the twoflume experiments were conducted for a train of four hill modules, one with a bare surfacewhile the other was covered with a densely arrayed rod canopy. The geometery of the trainof hills, the flow rate, and the mean water depth were identical in these two experiments.

2.1 Topography

In both experiments, the topography was constructed using a wavy stainless steel wall com-posed of four modules, each representing a cosine hill with a shape function given by

f (x) = H

2cos(β X), (1)

where X is the longitudinal distance, H (= 0.08 m) is the hill height, β = π/(2L) is the hillwavelength, and L ( = 0.8 m) is the hill half-length (Fig. 1). Hence, with H/L � 1, the hillcan be classified as gentle. The 5-m spacing between the flume entrance and the test sectionwas equipped with a honeycomb having a density that varied with the distance from the wallso as to impose an initial mean velocity profile similar to that expected in a fully developedboundary layer. Moreover, a 4-m flat surface between the first hill section and the channelentrance was covered with a dense canopy identical in height and density to the canopy on thehills. The longitudinal (u) and vertical (w) velocity time series were measured above the third

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L 4 L

z, Z

Flow direction

Inner Region

Canopy Sub-Region

Open channel

Xx

x = const

0

h

Hh

i

-Hc

z = 0

z = -Hc

z = -hi

Bare Surface Canopy

Fig. 1 Top: Lateral view of the flume facility showing the test section. Middle: The definition of the displacedcoordinate system (x; z) in relation to the canopy height (Hc) and hill dimensions (H ; L). The inner-layerheight (hi ) is also shown for reference. Bottom: photographs of the hill modules in the absence of a canopyand the model canopy used

hill module in both experiments. To check whether the turbulence was completely developed,preliminary measurements were conducted on the second, third, and fourth sections. Theseshowed that the u statistics acquired at four locations (and 10 vertical positions) around thecrest of the second and the fourth hills did not significantly differ from their analogous sta-tistics at the crest of the third hill for both surface covers. Furthermore, first-order closuremodel runs were conducted and they too demonstrated that the mean flow patterns on the

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third hill did not significantly differ from the mean flow patterns observed beyond the thirdhill (Poggi et al. 2007).

2.2 The Bare Surface

In contrast to the canopy case, one main limitation of the bare surface set-up pertains to thegeneration of an effective surface roughness necessary to achieve a sufficiently deep innerregion for meaningful comparison with the canopy case. To achieve a sufficiently deep innerregion, the roughness height must be small, which necessary implies that viscosity (ν) playsa role in a thin region close to the ground. For the experiments here, the roughness Reynoldsnumber Rek = u∗ ks/ν = 36, where the measured friction velocity u∗ = 0.018 m s−1, andks = 2 mm. The region in which 4 < Rek < 60 is classically referred to as dynamicallyslightly rough. However, the presence of a thin viscous region need not affect the analysisif the flow comparisons are restricted to the inner region well above the viscous sub-layer.Detailed analysis reported elsewhere (Poggi et al. 2007) demonstrated that this viscous regionwas sufficiently thin, and inner layer mean flow dynamics can thus be successfully explainedusing classical analytical theories. These assume a completely rough surface (Jackson andHunt 1975) provided variations in the ground shear stress are accounted for.

2.3 The Canopy

The canopy was composed of an array of vertical stainless steel cylinders (rods) having adiameter of 4 mm ( = dr ), and a mean height of 0.1 m ( = Hc); these were arc-welded intostainless steel sheets at equal spacing along the 12.8 m long and 0.9 m wide test section. Therod density (n) was 675 rods m−2, which was shown to be sufficiently dense to produce a dragcoefficient (Cd ≈ 0.2) typically observed in dense forest canopies (Poggi et al. 2004c). Also,at such a high rod density, dispersive stresses were shown to be negligible for the flat-terraincase (Poggi et al. 2004b). Hence, changes in the total stresses longitudinally can be assumedto originate entirely from turbulent stresses (and partly from viscous stresses close to theground for the bare surface case) but not dispersive stresses. The vertical distribution of therod frontal area was not constant with height but was concentrated in the top third and almostconstant below (see Fig. 1) resembling a hardwood canopy at maximum leaf area. For thisconfiguration, the mean adjustment length scale Lc = (Cda)−1 ≈ (Cdndr )

−1 is 1.85 m sothat the hill can be classified as narrow (Lc/L ≤ 1) and the canopy can be classified as dense(Hc/Lc < 2).

Due to the low aspect ratio (flow width to depth was <2), secondary currents in the flumemay be generated. In designing the experimental set-up here, we were confronted with a twindilemma. The aspect ratio had to be very large but, simultaneously, the flow depth had tobe sufficiently large to adequately resolve the flow field inside the canopy sub-layer. Hence,we had to select between two less than optimal solutions, and decided on a low aspect ratiobecause of our interest in exploring the canopy sublayer. Preliminary measurements showedthat, both close to the bottom and in the middle of the channel, the primary source of dragis the canopy resistance, perhaps due to the fact that the canopy is tall enough to dampensecondary currents.

2.4 Velocity Measurements

As earlier noted, the u and w time series were measured using two-component laser Doppleranemometry (LDA). The LDA measurements were performed at 10 positions to

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longitudinally cover one hill module, and along 35 vertical positions displaced along aspecified coordinate system that adjusts for topography, as defined later. This x − z sam-pling plane was located 0.40 m from the lateral wall in the spanwise direction. The sam-pling duration and frequency for each run were 300 s and 2,500–3,000 Hz, respectively, andwere deemed sufficient to ensure convergence of the flow statistics (Poggi et al. 2003). Thevelocity measurements were conducted at fully-developed turbulent flow conditions charac-terized by a bulk Reynolds number Reb > 1.3 × 105 (defined here using hw = 0.6 m andur = Qr/(bwhw) = 0.22 m s−1 as the boundary-layer depth and area-averaged flow speed,respectively).

2.5 Coordinate System and Data Aquisision

To simplify the acquisition procedure, the inclination of the two velocity components waskept constant and equal to the slope of the surface. However, the measurement path from thesurface followed the displaced coordinate system. Numerical post-processing of the acquireddata was then carried out to re-adjust the two components of the velocity with the theoreticalcoordinate system, thereby permitting direct comparisons between theory and observations.While several coordinate systems are possible (e.g. rectangular Cartesian or terrain follow-ing), a streamline coordinate system (hereafter referred to as a displaced coordinate system)that adjusts according to hypothetical mean flow dynamics is preferred. The advantages ofa displaced coordinate system is that it reduces to terrain-following near the ground and torectangular Cartesian well above the hill, hence retaining the advantages of both coordinatesystems in the appropriate regions (Finnigan 1983; Finnigan and Belcher 2004). The datapost-processing employed this displaced coordinate system; with respect to a rectangularcoordinate system (X and Z ), the displaced coordinates (x and z) are approximately givenby

x ≈ X + H

2sin(β X) e−k Z , (2a)

z ≈ Z − H

2cos(β X) e−k Z , (2b)

where H , L , and β are defined as before. This approximation is valid if H/L is small (as isthe case here). Also, note that within-canopy flows are bounded by 0 > z > −Hc so that thetop of the canopy is at z = 0.

2.6 Data Representation

The influence of topographic variations on the mean and turbulent flow statistics were quanti-fied by decomposing the flow variables into an unperturbed background state and a perturbedstate produced by topographic variations. This decomposition was employed because thebackground state is highly inhomogeneous in the vertical direction thereby masking smallervariations produced by the hill. The background state is not uniquely defined when dealingwith flows over complex terrain and a number of possibilities exist. From a Reynolds decom-position perspective, a background state that is defined as the average along a hill wavelengthis preferred because it ensures that the spatial mean of the hill-induced mean velocity per-turbations are identical to zero. More common to models of isolated hills is a backgroundstate referenced to the undisturbed upstream conditions well before the flow collides with thehill. This definition is unambiguous in the case of an isolated hill, but remains ambiguous incomplex terrain (Ayotte 1997; Pellegrini and Bodstein 2005). Based on a previous analysis

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(Poggi and Katul 2007b; Poggi et al. 2007), the former definition of a background state isadopted here, and hereafter, the measured mean flow speed U is decomposed as

U (x, z) = Ub(z) + �u(x, z), (3)

where the background state is defined as the spatial average along the third-hill wavelength.Hereafter, subscript b indicates a background state derived from planar averaging over one-hill wavelength.

3 Scaling Analysis and Dynamical Regions

Early analysis of boundary-layer flows over gentle hills decomposed the boundary layer intotwo distinct regions, known as the outer and inner regions (Jackson and Hunt 1975; Belcherand Hunt 1998), hereafter referred to as JH75. These two regions emerge from time scalearguments associated with the relative adjustment of the mean and turbulent flows to topo-graphic perturbations. In particular, the mean distortion (TD) and the Lagrangian integral(TL) time scales are often employed to explore how the mean flow and turbulence adjustto topographic variations within these two regions. The TD characterizes the distortion ofturbulent eddies due to the straining motion associated with the spatial variability in the meanflow caused by the hill, and represents the characteristic time that the mean flow field needsto stretch and destroy large eddies through work done by advection against the mean spatialvelocity gradients. An estimate of TD can be obtained from

TD = L

U= L

Ub + �u= L

Ub

(1 + �u

Ub

)−1

≈ L

Ub

(1 − �u

Ub

). (4)

On the other hand, TL characterizes the time scale of classical turbulent stretching (or relax-ation) of large eddies due to the action of a local mean-flow velocity gradient. Stated differ-ently, TL is the time that turbulent fluctuations need to reach equilibrium with the local meanvelocity gradient, and this can be estimated from

TL = u2∗ε

= u2∗u′w′ ∂U

∂z

= u2∗u3∗kv z

= kvz

u∗= kvz(

1 + �u∗u∗b

) ≈ kvz

u∗b

(1 − �u∗

u∗b

)

≈ kvz

u∗b

(1 + Kt,b

u∗b

∂�u

∂z

), (5)

where ε is the mean turbulent kinetic energy dissipation rate (assumed to balance shear pro-duction for scaling purposes), Kt is the turbulent diffusivity, and kv = 0.4 is the Von Karmanconstant. Note from Eqs. 4 and 5 that, while TD scales with �u, TL scales with ∂�u/∂z. Theratio TD/TL can be used to separate outer from inner regions, as briefly discussed below.

3.1 Outer Region

In the region where TD/TL � 1, local stretching of large eddies is much slower than thedistortion due to advection. This region is called the rapid-distortion region or the outerregion and is characterized by flow dynamics governed by the balance between advectionand the pressure gradient terms, with turbulent stresses playing a minor role. The turbulentflow is rapidly distorted and a direct proportionality between the hill shape and the flowstatistics can be assumed. These assumptions form the basis of the so-called rapid distortion

123


theory (RDT). Spatially, the outer region is defined for z > hm , where hm is known as themiddle-layer depth estimated by solving

hm

L∼

[ln

(hm

zo

)]1/2

, (6)

where zo is the aerodynamic roughness length (≈ ks/30). The statistics in this region areexpected to scale with the local velocity perturbation �u/Ub as predicted from RDT and thetime scale estimate in Eq. 4. This scaling will be used to analyze the second-order statisticsmeasured in this region.

3.2 Inner Region

When TD/TL � 1, the local stretching of large eddies is fast enough to compete with thedistortion due to the mean flow advection. This region is called the local equilibrium regionor inner region because the local eddies relax to equilibrium with the local mean velocitygradient before spatial advection can transport and stretch them. Because of this equilibriumin the inner region, K-theory can be used to predict perturbations in the turbulent stressesor �u∗, even if K-theory fails to describe the background state. Spatially, the inner region isdefined for z < hi , where hi is known as the inner layer depth estimated by solving

hi

L� 2k2

v

ln(

hiz0

) . (7)

In this region, the longitudinal variations of the local stastics are expected to scale with�u∗ = −Kt∂�u/∂z; this scaling will be also used to analyze the second-order statisticsmeasured in the inner region.

The so-called middle region, situated between the inner and outer regions, is generallyassumed to be inviscid but rotational. In this layer, the mean flow undergoes a transition fromits equilibrium state in the inner region to being rapidly distorted by advection in the outerregion.

3.3 Canopy Region

The presence of a canopy sub-layer modifies the inner layer arguments in JH75 by alteringthe no-slip lower boundary condition and by introducing the drag term in the mean momen-tum budget equation. Hence, the canopy region adds two new length scales to the meanmomentum balance; namely, Lc and Hc. Here, the second-order statistics and the spectra inparticular are complicated by numerous factors (e.g. wake production and short-circuiting ofthe cascade) that prevent an a priori velocity scale to be defined strictly based on mean flow(or shear stress) considerations.

3.4 Mean Flow Considerations in the Three Layers

Figure 2a shows measured �u(x, z) for the bare surface case together with computed hi andhm . Within the middle layer (i.e. hi < z < hm), an overspeeding region near the hilltop(on the windward side) and a deceleration region on the lee side are discernable, consis-tent with numerous experiments and analytical theories (Finnigan et al. 1990; Kaimal andFinni-gan 1994; Raupach and Finnigan 1997). These overspeeding and deceleration zones

123


Fig. 2 Top: The spatial variation of the mean velocity for the bare surface case, �u(x, z), shown as a colourmap along with the delineation of hi (dot-dashed line) and hm (dashed line). Middle: Same as top but forσ 2

u (x, z)/σ 2u,b(z). Bottom: Same as top but for σ 2

w(x, z)/σ 2w,b(z)

are amplified in the inner layer (z < hi ) and are almost in phase with the hill surface. Furtherdetails about the comparison between analytical models and mean velocity perturbationsare reported elsewhere for both bare (Poggi et al. 2007; Poggi et al. 2008b) and forested(Poggi et al. 2008b) surfaces. Figure 3a shows the spatial patterns of �u(x, z) for the for-ested surface case so as to contrast their features with Fig. 2a. Here, instead of normalizingwith hi , the canopy height is chosen as the characteristic length scale. While the spatialpatterns in Figs. 2a and 3a share some similarities in the middle layer, the regions near thesurface are considered first. Just above the canopy, and near the hilltop, an overspeedingregion is observed, within which �u reaches its maximum value. The spatial extent of thisregion is comparable to the region noted within the inner layer above the bare surface hill.Just above the canopy, and near the hill bottom, a slowdown region is also evident and issimilar to its counterpart region within the inner layer above the bare surface case. In thelower canopy region, the most striking feature is the large and near-maximum �u in a thinlayer situated on the upwind and part of the downwind side near the hill surface, which hasbeen explained as a balance between the longitudinal advective term, the pressure gradi-ent term, and the drag force (Poggi and Katul 2007b; Poggi et al. 2008b). On the lee sideof the hill, a recirculation zone has already been reported for this set-up (Poggi and Katul2007c), a region where U < 0 and is delineated with a continuous line in Fig. 3a. Thisregion, predicted to occur because of an interplay between the mean longitudinal pressuregradient and the nonlinear drag force (Finnigan and Belcher 2004), has no analogue in thebare surface case despite the similarity in topographic variability and mean flow conditions.Hence, particular attention is devoted to this region when evaluating the velocity variancesand the spectra.

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Fig. 3 Same as Fig. 2 but for the forested surface case

4 Results

Having reviewed the salient properties of the mean flow in the three layers, results pertinentto the turbulent intensities and the spectra in these layers as influenced by surface cover arepresented next. For the spectra, results are presented separately for the background state andfor the longitudinal variations in the three layers. The spectra in the recirculation zone, asdelineated by the mean flow inside the canopy, is also analyzed separately by contrasting itsshape and energetic scales with spectra outside the recirculation zone.

4.1 Turbulent Intensities

In the outer layer and the upper portion of the middle layer, the linearized RDT predicts areduction in σu and a concomitant increase in σw, given by (Britter et al. 1981; Athanassiadouand Castro 2001),

σ 2u (x, z)

σ 2u,b(z)

= 1 − 4

5

�u(x, z)

Ub(z), (8a)

σ 2w(x, z)

σ 2w,b(z)

= 1 +(

6

5− 2R

5

)�u(x, z)

Ub(z), (8b)

R = σ 2u,b

σ 2w,b

(8c)

where R is the degree of anisotropy existing in the background state (R = 1 is for an iso-tropic flow), which can be determined from measurements. Immediate consequences of thelinearized RDT equations is that σu and σw are in phase with �u. Figure 2b and c present

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0

0.5

1

1.5

2

( σu

(z,x

)/σ b)2

z/Hc=2

0

0.5

1

1.5

2

( σu(z

,x)/

σ b)2

z/Hc=−0.1

0 2 4 6 80

0.5

1

1.5

2

( σu(z

,x)/

σ b)2

x/L

z/Hc=−0.6

0

0.5

1

1.5

2( σ

u(z

,x)/

σ b)2

z/hi=5

0

0.5

1

1.5

2

( σu(z

,x)/

σ b)2

z/hi=3

0 2 4 6 80

0.5

1

1.5

2

( σu(z

,x)/

σ b)2

x/L

z/hi=1

Fig. 4 The comparison between measured and modelled σ 2u (x, z)/σ 2

u,b(z) for the bare (left panels) andforested surface case (right panels). Top panels are for a thin region close to the outer layer, middle panels forregions near the inner layer, and bottom panels for regions close to the surface

measured σ 2u (x, z)/σ 2

u,b(z) and σ 2w(x, z)/σ 2

w,b(z) for the bare hill surface and Fig. 3b and cpresent the same measured quantities but for the forested surface; the spatial behaviour of �uis also shown for reference in Figs. 2a and 3a, for bare and forested surfaces respectively. Inthe forested case, it is clear that as z → hm , the velocity variances become in phase with �u,as predicted by RDT, at least when compared to the bare surface case. For the bare surfacecase, however, these variances appear to be out of phase with �u by L/2.

To illustrate this phase lag quantitatively, Fig. 4 presents a comparison between mea-sured and linearized RDT modelled σ 2

u (x, z)/σ 2u,b(z) for both hill surface covers at three

heights, chosen near the outer layer, near the inner layer, and close to the surface. In Fig. 5the same analysis is repeated for σ 2

w(x, z)/σ 2w,b(z). For the forested case, σ 2

u (x, z)/σ 2u,b(z)

and σ 2w(x, z)/σ 2

w,b(z) appear to be in phase with �u as predicted by RDT when z → hm .However, for the bare surface case, the lag of about L/2 noted earlier persists for both velocitystatistics even near z → hm .

Just above the inner layer, σ 2u (x, z)/σ 2

u,b(z) and σ 2w(x, z)/σ 2

w,b(z) are not expected to bein phase with �u but should begin to scale with the local turbulent stresses (in phase with∂�u(x, z)/∂z as earlier mentioned). Interestingly, surface cover can modify these phaserelationships significantly. Figure 5 suggests that σ 2

w(x, z)/σ 2w,b(z) (and to a lesser extent

σ 2u (x, z)/σ 2

u,b(z) in Fig. 4) remain more in phase with �u for the forested surface when

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0

0.5

1

1.5

2

( σw

(z,x

)/σ b)2

z/Hc=2

0

0.5

1

1.5

2

( σw

(z,x

)/σ b)2

z/Hc=−0.1

0 2 4 6 80

0.5

1

1.5

2

( σw

(z,x

)/σ b)2

x/L

z/Hc=−0.6

0

0.5

1

1.5

2( σ

w(z

,x)/

σ b)2

z/hi=5

0

0.5

1

1.5

2

( σw

(z,x

)/σ b)2

z/hi=3

0 2 4 6 80

0.5

1

1.5

2

( σw

(z,x

)/σ b)2

x/L

z/hi=1

Fig. 5 Same as Fig. 4 but for σ 2w(x, z)/σ 2

w,b(z)

compared to the bare surface case, even as the inner layer is approached. This result appearsproblematic, since linearized RDT predictions should not hold in the inner layer and σ 2

u andσ 2

w should primarily scale with w′u′(x, z). Figure 6 confirms the inner-layer scaling with thelocal shear stress. When σ 2

u (x, z) and σ 2w(x, z) are normalized by w′u′(x, z), an unambigu-

ous collapse in the data occurs except for the canopy layer or the thin layers near the groundfor the bare surface case. The data collapse in Fig. 6 can be used to diagnose the thicknessof the inner layer and perhaps objectively delineate the transition from the inner layer to thecanopy layer.

In the canopy layer, w′u′(x, z) is no longer the only leading term for turbulent productionbecause of other mechanisms such as wake production. Wake production (Wp) by the mean

flow scales as Wp ∼ U3(x, z)/Lc (Finnigan 2000), which can exceed the shear production

w′u′(x, z)dU/dz in some regions of the canopy over flat terrain (Poggi et al. 2004a). Thedominance of wake production inside the canopy can explain why σ 2

w (and to a lesser extentσ 2

u ) might be in phase with �u (Fig. 2b) at least when compared to their bare surface counter-part (Fig. 2a). To illustrate this, it is convenient to consider the linear wake production term,expressed as

Wp,l(x, z) ≈ U 3

Lc=

U 3b

(1 + �u

Ub

)3

Lc∼ U 3

b

Lc

(1 + 3

�u

Ub

), (9)

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−1 0 1 2 30

2

4

6

8

10

σ u2(z

,x)/

uw(z

,x)

Z/H c

−1 0 1 2 30

1

2

3

4

5σ w

(z,x

)2 /uw

(z,x

)

Z/H c

100

1

2

3

4

5

6

7

8σ u2

(z,x

)/uw

(z,x

)

Z/hi

100

1

2

3

4

5

6

7

8

σ w2(z

,x)/

uw(z

,x)

Z/hi

Fig. 6 The ten profiles of σ 2u (x, z) (left panels) and σ 2

w(x, z) (right panels) normalized by the local shearstress w′u′(x, z). Top and bottom panels are for the bare and forested surface cases, respectively

which suggests that wake production is in phase with �u (for small �u/Ub). Shear pro-duction, which is the other main source of turbulent energy inside the canopy, scales with(∂�u/∂z)2. If wake production becomes the dominant source of turbulent energy (as may beanticipated for σ 2

w and to a lesser extent σ 2u inside canopies), its action is to re-align velocity

variances to be in phase with �u in those layers, a mechanism completely absent for thebare surface case. This may partly explain why the phase agreement between �u and σ 2

w

emerges throughout the entire boundary layer over the forested cover when compared to thebare surface cover. However, it should be emphasized that this phase agreement is not linkedwith RDT in the inner or canopy layers. Are there sufficient wake production signatures inthe energy spectra, and how does such wake production affect the scales contributing to σ 2

uand σ 2

w for the forested hill surface?

4.2 Velocity Spectra

Figure 7 shows the background longitudinal and vertical velocity spectra (Eu and Ew) for thetwo surface covers. The background spectra at a given height are computed by longitudinally

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Fig. 7 The velocity spectra (Eu in the left panels and Ew in the right panels) as a function of the wavenumberk normalized by z. The velocity spectra are all reported as frequency weighted (i.e. k E(k)) at all heights. Toppanels are for the bare surface case and bottom panels are for the forested surface case. The computed spectraare vertically shifted to facilitate comparisons

averaging the 10 individual spectra, each of which was computed using Welch’s averagedmodified periodogram method in which a given velocity time series was divided into sixoverlapping sections, de-trended, then windowed using Hanning’s method. The computedspectra are vertically shifted relative to each other to facilitate comparisons. The wavenumberk in Fig. 7 was determined from Taylor’s frozen turbulence hypothesis then normalized byz. The velocity spectra are all reported as frequency weighted (i.e. k E(k)) at each height.The canonical shapes of these spectra for the background state and different regions arecontrasted with their counterparts over flat terrain. Next, variations in the individual spectraat each longitudinal position on the hill are presented.

Middle layer region: For the bare surface case, both background Eu and Ew spectra neverattained their canonical flat boundary-layer shapes. In particular, the power-law exponent ofthe slope was always less steep than −5/3. To the contrary, both component spectra collectedover the forested surface exhibit an extensive −5/3 scaling, suggesting that, for the bare sur-face case, hill-induced distortions persist in the background spectra even at wavenumberscommensurate with inertial subrange scales. For the canopy case, these distortions remain

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restricted to the lower-wavenumber regions of Eu , where large-scale variability exists. The−5/3 scaling appears to hold within the inertial subrange, unlike the bare surface case.

Inner region: For the bare surface case, the Eu exponents become less steep than −5/3within the inertial subrange, and appear to approach −1, as expected at lower wavenumbersfrom flat-terrain experiments and recent LES (Bradshaw 1967; Antonia and Raupach 1993;Katul and Chu 1998; Nikora 1999; Hogstrom et al. 2002; Drobinski et al. 2007). For Ew ,however, these spectra appear to approach a −1 exponent over a narrow range of wavenum-bers, but boundary-layer depth restrictions force this exponent to approach a near-constantvalue (or zero exponent). Recall that the boundary-layer depth is 0.6 m, much smaller thanthe hill half-length (hw/L < 0.2). Moreover, the scale separation between low-wavenumberproduction and its dissipation appears too narrow to permit the onset of the inertial subrangein Ew for the lowest inner layer levels. In these layers, the distance from the surface boundsthe scale at which energy is produced, as expected from flat boundary-layer theory. On thecontrary, the forest surface still exhibits an extensive inertial subrange throughout the innerlayer, given the high porosity of the canopy. To illustrate, recall that the average canopyporosity (n p) for a uniform cylindrical canopy is,

n p = 1 − π

4nd2

r , (10)

where n = 675 rods m−2 and dr = 0.004 m, resulting in n p < 1%. Clearly, such a highporosity cannot impose a significant no-slip at the canopy/inner-layer interface when com-pared to a solid boundary. To sum up, as in the middle layer, the spectral distortions bytopography remain significant in the background of the inner layer for the bare surface casebut not the forested surface case.

Canopy layer: For the canopy layer, the measured background Eu and Ew spectra approachtheir canonical shapes (Finnigan 2000), with a secondary peak at wavenumbers larger thanthe shear production length scales. This secondary peak, already reported for the flat-terraincase for a similar flume set-up (Poggi et al. 2004a,c; Poggi and Katul 2006), is connectedwith wake production and is well described by frequencies commensurate with Von Karmanstreets ( fvs = 5 dr/U ). Figure 8 shows the background Eu and Ew for a large numberof vertical positions within the canopy and up to two times the canopy height. Note thatthe frequency domain is used from this point onwards and Taylor’s hypothesis is no longeremployed, given the recirculation and negative mean velocities. Superimposed on these fig-ures are vertical variations of two primary frequencies—the wake production frequency fvs

and the shear production frequency (∼Ub z−1). From this Figure, it is evident that the sec-ondary peaks are associated with Von Karman vortices, and persist to levels just above thecanopy. More importantly, their overall contribution to Eu and Ew is dynamically differ-ent. Near the ground, the Von Karman vorticity is the main source of energy for Ew (andhence σ 2

w) but not for Eu . For Eu large-scale eddy motion (i.e. motion comparable to theboundary-layer depth) contributes a significant source of energy.

The focus up to this point has been on how topographic perturbations alter the verti-cal variations of the ‘background’ Eu and Ew spectra within the canopy sublayer, and itwas shown that gentle hills do not appreciably modify the background spectra from theircanonical flat-terrain shapes for the dense canopy surface. Nevertheless, Fig. 2a and exper-iments and model runs reported elsewhere (Finnigan and Belcher 2004; Poggi and Katul2007a,b,c; Poggi et al. 2007) demonstrated that the recirculation zone inside the canopy hasa significant impact on some aspects of canopy turbulence. More specifically, it was shownvia flow visualization experiments that this recirculation zone is not characterized by a sta-ble vortex (or rotor-like motion) but appears to be dominated by intermittent positive and

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Fig. 8 The background Eu (left panel) and Ew (right panel) for a number of vertical positions within thecanopy and up to 2 Hc . The vertical variation of the wake production frequency fvs and the shear productionfrequency (∼ Ub z−1) are shown for reference

negative velocity excursions negatively skewed around U < 0. Because the wake produc-tion scales with �u, the re-circulation zone can significantly modify the spectra. Figure 9presents these spectral modifications by comparing Ew inside and above the recirculationzone with Ew elsewhere on the hill. We choose Ew instead of Eu for demonstration becauseit is less sensitive to low frequency energetic eddies (note that the frequency domain is usedfrom this point onwards and Taylor’s hypothesis is no longer employed, given the presenceof recirculation and negative mean velocities). The six panels in Fig. 9 present two spectravertically averaged around the labeled height. One spectrum is for sections inside the recir-culation zone (Sect. 4 to Sect. 7) while the other is for sections outside the recirculationzone (Sect. 1 to Sect. 3 and Sect. 8 to Sect. 10). Above the recirculation zone, these twospectra converge and the recirculation zone has minor effects on Ew . However, within therecirculation zone, the two spectra differ in several ways: (i) inside the recirculation zone,the scales commensurate with wake production are ‘broader’ when compared to outside therecirculation zone, and (ii) inside the recirculation zone, there is significantly more energyat higher frequencies when compared to outside the recirculation zone. Intermittent direc-tional shifts in von Karman vortex generation frequency fvs can lead to spectral broadeningin the recirculation zone. Because there is no sustained mean velocity producing fvs in therecirculation zone, these directional shifts in the velocity inject energy at a broad range offrequencies (at least when compared to cases where there is a significant mean flow speedpresent).

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Fig. 9 The Ew inside and above the recirculation zone (thin line) compared with Ew elsewhere on the hill(thick line). The six panels present the two spectra, inside and outside the recirculation region, verticallyaveraged around the labelled height

5 Discussion

Having presented the background spectra and the spectra inside the recirculation zone, wenow discuss the longitudinal and vertical variations of the locally measured spectra acrossdifferent layers for the two experiments. To ‘compress’ the information from these func-tions into a ‘scalar measure’ that can be readily visualized and diagnosed in two-dimensionalspace, the spectral slope of each time series is selected. Because the spectral slope is a rea-sonable indicator of whether the background state averages out topographic perturbations (asshown earlier), it is a logical choice to assess local departures from canonical spectral shapesexpected over flat terrain (Fig. 10).

Figures 10a and b show the spectral slopes for Eu , clearly changing both vertically andlongitudinally for both surface cover types, and exhibiting spatial coherency. For the baresurface, regions where the spectral scaling is far from −5/3 appears to propagate all theway up to the middle layer as evidenced on the lee side of the hill. Conversely, near the hillsummit, most of the middle layer and outer layer spectral scaling is close to −5/3. In theinner layer, the departure from −5/3 appears to be significant at all locations. Moreover, thelargest departure from −5/3 scaling appears to be on the lee side of the hill, as for the middlelayer. Hence, the spectra over the bare surface are significantly perturbed when compared totheir canonical flat-terrain form and these perturbations are spatially correlated with the hillsurface (and do not average out as evidenced by the background analysis).

For the canopy surface, the scaling in the spectra within the inner region and middle layerappears spatially homogeneous and close to −5/3. The largest excursions from −5/3 scalingare inside the canopy layers, where the effect of short-circuiting and wake production playa significant role, as expected. At these locations, a constant spectral scaling exponent does

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Fig. 10 The spatial patterns of the spectral slopes of Eu for the bare surface case (top panel) and for forestedsurface case (middle panel). In the bottom panel, the spatial pattern of the ratio between the power content atfvs ( = fvs Ew( fvs )) and the power content at large scales (taken as the maximum of f Ew( f )) is shown

not exist because of wake production and energy short-circuiting. Hence, these exponentsmust be viewed as ‘effective’ values, derived by approximating the spectra at small scalesby a single exponent (Poggi et al. 2008a). These effective exponents, while not physicallyrepresenting a cascade in the inertial subrange, provide one metric to assess the importanceof wake production and energy short-circuiting. As shown elsewhere (Poggi et al. 2008a),wake production tends to decrease this ‘effective’ exponent, while energy short-circuitingtends to increase it. As discussed in a number of studies (Finnigan 2000; Poggi and Katul2006; Cava and Katul 2008), the short-circuiting of the energy cascade modifies the standardinertial cascade by an exponential function that decays to unity with increasing wavenum-ber. In a limited range of wavenumbers bounded by shear and wake-producing scales, thisexponential modulation leads to a spectral exponent closer to −7/3. Wake production, onthe other hand, does not lend itself to any form of power-law scaling because it introduces asecondary peak in the spectrum. A power-law fit to this function would yield an exponentialslope flatter than −5/3. With this interpretation in mind, the upwind side of the hill exhibitsslopes less negative than −5/3 inside the canopy (i.e. dominated by wake production), andconversely for the lee side. Recall from Fig. 3a that for the upwind conditions inside thecanopy, the flow speeds up (i.e. wake production is more significant), and conversely on thelee side.

A more rigorous assessment of the importance of wake production referenced to thelarge-scale production can be derived directly from the power content at fvs (= fvs Ew( fvs))in relation to the power content at large scales (taken as the maximum of f Ew( f )). Again,

123


we choose Ew for this analysis for the same reasons as in Fig. 9. Figure 10 shows the spatialbehaviour of this relative energy content ratio at all measurement points. At the upwind sideof the hill in the lower canopy layers, this ratio exceeds unity suggesting that wake productionis important. For the lee side of the hill and in the lower canopy layers, this ratio is below unitysuggesting that wake production is less important. The spatial pattern of this ratio correlateswith the hill shape, as did the effective exponent, thereby independently confirming the roleof wake production.

6 Summary and Conclusions

Recent comparisons of observations with numerical models using RANS (Bitsuamlak et al.2004; Ross et al. 2004) demonstrated that σ 2

u and σ 2w are not well reproduced by existing

higher-order closure schemes, partly motivating the present study. In particular, exploringhow σ 2

u and σ 2w, and more importantly, their spectra are modified by the simultaneous action

of the hill and canopy is a logical first step towards progress on a new generation of RANS clo-sure schemes and possible modifications to their empirical length or time scales. To addressthis objective, two flume experiments were carried out on gentle hills with bare and forestedsurfaces. The observations from these two experiments were then compared in the middle,inner, and near-surface layers. The conclusions from these comparisons are:

(1) In the middle layer, and for a canopy surface, velocity variances were in phase with �uas predicted by rapid distortion theory, at least when compared to the bare surface case,where variances were out of phase with �u by L/2.

(2) In the inner layer, with σ 2u (x, z) and σ 2

w(x, z) normalized by the local shear stresses,an unambiguous ‘collapse’ of data occurred, except for the canopy layer and the thinviscous layer near the ground for the bare surface case. Such a collapse provides adiagnostic method strictly from the data to determine the thickness of the inner layerand transition to the canopy layer. One surprising result that emerged from this analysisis that in the inner layer, σ 2

w(x, z)/σ 2w,b(z) remained in phase with �u for the canopy

surface (at least when compared to the bare surface case). In the canopy layer, wakeproduction was shown to be a significant source of energy for σ 2

w (and to a lesser extentσ 2

u ), and its action was to re-align velocity variances with �u in that layer, a mechanismabsent in the bare surface case. This mechanism provides some explanation as to whyσ 2

w(x, z)/σ 2w,b(z) remained more in phase with �u.

(3) In the inner and middle layers, the distortions of the energy spectrum by the hill remainedsignificant for the background state of the bare surface case but not the canopy surfacecase. The persistence of this distortion implies that the background of the bare sur-face case does not ‘average’ out hill perturbations, and need not follow its flat terraincounterpart.

(4) Detailed analysis of the spectra inside the canopy on the lee side of the hill, wherea recirculation zone was delineated from mean velocity considerations, revealed thatscales comparable with von Karman vorticies were ‘broader’ when compared to theircounterpart outside the recirculation zone. Inside the recirculation zone, there was sig-nificantly more energy at higher frequencies when compared to outside the recirculationzone. Strong directional shifts in velocity perturbations around a near-zero mean leadsto enhanced broadening of fvs . However, these two effects were local to the recircula-tion zone, and the spectra in layers adjacent to the recirculation zone were unaffectedby the recirculation.

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(5) In the inner and middle layers, the effective exponents of the power spectra departedfrom −5/3 and these departures spatially correlate with the hill surface for the baresurface case. However, for the canopy surface case, the exponents were near −5/3above the canopy though the differences from −5/3 also appear correlated with the hillsurface. Inside the canopy, the departures from −5/3 were due to wake production andshort-circuiting of the energy cascade. These departures from −5/3 correlate with thehill surface variations. Because the hill surface affects the longitudinal variation in �u,and because wake production scales with �u, these correlations are expected if wakeproduction is a large source of energy inside the canopy.

More broadly, the phase relationships and the exponents presented here for the two sur-face covers provide benchmark observations for assessing how RANS higher-order closureschemes and large-eddy simulation capture the simultaneous effects of hills and canopy onthe second-order statistics and spectra. Such benchmark data are a necessary first step toprogress on the more difficult cases of multiple modes of terrain variability, density stratifiedflows, and inhomogeneities in the canopy density in both vertical and planar directions.

Acknowledgements This research was supported, in part, by the National Science Foundation (NSF-EAR06-35787 and NSF-EAR-06-28432), the United States-Israel Binational Agricultural Research and Develop-ment (BARD, Research Grant No. IS3861-06), and the US Department of Energy (DOE) through the officeof Biological and Environmental Research (BER) Terrestrial Carbon Processes (TCP) program (Grants #10509-0152, DE-FG02-00ER53015, and DE-FG02-95ER62083).

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turbulent intensities and velocity spectra for bare and ......turbulent intensities and velocity...

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