Equilibrium hydrodynamics concept for developing dunes

PHYSICS OF FLUIDS 18, 105104 共2006兲

Equilibrium hydrodynamics concept for developing dunes S. E. Colemana兲 Department of Civil and Environmental Engineering, The University of Auckland, Private Bag 92019, Auckland, New Zealand

V. I. Nikora Department of Engineering, University of Aberdeen, Aberdeen AB24 3UE, United Kingdom

S. R. McLean Ocean Engineering Laboratory, University of California at Santa Barbara, Santa Barbara, California 93106-1080

T. M. Clunie, T. Schlicke, and B. W. Melville Department of Civil and Environmental Engineering, The University of Auckland, Private Bag 92019, Auckland, New Zealand

共Received 19 December 2005; accepted 23 August 2006; published online 10 October 2006兲 Experiments utilizing two-dimensional fixed dune profiles and varying flow depth 共dune regime flows兲 highlight the equilibrium 共self-similar兲 nature of the near-bed boundary layer over developing dunes with flow separation in the dune lee. The negligible variation in roughness layer 共comprising the interfacial and form-induced layers兲 flow structure for developing dunes was confirmed in terms of spatial fields of time-averaged velocities and stresses; and vertical distributions of: 共a兲 double-averaged 共in time and space兲 longitudinal velocity, 共b兲 double-averaged normal stresses, and 共c兲 the components of the momentum balance for the flow. The finding of an equilibrium nature for the near-bed flow over developing dunes is significant in its centrality to understanding the feedback loop between flow, bed morphology, and sediment transport that controls erodible-bed development. Further research is required into the form of the distribution of double-averaged velocity in the form-induced layer above roughness tops, and also to complete generalization for varying dune steepness of the universal expression for double-averaged longitudinal velocity 共varying linearly with elevation兲 determined herein for the interfacial layer 共below roughness tops兲. Work is presently focusing on the additional effects on flow structure due to sediment transport and three-dimensional flow and bed morphology, although it is expected that the equilibrium boundary layer flow structure patterns identified herein will still be evident for these more complex systems. © 2006 American Institute of Physics. 关DOI: 10.1063/1.2358332兴 I. INTRODUCTION

Waves in sand beds beneath tranquil water flows are typically described as ripples or dunes; ripples being small fine-sediment bed waves that do not influence the water surface, and dunes being bed waves that occupy a significant portion of the flow depth, causing the water surface to be disturbed. Ripples and dunes are common features of river beds and many hydrotransport engineering systems. For subcritical unidirectional flow over an initially plane sediment bed, dunes grow 共Fig. 1兲 at rates reducing with time to attain lengths and heights in equilibrium with the applied flow. To avoid confusion with “equilibrium boundary layer” concepts, dunes reaching equilibrium with a flow are referred to hereafter as “steady-state” dunes 共Fig. 1兲. Studies of key flow structures 共e.g., velocity and stress fields兲 controlling or associated with dunes have typically considered steady-state bedform magnitudes, with recent intense focus1 on the simplified case of flow over fixed two-dimensional bedform shapes including the works of van Mierlo and de Ruiter,2 a兲

Author to whom correspondence should be addressed. Electronic mail: [email protected]

1070-6631/2006/18共10兲/105104/12/$23.00

Lyn,3 Nelson et al.,4 McLean et al.,5,6 Nelson et al.,7 Bennett and Best,8 Kadota and Nezu,9 Venditti and Bennett,10 and Cellino and Graf.11 These studies, and others, have highlighted that dominant features of flow over a dune include 共Fig. 1兲 a separating shear layer 共with internal recirculation but also active mixing with the outer flow兲 in the lee of the dune crest, reattachment of the separated flow to the bed approximately 4–6 dune heights downstream of the crest, and development of an internal boundary layer along the dune stoss slope, with near-bed turbulence over much of the dune stoss slope differing markedly from either classical boundary-layer or wake turbulence.1 Control of ripples has similarly been attributed to near-bed flow separation dynamics,12–15 although of flow magnitudes insufficient to generate the macroturbulence required to develop dunes, and with no interaction between bedform morphology and outerflow structures. A dune, in contrast, dominates the entire flow field throughout its depth,15 and dunes scale with outer-flow lengths.16 Despite recognizing the centrality of the feedback between flow and bed morphology on controlling evolution of an alluvial flow system,14,15,18,19 no work to date has focused on the evolution of hydrodynamics for dunes developing

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© 2006 American Institute of Physics

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Coleman et al.

FIG. 1. Schematic of dune development from plane-bed conditions 关after Coleman and Eling 共Ref. 17兲兴.

from plane-bed conditions for a steady flow. The present study provides such a focus, drawing on the findings of studies 共including those above兲 of steady-state dunes, along with the concepts of equilibrium boundary layers. In particular, using an innovative experimental approach to represent developing dunes, the goal of the present work is to advance understanding of the least known, but dynamically most important, near-bed flow region, largely overlooked in previous studies. The work utilizes the recently developed doubleaveraging methodology 共DAM兲 共Refs. 20–24兲 to highlight understanding and interpretation of these rough-bed flows. In the following sections, equilibrium boundary layers and their relation to developing dunes are discussed, the doubleaveraging methodology framework used is outlined, the tests undertaken are presented, and results are discussed and conclusions drawn. II. EQUILIBRIUM BOUNDARY LAYERS A. Conventional spatially varying boundary layer

A boundary layer is conventionally defined as an equilibrium 共or self-similar兲 boundary layer if its structure at different spatial sections: 共a兲 differs only in characteristic boundary 共e.g., roughness兲 and flow 共e.g., velocity兲 scales, and 共b兲 is described by self-similar solutions of the general boundary-layer equations. These 共asymptotic兲 solutions can be presented as Pn / un* = f 共z / L, additional flow and geometry parameters兲, where Pn is an n-order statistical moment of a velocity field 共first and second moments are considered herein兲, uⴱ is shear velocity, z is distance from the boundary, and L is a characteristic scale of the boundary roughness.25 With a self-similar 共Blasius兲 nondimensional velocity profile ¯u / U = f共z / ␦兲, laminar boundary layers provide a straightforward example of equilibrium boundary layers, where ¯u is mean 共time-averaged兲 velocity at level z, U is free stream velocity 共at the outer limit of the boundary layer兲, ␦ is boundary-layer thickness, f共z / ␦兲 does not change along the flow, and uⴱ can be used in lieu of U. Turbulent boundary layers similarly exhibit a “near universality” in terms of ve-

Phys. Fluids 18, 105104 共2006兲

locity, shear, energy, turbulence scales, and velocity variances, that is considered satisfactory for all but the most precise measurements.20,26 The universal nondimensional velocity profile for turbulent smooth-wall boundary layers is in terms of normalized velocity defect 共von Kármán’s velocity ¯ − U兲 / u* = f共z / ␦兲, and ¯u / u* defect law兲 in the outer layer 共u = f共zu* / ␯兲 in the near-wall inner layer 共Prandtl’s universal law of the wall兲. The outer-layer scales are U, uⴱ, and ␦, with scales of uⴱ and ␯ / uⴱ for the inner layer, where ␯ is fluid kinematic viscosity. For the overlap region between these two layers, the scales of the bounding zones imply a logarithmic velocity profile, i.e., ¯u / u* = A log共zu* / ␯兲 + B, where A and B are coefficients. For a rough-wall boundary with roughness scale 共e.g., height兲 k, the universal smooth-wall turbulent-flow profiles are retained, albeit with a zero-plane displacement distance for the logarithmic profile, and with an additional principal scale of k for the inner layer. This boundary roughness is thereby identified as “k-type” roughness. Perry et al.27 argue that for some “d-type” roughnesses the additional scale is pipe diameter d 共or an outer-flow length scale兲 in lieu of k, with flow resistance independent of k, although the existence of this separate roughness type is yet to be accepted.20,28,29 The logarithmic 共inertial兲 sublayer has been found to extend from a lower bound of 1–4 roughness heights 共h兲 above roughness crests20 to an upper limit of 0.1–0.2␦ 共Refs. 20 and 26–28兲. B. Temporally varying boundary layer

The equilibrium boundary-layer framework that is conventionally applied to spatially developing boundary layers can also be applied in concept to a temporally varying boundary layer. For this latter case, the flow structure at different times, rather than spatial sections, is self-similar and differs only in characteristic flow and boundary scales, i.e., Pn / un* = f共z / L兲, where f共z / L兲 does not change as the boundary and flow scales 共i.e., L and uⴱ兲 vary in time. The potential equilibrium 共self-similar兲 nature of the near-bed flow associated with the present developing dunes 共Fig. 1兲 is indicated by the geometric similarity of the dune roughness elements as these grow 共Fig. 1兲, and the geometric and dynamic similarity of the separation zone for a growing dune 共Fig. 1兲, this latter zone controlling boundary-layer flow dynamics for steady-state ripples and dunes,2–15 and therefore presumably for growing sand waves. The novelty of the growing-duneassociated equilibrium boundary layer studied herein is therefore self-similarity in time as the bedforms grow, similarity being tested using the first and second velocity-field statistical moments for the present study.

III. DOUBLE-AVERAGING METHODOLOGY „DAM… FRAMEWORK

ASCE 共Ref. 1兲 questions whether attempts to date at understanding dynamically complex flow fields over dunes have had any impact, qualitative or quantitative, on illuminating the basic understanding of dune mechanics. The ASCE 共Ref. 1兲 committee notes that spatial averaging is a practical necessity for providing more tractable systems, and

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this is entirely the framework, and advantage, that the double-averaging 共in time and space兲 approach23,24,30 provides. Velocities varying in space and time above a rough bed can be decomposed as20,23,24 ¯ i典 + ˜ui + ui⬘ , ui = 具u

共1兲

where ui is the ith component of point velocity 共in direction xi兲; the straight overbar and angle brackets denote the time and spatial averaging of flow variables, respectively; the wavy overbar denotes the spatial fluctuations in the time¯ i典 + ˜ui; and ui⬘ are turbulent averaged flow variable, i.e., ¯ui = 具u fluctuations. Here the spatial averaging is carried out over a thin slab parallel to the mean bed having small thickness dz. The planar extent of the thin-slab averaging volume Vo is typically chosen to be much larger than the roughness or flow geometry scales, but smaller than the larger geometric features, such as channel curvature, widening, or narrowing. Based on Eq. 共1兲, boundary-layer self-similarity for developing dunes will be considered herein in terms of vertical ¯ 典, spatial fields of ¯ui and ˜ui, vertical distridistributions of 具u ˜ i˜ui典, and vertical butions of the normal stresses 具ui⬘ui⬘典 and 具u distributions and spatial fields of the components of the momentum balance for the flow. In regard to the latter quantities, for steady uniform two-dimensional high-Reynolds number water flow over fixed boundaries with no secondary currents 共as herein for fixed two-dimensional dunes兲, the double-averaged 共in time and space兲 momentum conservation equation can be presented in an integral form as21,24

␳gSb

冕

zws

␾dz

z

˜w ˜ 典共z兲 = − ␳␾具u⬘w⬘典共z兲 − ␳␾具u

冕 /冋 zc

−

z

1 Vo

Sint

¯pni=1 − ␳␯

册

⳵¯u ni dSdZ, ⳵xi

共2兲

where viscous fluid stress is not shown, as this stress is orders of magnitudes less than turbulent stress for high˜w ˜ 典共zws兲 Reynolds number water flows, and 具u⬘w⬘典共zws兲 = 具u = 0. This integral equation can be thought of as simply summing from the water surface of level zws the thin slabs used in spatial averaging. Equation 共2兲 can be seen to describe the flux of gravity-induced momentum 共left-hand side of the equation兲 via fluid stresses 共i.e., Reynolds and form-induced stresses兲 to the boundary where the momentum is removed through form drag 共the integral of pressures over the boundary surface兲 and skin friction 共the integral of tangential shear stresses over the boundary surface兲. The terms of Eq. 共2兲 will then also be used herein to assess boundary-layer selfsimilarity for developing dunes. In the above equations, Sb is the mean bed slope, ␾ = V f / Vo is the roughness geometry function 共1 艌 ␾ 艌 0兲, V f is the volume of fluid within the averaging volume Vo, Sint is the surface area of roughnessfluid interface within the thin-slab averaging volume, p is point pressure, ni is the ith component of the unit vector normal to the surface element dS and directed into the fluid, ␳ = fluid density, ␯ = fluid kinematic viscosity, g is gravitational acceleration, zc is the uppermost 共crest兲 level of the

FIG. 2. Relative stages of dune development.

boundary, ␳具u⬘w⬘典 is a spatially averaged Reynolds stress, ˜w ˜ 典 is a form-induced 共dispersive兲 stress that is analoand ␳具u gous to Reynolds stress but due to spatial variations rather than temporal fluctuations. The roughness geometry function ␾, which varies from unity above the roughness crests 共z ⬎ zc兲 to a minimum within the channel bed 共refer to example functions given in the discussion of momentum fluxes兲, is a statistical measure of both the random geometry of the bed surface and the porosity, becoming analogous to the conventional porosity coefficient when evaluated within a gravel bed. For details of the derivation of Eq. 共2兲, and further discussion of the terms of this equation, the reader is referred to Nikora et al.23,24 Within the postulated temporally varying equilibrium boundary layer, flow structure does not change as dunes develop in time for a steady flow, even though overall flow structure must change as dunes develop with the outer and overlap regions continually compressed, and even potentially destroyed. In terms of the vertical limit of the equilibrium boundary layer above developing dunes, Nikora et al.21,22 propose that rough-wall flow of ␦ / k Ⰷ 1 may be subdivided into four regions: 共1兲 the interfacial 共wall兲 sublayer between roughness crests and troughs 共that is influenced by individual roughness elements兲, 共2兲 a form-induced sublayer just above the roughness crests 共where the flow may still be influenced by roughness elements兲, 共3兲 the overlap 共logarithmic兲 layer, and 共4兲 the outer layer, where viscous effects and forminduced momentum fluxes are negligible and the spatially averaged equations are identical to the time 共ensemble兲averaged equations. Together the form-induced and interfacial sublayers form the roughness layer. The equilibrium nature of the developing-dune boundary layer will be tested within the roughness layer 共up to one sand-wave height above the dune crest兲 for the present study, with free surface and inertial effects being expected to influence the boundary layer above this level. IV. TESTS OF DEVELOPING DUNES

Nikora and Hicks31 propose a common scaling relationship for the general growth 共Fig. 2兲 of sediment-wave lengths and heights from plane-bed conditions, namely 共P / Pss兲 = 共t / tss兲␥, where P is the average value of a sedimentwave parameter 共length ␭ or height h兲 after time t, tss is the

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Coleman et al. TABLE I. Test flows.

Run

H 共m兲

U 共m/s兲

Sb 共⫻10−3兲

共uⴱ兲HR 共m/s兲

共uⴱ兲Re = uⴱ 共m/s兲

共uⴱ兲V 共m/s兲

F

fT

t / tss

12

0.24

0.588

1.76

0.0446

0.0387

0.0438

0.38

0.037

0.16-0.13

11 10

0.18 0.15

0.578 0.552

2.30 2.61

0.0473 0.0478

0.0412 0.0424

0.0444 0.0462

0.44 0.45

0.053 0.060

0.34-0.27 0.56-0.43

9

0.12

0.503

2.77

0.0459

0.0424

0.0425

0.46

0.067

1.00-0.78

time to achieve steady-state magnitude Pss, and 0.01⬍ t / tss ⬍ 1. Based on a series of six experiments of two sediment sizes, the authors find that the growth exponent ␥ = 0.28 for both sand-wave heights and lengths, implying invariant bedform steepness 共共h / ␭兲 / 共hss / ␭ss兲 = 共h / hss兲共␭ss / ␭兲 = 共t / tss兲共0.28-0.28兲 = 1兲 for the developing bedforms. Coleman et al.32 confirm the form of the above relation for a greater number of experiments of wider ranges of flows and sediments, with ␥␭ = 0.32 共on average兲 for bedform lengths, and ␥h = 0.37 共on average兲 for bedform heights, again implying a very rapid initial growth in bedform steepness 共h / ␭ ⬀ 共h / hss兲共␭ss / ␭兲 ⬀ 共t / tss兲共␥h−␥␭兲 ⬀ t0.05兲 to a value approximately maintained over the major period of bedform growth. Recognizing that dune steepness 共and thereby dune geometric similarity兲 can be taken to be essentially invariant during the major period of growth, a series of experiments was designed to simulate flow-dune interaction over the growth of a dune for the same flow 共in contrast to earlier investigations33,34 of sand-wave variations for changing flows兲. For the tests, a single fixed two-dimensional dune profile was adopted, measuring 0.75 m in length and 0.04 m in height 共crest to trough兲, with a 30° lee slope and a cosineshaped stoss slope, and with glued sand of size d = 0.8 mm covering the bedform surface. For the present flows of relatively low strengths and mobile-bed transport rates, the ratio of rate of boundary change 共through dune migration and growth via sediment transport兲 to rate of flow change 共through turbulent velocity fluctuations兲 would be very small for the mobile bed, such that a fixed-bed model can be justified for flow simulation, with the bed essentially stationary with respect to the flow. The focus bedform investigated was twelfth in a series of 15 identical bedforms. All four tests of flow over the dune train were carried out in a glass-sided recirculating tilting flume measuring 0.38 m ⫻ 0.44 m 共wide兲 ⫻12 m, with slopes monitored to maintain uniform flow conditions for the tests 共hence varying mean bed slope Sb in Table I兲. Threedimensional centerline point velocities were measured, using a downward-looking acoustic Doppler velocimeter 共ADV兲, along the focus bedform wavelength at 5 mm elevation increments and 20 mm 共particularly in the lee region兲 to 40 mm horizontal spacings. The measurements, conducted at 50 Hz for 120 s at each point, were focused in the near-bed region of notable velocity and stress variations 共variations being less notable in the flow above兲, namely up to two dune heights above trough levels. Using a differential pressure meter 共accurate to 0.1 mm water head兲, centerline bed pressures were measured against a static head via 1 mm diameter

tappings at 5 mm elevation intervals over the focus bedform surface. Centerline water surface elevations were measured along the flume at 12.5 mm horizontal spacings using an upward-looking ultrasonic transducer. Test flows are summarized in Table I, where H is mean flow depth, U is depth-averaged velocity, centerline Froude number F = U / 共gH兲0.5, friction factor f T = 8共uⴱ / U兲2, and bed shear velocity uⴱ was determined from extrapolation of the Reynolds stress profile to the mean bed level 共uⴱ兲Re, from a logarithmic fit to the velocity profile above the dune crest 共uⴱ兲V, and from hydraulic radius m as 共uⴱ兲HR = 共gmSb兲0.5 共with sidewall correction procedures not applied to the uⴱ estimates兲. From assessment of the results shown herein, the overall value of uⴱ adopted for normalization of results was that determined from the Reynolds stress profile uⴱ = 共uⴱ兲Re 共consistent with McLean et al.,6 Cellino and Graf,11 and others兲, shear velocity for the different assessment methods varying ±13% in Table I. Friction factor f T can be seen to increase as the flow depth decreased for the above tests, reflecting increasing boundary hydraulic roughness as relative exposure increases with bedform growth. For the deepest flow, of the lowest channel aspect ratio and the greatest sidewall effects, measured 共up to two dune heights above trough levels兲 centerline vertical velocities were generally negative from above the crest to midway along the stoss slope, and positive in between. Based on this trend, low measured centerline transverse velocities for the tests 共on average ±2 mm/ s兲, the agreement of measured Reynolds stress trends with theoretical expectations 共discussed later under momentum flux considerations兲, and the relatively significant magnitudes of the two-dimensional bed roughness, the tested flows can be taken to be essentially two-dimensional in nature, at least in the studied central part of the flow. Consideration of earlier works confirms the natures of the chosen dune bedforms and flows 共Table I兲 for the present purposes. For the sediment coating the bedform surface 共median size d50 = 0.8 mm, sediment specific gravity s = 2.65, critical shear velocity for entrainment uⴱc = 0.0206 m / s from Shields diagram兲, the flows adopted are of transport stage parameter T = 1.9– 2.6 and dimensionless particle diameter Dⴱ = d 关g共s − 1兲 / 共␯2兲兴1/3 = 20.2, indicating dune regime flows.16 The present flows would similarly be predicted to generate dunes16 for any sediments of Dⴱ 艌 10 共d50 艌 0.4 mm for a quartz sand in water flows兲 and uⴱ ⬎ uⴱc. The dune steepness of h / ␭ = 1 / 18.75 adopted is also consistent with the common perceptions of steady-state bedforms of height35 h ⬇ H / 3 and wavelength36 ␭ ⬇ 6.25H 共h / ␭ = 共h / H兲共H / ␭兲 = 1 / 18.75= 0.053兲. Confirming the dune steep-

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FIG. 3. Scaling of the present tests.

ness adopted, based on an extensive data set, van Rijn16 predicts steady-state steepnesses of h / ␭ ⬇ 0.044–0.049 for the tests of T = 1.9–2.6. Van Rijn16 also predicts steady-state dune heights of h / H = 0.32–0.36 for the present tests, confirming the maximum h / H ratio tested herein as being appropriate to steady-state conditions. In regard to these van Rijn predictions, Julien and Klaassen36 note examples of underprediction of dune steepness and height for T ⬍ 5 using van Rijn’s method. In contrast, Yalin37 concluded that sandwave 共dune兲 heights on average cannot exceed H / 6, and Yalin and Karahan38 predict lower steepnesses of h / ␭ ⬇ 0.02–0.045 for the present tests of 共uⴱ / uⴱc兲2 = 3.5– 4.2. The latter authors also indicate bedform steepness to be independent of H 共as desired兲 for the present large depths of H / d50 艌 150. Based on these considerations, both the present dune steepness and also the maximum 共steady-state兲 h / H ratio tested can be seen to be of appropriate magnitudes. The present turbulent flows 共Reynolds number Re⬎ 60 000 for all runs兲 were designed to be of similar strength, essentially having only relative boundary roughness 共h / H, or flow depth H for constant h兲 varying to represent bedform growth 共Figs. 1–3兲 for the same flow. With steadystate dune length scaling with flow depth 共␭ss ⬇ 6.25H兲, the varying flow depths of Table I together with the fixed bedform length of 0.75 m represent varying ␭ / ␭ss and thereby varying stages of dune development t / tss according to the recognized power-law relation ␭ / ␭ss = 共t / tss兲␥. The relative stages of development 共t / tss兲 of the present tests are indicated in Table I and Fig. 2 for ␥ = 0.14D0.33 ⴱ = 0.38 共Coleman et al.32兲. To provide bounds on the stages of dune development 共the system scales兲 represented by the present tests, the lower limit t / tss values given in Table I and Fig. 2 were obtained by allowing for a potential 10% increase in flow depth during dune development; this estimated by the writers as a potential upper limit in nature for such a free subcriticalflow system of increasing boundary roughness. Testing of deeper flows, and lower t / tss ratios, although desirable, was not undertaken for the present flume cross section. Demonstration of any equilibrium 共self-similar兲 nature for the near-bed flow associated with developing dunes, as desired herein, can be achieved by measuring flow fields over repeated dunes for a single flow, where the sizes of the dunes are varied between tests. Universality of the resulting framework can be proven by repeating these tests for a range of flows. The present tests provide a succinct proof-ofconcept approach to such an investigation through testing a single-dune profile with varying flow depths 关Fig. 3共a兲兴; the

key to interpreting these tests lying in dune scaling with growth, with dune geometry remaining essentially invariant 共shown above兲, resulting in these tests equating to varying dune size for a single flow stage. In order to highlight this approach, Fig. 3共a兲 shows the present single-dune-profile tests for the varying flow depths, including expected steadystate dune magnitudes for the runs. Also shown is the degree of dune development 共␭ / ␭ss兲 for each run recognizing that dune steepness 共and thereby geometric similarity兲 can be taken to be essentially invariant during the major period of growth to the steady-state magnitudes of h = H / 3. Figure 3共b兲 shows that runs 12–9 are geometrically equivalent to varying dune size for a single flow stage, where degree of dune development, including t / tss values based on Fig. 2, are confirmed for each case. With Froude number scaling dominating for the present free-surface flows of high-Reynolds numbers, velocities scale as the square root of the length scale for dynamic similarity of flows, meaning that the present run 12 velocities would need to be 冑2 times those of run 9 for dynamic similarity of the runs when considered as runs of the same flow depth 共run 12 lengths and velocities reduced by factors of 2 and 冑2, respectively兲. In lieu of this scaling, normalizing velocities by the associated bulk shear velocity uⴱ facilitates assessment of the dynamically equivalent Fig. 3共b兲 flow fields, for a single flow or a range of flows, from the present 关Fig. 3共a兲兴 tests. Normalization of flow results by uⴱ and the primary inner- and outer-flow lengths of h and H then enables comparison of the present results across the runs and also facilitates the desired Fig. 3共b兲 assessment of any near-bed boundary-layer selfsimilarity with bedform growth for the same flow, and also for varying dune-regime flows. Such assessments are carried out in the following sections on this basis.

V. VELOCITY DISTRIBUTIONS

Each of the present time-averaged flow fields over the fixed dune profile is shown in Fig. 4, along with associated fields of normalized Reynolds stresses. The location of the reattachment point 共defined by the intersection of the timeaveraged dividing streamline with the bed, Fig. 4兲 was determined as approximately 4h downstream of the crest for the tests 共3.75± 0.5h for runs 12 and 11, and 4.0± 0.5h for runs 10 and 9兲. This result is consistent with previous studies of flows over fixed dunes,2,3,5,8,9,15 for which reattachment was 3–7h downstream of the crest, with a mean value of about

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FIG. 4. Velocity vectors and contours of normalized 共by ␳u2ⴱ兲 Reynolds stresses 共separation zone in dune lee is highlighted兲.

4.2h. Similarity of the near-bed spatial flow structure over developing dunes is apparent in the velocity fields of Fig. 4. Vertical distributions of double 共time- and space-兲averaged velocity for the runs are shown in Fig. 5. The collapse of the curves indicates negligible variation in the form

of this distribution as dunes grow 共particularly below roughness tops, z / h ⬍ 0.5, where z = 0 is mean bed level兲. The overall form of the velocity profiles displays an approximately linear distribution below roughness tops and a potentially logarithmic distribution 共Fig. 6兲 above roughness tops.

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FIG. 5. Vertical distributions of double-averaged velocity.

FIG. 6. Logarithmic representation of form-induced layer velocity profiles.

The change in the nature of the profile occurs immediately above the roughness tops, at the same level as Reynolds stress begins to deviate from a linear profile with depth 共refer below兲. This is consistent with flow over “k-type” roughness, involving interaction between the outer flow and flow between roughness elements 关Perry et al.27 indicating a change in profile nature for these flows occurring at 共1 + ␣兲h above the roughness trough兴. For the present tests, the change in profile occurs 5–7 mm 共␣ = 0.15兲 above the roughness tops, with no apparent trend with flow strength. Within the interfacial layer 共below roughness tops兲, Nikora et al.22 propose that the double-averaged 共in time and space兲 velocity profile can take several forms dependent on flow conditions and roughness geometry: constant, exponential 共also Raupach et al.20 for within-canopy wind profiles兲, linear, or a combination of these. Figure 5 confirms the approximately linear nature of the inner-flow profile below the roughness tops for the present results. For the present natural dune steepness of approximately 0.05, the linear velocity profile shown in Fig. 5 of

sublayer however. Velocity measurements above dunes by previous authors3,4,6 support a logarithmic profile immediately above roughness crests. The present results for a natural dune steepness of 0.05 indicate possibilities of linear 共Fig. 5兲 or logarithmic 共Fig. 6兲 profiles in this layer above dunes. In line with conventional logarithmic descriptions, the measured velocity profiles can ¯ 典 / u* = 2.5 ln共z / z0兲, where z0 be approximated as 具u = 0.23–0.33 mm 共⬃d50 / 3兲, and the bed origin is at the mean ¯ 典 / u* = 2.5 ln共z / k兲 bed elevation. Expressing this profile as 具u + 8.5 typical of fully rough flows gives normalized roughness heights k / h for runs 12–9 of 0.17, 0.19, 0.22, and 0.25. The present results 共Fig. 6兲 highlight the increase in boundary ¯ 典 / u* deroughness with increasing relative exposure h / H 共具u creasing, with increasing roughness scales z0 / h and k / h兲. This dependence 共k / h ⬃ h / H, and zo / h ⬃ h / H兲 is contrary to the two options that are conventionally expected for the genuine logarithmic layer: 共1兲 k / h does not depend on relative exposure, as for the k-type roughness 共of flow resistance proportional to roughness size h and independent of H兲, and 共2兲 k / h ⬃ H / h, as for the d-type roughness 共of flow resistance proportional to flow depth H and independent of roughness size h; the existence of this roughness type still debatable29兲. Most probably, the reason for this discrepancy is low flow submergence for the dune roughnesses in comparison to typically considered roughness submergences, and thus, inapplicability of the conventional log law. However, as noted above, our and other results indicate that the logarithmic function may serve as a good fit for the double-averaged ¯ 典 / u* = f共z / h , h / H , h / ␭兲 velocity profile in this layer, e.g., 具u where boundary roughness might be expected to increase and then decrease with increasing steepness 共h / ␭兲 共e.g., Davies18兲. Vectors of spatial fluctuations in time-averaged veloci˜ ,w ˜ 兲 are presented in Fig. 7. The time-averaged velocties 共u ¯ ,w ¯ 兲 of Fig. 4 are given by the superposition of ity vectors 共u

¯ 典/u* = 11.0共z/h兲 + 4.1, 具u

共3兲

where z = 0 is mean bed elevation, defines the universal equilibrium boundary-layer function for the interfacial layer. Raupach et al.20 note that a crucial condition for existence of any overlap logarithmic 共inertial兲 sublayer above roughness crests is that the boundary-layer thickness ␦Ⰷ 共␯ / uⴱ, h, other roughness length scales兲. For dunes of roughness heights35 up to h = H, the existence of a genuine overlap logarithmic layer is thereby debatable, particularly for roughness layers extending to 2–5h, as noted by Raupach et al.20 In general, these authors comment that as elevation decreases to within the form-induced sublayer 共above roughness crests兲, the velocity profile begins to deviate from any logarithmic distribution existing above. They provide no indication of a likely velocity profile for the form-induced

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˜ ,w ˜ 兲 and contours of normalized 共by ␳u2ⴱ兲 form-induced stresses. FIG. 7. 共u

the global double-averaged flow of Fig. 5 with the spatial ¯ i典 + ˜ui. Figure 7 highlights fluctuations of Fig. 7, i.e., ¯ui = 具u “coherent” organized spatial structures in the mean-flow disturbances introduced by dunes. The structure of flow over dunes then consists of a pair of counter-rotating vortices set up by the dune profile 共Fig. 7兲 superimposed on an overall

average profile of linear and potentially logarithmic sections 共Fig. 5兲. Figures 4 and 7 highlight similarity of the near-bed spatial velocity fields over developing dunes. The distributions of double-averaged longitudinal velocity shown in Figs. 5 and 6 confirm an equilibrium boundary layer nature, i.e.,

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Phys. Fluids 18, 105104 共2006兲

FIG. 8. Normalized spatially averaged momentum flux components.

¯ 典 / u* = f共z / h , h / H , h / ␭兲, within the roughness layer 共par具u ticularly within the interfacial layer兲 for developing-dune hydrodynamics. Further research needs to be carried out, however, to complete generalization for varying dune steepness ¯ 典 / u* of the universal equilibrium boundary-layer function 具u = f共z / h兲 for the interfacial layer below crests. Additional work is also required to clarify both the theoretical basis of the velocity distribution above dune crests, and also any detailed universal relation that can be adopted for this region. VI. MOMENTUM FLUX COMPONENTS AND LONGITUDINAL NORMAL STRESSES

For each of the present tests, spatial fields of Reynolds ˜ , Fig. stresses 共␳u⬘w⬘, Fig. 4兲 and form-induced stresses 共␳˜uw 7兲 were determined from the measured point velocities. Figures 4 and 7 highlight that there is negligible variation in the forms of these fields as dunes grow. Peak Reynolds stresses occur in the shear layer associated with the separation zone 共in the vicinity of the reattachment point at x = 830 to 840± 20 mm, Fig. 4兲, consistent with shear-layer instabilities and vortex shedding off this layer.15 Elevated Rey-

nolds stresses extend from this region into the outer flow, corresponding to the turbulence generated in this region propagating to the water surface.39,40 Figure 7 highlights that form-induced stresses are maximum adjacent to boundary surfaces, so that accurate assessment of these stresses and their role in momentum flux requires concentrated measurements in these regions. Figure 8 presents the variation with elevation of each of the spatially averaged momentum flux components of Eq. 共2兲, namely spatially averaged Reynolds stress 关−␳␾具u⬘w⬘典, ˜w ˜ 典, Fig. 8共b兲兴, form Fig. 8共a兲兴, form-induced stress 关−␳␾具u zc drag 关−兰z 1 / Vo/Sint¯pni=1dSdz, Fig. 8共c兲兴, and skin friction 关兰zzc1 / Vo/Sint␳␯⳵¯u / ⳵xinidSdz, Fig. 8共d兲兴, with all flux components normalized by ␳u2ⴱ. Summations of the momentum flux components are given in Fig. 9, with equivalent plots of normalized and double-averaged longitudinal turbulent ˜˜u典, Fig. 10共b兲兴 关␳具u⬘u⬘典, Fig. 10共a兲兴 and form-induced 关␳具u normal stresses given in Fig. 10. The skin friction terms of Fig. 8 were calculated 共despite questions as to applicability for flow immediately adjacent to the dune surface兲 by assuming a rough-wall logarithmic velocity profile below the low-

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FIG. 9. Normalized 共by ␳u2ⴱ兲 total momentum flux.

est velocity measurement point 共adopting a realistic value of z0 = 0.03 mm for the near-bed grain roughness兲. The formdrag terms were assessed from integration of the measured pressure distributions. For the present tests, however, measured pressures appeared to give form drag significantly larger than that expected based on other measured flow properties. These anomalies in measured pressures are attributed by the writers to difficulties experienced in measuring the fine pressure differences using barely legible scales for the system used. Pressures for form-drag calculations were then provided from measured flow velocities for these tests by vertically integrating the vertical component of the steady¯ / ⳵xi = gz − 1 / ␳⳵¯p / ⳵z flow Reynolds equation, i.e., ¯ui⳵w ¯ / ⳵xi兲, from the water surface to the − ⳵w⬘u⬘i / ⳵xi + ⳵ / ⳵xi共␯⳵w bottom boundary 共where gz is the component of g in the z direction兲. The validity of this approach was supported by

Phys. Fluids 18, 105104 共2006兲

agreement of equivalently calculated pressures with measured pressures for runs 2–7 of McLean et al.6 For the velocity-field second moments P2 of Figs. 4, 7, 8, and 10, P2 / u2ⴱ = f共z / h兲, which, as highlighted above, is appropriate to boundary layers that are self-similar in nature. The equilibrium boundary-layer nature of the near-bed flow over dunes as they grow is then clearly highlighted by the lack of any appreciable variation in each of the normalized distributions of Figs. 4 and 7 共stresses兲 and 8–10 as dunes grow 共Fig. 3兲. Figure 8 shows that Reynolds stress dominates other momentum flux components above dune crests 共z / h ⬎ 0.5兲. Trends in total momentum flux given in Fig. 9 then highlight that Reynolds stress 共⬃total flux兲 varies linearly with depth from zero at the free surface 共z / H = 1兲 to immediately above the dune crests 共Fig. 8兲; the change in profile occurring at this level 共5 – 7 mm above the roughness tops, Fig. 8兲 coinciding with the earlier-discussed 共Fig. 5兲 change in velocity profile at this level 共with ␣ = 0.15兲. Interestingly, increases in double-averaged Reynolds stress in the vicinity of the dune crest are balanced by equivalent negative form-induced stresses at these levels 共Fig. 8兲. Figures 8 and 9 also reveal that the relative contribution of each flux component to the overall momentum flux balance does not change significantly for dunes at different stages of development 共consistent with the equilibrium-boundary-layer nature of the near-bed flow over growing dunes兲. For the normalized gravity-induced momentum flux 关left-hand side of Eq. 共2兲兴 equivalent to the sum of momentum flux components, Eq. 共2兲 predicts a straight line from zero at the mean water surface to the roughness crest level, with this line then curving 共due to the influence of the roughness geometry function ␾, Fig. 9兲 to give unity at the roughness trough. The upper flow straight line profile for the gravity term extrapolates 共Fig. 9兲 to give unity at the mean bed level. For each test, the theoretical total gravity-induced momentum flux 关“Gravity,” left-hand side of Eq. 共2兲兴 is shown on Fig. 9, along with the corresponding roughness geometry function ␾ 共“Phi”兲, with the influence of increasing relative

FIG. 10. Normalized longitudinal normal stresses.

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clusion. The invariance of the bed-stress profile as dunes grow is consistent with the conceptual framework presented herein of an equilibrium nature for the temporally varying boundary layer of developing dunes, i.e., ␶0 / u2ⴱ = f共z / h兲 共=f共x / ␭兲 for invariant bedform shape兲. VIII. CONCLUSIONS

FIG. 11. Normalized bed shear stresses.

exposure h / H 共tests 12–9兲 highlighted for each set of curves. The total of the momentum flux components of Eq. 共2兲 coincides 共Fig. 9兲 with the gravity-induced flux for each of the present runs, as expected. VII. BED SHEAR STRESSES

Theories of dune formation41–45 recognize that phase shifts between flow, bed, and sediment transport are necessary for dune formation and growth, particularly phase shifts between local bed level and local bed shear and between local transport and local bed shear. Although sediment transport was not studied in the present tests, the former phase lags and their variation with dune growth 共Fig. 3兲 can be assessed using the present results. For each test, local bed 2 at each point along the dune profile was shear ␶0 = ␳u*0 evaluated by assuming a rough-wall logarithmic profile, uⴱ0 = ¯u / 关2.5 ln共z / z0兲兴, below the lowest velocity measurement point ¯u共z兲 and adopting a realistic value of d50 / 30 = 0.03 mm for the hydrodynamic roughness height z0. The results are presented in Fig. 11 along with earlier results for ripples12 and dunes,5 respectively. For this figure, stress results have been normalized by bulk shear stress consistent with the earlier works. With varying bed profiles for the different works 共the present dune profile is shown兲, the results of Fig. 11 have also been aligned for a common bedform crest position. The present bed shear trends are consistent with the earlier results,5,12 particularly allowing for potential uncertainties in determinations of bulk shear stresses and also the different bed profiles tested. From Fig. 11, bed shear stress can be seen to peak upstream of the bedform crest, consistent with linear-analysis expectations for bed instability and bedform initiation and growth from plane-bed conditions.42,43 It needs to be noted, however, that linear-theory description of the roles of phase lags is not strictly applicable to asymmetric finite-amplitude bedforms such as the present fixed dunes, for which nonlinear bed-flow interactions act to dominate bed growth to 共stable兲 steady-state bedforms. The present tests indicate a generally negligible change in the bed-stress profile as dunes develop 共runs 12-9, Fig. 3兲, although uncertainties due to the means of determining ␶0 values and the sparseness of results are a factor in this con-

An equilibrium boundary layer is conventionally characterized by structure at different spatial sections differing only in characteristic scales, with self-similar expressions for velocity-field statistical moments. Flow within 共up to one sand-wave height above the dune crest兲 the near-bed boundary layer of growing two-dimensional sand dunes is shown herein to be equivalently self-similar in time, rather than space, with free surface and inertial effects influencing the boundary layer above this level. The equilibrium boundary layer nature of the near-bed flow for developing dunes is confirmed for the present tests in terms of all of: 共a兲 vertical distributions of double-averaged 共in time and space兲 longitudinal velocity; 共b兲 spatial fields of near-bed time-averaged velocities; 共c兲 vertical distributions of double-averaged turbulent and form-induced normal stresses; and 共d兲 vertical distributions and spatial fields of the components of the momentum balance for the flow. In regard to these findings, the double-averaging methodology adopted herein to describe flow over dunes is shown to provide a particularly valuable framework for deriving understanding and interpretation of these rough-bed flows. Additional interesting findings of the present work include: 共a兲 a linear double-averaged velocity distribution below dune crests; 共b兲 invariance of the bed-stress profile as dunes grow; 共c兲 increases in double-averaged Reynolds stress in the vicinity of the dune crest being balanced by equivalent negative form-induced stresses at these levels; and 共d兲 the relative contributions of each of Reynolds stress, form-induced stress, skin friction, and form drag to the overall momentum flux balance not changing significantly for developing dunes, with the sum of these components balancing the streamwise component of the weight of the fluid volume. Further research is found to be needed to complete generalization for varying dune steepness of the universal linear expression for double-averaged longitudinal velocity determined herein for the interfacial layer below dune crests. Additional work is also required to clarify both the theoretical basis of the velocity distribution in the form-induced layer above dune crests, and also a detailed universal relation that can be adopted for this region. The finding of an equilibrium boundary layer nature for near-bed flow over developing dunes is significant in its centrality to the feedback loop between flow, bed morphology, and sediment transport that controls the mechanics of sandwave development, and, more generally, controls erodiblebed development. By measuring flows over beds of mobile sediment developing natural dune profiles, work is presently focusing on additional effects on flow structure due to sediment transport and three-dimensional flow and bed morphology. It is expected that the equilibrium boundary-layer flow structure patterns identified herein will still be evident for

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these more complex systems, although flux component magnitudes may be altered 共particularly reduced兲 due to the additional flow complexities. ACKNOWLEDGMENTS

This research was partly funded by the Marsden Fund 共No. UOA220兲 administered by the Royal Society of New Zealand. The authors are grateful to J. Best, D. Lyn, G. Parker, and M. García for useful discussions. The guidance of the anonymous reviewers to improve the paper is gratefully acknowledged. 1

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20

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