An experimental investigation of turbulent flows over a hilly surface

PHYSICS OF FLUIDS 19, 036601 共2007兲

An experimental investigation of turbulent flows over a hilly surface D. Poggia兲 Dipartimento di Idraulica, Trasporti ed Infrastrutture Civili Politecnico di Torino, 10129 Torino, Italy and Nicholas School of the Environment and Earth Sciences and Department of Civil and Environmental Engineering, Pratt School of Engineering, Duke University, Durham, North Carolina 27708

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

J. D. Albertsonc兲 Department of Civil and Environmental Engineering, Pratt School of Engineering and Nicholas School of the Environment and Earth Sciences, Duke University, Durham, North Carolina 27708

L. Ridolfid兲 Dipartimento di Idraulica, Trasporti ed Infrastrutture Civili Politecnico di Torino, 10129 Torino, Italy

共Received 18 August 2006; accepted 9 January 2007; published online 8 March 2007兲 Gentle topographic variations significantly alter the mass and momentum exchange rates between the land surface and the atmosphere from their flat-world state. This recognition is now motivating basic studies on how a wavy surface impacts the flow dynamics near the ground for high bulk Reynolds numbers 共Reh兲. Using detailed flume experiments on a train of gentle hills, we explore the spatial structure of the mean longitudinal 共u兲 and vertical 共w兲 velocities at high Reh. We show that classical analytical theories proposed by Jackson and Hunt 共JH75兲 for isolated hills can be extended to a train of gentle hills if the background velocity is appropriately defined. We also show that these theories can reproduce the essential 2D structure of the Reynolds stresses. The basic assumptions in the derivation of the JH75 model are also experimentally investigated. We found that the linearization of the advective acceleration term and the mixing length proposed in JH75 are reasonable within the inner layer. We also show that the measured variability in the linearized mean longitudinal advective acceleration term can explain much of the measured variability in the entire nonlinear advective acceleration term for the longitudinal mean momentum balance. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2565528兴 I. INTRODUCTION

Turbulent flow over complex surfaces remains an active research area in modern engineering, geophysical and environmental fluid dynamics. The impact of a wavy surface on bulk properties of the flow field is of interest in a large number of applications spanning a broad range of spatial scales, from micromaterials used for drag reduction to drag parametrization of mountains within large-scale weather models, from designs of optimal roughnesses for heat exchangers to the evolution and migration of desert sand dunes. The length and time scales characterizing the flow over wavy surfaces in such applications can range from microscopic 共␮m, 100−1 s兲 to macroscopic 共km, h兲 thereby resulting in Reynolds numbers that vary from laminar to completely developed turbulent flows 共Reh = 300− 106兲 and from hydraulically smooth to completely rough flows 共Rek = 0 − 106兲. Here, the two Reynolds number definitions follow standard convention with Reh = h ubu / ␯, where h is the depth of the boundary layer, ␯ is the kinematic viscosity, and ubu is the time- and depthaveraged bulk velocity, and Rek = ks u* / ␯, where ks is the a兲

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1070-6631/2007/19共3兲/036601/12/$23.00

characteristic roughness of the wavy surface 共representing both the equivalent sand roughness of the surface and the global drag induced by the terrain variability on the flow兲 and u* is the friction velocity. While still lacking, any general theory for flow over wavy surfaces must reproduce the flow dynamics for the widest possible Rek-Reh configurations. Experimentally, however, much of the data collected over the past 40 years is clustered within two broad categories: low to moderate Reh and low Rek 共i.e., hydrodynamically smooth or transition兲, and very high Reh and Rek. Low to moderate Reh and low Rek experiments were primarily intended to understand the initial phases of wave generation, coherent structures formation and development,1–9 or mechanisms inducing passive drag reduction.10,11 The Reh and Rek that characterize the flow over wavy surfaces interacting with the atmospheric boundary layer 共ABL兲 are much higher and often necessitate different experimental and theoretical treatment, the subject of this investigation. During the past two decades, interest in atmospheric boundary layer 共ABL兲 flows over complex terrain significantly increased in micrometeorology and surface hydrology12 though they did not receive the same level of attention as flows over uniform flat terrain. However, the urgent need to link long term CO2 turbulent flux measurements with ecophysiological processes across different

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© 2007 American Institute of Physics

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biomes,13,18 often situated on complex terrain, is now elevating the interest in ABL flows over hills.16,17 This interest is partly driven by the recognition that gentle topographic variations can lead to significant changes to the climate near the ground surface, and thereby on exchanges of energy, mass, and momentum between the land surface and the atmosphere.16 More significant is the recognition that for the vertically integrated mean scalar continuity equation, the local balance between turbulent fluxes above the canopy and integrated ecophysiological sources and sinks within the canopy can be significantly disrupted by advective terms originating from topographic variability.14,15,17–20 One possible “fix” to this “imbalance problem” is to directly measure the vertically integrated local advective terms. Unfortunately, measuring vertically integrated scalar advective terms at a single tower remains difficult in long-term flux monitoring experiments. Another solution is to model the advective terms and correct 共or filter兲 flux measurements when advective terms are predicted to play a significant role in the mean scalar conservation equation. This modelling approach may be promising as high resolution digital elevation maps that drive such computations are now available. The main limitation here is the availability of a simplified theory that links topographic variations to the mean scalar advective terms. Clearly, to progress on this latter point, ABL flows over the “nonflat” terrain must now be confronted with laboratory and field experiments, detailed numerical simulations, and development of simplified theories that retain clear analytical tractability. Interestingly, substantial progress on high Reh flow over complex terrain did not come initially from laboratory experiments but from analytical theories, the first attributed to Hunt and co-workers21 共hereafter referred to as the JH75 theory兲. Although these theoretical results are limited in applicability to hills with low slopes 共but a wide range of Rek兲, they were fundamental to stimulating several field campaigns as early as 197922–29 共see Mammarella et al.29 for a recent review兲. The field experiments, primarily conducted for high Rek and Reh, supported predictions by the linear theory even for conditions when the theory was not strictly applicable. Not surprisingly, the spatial resolution of these field experiments were highly restricted given that tower measurements were often conducted only at the crest of the hill. To overcome this spatial sampling problem, wind tunnel experiments have also been conducted.1,30–35 These experiments were successful at resolving the spatial sampling problems that plagued field experiments at the expense of other limitations. For example, to aerodynamically simulate a real rough canopy on a hill, the inner layer depth becomes comparable to the roughness height resulting in wind-tunnel experiments having little relevance to inner layer studies of the ABL.30,36 The inner region is the region of immediate interest here because it exerts the most influence on the environment near the ground surface, and thereby on the exchange rates of energy, mass, and momentum between the land surface and the atmosphere. In short, both field and wind tunnel experiments contributed crucial data on ABL flow over a wavy surface for high Rek and Reh. However, the former suffered from severe spa-

Phys. Fluids 19, 036601 共2007兲

tial resolution issues while the latter did not provide a realistic description of the inner layer dynamics most relevant to biosphere-atmosphere exchange within the ABL. What is lacking is a data set with sufficiently thick inner layer depth and of sufficient spatial resolution to assess closure assumptions 共e.g., eddy diffusivity兲, linear simplifications of advective terms, and background velocity estimation when extending this theory to situations diverging from an isolated hill. JH75, along with such assumptions, are now forming the basic foundation for more detailed analytical models of flow inside tall canopies on hilly terrain.17,20 Hence, this study is motivated, in part, by the clear need for both simplified models describing flow over hilly terrain, and high spatial resolution experiments that primarily focus on inner layer dynamics under controlled laboratory conditions. Numerous factors such as thermal stratification, unsteady flow conditions 共including shifts in wind directions兲, and spatial variability in surface roughness prohibit the immediate extrapolation of these flume experiments to the ABL. However, resolving the joint effects of all these factors, in addition to topographic variability, is well beyond the scope of a single study. Hence, the compass of this work is restricted to the interplay between the topographic variability and the mean flow field for a train of gentle hills within the framework of JH75 and the flume experiments described next. We choose the JH75 framework as a starting point for our work because it provides the most parsimonious balance between the resolved physical processes and the “lowdimensionality” in linking topographic variability to mean flow variability across hilly terrain. II. EXPERIMENTAL FACILITIES

The experiments were conducted in the OMTIT recirculating channel at the Giorgio Bidone Hydraulics Laboratory, DITIC Politecnico di Torino. The flume has an 18 m long, 0.90 m wide, and a 1 m deep working section 共Fig. 1兲, and a recirculating flow rate up to 360 l s−1. At the entrance, the flow passes through a 1.5 m long diffuser and then into a honeycomb with a hexagonal cell structure made of reinforced plastic elements 共length-to-cell-size ratio of 6兲. The channel sides are made of glass to permit optical access.37,38 The topography is constructed using a removable wavy stainless steel wall composed of four modules, each representing a sinusoidal hill with a shape function given by ˜f 共x兲 = H / 2 cos共kX兲, where X is the longitudinal distance, H共 =0.08 m兲 is the hill height, k = ␲ / 共2L兲 and L共=0.8 m兲 is the hill half length as shown in Fig. 1. This section begins at 4 m downstream from the channel entrance. The longitudinal 共u兲 and vertical 共w兲 velocity measurements were performed above the third hill module. To check whether the turbulence was completely developed, preliminary measurements were conducted on the second, third, and fourth sections. These preliminary measurements showed that the u statistics acquired at four locations 共and 10 vertical positions兲 around the crest of the second and the fourth hills do not significantly differ from their analogous statistics at the crest of the third hill 共overall R = 0.95 and m = 1.08, where R is the correlation coefficient and m is the regression slope兲. During the experi-


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FIG. 1. 共Color online兲 Experimental setup: The test section along with the train of hills 共top right兲. The definition of the displaced coordinate system 共x , z兲 in relation to the inner layer height 共hi兲 and hill dimensions 共H , L兲 is also shown for reference.

ment planning phase, we conducted model calculations using first order closure principles and found that the mean u and w variations above the third hill do not differ from their counterpart above the fourth hill 共overall R = 0.85 and m = 1.18兲. The velocity measurements were carried out using a two-component Laser Doppler Anemometry 共LDA兲 operated in a forward scattering mode. A key advantage of LDA is its nonintrusive nature and its small averaging volume. Two separated manual traverse systems were used to position the LDA apparatus and the receiving optics, allowing a measurement precision on the order of ±0.025 mm. The measurements were performed at ten positions to longitudinally cover one hill module, and at 0.40 m from the lateral wall in the spanwise direction. These measurements were performed along a large number of vertical positions 共⯝35兲 displaced along a specified coordinate. When modeling or measuring the flow over complex surfaces, the reference coordinate systems can be externally imposed 共e.g., rectangular Cartesian systems兲, related to the geometry of the surface 共e.g., terrain following systems兲, or allowed to adjust according to the flow dynamics 共e.g., streamline coordinates兲. As discussed in FB04,17,39 the latter choice is the most preferred for flow over hills and is adopted here. For our experiment, the hill shape function is a cosine surface with respect to the rectangular coordinate system 共X and Z兲 resulting in vertical and longitudinal streamline coordinates, z and x, given by x = X + H/2 sin共kX兲e−kZ;

共1兲

z = Z − H/2 cos共kX兲e−kZ .

共2兲

The key advantage of a displaced coordinate system is that it reduces to terrain-following near the ground, and to

the rectangular Cartesian system well above the hill. Hence, it retains the advantages of both coordinate systems in the appropriate regions. We sampled 45 cm 共of the 60 cm water level兲 in the displaced vertical direction. We concentrated on the vertical measurement array close to the ground so as to zoom into the inner layer dynamics. The vertical arrangement of each measurement point was chosen to have an approximate logarithmic vertical distribution. To simplify the acquisition procedure, the inclination of the two velocity components was kept constant and equal to the slope of the surface. However, the measurement path from the surface follows the displaced coordinate system. Numerical postprocessing of the acquired data was then carried out to readjust the two components of the velocity with the theoretical coordinates thereby permitting direct comparisons between theory and data. The water depth was retained at a mean steady state value of 60 cm throughout the experiment. We are aware that the ratio between a 60 cm water depth and a 90 cm channel width does not guarantee that the lateral walls do not influence the statistics of u and w. We carried out a sensitivity analysis in which the velocity statistics were acquired at several spanwise positions 共ranging from 20 to 40 cm from the side wall兲 within the inner region and we found no significant difference between them for the first and second moments. Therefore, for the purposes of modelling the mean flow in the context of JH75, the effects of the lateral walls were neglected. One technical challenge to quantifying the flow statistics within the inner layer was the use of LDA to acquire w near the ground surface. Tilting the laser beams cannot be employed because such tilting leads to inadequate spatial co-


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location between the horizontal and vertical velocity. To avoid such a problem, we used a technique previously developed and tested in this flume.40 A series of ten narrow slits 共0.1 mm wide兲 were made in the stainless steel sheet for each longitudinal position to permit vertical passage of the laser. To guarantee the same optical path for every beam, the lowest part of the sinusoidal surface, which is at 0.15 m above channel bottom, was filled with water. The presence of water 共instead of air兲 allows for a decrease in the pressure difference between the region above and below the thin steel sheet leading to a decrease in stress and vibrations. To avoid bubble formation, a low hydraulic discharge of 2 l / s under the test section was also imposed. Finally, to reduce every interference between the water below and above the test section, a “pocket” was projected ad hoc and installed below the steel sheet for each slit. By injecting a fluorescent dye solution 共Red Rhodamine兲 in the fluid under the hill, we confirmed that no interaction between the fluid below and above the test section occurred. As a final check, we compared the w measurements with direct predictions from the continuity equation using the u measurements as input 共described later兲. The u共x , z兲 and w共x , z兲 measurements were conducted at Reh = 1.5⫻ 105 共fully developed turbulence兲. The sampling duration for each run was 300 s, which was shown elsewhere to be sufficiently long to ensure convergence of the statistics.41 The sampling frequency for each run ranged from 2500 to 3000 Hz. The analog signals from the processor were checked by an oscilloscope to verify the Doppler signal quality at every run. The steadiness of the flow during the very long experimental times 共about 8 h兲 was verified by continuously monitoring the flow rate. No artificial seeding of the channel was employed. The signal processing was performed by two Dantec Burst Spectrum Analyzer 共BSA兲 processors. The coincidence mode was used to obtain reliable measurements of the Reynolds shear stress. To preserve the correlation coefficient between w and u, all data points not exactly temporally coincident were discarded. Further details about the LDA configuration and signal processing can be found elsewhere.40 One limitation of this experimental setup pertains to the surface roughness. To achieve a sufficiently deep inner region, we reduced the roughness height, which necessarily implies that the viscous effects can play a role in a thin region close to the ground. For our experiments, the roughness Reynolds number is Rek = u*ks / ␯ = 36, where the measured u* = 0.018 m / s, and ks = 2 mm. The region in which 4 ⬍ Rek ⬍ 60 is classically referred to as dynamically slightly rough.42 However, the presence of a thin viscous region need not affect the data-model comparison if we restrict the comparisons to the inner region well above the viscous sublayer. Hence, in the model-data evaluation, we discarded all the measurements within this viscous region but retained them when graphically presenting the full data sets. In terms of impact on the JH75 framework, the presence of a viscous region necessitates a re-examination of the lower boundary condition, especially the constant stress assumption, even if this region is not included in the data-model comparison. We elaborate on this issue in the results and discussion section.

III. THEORY

According to Jackson and Hunt,21 the domain above a 2D gentle hilly surface can be decomposed into an inner and an outer region. In the outer region, the perturbations associated with the shear stress gradient induced by the flow over the hill are of no dynamical significance and the flow can be treated as inviscid. In the inner region, these perturbations in the shear stress are comparable with inviscid processes. Belcher and Hunt48 共hereafter referred to as BH93兲 estimate the inner region depth as the solution of

␥ hi ⯝ , L ln共hi/z0兲

共3兲

where z0 is the momentum roughness height 关related to ks through the mean longitudinal profile as z0 = ks exp共−8.5kv兲兴, ␥ = 2k2v, and kv = 0.4 is the von Karman constant. In the inner layer and in a thin layer just above 共i.e., the middle layer as defined in the Appendix兲 the pressure perturbation induced by a sinusoidal surface can be expressed as ˜p共x兲 = − U20H/2k cos共kx兲.

共4兲

Hereafter, tilde denotes solutions for sinusoidal hills. For analytic tractability, it is convenient to decompose the mean velocity into an unperturbed or background state 共i.e., assuming the flow is over flat terrain兲 and a perturbation induced by the hill, given by u共z,x兲 = Ub共z兲 + ⌬u共x,z兲; w共z,x兲 = ⌬w共z,x兲;

共5兲

␶共x,z兲 = ␶b共z兲 + ⌬␶共x,z兲. JH75 suggested that, in the case of gentle hills 共H / L 1兲, 共a兲 the perturbation terms are small compared to the background terms, and 共b兲 the nonlinear terms can be neglected. Based on these assumptions, the mean-momentum equation can be written as Ub共z兲

⳵ ⌬u共x,z兲 ⳵ Ub共z兲 ⳵ ⌬p共x兲 ⳵ ⌬␶共x,z兲 + ⌬w共x,z兲 =− + ; ⳵x ⳵z ⳵x ⳵z 共6兲

the robustness of these assumptions will be tested in the next section. Moreover, the above equations imply that the background velocity plays an important role in the estimation of the perturbation terms. Hence, an unambiguous definition for Ub becomes critical. Instead of writing a budget equation for ␶共x , z兲, JH75 introduced another controversial assumption: the eddy viscosity closure with a linear mixing length throughout the inner layer.47 With this assumption, the equation for the perturbed Reynolds shear stress can be written as a function of ⌬u共x , z兲 using ⌬␶共x,z兲 = 2kvu*z

⳵ ⌬u共x,z兲 , ⳵z

共7兲

where the logarithmic mean velocity profile for the background velocity was used 关i.e., Ub共z兲 = u* / kv log共z / z0兲兴.


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While this background velocity assumption may be reasonable for an isolated hill, its validity for more complex terrain 共e.g., a train of hills兲 remains to be investigated. For the particular case of a flow above a sinusoidal hill, 共x , z兲 is the solution of ⌬u ⌬u共x,z兲 = −

u* UI0 kv

冋冉

1 + ln

册

冊

h2i cos共kx兲 − Re共4B0e−ikx兲 , z 0z 共8兲

where UI0 is the dimensionless scaling for the streamwise velocity in the inner region, UI0 = 共Hk / 2兲共U20 / UI2兲, UI is the background velocity defined at the inner region height 共Ub共hi兲 = UI兲, B0 = K0共2冑ikLz / hi兲, and K0共.兲 is the modified Bessel function of the zeroth order. Likewise, the perturbation induced by a sinusoidal hill on the vertical velocity in real space can be evaluated as 共x,z兲 = − kv␲u*UI0 ⌬w

z sin共kx兲. hi

共9兲

Following this derivation, it is clear why the JH75 is a logical starting point to investigate the connections between topographic variability and mean flow variability. In the JH75 framework, topographic variability produce pressure perturbations that induce mean flow perturbations, which in turn require that the turbulence adjusts to these perturbations via the turbulent stress gradients. The model simplifications for linking topographic variability to the pressure perturbations and how the turbulent stresses adjust to the mean flow perturbations 共i.e., K-theory兲 will be experimentally explored in greater detail here. We note that these model simplifications often go beyond the JH75 framework and are applied to almost all closure approaches in Reynolds-Averaged NavierStokes 共RANS兲 approaches. The additional simplification in JH75 共beyond standard first order closure principles兲 is the linearization of the advective acceleration term, which we explore in greater depth as well using the data. From the dynamics perspective, the JH75 solution reveals several testable hypotheses about the phase relationships between the velocity perturbations and the hill shape 共x , z兲 includes a term in function. For example, while ⌬u phase with the hill surface and a term that depends on the 共x , z兲 is always in phase imaginary and real part of B0, ⌬w with the pressure gradient and hence out-of-phase with the hill surface. Moreover, B0 tends to zero exponentially far above the surface leading to a ⌬u共x , z兲 in-phase with the surface and out-of-phase with ⳵⌬p共x兲 / ⳵x for large z. Furthermore, the solution for the perturbed shear stress above a sinusoidal hill is given by

再

冋

⌬␶共x,z兲 = 2u2*UI0 cos kx − Re

共k兲B ikA 1 1

冑hiik/z

e−ikx

册冎

, 共10兲

where B1 = K1共2冑ikLz / hi兲, and K1共.兲 is the modified Bessel function of the first kind. Above the surface, similar to ⌬u共x , z兲, ⌬␶共x , z兲 tends 共exponentially兲 to be in-phase with the surface and out-of-phase with ⳵⌬p共x兲 / ⳵x. But before testing these hypotheses and simplifications, it is imperative

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

IV. RESULTS A. The background velocity and the velocity perturbations

Background velocity: Recall that while the definition of Ub共z兲 is unambiguous for an isolated hill, being the upstream mean longitudinal velocity profile, its definition remains ambiguous for complex terrain.39,43 For example, for a train of ridges, the upstream velocity profile colliding with the nth hill is also a function of the previous nth− 1 hill. When experimentally studying periodically varying flows, it is plausible to assume that Ub共z兲 can be determined from the mean velocity averaged over the hill wavelength. On the other hand, it is theoretically desirable that Ub only be determined from flow over flat terrain because 共1兲 these profiles are independent of the hill configuration 共except through a surface roughness兲 and they have been tested in numerous experiments, and 共2兲 the analytical approach in JH75 a priori assumes a logarithmic shape for Ub, known to be the most robust parametrization of the mean velocity profile for flat boundary layers. A logical question to explore is whether the spatially averaged velocity profile, strictly determined from the data here, is related to the velocity profile determined from “flat-world” formulations. If the two Ub estimates collapse, then the ambiguity in the definition of Ub is not critical for our purposes.44,45 This question is explored next in Fig. 2. Figure 2 shows all ten measured longitudinal velocity profiles across the hill together with the Ub共z兲 determined from spatial averaging and its best fit logarithmic velocity profile 关given by Ul共z兲 = u* / kv log共z / z0兲兴. In Ul共z兲, u* = 0.018 m s−1 was evaluated using the maximum value of ␶b共z兲, and the roughness length z0 = 0.062 mm was computed by minimizing the root-mean squared error between Ub共z兲 estimated from spatial averaging and Ul共z兲. The comparison between Ub共z兲 and Ul共z兲 suggests that the background velocity retains its logarithmic shape. The ten shear stress profiles measured along the hill are also presented in Fig. 2 together with the background shear stress, ␶b共z兲. Within the inner region, it is clear that ␶b共z兲 is approximately constant in most of the inner region consistent with the logarithmic background velocity profile assumption. The combination of the logarithmic Ub共z兲 and constant ␶b共z兲 already hints that the eddy viscosity model, a major assumption of the JH75 theory, may be reasonable in the inner region. Furthermore, Fig. 2 shows a thin region close to the ground where ␶b共z兲 decreases toward the origin. This behavior is often used as a robust identification of the viscous sublayer.46 Here, this layer is limited to a thin region 共z / hi ⬍ 0.2, where hi = 0.08 m is the inner layer depth兲 and only affects the first two or three sampling points. Recall that the JH75 solution to the mean longitudinal momentum balance neglects viscous effects altogether. However, the presence of a thin viscous subregion does not alter the formulations proposed in JH75. The JH75 model admittedly assumes a


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FIG. 2. 共Color online兲 The measured longitudinal mean velocity u共x , z兲 and turbulent shear stress ␶共x , z兲 across the hill. Top panel: Color map of the 2D structure of the measured u共x , z兲 along with the hill surface and the inner region boundary 共dotted-dashed兲. Bottom-left panel: The profiles of the normalized u at the 10 sections 共s1– s10兲 marked in the top-left figure 共vertical lines兲. The logarithmic profile 共dotted-dashed line兲 and the spatially averaged values 共open triangles兲 used to define Ub are also shown. The velocity is normalized by Uh = Ub共hi兲. Bottom-right panel: Same as bottomleft but for the shear stress. The normalizing variable is the maximum horizontally averaged u*.

completely rough surface as a bottom boundary condition, but this assumption was only invoked to force the turbulent shear stress to be constant near the wall. In most of the lower levels of the inner region of this experiment 共but above the thin viscous sublayer兲, the shear stress appears to be approximately constant with respect to z 共see Fig. 2兲.

The measured 2D spatial structure of the velocity components and Reynolds stress: Having defined the Ub共z兲 and ␶b共z兲 profiles, we show in Fig. 3 the measured 2D spatial structure of the mean flow perturbations induced by the hill in the inner region. We focus on three discernible zones within the inner region of Fig. 3: upwind, summit, and wake.

FIG. 3. The 2D structure of the flow perturbations around the background state: Top panel is ⌬u共x , z兲, middle panel is ⌬w共x , z兲, and bottom panel is ⌬␶共x , z兲. The inner layer depth is also shown for reference 共dotted-dashed兲.


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FIG. 4. 共Color online兲 Profile comparisons between kvz 共solid line兲 and the effective mixing length l共z兲 estimated from the measured u⬘w⬘ and ⳵u / ⳵z using the eddy-viscosity model 共circle-line兲 across the hill.

In the upwind zone, ⌬u共x , z兲 progressively increases with increasing x. Initially, this increase is confined to a zone very close to the wall 共at x / L = −1.5兲 but extends to the whole inner layer as the summit of the hill is approached 共x / L = 0兲. Here, ⌬w共x , z兲 remains slightly negative throughout. Also, a decrease in ⌬␶共x , z兲 is noted. Just after the summit, the two noticeable features are a decrease in ⌬u共x , z兲 compared to its peak value in the upwind region, and a reversal in sign of ⌬␶共x , z兲. Here, ⌬w共x , z兲 remains small in magnitude. In the so-called wake region, a strong velocity deficit that extends throughout hi from x / L = 1 to the middle of the upwind side 共x / L = 2.5兲 is evident. In this region, the magnitude of the Reynolds stress is enhanced, while ⌬w共x , z兲 is slightly negative. Next, we compare the measured individual velocity and stress profiles with the JH75 model results and further explore their assumptions and simplifications. B. The analytical model

Closure assumption: Before presenting the comparison between measured and modelled flow statistics, we first analyze the basic assumptions used in the previous sections to derive the JH75 analytical model. The most controversial assumption used in JH75 is the eddy viscosity closure with a linear mixing length throughout the inner layer.47 In Figs. 4, we show the mixing length determined from measured u⬘w⬘共x , z兲 and the gradients in u共x , z兲. We found that a linear mixing length 共=kvz兲 agrees reasonably well with the measurements. Nevertheless, a minor disagreement between the modelled and the measured mixing length is noticeable on the lee side of the hill where l = kvz tends to overestimate the measured mixing length.


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

Another simplification in the JH75 model is the linearization of the advective acceleration terms. We compared the measured nonlinear advective terms 共u ⳵ u / ⳵x + w ⳵ u / ⳵z兲 with their linearized counterpart 共Ub ⳵ ⌬u / ⳵x + ⌬w ⳵ Ub / ⳵z兲 in Fig. 5共a兲. It is clear from Fig. 5共a兲 that the linearization proposed by JH75 is quite reasonable 共R = 0.997 and m = 1.01, where R is the correlation coefficient and m is the regression slope, as before兲. Also, the data permitted us to further explore other simplifying assumptions regarding the advection in the longitudinal momentum balance. We found that the measured Ub ⳵ ⌬u / ⳵x explains more than 90% of the measured 共nonlinear兲 advective acceleration term as shown in Fig. 5共b兲共R = 0.992, m = 1.09兲. Other assumptions in the JH75 model such as neglecting ⳵⌬p / ⳵z and ⳵u⬘u⬘ / ⳵x can be a source of error in modeling the velocity statistics. To explore this point further, we conducted a separate experiment on the variation of p at the hill surface and found that p共x兲 is well approximated by a local hydrostatic assumption as we moved along the hill 共data not shown here兲. Furthermore, we conducted preliminary analysis on ⳵u⬘u⬘ / ⳵x and found it to be small 共in a global sense兲 when compared to ⳵ p / ⳵x 共data not shown兲. Comparison between measured and modelled flow variables: We first present the overall comparisons between measured and modelled u, w, and ␶ in Fig. 6 followed by a detailed analysis of the spatial structure of the error in ⌬u, ⌬w, and ⌬␶ in Figs. 7–9. From Fig. 6, the best agreement in terms of regression slope m and correlation coefficient R is for u 共R = 0.98, m = 1.11兲, followed by ⌬w 共R = 0.84, m


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

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

= 0.96兲, and the worst agreement is for ⌬␶ 共R = 0.77, m = 0.74兲. What is clear from this comparison is that the model biases, for all three variables, are not entirely random. To explore the 2D structure of the differences between model and measurements across the hill, we show the profiles of the perturbed flow statistics along the vertically displaced coordinate 共z兲 at ten stations uniformly spaced in the longitudinal direction 共from s1 at x / L = 0 to s10 at x / L = 3.64兲 and their 2D spatial structure. Figure 7 共s1–s10兲 shows the measured 关⌬u共x , z兲兴 and ˜ 共x , z兲兴 velocity perturbation for the ten vertical modelled 关⌬u ˜ 共x , z兲 and ⌬u共x , z兲 profiles. The comparison between ⌬u shows that the model is capable of describing the main features of the 2D perturbed longitudinal velocity within the three zones, at least in terms of signs and order of magnitude of ⌬u共x , z兲. However, the model clearly underestimates the extremes. Figure 8 共s1–s10兲 shows the profile comparison between ˜ 共x , z兲. From Fig. 8 it is clear that the model ⌬w共x , z兲 and ⌬w predicts an approximate linear profile for ⌬w共x , z兲 through˜ 共x , z兲 captures the basic features of ⌬w共x , z兲. out, and that ⌬w

Nevertheless, some disagreement in the phase behavior is noticeable from Fig. 8. Given the small magnitude of ⌬w共x , z兲 and possible measurement error, we estimated ⌬w共x , z兲 from the measured ⌬u共x , z兲 using the continuity equation by imposing a zero vertical velocity at the hill surface. The comparison between measured and estimated ⌬w共x , z兲 are also presented in Fig. 8. The difference between these two estimates is not different from the model-data dif-

FIG. 8. 共Color online兲 Same as Fig. 7 but for the vertical velocity. In addition, estimates of ⌬w共x , z兲 obtained from the measured ⌬u共x , z兲 via the continuity equation are also shown 共squares兲.


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FIG. 9. 共Color online兲 Same as Fig. 5 but for the shear stress. Both modelled ⌬˜␶共x , z兲 and modelled ⌬˜␶共x , z兲 − ⌬˜␶共x , 0兲共i.e. adjusting the lower boundary condition to assure a zero-stress in the viscous sublayer兲 are compared to measured ⌬␶共x , z兲. Note the agreement between modelled ⌬˜␶共x , z兲 − ⌬˜␶共x , 0兲 and measured ⌬␶共x , z兲.

ferences, especially on the upwind side of the hill. The largest difference between the two estimates of ⌬w共x , z兲 and the model appears to be also on the lee-side of the hill 共Secs. IV and V兲. Figure 9 shows the comparison between the profiles of measured and modelled ⌬␶共x , z兲. From this figure, while ⌬˜␶共x , z兲 is able to reproduce the basic behavior of ⌬␶共x , z兲, the agreement between the linear model and the data is less encouraging, especially near the ground. Recall that near the viscous sublayer, the ⌬␶共x , z兲 is approximately zero, while the ⌬˜␶共x , z兲 remains finite because the layer is assumed to reside in a fully developed turbulence zone. If this lower boundary condition “contamination effect” is minimized by comparing ⌬˜␶共x , z兲 − ⌬˜␶共x , 0兲 with ⌬␶共x , z兲, then the agreement is significantly improved as shown in Fig. 9.


defining appropriate u* and z0, as a background velocity to predict the velocity perturbation above gentle terrain. Moreover, Ub, retaining a logarithmic profile, permits us to extend the applicability of analytical theories previously proposed for isolated hills 共e.g., JH75兲 to more complex, though gentle, terrain. This study is the first to experimentally investigate the linearization of the advective acceleration terms proposed in JH75 and found it to be reasonable. However, the often cited main criticism to the application of analytical theories and even first order closure models to flows over complex terrain remains the eddy viscosity models 共or their mixing length variant兲. Using the data from this experiment, we found that a linear mixing length describes reasonably well the data throughout the inner layer. Hence, this comparison suggests that eddy-viscosity models may be sufficiently robust in modelling topographically induced perturbations in the mean flow and turbulent stresses, at least for gentle topography. When these two findings are taken together, it is clear that approximations “endogenous” to the flow dynamics in JH75 are reasonable. Note that these conclusions go beyond JH75 and apply to other approaches such as first 共or even higher兲 order closure models in RANS that employ the mixing length concept. The linearization findings here also go beyond the particulars of the JH75 model. For example, the linearized approximation can permit explicit tracking of how individual energetic modes in the topography influence the mean flow field variability. This finding is particularly timely given the availability of high resolution digital elevation maps from remote sensing products.49 Finally, the experiment here permitted us to explore new simplifications to the mean longitudinal momentum balance for a train of hills not previously explored. We found that the measured variability in Ub ⳵ ⌬u / ⳵x captures most of the variability in the nonlinear advective acceleration within the inner layer. The significance of this simplification is that a new univariate linear PDE may form the basis for future analytical solutions to the 2D structure of the mean flow field. The generality of this finding remains to be investigated in the future for terrain variability possessing more than one energetic wave number 共or more complex features兲.

V. SUMMARY AND CONCLUSIONS

APPENDIX: REVIEW OF THE LINEAR ANALYTICAL MODEL

A new data set was collected above a train of gentle hills to explore experimentally and theoretically the 2D structure of the mean velocity. Our choice of a train of sinusoidal hills here is a logical progression from isolated hills so as to begin confronting the problem of flow over complex terrain encountered in nature. In extending the analysis from an isolated hill to a train of hills, several issues must be addressed. The first is that the background velocity is no longer unambiguously defined and its shape cannot be a priori assumed. Nevertheless, the experiment here suggests that the horizontally averaged velocity is a reasonable and unambiguous interpretation of the background velocity, Ub, given its logarithmic shape. From a prognostic point of view, this result suggests that a logarithmic velocity profile can be used, after

According to Jackson and Hunt,21 the domain above a 2D gentle hilly surface can be decomposed into an inner and an outer region. In the outer region, the perturbations associated with the shear stress gradient induced by the flow over the hill are of no dynamical significance and the flow can be treated as inviscid. In the inner region, these perturbations in the shear stress are comparable with inviscid processes. The analytical derivation commences with scaling arguments reviewed by BH93. BH93 introduced two time scales: the advection-distortion time scale, TD = ␥L / U共z兲, and the Lagrangian integral time scale, TL = kvz / u*, and ␥ is a similarity constant. TD characterizes the advection and the distortion of turbulent eddies versus the straining motion associated with perturbations induced by the hill on the mean


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flow. TL characterizes the decorrelation time scale of the large energy-containing eddies and the time scale at which the turbulence comes into equilibrium with the surrounding mean-flow velocity gradient.50,36 Note that while TD decreases with increasing z, TL increases. Hence, the inner region depth is often defined by the height z = hi at which these two time scales become equal. BH93 estimated this height as the solution of

␥ hi ⯝ . L ln共hi/z0兲

共A1兲

Hunt et al.47 introduced two additional layers: the inner surface layer and the middle layer. The inner surface layer corresponds to a thin layer that was added to take into account the surface boundary conditions. The middle layer is the lowest region of the outer layer and is introduced to match inner and outer layer dynamics. In the middle layer, the Reynolds stress gradient, as in the upper part of the outer region, is small enough to be neglected but the flow is now rotational. The depth of the middle layer, hm, differs depending on the ratio between L and the ABL height, hBL. For elongated hills 共L hBL兲, hm may be assumed to be equal to the hBL.51 Otherwise, for short hills 共L hBL兲, hm is defined as the solution of hm = L ln1/2共hm/z0兲.

共A2兲

Having defined the inner and outer layer depths and their respective time scale arguments, we proceed to the next scaling arguments; the pressure perturbations at the interface between the middle layer and the upper part of the outer layer, pm共x兲 = p共x , hm兲. The pressure perturbation induced by the surface can be expressed as a representative magnitude, po, and a dimensionless longitudinal function, ␴共x兲, using pm共x兲 = po␴共x兲,

共A3兲

where ␴共x兲 is a function of the hill shape and po is the forcing due to the flow field in the outer region given by po = U2b共hm兲 = U20, where Ub共z兲 is the background velocity 共i.e., the velocity before the hill is encountered兲. Here U0 is both the characteristic velocity in the outer layer and the velocity scale for the pressure perturbation. Again, our interest in the regions above the inner layer are presented because these regions are necessary to formulate upper boundary conditions for the inner layer. Because the middle layer is thin compared to L 共i.e., hm / L 1兲, the pressure perturbations in both the middle and inner layers can be considered constant21,47 with respect to z and given by pm共x兲 = po␴共x兲. For a two-dimensional hill, ␴共x兲 can be computed from the hill shape function,52 f共x兲. In the case of sinusoidal hills, where the hill shape function is described by ˜f 共x兲 = H / 2 cos共kx兲, the terms ␴共x兲 and p共x兲 become ˜␴共x兲 = − H/2k cos共kx兲,

共A4兲

˜p共x兲 = po˜␴共x兲 = − U20H/2k cos共kx兲.

共A5兲

To derive the solution for the mean flow, we consider the stationary mean longitudinal momentum equation for a 2D turbulent flow along with the continuity equation, given by u共x,z兲

⳵ u共x,z兲 ⳵ u共x,z兲 ⳵ p共x兲 ⳵ ␶共x,z兲 + w共x,z兲 =− + ; ⳵x ⳵z ⳵x ⳵z

⳵ u共x,z兲 ⳵ w共x,z兲 + = 0. ⳵x ⳵z

共A6兲

As stated earlier , it is convenient to decompose the velocity into an unperturbed or background state 共i.e., assuming the flow is over flat terrain兲 and a perturbation induced by the hill, given by u共z,x兲 = Ub共z兲 + ⌬u共x,z兲; w共z,x兲 = ⌬w共z,x兲; p共x,z兲 = pb共z兲 + ⌬p共x,z兲;

共A7兲

␶共x,z兲 = ␶b共z兲 + ⌬␶共x,z兲. When this decomposition is applied to gentle hills 共H / L 1兲 and assuming that the hill perturbations are small compared to the background terms, JH75 derived the linearized mean-momentum and continuity equations as Ub共z兲

⳵ ⌬u共x,z兲 ⳵ Ub共z兲 ⳵ ⌬p共x兲 ⳵ ⌬␶共x,z兲 + ⌬w共x,z兲 =− + ; ⳵x ⳵z ⳵x ⳵z

⳵ ⌬u共x,z兲 ⳵ ⌬w共x,z兲 + = 0. ⳵x ⳵z

共A8兲

The perturbed quantities are then expanded as a power series in a asymptotically small dimensionless parameter ␦ given by ⌬u共x,z兲 = − po/Ub共h兲共u关0兴 + ␦u关1兴 + ¯ 兲; ⌬w共z,x兲 = − po/Ub共h兲共␦w关1兴 + ␦2w关2兴 + ¯ 兲; ⌬p共x,z兲 = po共␴关0兴共x兲 + ␦2␴关2兴共z,x兲 + ¯ 兲;

共A9兲

⌬␶共x,z兲 = − 2po/U2b共h兲共␶关1兴共x,z兲 + ␦␶关2兴共x,z兲兲, where all the terms of the power series are dimensionless. Note that both Ub共h兲 and ␦ have to be separately specified for each layer depending on the characteristic length and velocity scales. Once the appropriate length and velocity scales are determined for each layer, the linear approach can be employed in the solution. Only terms belonging to the first order expansion, linear in ␦, are retained. According to Hunt et al.,47 the asymptotically small dimensionless parameter in Eq. 共A9兲 for the inner layer, ␦, is determined from z0 and hi using ␦ = ln−1共hi / z0兲. Furthermore, the characteristic velocity Ub共h兲 can be approximated using the background velocity defined at the inner region height 关Ub共hi兲 = UI兴. Using these two assumptions, the perturbed velocity quantities 共for any 2D hill shape function兲 reduce to ⌬u共x,z兲 = − U20/UI共u关0兴 + ln−1共hi/z0兲u关1兴兲;

共A10兲


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⌬w共x,z兲 = − U20/UI共ln−1共hi/z0兲w关1兴兲;

共A11兲

⌬p共x兲 = po␴关0兴共x兲.

共A12兲

The above equations imply that the background velocity plays an important role in the estimation of the perturbation terms. Hence, an unambiguous definition for Ub becomes critical. Instead of writing the equation for ␶共x , z兲 derived from 共A9兲, we introduce another controversial assumption used in JH75—the eddy viscosity closure with a linear mixing length throughout the inner layer.47 We note again that the JH75 formulation assumes that the turbulent viscosity is much larger than the molecular viscosity within the inner layer so that the total stress term is dominated by the turbulent stress. With this assumption, the equation for the perturbed Reynolds shear stress can be written as a function of ⌬u共x , z兲 using ⌬␶共x,z兲 = 2kvu*z

⳵ ⌬u共x,z兲 , ⳵z

共A13兲

where the logarithmic mean velocity profile for the background velocity was used. Upon replacing the previous three equations in the linearized momentum budget and continuity equations,47 all the unknown terms can be evaluated, albeit in the Fourier domain, using

␴关0兴共k兲; uˆ关0兴共k,z兲 = ˆ

冋

ˆu关1兴共k,z兲 = ˆ ␴关0兴共k兲 1 − ln

册

z − 4B0 ; hi

共A14兲

z ˆ 关1兴共k,z兲 = 2ˆ ␴关0兴共k兲ikLk2v , w hi

␴共k兲 is the Fourier transform of ␴共x兲. For the particuwhere ˆ lar case of a flow above a sinusoidal hill, the solution of 共x , z兲 is ⌬u ⌬u共x,z兲 = −

u* UI0 kv

冋冉

1 + ln

册

冊

h2i cos共kx兲 − Re共4B0e−ikx兲 , z 0z 共A15兲

where UI0 is the dimensionless scaling for the streamwise velocity in the inner region, UI0 = 共Hk / 2兲共U20 / UI2兲. Likewise, the perturbation induced by a sinusoidal hill on the vertical velocity in real space can be evaluated as 共x,z兲 = − ⌬w

=− kv␲u*UI0 1

冋

册

2HU20k2Lk2v z Re i e−ikx hi ln共hi/z0兲 2UI z sin共kx兲. hi

共A16兲

共A17兲

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