ORIGINAL

Hydraulic roughness over simple subaqueous dunes

Jesper Bartholdy & Burg W. Flemming &

Verner B. Ernstsen & Christian Winter &

Alex Bartholomä

Received: 14 July 2009 /Accepted: 23 July 2009 /Published online: 14 August 2009# Springer-Verlag 2009

Abstract Detailed studies of flow over subaqueous dunesin laboratory flumes were used to suggest a virtual near-bedlayer of twice the dune height in which the mean velocity isaccelerated towards the crest by contraction. The mean flowvelocity in this layer above the crest, transformed intofriction velocity by means of the surface skin roughness, isshown to give values consistent with measured values. Theresulting dimensionless shear stress due to skin friction isdepth-independent, in contrast to that derived by means ofoften cited traditional methods. As a result of therelationship between dune height and the thickness of thenear-bed layer, an expression for the expansion loss behinddunes was formulated and used to relate form resistancedirectly to dune height.

Introduction

A general observation related to the flow over subaqueousdunes is that converging flow over the gently sloping stossside is accelerated and subject to flow separation at the

downstream crest. This results in expansion loss and eddies(e.g. Raudkivi 1966; Jackson 1976; Middleton andSouthard 1978; Engelund and Fredsøe 1982; Müller andGyr 1982; McLean and Smith 1986; McLean 1990;Kostaschuk and Church 1993; Bennett and Best 1995;Carling 2000; Schindler and Robert 2005; Sukhodolov etal. 2006), and contributes to the hydraulic resistance as so-called form drag (e.g. Engelund 1966; White et al. 1980;van Rijn 1984a, 1984b; Yang et al. 2005).

Early attempts to calculate the form-corrected bed shearstress and separate it from skin friction—related directly tothe bed surface roughness—have built on the approach ofEinstein (1950) and Einstein and Barbarossa (1952). This isthe case for the often cited method of Engelund (1966),described in detail in Engelund and Hansen (1967) andwhere, based on a so-called similarity principle, they alsodeveloped a way of relating form-corrected dimensionlessshear stress to the overall dimensionless shear stress.Another often cited method of calculating the skin frictionis that of van Rijn (1984a). Both methods use the meanflow velocity over water depth as their basis. Thisintroduces problems when working in areas where waterdepths are larger than the shallow flows in which they werecalibrated. McLean et al. (1999a) use a drag coefficient andthe velocity one bedform height above the crest to calculatethe form drag. The latter is found from a proceduresuggested by McLean et al. (1999b) which, even if itrepresents a simplification of a model originally suggestedby McLean and Smith (1986), is still rather complicated touse.

The approach presented in this paper relies on a calibratedexpression of the expansion loss (Carnot loss; Engelund 1966)in a suggested near-bed layer related to bedform height. Thethickness of the near-bed layer is found from flume studies,and supported by a field example.

J. Bartholdy (*)Department of Geology and Geography, University of Copenhagen,Øster Voldgad 10,1350 Copenhagen K, Denmarke-mail: jb@geogr.ku.dk

B. W. Flemming :A. BartholomäMarine Science Department, Senckenberg Institute,Suedstrand 40,26382 Wilhelmshaven, Germany

V. B. Ernstsen :C. WinterMARUM, University of Bremen,Leobener Str.,28359 Bremen, Germany

Geo-Mar Lett (2010) 30:63–76DOI 10.1007/s00367-009-0153-7

The primary objectives of the present paper are (1) tointroduce a method of calculating skin friction shear stresswithout relying on depth-averaged flow velocity and (2) torelate bedform dimensions to form-induced hydraulicroughness.

Materials and methods

Data sources

The flume datasets used in this study consist of thosepublished by Guy et al. (1966) and McLean et al. (1994). Inthe former case, only datasets from the widest flume werechosen (8 ft/2.43 m wide) in order to avoid effects resultingfrom too narrow boundaries. Also, only runs with cleanwater were used, avoiding those where clay had been addedto the flume. This leaves data from the following tables inthe report of Guy at al. (1966): Table 2 (pp. 62–63), Table 3(pp. 64–65), Table 4 (pp. 64–65), Table 5 (pp. 66–67) andTable 6 (pp. 68–69), of which 138 datasets are reportedwith information on bedform dimensions, 76 are classifiedas dunes or transition between dunes and plane bed upperregime, and 59 are classified as dunes alone. For thepurpose of this study, only the latter data were used, theresults being thus restricted to flow over dune-coveredbeds. The reported friction velocity was based on measuredwater slope and mean water depth. The former wasassessed by means of two continuously recording gaugesmeasuring the difference in elevation of the water surface towithin 0.0003 m. The gauge connections were 30.5 mapart. The latter was based on measured water depths atpre-selected intervals observed in the part of the flumehaving uniform slope. The selection of the intervaldepended upon the bed configuration. The reported meanvelocity was calculated by dividing the observed discharge(measured with an accuracy of ±2% by a calibrated orificemeter) by the cross-sectional area. The length, height andvelocity of the bedforms were evaluated by (1) observationsthrough the wall of the flumes marked by a grid, (2)observations in the centre of the flume by point gauge andfoot attachment and (3) observations by sonic sounder. Thenumber of observations made for each run was generallyconsidered adequate to define the mean values reported towithin 10%.

McLean et al. (1994) placed 20 rigid cement bedforms,each 0.8 m long and 0.04 m high, into a 22-m-long and 0.9-m-wide flume. Flow measurements were made over the16th bedform (numbered from upstream) using a two-component, laser-Doppler velocimeter. Silver-coated glassparticles from 10 to 40 μm in diameter were used to seedthe flow. The pressure was specified from measurements ofbottom pressure from a series of small holes drilled into the

bedform and connected through a manifold of valves to apressure transducer. The reader is referred to McLean et al.(1994, 1996) for more details.

Estimating effective bed shear stress

The bedform-induced shear stress represents the major partof energy expenditure in flow over bedforms and isgenerally not expected to participate in driving the bedload. The part participating in this bed load transport isoften referred to as the grain-related bed shear stress or theeffective bed shear stress, and is due to the skin frictionacting directly on the bed surface. Therefore, the hydraulicroughness of the bed surface and, thus, the grain size isimportant for calculating this part of the shear stress.Various methods have been proposed for calculating thegrain-related dimensionless bed shear stress (θ′), of whichthose of Engelund and Hansen (1967) and van Rijn (1984a)are often cited. Van Rijn suggests an equivalent skinroughness, kskin, to be equal to 3d90 with d90 being thegrain size of which 90% of the bed material is finer.Engelund and Hansen suggest a relation to the mean grainsize (or rather median grain size), d50, where kskin=2.5d50.This latter term agrees well with Yalin (1992) whoanalysed data of Wilson (1988, 1989) and Wilson andNnadi (1990). He found that kskin equals 2d for smallvalues of θ. Based on Yalin's analysis, it can bedemonstrated that kskin remains between 2d and 3d forflat bed values of θ<1.44, which keeps the range wellinside the dynamic regime of interest in this study.Following the idea of Fredsøe (1982), who stated that θ′can be directly related to the local Shields parameter dueto skin friction acting on the crest of bedforms (θcrest), thisparameter seems to be a logical choice when searching fora suitable dynamical parameter able to relate bedformmigration to flow dynamics.

Results

Theoretical derivations

Under the assumption that a near-bed layer exists in which,during the passage of bedforms, the mean current velocityaccelerates and decelerates according to a constant dis-charge in the layer, q′=D′V′, where V′ and D′ are meanvelocity and mean height of the layer respectively (Fig. 1),the mean current velocity over the crest of bedforms can becalculated as

V 0crest ¼

V 0D0

D0 � 0:50Hð1Þ

where H is the bedform height.

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As discussed below, the accelerated flow conditions over thebedform crest can be regarded in terms of the shear stress closeto the bed. It is well known from experiments (e.g. McLean etal. 1994) that the flow manages to establish equilibriumconditions with the surface roughness only in the lowermostpart of the water column (a few cm) over the bedform crest.Also, from detailed measurements of the velocity pattern overbedforms (e.g. Bennett and Best 1995), it is well known thatdistinct flow layers are hard to identify, these appearing asdiffuse zones which are only semi-horizontal. McLean et al.(1999b) describe the near-bed flow pattern controlled by aboundary layer developing beneath a wake region risingtowards the bedform crest downstream of the reattachmentzone. The proposed near-bed layer and the postulatedcontinuity in Eq. (1) are therefore to be regarded as “virtual”entities in the sense that they can be used to develop a workinghypothesis in which such a virtual D′ can be found to satisfymeasured conditions close to the bed at the bedform crest.

As described by, e.g. McLean et al. (1999b), the velocityin the near-bed layer over the bedform crest is distributedover depth according to the logarithmic velocity profile(e.g. Yalin 1977). If the hydraulic roughness correspondingto the skin friction of the bed is determined in agreementwith Engelund and Hansen (1967), then the frictionvelocity at this point of the bedform can be found from

V 0crest

uf crest¼ B� 2:5þ 2:5 ln D0 � 0:5Hð Þ=2:5d½ �

)

uf crest ¼ V 0crest

B� 2:5þ 2:5 ln D0 � 0:5Hð Þ= 2:5dð Þ½ � ð2Þ

where uf crest is the friction velocity at the bedform crest,and d is the mean grain size of the bed material.

The dimensionless bed shear stress due to skin frictionon the bedform crest is then

qcrest ¼ uf crest2

s� 1ð Þgd ð3Þ

Most flow conditions over the crest of bedforms belongto the transitional regime between rough and smooth flow.Therefore, the usual constant 6 added to the logarithmicterm in the theoretical velocity equation is, in Eq. (2),replaced by B-2.5 where 2.5 is a constant stemming fromintegrating the logarithmic velocity distribution. For 1.6<R<70, where R=ufks/ν in which ν is the kinematicviscosity, B can be expressed in form of a third-degreepolynomial approximation:

B ¼ 4:52þ 11:19 logR� 7:83 logRð Þ2

þ 1:59 logRð Þ3 ð4ÞThis approximation closely follows the experimental

points originally obtained by Nikuradse (1933) as plotted inYalin (1977, p. 28). For R>70, B becomes a constant equalto 8.5. Equation(2) can be solved for uf crest using Eq. (4)by means of iteration (see the example in Appendix 1).

In order to be able to work with Eq. (2), it is necessary todetermine D′. In flume studies this has been assumed to beequal to the water depth (e.g. Fredsøe 1982, Fredsøe andDeigaard 1992; Tjerry and Fredsøe 2005), and results fromdepth-limited flumes supposedly being able to be scaled upto much larger water depths in nature. Depending on grainsize, the bedforms present in flumes would indeed grow insize if the flume depth was increased under constantdynamics, but only to a much smaller size than wouldtheoretically be possible in deep water. In deeper waterflow, the correspondingly larger dunes would still scalewith water depth (at least in relatively shallow water) but,as a rule, display smaller dunes superimposed on their back(e.g. Bartholdy et al. 2002; Bartholomä et al. 2004;Ernstsen et al. 2005). In larger water depths—say, severalhundreds of metres—dunes of more than one size class canstill be present (e.g. Flemming 1978; Kuijpers et al. 2002)but the depth-scaling argument will no longer apply evenfor the largest bedforms. For the smallest bedforms, it wasshown in Bartholdy et al. (2002, 2005) that depth-independent examples varied in size with dynamics andgrain size in the same manner as reported from flumestudies, only on a somewhat larger scale. Depth-independent bedforms scaled down to about 1/4 of theirheight in water depths of about 0.3 m. This supports theidea put forward by, e.g. Middleton and Southard (1978)that all bedforms are controlled by a boundary layer andthat many such boundary layers may exist simultaneously.It can therefore not be claimed offhand that water depth isequal to the local boundary-layer thickness controlling thedunes, even in shallow flumes.

The velocity distribution over the mean bed level abovean artificial concrete-flume bedform midway betweentrough and crest, and over the crest, is shown in Fig. 2.The data are from McLean et al. (1994 , run 2). Also shown

Fig. 1 Definition sketch of an assumed (virtual) near-bed layer(between dashed lines) over bedforms and the corresponding variationin bed load and mean velocity above the mean bed level at α=0towards the bedform crest at level α=0.5

Geo-Mar Lett (2010) 30:63–76 65

is the theoretical velocity distribution based on meanconditions in the flume, and the logarithmic velocitydistribution above the bedform crest based on the frictionvelocity found from measurements in the lowest 3 mmabove the bed (McLean et al. 1996). The velocitydistribution can obviously be subdivided into two sections,as already pointed out by Smith and McLean (1977): (1) alower section, where the non-uniform flow at the mean bedlevel accelerates up the stoss side of the bedform andmanages to establish a “correct” logarithmic velocity profilevery close to the bed up to about (in this case) 2 cm abovethe bed (12 cm under the water surface); and (2) an uppersection where the overall velocity gradient created by themean conditions in the flume is more or less preservedunder acceleration of the water column induced by thechange in flow conditions near the bed. The 0.035 ms−1

added to the theoretical curve (in Fig. 2) is an arbitraryvalue chosen to illustrate this reflection. Based on theargument that primary dunes of size similar to those influme studies also form in deeper water, a reasonablesuggestion of a scaling factor for the thickness of a near-bedlayer controlling flow acceleration up the stoss side is thebedform height itself. Thus, the trends in Fig. 2 lendsupport to the idea that, by using the theoretical velocitydistribution under average conditions (which often is ouronly choice), we might be able to define the thickness of a

virtual near-bed layer relative to the bed form height, intowhich the flow accelerates and controls the shear stress atthe bedform crest. When the shear velocity associated withlogarithmic profiles based on the spatially averagedvelocity conditions is used to calculate the total boundaryshear stress, it (according to McLean et al. 1999a) isaccurate to ±30%, and corresponds to an accuracy of thevelocity of about ±15%.

In Fig. 3, the measured friction velocity after McLean etal. (1996 , run 2) is compared with friction velocities at thebedform crest using boundary layers of different thickness(D′) in Eq. (1), to obtain V′ from which uf crest was foundusing Eq. (2). The overall mean velocity in the boundarylayer was calculated by means of the logarithmic expres-sion for mean velocity over depth in rough turbulent flow(e.g. Yalin 1977):

V 0 ¼ 6þ 2:5 lnD0

ks

� �� �uf ð5Þ

with uf and ks as mean friction velocity and roughness inthe flume respectively. As shown, this experiment suggestsa boundary layer of about 2H.

If a bedform is to retain its shape, it follows from theequation of continuity that form-related bed load on thestoss side at a certain level below the crest will remain

Fig. 2 Current velocity over mean bed level at α=0 (solid thick lineto the left) and over the crest of the bedform at α=0.5 (solid thickline to the right) from McLean et al. (1994, run 2). The left stippledline describes the theoretical velocity distribution at mean bed levelunder mean conditions in the flume. The right stippled line describesthis distribution when adding 0.035 ms−1 at all depths. The thinsolid line describes the theoretical distribution above the bedformcrest based on the method outlined in McLean et al. (1996) withfriction velocity found from the measured velocity below 3 mmabove the bed

Fig. 3 Friction velocity over a bedform crest (McLean et al. 1994, run2) determined by means of the theoretical mean velocity (V′) in ahorizontal boundary layer (using average flume conditions) whichaccelerates into the contracted boundary layer at the bedform crest(V′crest=V′[D′/(D′−½H)]). The friction velocity at the crest is deter-mined by means of the reported surface skin roughness. The measuredfriction velocity at the bedform crest is based on the method outlinedin McLean et al. (1996) with friction velocity found from themeasured velocity below 3 mm above the bed

66 Geo-Mar Lett (2010) 30:63–76

constant relative to that at the crest throughout the historyof the bedform (Fig. 1). In fact, under these circumstances,the decrease in bed load varies linearly with the decrease inbed level below the crest, so that form-related transport onthe back of a bedform relates to the transport at the crest as

qba ¼ 0:5þ að Þqb crest ð6Þwhere qb α is the form-related bed load and α varieslinearly from 0 halfway between trough and crest (at orclose to the mean bed level) to 0.5 at the bedform crest.Using the bedload formula of Meyer-Peter and Müller(1948), McLean et al. (1996) investigated the theoreticalsediment transport from near-bed velocity and pressuremeasurements over their concrete dune. They found nooverall agreement between “measured” and estimatedtransport (the former inferred from the local elevation ofthe bed) but, in the upper half of the bedform, therelationship seems to be fair when the theoretical transportis based on time-averaged instantaneous local fluxes fromthe near-bed velocity profile (McLean et al. 1996, theirFig. 10.1, p. 206). Using the average shear stress forcalculating the local sediment transport on the bedformback, as in the present approach, one would expect anincreasing underestimation when moving from the cresttowards the trough. This is because turbulence, whichincreases in that direction, in combination with thenonlinear relationship between the friction velocity andthe sediment transport, will cause the actual transport rate toincrease, even if the friction velocity retains a constantmean value. If the form-related transport is estimated to beproportional to uf

2 (uf–uf c) (Bartholdy et al. 2009), thenfrom Eq. (6) we obtain:

g � ufa2 ufa � ufcð Þ ¼ 0:5þ að Þuf crest2 uf crest � ufcð Þ

)

0:5þ a ¼ guf a2 uf a � ufcð Þ

uf crest2 uf crest � ufcð Þ ð7Þ

where γ is an unknown factor related to the specific α-leveland which compensates for the turbulence-enhanced trans-port, and uf c is the critical friction velocity for initiation ofsediment transport.

The expression used as a proxy for the form-relatedbedload transport in Eq. (7) corresponds to the Meyer-Peterand Müller approach. It is similar in the sense that itprescribes a proportionality between qb and uf

3 forconditions where uf>>uf c. The critical friction velocityvaries as a function of grain size (grain Reynolds number)according to the Shields curve. The Shields curve used hereis that suggested by Yalin and Karahan (1979). At thisstage, it is not possible to say whether the adjustment of thetheoretical transport on the bedform stoss side can be

achieved by a simple factor (γ) or not. As will be shown,the use of a constant value of γ in calibrating an expressionfor the thickness of a near-bed layer turns out to have arelatively high coefficient of determination (R2), a featurewhich supports this simple approach.

While the velocity distribution at mean bed level isassumed to be controlled by the overall friction velocity androughness, the corresponding local friction velocity andShields parameter within the boundary layer (changed byacceleration up the stoss side) are currently taken to becontrolled by the skin friction at 2.5d, as stated above.Based on this, the local friction velocity and Shieldsparameter on the bedform back above the mean bed levelcan be expressed as (cf. Eqs. 2 and 3 respectively)

ufa ¼ V 0D0= D0 � aHð ÞB� 2:5þ 2:5 ln D0 � aHð Þ= 2:5dð Þ½ � ð8Þ

qa ¼ ufa2

s� 1ð Þgd ð9Þ

The mean velocity of the water flowing into theboundary layer from upstream (as illustrated above) basedon Eq. (5) expresses the mean velocity over depth in roughturbulent flow.

This procedure utilizes the logarithmic velocity distributionin non-uniform flow and, as such, may not be physicallycorrect (McLean et al. 1999a, 1999b); it is nevertheless inrelatively good agreement with the results of McLean et al.(1996, p. 216). They determined the mean bed shear stressover the investigated bedform in three different ways: (1)Reynolds stress, (2) a control volume approach incorporatingmeasured pressure, mean velocity and turbulent stresses, and(3) from the slope of a logarithmic velocity profile near thebed. Towards the crest of the bedform (from about α=0), allthe stress estimates yielded similar values.

Testing the average percentage error between measuredand calculated (Eq. 8) friction velocity at the bedform crestas a function of boundary-layer thickness (on the basis ofthe runs from McLean et al. 1996) shows that a boundarylayer of D′=2H gives the best result (Fig. 4a). As this isrelatively insensitive to the exact choice of D′ in a regionaround this value (an error of less than 5% is achieved for1.75H<D′<3H), it seems pointless to try to further calibratethis value by incorporating more flume results at this stage.All other possible test results are therefore used to test thefinal result on as many independent datasets as possible.

The boundary-layer thickness in all flume runs of Guy et al.(1966 , wide flume) which had dunes on the bed werecalculated with α=0.25 and varying values of γ in Eq. (7)until a regression between dune height and boundary-layerthickness satisfied D′=2H. The result of this procedure gaveγ=1.46. As shown in Fig. 4b, the boundary layer defined in

Geo-Mar Lett (2010) 30:63–76 67

this way correlates well with dune height (R2=0.98).Furthermore, it is located at the lower limit of the reportedwater depths, which was to be expected because the waterdepth represents the upper possible limit for the boundarylayer in nature. With a few exceptions, and within an errorwindow of about 20%, the thickness of the boundary layerderived in this way is smaller than the measured waterdepth.

Field experiment

The proposed method for predicting near-bed flow veloc-ities at bedform crests is here tested on a high-resolutiondataset acquired during a recent field experiment in the Jadetidal inlet channel (see Svenson et al. 2009 for details). Thestudied object is an about 190-m-long and 3.9-m-high dunein a water depth of about 15 m (Fig. 5). The grain size atthe dune crest was d=0.7 mm. The measurements wererecorded under quasi-stationary flow conditions aroundmaximum ebb tidal current on 10 April 2008. Currentvelocities were measured along the dune perpendicular tothe dune crest using a ship-borne Workhorse Rio Grande™(Teledyne RD Instruments) acoustic Doppler current pro-filer (ADCP) operating at 1,200 kHz, and the WinRiver™(Teledyne RD Instruments) software package for acquisi-tion. The vertical resolution of the ADCP was set to 0.2 m.The lateral resolution of the recordings was≈2 m at a pingrate of 0.5 Hz and a vessel speed of≈1 ms−1.

Fig. 4 a The average percentage error between measured and calculatedfriction velocities at the bedform crest as a function of boundary-layerthickness from two detailed flume studies (McLean et al. 1996). bRelationship between calculated boundary-layer thickness (γ=1.46 at α=0.25) and bedform height for all runs of Guy et al. (1966, wide flume,clean water) with dunes. The open symbols are the respective waterdepths. The regression (forced through origin) between dune height (H)and thickness of the boundary layer (D′) is D′=2.0H, with R2=0.98

Fig. 5 Measured and predictedvelocity distributions above adune in the Jade tidal inletchannel (German Bight, south-eastern North Sea) during max-imum ebb tidal current on 10April 2008 (WGS84 UTM32:442 030 E, 5 945 420 N). BelowLongitudinal profile of the dune.Upper right Measured andfitted velocity distribution abovethe mean bed level, midwaybetween trough and crest(mean over ±5 m). Upper leftMeasured and predicted velocitydistribution above the dune crest(mean over ±5 m)

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The mean velocity in the zone 2H above the mean bedlevel, midway between trough and crest, was fitted to thelogarithmic velocity distribution (uz=[8.5+2.5 ln(z/1.10)]0.077; R2=0.85), giving a mean velocity in the zone ofV′=0.84 ms−1. By means of Eq. (8), this gives uf crest=0.043 ms−1 and a near-bed velocity distribution as shownwith a dashed line in Fig. 5 (upper left).

Because of limitations in the ADCP technique, data fromthe last 6% of the range to the seabed can be contaminated(RDI 1996), which means that in the present case it was notpossible to measure reliable velocities closer to the bedthan≈0.9 m. This is not as close to the actual crest asdesired, and prevents the procedure from being tested forsmaller dunes. On the other hand, the example shows thatthe procedure is able to predict the near-bed velocity at thecrest within an accuracy of ±10% after acceleration overthe stoss side of the dune adding≈50% to the near-bedvelocity. This supports the proposed procedure and indi-cates that the flume-based results are not biased when testedon this type of bedform in water depths at decametre scale.

Form resistance

An important implication of a known boundary-layer thicknessrelated to the bedform height is that it enables us to calculate theexpansion loss (ΔHE) behind bedforms and, in this way, relatethe bedform dimensions to hydraulic resistance. Based on thevelocity before and after flow expansion, the so-called Carnotloss (e.g. Rouse 1950; Engelund 1966; Engelund andPedersen 1974) can be used to calculate ΔHE by means ofthe velocity in the boundary layer above the bedform crest V′1and the bedform trough V′2,

ΔHE ¼ xV 0

1 � V 02

� �22g

ð10Þ

where ξ is a constant which is dependent on the expansionangle. With a horizontal near-bed layer of D′=2H over themean bed level, the mean velocity in this layer over thebedform crest is 1.3333V′ and over the trough 0.80V′, sincethe corresponding “layer depths” are 1.5H over the crest and(more or less) 2.5H over the trough. Even if this layer isvirtual in the sense that the velocity does not reach itstheoretical value throughout the layer, the constant value of D′in relation to bedform height suggests a possible link whichcould form the basis for a calibration.

From this, Eq.(10) reduces to

ΔHE ¼ x0:01448V 02 ð11ÞIf we regard the flow in the boundary layer as isolated

(in accordance with the discussion of Fig. 2), it enables us toexpress the friction velocity related to form resistance, uf′′, bymeans of the known flow depth D′=2H, and the energy

gradient related to this resistance (I′′). From the logarithmicvelocity profile for rough turbulent flow (Eq. 5), we thenobtain

V 0

uf 00¼ V 0

2HgI 00ð Þ0:5 ¼ 6þ 2:5 ln2H

kform

� �ð12Þ

where kform is the hydraulic roughness related to the form-induced expansion loss. The roughness due to surfaceroughness is (in the general example) orders of magnitudesmaller than this.

The energy gradient related to form resistance is linkedto bedform length, L, after

I 00 ¼ ΔHE

L

which by means of Eq. (11) gives

I 00 ¼ x0:01448V 02

L

Inserting this expression for I′′ into Eq. (12), we obtain

V 02Hgx0:01448V 02

L

� �0:5 ¼ 6þ 2:5 ln 2Hkform

� �

) 0:2844xHL

� ��0:5¼ 6þ 2:5 ln 2H

kform

� �)

kform ¼ 2H

exp0:2844xH

Lð Þ�0:5�6

2:5

� � ð13Þ

As the total hydraulic roughness, ks, is expected to be thesum of form and skin roughness (e.g. van Rijn 1993), andbecause the latter—as pointed out above—can be definedas 2.5d, the final expression of ks becomes:

ks ¼ kskin þ kform ¼ 2:5d þ 2H

exp0:2844xH

Lð Þ�0:5�6

2:5

� � ð14Þ

In this equation, ξ is expected to depend on bedformsteepness. However, by using the dataset of Guy et al.(1966), the simplest and best correlated relationship wasfound by plotting ξ (isolated from Eq. 14) against theinverse steepness ratio (L/H). The result fits a simplefunction through the origin with R2=0.81 (Fig. 6):

x ¼ 0:042L

Hð15Þ

As part of the calibration procedure of the hydraulicroughness, this parameter is affected by several assump-tions, including the use of the total bedform height for

Geo-Mar Lett (2010) 30:63–76 69

the expansion width. This result should therefore not betaken as physical evidence for ξ but merely as acalibration result of the flow resistance model. It shouldbe mentioned, however, that with typical L/H values of17–28 for dunes below 1 m in height (e.g. Flemming1988; Ashley 1990), Eq. (15) gives values of ξ in aninterval between 0.7 and 1.2. This range corresponds to

realistic values of ξ for flow expansions above 12° (e.g.Engelund and Pedersen 1974). Thus, the larger expan-sion loss behind relatively long bedforms could be dueto a more fully utilized expansion height, when thebedform in front of the negative slope rises moregently (large L). When inserting ξ from Eq. (15) into Eq.(14), the surprisingly simple relationship

ks ¼ kskin þ kform ¼ 2:5d þ 0:57H ð16Þ

results, reflecting a direct relationship between bedformheight and form resistance.

In Fig. 7a, the calculated hydraulic roughness has beenplotted against the measured roughness in form of theManning number, M ¼ V= D2=3I1=2

� �. The reason for using

the Manning number is that this parameter is directlyrelated to mean velocity, in contrast to the roughnessparameter which is sensitive to small changes and of littlerelevance to the result. Also included in Fig. 7a are theresults based on the method suggested by van Rijn (1984b).These show a similar spread around the line of perfect fit,with the calculated Manning number kept within an error of±15% (with the exception of a few outliers). This is also thecase for the method suggested by Yang et al. (2005 , notshown). The open symbols are from runs with H/L above0.04 and the closed symbols from runs with steepness ratiosbelow this value (datasets of Guy et al. 1966). There doesnot seem to be any obvious tendency in the relationshipswhen discriminating on the basis of steepness. Note thatmost of the relatively high dunes in the dataset also have arelatively large steepness.

Fig. 6 The expansion loss coefficient (ξ) as a function of the relationbetween bedform length and height. The linear regression through origin(ξ=0.042L/H) has an R2=0.81. If the two datasets with large values ofL/H are removed, the regression constant changes by less than 10%.

Fig. 7 a Relationship between measured and calculated Manningnumbers, whereby M=V/(D2/3I1/2), from this study (diamonds) andfrom van Rijn (1993; circles). Open symbols represent results with H/L>0.04 and closed symbols those with H/L<0.04. b Relationshipbetween friction velocity calculated on the basis of water depth, mean

velocity over depth and hydraulic roughness based on bedformdimensions and bed material grain size from this study and from thatof van Rijn (1984b). All data points are derived from Guy et al. (1966,wide flume, clean water with dunes on the bed)

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In Fig. 7b, friction velocities derived from the hydraulicroughness calculated by the method suggested by van Rijn(1984b) and that of Eq. (16) are compared. The frictionvelocities based on the two methods correlate very well (R2=0.97), essentially giving the same result. The slightly largerfriction velocity (≈5%) obtained from the method of van Rijn(1984b) is most likely due to the exclusive selection ofresults in the current study from the wide flume of Guy et al.(1966) which more directly represent bedform roughness.Van Rijn used both flume and field data where other factorsalso influence the hydraulic roughness. For a bedformsteepness of 0.03, the two methods give identical results.

Even though it is intuitively reasonable that hydraulicroughness should increase with bedform steepness, it is unclearhow this is reflected by the algorithm of van Rijn; so, there is nogood theoretical reason why the van Rijn method should be

given preference over the much simpler approach suggestedhere. The observed variability between measured and calculat-ed hydraulic roughness suggests, however, that we still havemuch to learn about form resistance.

Discussion

The model outlined above is illustrated in Fig. 8 on thebasis of a comparison with the results of the flume study byMcLean et al. (1994 , run 2 and run 3). Flow velocity of thewater flowing into the upper end of the layer (D′=2H=0.08 m) as a function of height above the bed at mean bedlevel was calculated on the basis of the logarithmic velocityprofile with uf determined from mean flow over depth,mean depth, and a hydraulic roughness based on Eq. (16).

Fig. 8 Velocity distributions over a 0.04-m-high concrete bedformin a flume (a, b run 2, c, d run 3 from McLean et al. 1994).Measured velocity (solid line) and calculated velocity (stippledline) over mean bed level at α=0 (a, c) and over the bedform crest

at α=0.5 (b, d). Mean velocities and mean water depths are0.377 ms−1/0.284 ms−1 and 0.158 m/0.546 m respectively for run 2and run 3

Geo-Mar Lett (2010) 30:63–76 71

The calculated velocity distribution above the crest wasfound on the basis of the discharge into the layeraccelerated as a result of contraction over the bedformcrest, whereby V 0

crest ¼ V 0 2=1:5ð Þ. From this, uf crest and thevelocity distribution were calculated using the surface skinroughness alone. The surface skin roughness used here is0.0003 m, which is equal to the actual surface skinroughness suggested by McLean et al. (1994).

At the upstream end (α=0) of both runs, the amount ofwater transported into the upper end of the boundary layerat mean bed level is relatively well described by thisapproach. The measured mean velocities for the two runsover D′ at α=0 were 0.329/0.195 ms−1, and the similarlycalculated mean velocities were 0.319/0.189 ms−1 respec-tively, an error of less than 5%. This suggests that Eq. (16)describes the total roughness (form+skin) fairly well, andthat the mean velocity profile at this point of the bedform isclose to being controlled by the overall roughness andfriction velocity. At the downstream end at the bedformcrest, only the lowest part of the velocity profile manages toadjust the “new” accelerated velocity to the surfaceroughness of the bed. The velocity in the upper part lagsbehind, while the near-bed zone is close to the adjustmentwhich would be expected if the whole boundary layer wasin equilibrium with a logarithmic velocity profile. Thedetailed measurements carried out by McLean et al. (1994)thus support the assumptions on which the methodproposed in this paper is based.

In Fig. 9, the calculated friction velocities at thebedform crest (based on the method presented here) arecompared with measured values (circles) for the flumestudy described by McLean et al. (1994) and Bennett andBest (1995). The parameters used for calculating the latterwere taken from Bennett and Best (1995, their Table 2,p. 496, and pp. 506–507) with a surface skin roughness of0.00025 m. The local skin friction shear stress immedi-ately upstream of the dune crest is given as having a valueof 1.08 Pa (uf=0.0329 ms−1). The entrance parameters forthe other dataset are from McLean et al. (1994 , theirTable 1, p. 12732). The measured values were obtainedfollowing the procedure described in McLean et al.(1996). Figure 9 also includes a comparison betweenmodelled crest friction velocities from McLean et al.(1999b) and similar results obtained by means of themethod presented here using the listed total frictionvelocities and a surface skin roughness of 0.0003 m. Thefit presented in Fig. 9 has a mean error of less than 9% forall data points.

In cases where bedforms constitute the only form-relatedroughness parameter, uf crest values are directly related tothe hydraulic roughness and grain size as

uf crest ¼ A � uf ð17Þ

where the dimensionless parameter A, for different Man-ning numbers (equal to the reciprocal Manning’s n and

Fig. 10 The dimensionless factor A connecting friction velocity at thebedform crest to the total friction velocity (uf crest=A uf) as a functionof grain size and Manning number. The regression line is representedby an exponential fit, A=C exp (Edphi)

Fig. 9 Results from measured friction at the bedform crest (McLeanet al. 1994; Bennett and Best 1995) circles and modelled friction at thebedform crest (McLean et al. 1999b) diamonds, compared with resultsof the method described in this study

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25.4/ks1/6) correlates with an exponential function of the

grain size measured in phi (Fig. 10),

A ¼ C exp Edphi� � ð18Þ

where the dimensionless constants C and E are linearfunctions of the Manning number,

C ¼ 0:01495M þ 0:1361 ð19Þ

E ¼ �0:00196M � 0:02691 ð20Þ

This procedure, described by Eqs. (17) through (20),can be used as a quick reference and is valid for mostpractical purposes. The term A is slightly dependent onthe kinematic viscosity (here taken as 10−6m2s−1), andmore precise calculations should use the proceduredescribed in Appendix 1.

The values of θcrest calculated by means of thesuggested method for the dataset of Guy et al. (1966),in comparison with the dimensionless shear stress due toskin friction proposed by van Rijn (1984a, θ′vR),

Engelund (1964, θ′E) and Engelund and Hansen (1967,θ′E&H), are illustrated in Fig. 11a–c. There is anextremely good correlation between θcrest and the twoother primary methods (Fig. 11a, b), whereas thecorrelation is slightly weaker for the indirect method ofEngelund and Hansen (1967; Fig. 11c). The goodcorrelation between the other methods and the methodsuggested here justifies the derivation of a directrelation between them (θ′vR=0.68θcrest and θ′E=0.73θcrest) but raises the question of what is achieved byyet another method of form correction of the bottomshear stress. The answer is twofold: (1) by aimingdirectly at the point where the skin friction-related shearstress is generated (the bedform crest)—instead ofaccepting the more diffuse θ′ concept—it greatly sim-plifies attempts of developing a more accurate bedformmigration model (Bartholdy et al. 2009) and (2) theproposed model has the advantage of being independentof water depth.

At this stage, it must be reiterated that the two otherdirect methods are not depth-independent. Their approachesrely on mean velocity over water depth, which can notsimply be transferred from shallow-water environments in

Fig. 11 Relationships betweendifferent methods of calculatingthe dimensionless shear stress. Ina–c, calculations were conductedon the basis of the selected datasetfrom Guy et al. (1966). a Thisstudy (θcrest) versus that of vanRijn (1984a; θ′vR, R

2=0.98,Δ (slope of regression linethrough origin)=0.67). b Thisstudy versus that of Engelund(1964; θ′E, R

2=0.97, Δ=0.73).c This study versus that of Enge-lund and Hansen (1967; θ′E&H,R2=0.73, Δ=0.81). d All fourmethods as a function of waterdepth with θ=0.50, d=0.430•10−3m, H=0.24 m,L=8.22 m. The two horizontallines at the bottom of the diagram(cf. arrows) refer to θ′E and θ′vRtransformed from θcrest based onthe regressions given above. Theform-related roughness is 0.137 mbased on the method of van Rijnas well as on the method sug-gested in the present study. Thisroughness served to calculate themean depth-averaged velocityused to calculate θ′E and θ′vR

Geo-Mar Lett (2010) 30:63–76 73

which they were tested to deepwater flows where waterdepth demonstrably does not correlate with dune height(e.g. Flemming 2000). As shown in Fig. 11d, for the samenear-bed conditions, increased water depth significantlyincreases the value of θ′ when based on the older methods.Sediment transport models developed on the basis of theseform-corrected bed shear stress expressions, and calibratedbased on flume studies, should therefore be expected tooverestimate transport when used under “natural” deeperwater conditions. Presumably, when used under such flowconditions, models depending on θ′vR and θ′E would givemore realistic results if they were based on θcrest andsubsequently reduced to θ′vR and θ′E by the above-statedrelations. An example applying the suggested method isshown in Appendix 1.

Conclusions

The following conclusions are drawn on the basis of thisstudy of flow over simple subaqueous dunes:

– The hydraulic form roughness (kform) is directly relatedto dune height, H, as kform=0.57H.

– In the method outlined in this paper, the additionalsurface roughness related to the surface of moving sandgrains is adequately described by ks=2.5•d50, assuggested by Engelund and Hansen (1967).

– In a (virtual) flow layer over mean bed level twice as highas the dunes (D′=2H), the mean velocity midway betweentrough and crest (V′) is distributed in accordance with thelogarithmic velocity distribution and is controlled by thecombined hydraulic roughness, ks=kskin+kform.

– The local friction velocity at the dune crest, uf crest, canbe found from an acceleration of the mean velocity inthe near-bottom flow layer related to dune geometry, asuf crest=V′crest/[6+2.5 ln(1.5H/kskin)], in which V′crest=V′(2.0/1.5). In the case of transitional flow, the constant 6added to the logarithmic term in this equation isexchanged with B-2.5 where B=4.52+11.19 logR−7.83 (logR)2+1.59 (logR)3.

– The friction velocity calculated in this way, and theassociated dimensionless shear stress acting on thecrest of simple subaqueous dunes, are suggested asadequate (depth-independent) measures of the form-corrected bed shear stress in flow over simple sub-aqueous dunes.

Acknowledgements This study was carried out under a projectfunded by the Danish Research Council for Nature and Universe:grant FNU (272-06-0225). We greatly appreciate the generosity ofSteve McLean for putting his detailed flume data as well as modelledresults at our disposal. The manuscript benefited from valuablecomments by two anonymous referees for which we would like toexpress our gratitude.

Appendix 1

ExampleIn an ebb-dominated tidal channel with a corresponding

flow velocity 4 m above the bed of 1.0 ms−1, the bed iscovered by depth-independent simple subaqueous dunes.Their mean heights (H) and lengths (L) are 0.24 m and10.10 m respectively. The bedforms are developed in sandwith a mean grain size of 0.430 mm and in a mean waterdepth of approximately 12 m. These data are consistent withthe dynamics and bedform dimensions in the tidal inletGrådyb on the Danish west coast where these simplebedforms are superimposed on very large compound dunes(L≈100 m; Bartholdy et al. 2002, 2005). The flow is regardedas being in equilibrium with the dimensions of the simpledunes on the upper stoss side of the compound features.

Surface skin roughness, kskin=2.5d=2.5•0.430•10−3=

1.08•10−3mForm roughness, kform=0.57H=0.57•0.24=0.137 mOverall hydraulic roughness, ks=0.138 mFriction velocity, uf ¼ u4= 8:5þ 2:5 ln 4=ksð Þ½ �

¼ 1:0= 8:5þ 2:5 ln 4=0:138ð Þ½ � ¼ 0:0591m

Mean velocity into the layer, D0 ¼ 2H ;V 0 ¼ 6þ 2:5½ln 2H=ksð Þ�uf

¼ 6þ 2:5 ln 0:48=0:1379ð Þ½ �0:0591 ¼ 0:539m s�1

Mean velocity in the virtual layer 1.5H above the crest(2H above the mean bed level),

V 0crest ¼ V 0 � 2=1:5¼ 0:539 � 21:5 ¼ 0:719m s�1

Friction velocity at the crest, ¼ ufcrest= 6þ 2:5 ln 1:5H=ð½:kskinÞ�uf crest ¼ 0:719

6þ 2:5 ln 0:36

1:075 � 10�3

� � � ¼ 0:0350m s�1

A test with ν=10−6m2s−1 gives R=uf•2.5d/ν=15.05. Asthis is <70, the flow has to be adjusted for conditions in thetransitional regime, exchanging the constant 6 in theequation above with B-2.5:

B ¼ 4:52þ 11:19 logR� 7:83 logR2� �þ 1:59 logRð Þ3

First iteration gives B=8.93 and the constant is replacedby 8.93−2.5=6.43. After a small number of iterations, aconstant value of B=8.95 and of uf crest=0.0343 areobtained.

Dimensionless shear stress at the crest, qcrest ¼uf crest2

s� 1ð Þgd½ �

¼ 0:03432

1:65 � 9:82 � 0:430 � 10�3� � ¼ 0:17

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List of symbols

A Empirical dimensionless constantB Constant in the logarithmic velocity equation

(dimensionless)C Empirical dimensionless constantD Flow depth (m)D′ Height of the boundary layer above mean bed

level (m)d Mean grain size of sediment transported at the bed

(m)d50 Median grain size of sediment transported at the

bed (m)dphi -log10grain size in mm/log102 mm (dimensionless)E Empirical dimensionless constantH Bedform height (m)I Slope of the water surface (energy line)

(dimensionless)I′′ Energy gradient related to form resistance

(dimensionless)ks Overall hydraulic roughness (m)kskin Surface skin hydraulic roughness (m)kform Form-related hydraulic roughness (m)L Bedform length (m)M Manning number, 1/n, where n is the Manning n

(m1/3s−1)V Mean velocity over depth (m s−1)V′ Mean velocity in the near-bed boundary layer

(m s−1)V′crest Mean velocity in the near-bed boundary layer over

the bedform crest (m s−1)q′ Discharge per unit width in the near-bed boundary

layer (m2s−1)qb Form-related bedload transport (m2s−1)qb crest Form-related bedload transport at the bedform

crest (m2s−1)R Reynolds number (R=uf•ks/ν) or correlation

coefficient (dimensionless)uf Friction velocity (m s−1)uf′′

Friction velocity related to form resistance (m s−1)

uf crest Friction velocity at the bedform crest (m s−1)uf c Critical friction velocity for initiation of sediment

transport (m s−1)ΔHE Expansion loss (Carnot loss) behind bedforms (m)θcrest Dimensionless shear stress at the bedform crest,

this studyθ′ Dimensionless shear stress due to skin frictionθ′E Dimensionless shear stress due to skin friction

proposed by Engelund (1964)θ′E&H Dimensionless shear stress due to skin friction

proposed by Engelund and Hansen (1967)θ′vR Dimensionless shear stress due to skin friction

proposed by van Rijn (1984a)s Relative density of sediment (ρs/ρ)g Acceleration due to gravity (m s−2)α Location parameter varying linearly with level

above mean level. At mean bed level α=0 and atthe bedform crest α=0.5 (dimensionless)

β D′/H (dimensionless)Δ Slope of regression line through originρ Density of water (kg m−3)ρs Density of sediment (kg m−3)γ Bedload-enhancing factor (dimensionless)ξ Constant in the Carnot formula dependent on the

expansion angleν Kinematic viscosity (m2s−1)

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