Flow separation and roughness lengths over large bedforms in a tidal environment: A numerical investigation

Continental Shelf Research 91 (2014) 57–69

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Research papers

Flow separation and roughness lengths over large bedforms in a tidal environment: A numerical investigation A. Lefebvre a,n, A.J. Paarlberg b, V.B. Ernstsen c, C. Winter a a

MARUM – Center for Marine Environmental Sciences, Leobener Str., 28359 Bremen, Germany HKV Consultants, Lelystad, The Netherlands c Department of Geosciences and Natural Resource Management, University of Copenhagen, Denmark b

art ic l e i nf o

a b s t r a c t

Article history: Received 20 December 2013 Received in revised form 11 August 2014 Accepted 2 September 2014 Available online 16 September 2014

This study characterises the shape of the flow separation zone (FSZ) and wake region over large asymmetric bedforms under tidal flow conditions. High resolution bathymetry, flow velocity and turbulence data were measured along two parallel transects in a tidal channel covered with bedforms. The field data are used to verify the applicability of a numerical model for a systematic study using the Delft3D modelling system and test the model sensitivity to roughness length. Three experiments are then conducted to investigate how the FSZ size and wake extent vary depending on tidally-varying flow conditions, water levels and bathymetry. During the ebb, a large FSZ occurs over the steep lee side of each bedform. During the flood, no flow separation develops over the bedforms having a flat crest; however, a small FSZ is observed over the steepest part of the crest of some bedforms, where the slope is locally up to 151. Over a given bedform morphology and constant water levels, no FSZ occurs for velocity magnitudes smaller than 0.1 m s
1; as the flow accelerates, the FSZ reaches a stable size for velocity magnitudes greater than 0.4 m s
1. The shape of the FSZ is not influenced by changes in water levels. On the other hand, variations in bed morphology, as recorded from the high-resolution bathymetry collected during the tidal cycle, influence the size and position of the FSZ: a FSZ develops only when the maximum lee side slope over a horizontal distance of 5 m is greater than 101. The height and length of the wake region are related to the length of the FSZ. The total roughness along the transect lines is an order of magnitude larger during the ebb than during the flood due to flow direction in relation to bedform asymmetry: during the ebb, roughness is created by the large bedforms because a FSZ and wake develops over the steep lee side. The results add to the understanding of hydrodynamics of natural bedforms in a tidal environment and may be used to better parameterise small-scale processes in largescale studies. & 2014 Elsevier Ltd. All rights reserved.

Keywords: Tidal dunes Recirculation zone Numerical modelling

1. Introduction In shallow water environments, the movement of sandy sediment under the action of currents and waves commonly generates rhythmic wavy features on the bed. The size and shape of these bedforms is usually considered to depend on hydrodynamic forcing, water depth and sediment size (Ashley, 1990). In estuaries, tidal inlets and rivers where the currents are strong and there is a high availability of sand, bedforms are commonly large, with a complex three-dimensional morphology involving crest line bifurcations and lateral variations of bedform dimensions, presence of superimposed bedforms and along and across-bedform sediment variations

n

Corresponding author. Tel.: þ 49 421 218 65680. E-mail address: [email protected] (A. Lefebvre).

http://dx.doi.org/10.1016/j.csr.2014.09.001 0278-4343/& 2014 Elsevier Ltd. All rights reserved.

(Dalrymple and Rhodes, 1995). Furthermore, these bedforms usually present some degree of asymmetry (having both a gentle and a steep side, Fig. 1a). Flow over large asymmetric bedforms has been widely studied (see reviews by Best, 2005; Venditti, 2013). Bedforms exert a strong influence on the flow within what is here referred to as the “form-influenced flow field”. Over asymmetric bedforms having a steep lee slope, this form-influenced flow field consist of a Flow Separation Zone (FSZ) and associated wake region (Fig. 1b). The FSZ is composed of a recirculating eddy generated by the strong pressure gradient over the bedform steep lee slope. A wake region, characterised by high turbulence intensities, grows along the upper boundary of the FSZ and extends downstream. The turbulence generated by the shear layer bounding the flow separation zone is dissipated in this region. The resistance exerted by bedforms on the flow is generated within the form-influenced flow field through energy loss due to

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A. Lefebvre et al. / Continental Shelf Research 91 (2014) 57–69

EBB

FLOOD crest

flood lee side e

trough

ee

le sid

st

trough

gent

p

sid

e

ebb lee side

form-influenced flow field flow wake region

sh

ear layer

crest

FSZ trough Fig. 1. (a) Nomenclature of asymmetric bedforms (here ebb-oriented) in a tidal environment and (b) detail of the form-influenced flow field (comprising the Flow Separation Zone (FSZ) and the wake region) during the ebb tidal phase.

turbulence in the wake region (Nezu and Nakagawa, 1993; Vanoni and Hwang, 1967). Furthermore, the form-influenced flow field is a region of complicated sediment transport patterns due to the presence of reverse flow within the FSZ, which alters the direction of bedload transport, and the generation of coherent flow structures along the shear layer, which controls the suspension of sand over large dunes (Kostaschuk, 2000; Kwoll et al., 2013; Venditti and Bennett, 2000). Therefore, a good knowledge of the dynamics of the form-influenced flow field is necessary for the understanding and modelling of hydro- and sediment dynamics in coastal environments where bedforms are present. Bedforms with lee side angles close or equal to the angle-ofrepose ( 301) commonly occur in unidirectional flow and the form-influenced flow field over such bedforms has been widely studied, especially through laboratory studies (Bennett and Best, 1995; Engel, 1981; Fernandez et al., 2006; McLean et al., 1999; Nelson et al., 1993; Venditti and Bennett, 2000). Over such bedforms, a permanent flow separation zone develops which results in an extended wake region showing high turbulence production and dissipation. However, bedforms with a lee side angle smaller than the angle-of-repose have also been observed in fluvial (e.g., Best and Kostaschuk, 2002) and tidal environments (e.g., Lefebvre et al., 2011b). The FSZ over lee sides gentler than the angle-ofrepose is thought to be non-existent or intermittent. However, the exact slope at which the FSZ becomes permanent has not yet been determined. Paarlberg et al. (2009) assumed flow separation as permanent for slopes larger than 101. Kostaschuk and Villard (1996) suggested that intermittent flow separation only occurs for lee side slopes smaller than 191, and Best and Kostaschuk (2002) found that intermittent flow separation was present for about 4% of the time over bedforms with maximum lower lee side slopes of 141. Despite flow reversal in tidal environments, large bedforms usually maintain their asymmetry, typically being oriented with the residual flow direction (e.g. Bartholdy et al., 2002; Ernstsen et al., 2006). Therefore, lee sides may be steep or gentle depending on the tidal phase (Fig. 1a) which implies that the presence of a FSZ depends on flow direction (Lefebvre et al., 2013a). Furthermore, the angle of the steep side of tidal bedforms is often smaller than the angle-of-repose, typically between 101 and 201, i.e., in the range of angles over which the presence of a permanent FSZ is still under debate. In addition, the influence of tidally-induced variations of flow velocity, water level and bathymetry on the form-

influenced flow field is still to be determined; in other words, the effect on the FSZ and wake region of tidal flow acceleration and deceleration, tidal range and variations of bathymetry due to sediment movement during a tidal cycle remain to be understood. It has been demonstrated that the presence or absence of a FSZ and associated wake region result in a change of hydraulic roughness, with the total flow resistance being an order of magnitude higher during the flow phase with steep lee sides, e.g. the ebb phase over ebb-oriented bedforms and vice versa (Hoitink et al., 2009; Lefebvre et al., 2011b, 2013a). Lefebvre et al. (2013a) assessed the presence of a FSZ together with estimates of total roughness along profiles covering large ebb-oriented bedforms in the Knudedyb tidal inlet (Danish Wadden Sea, North Sea). They concluded that the total hydraulic roughness was larger during the ebb, when there was evidence of a FSZ behind the large bedforms, than during the flood when no FSZ was detected over the bedforms gentle lee side. In that study the difficulty in measuring the near-bed flow and thus the evaluation of the presence and size of a FSZ in a natural environment was pointed out. Furthermore, turbulence along the transect lines could not be measured and the wake region could not be characterised. Another limitation of their study was the restriction due to flow acceleration/deceleration during the tidal cycle which reduced the amount of time during which the flow was steady enough to calculate reliable estimates of roughness. Recently it has been shown that numerical models can be used to simulate the interaction of bedforms and hydrodynamics (El Kheiashy et al., 2010; Lefebvre et al., 2014; Omidyeganeh and Piomelli, 2011; Stoesser et al., 2008). These may complement high-resolution measurements to compensate for limitations in field data as they allow simulations of near-bed flow fields, provide estimate of turbulence and can be used with tidal or steady boundary conditions. Using numerical modelling, we aim at investigating the dynamic behaviour of the form-influenced flow field and roughness lengths over large bedforms during a tidal cycle. In this paper we will (1) characterise the shape of the FSZ and wake region over natural asymmetric bedforms during a tidal cycle, (2) analyse how the shape of the FSZ and wake region vary with changing flow velocities, water levels and bathymetry and (3) explain how bedform roughness varies during the tidal cycle and with the presence/absence of a FSZ.

2. Methods 2.1. Study area and field data The Knudedyb tidal inlet channel, being  8.5 km long and  1 km wide and with an average water depth of  15 m, connects a tidal basin of the Danish Wadden Sea to the adjacent North Sea. The tides in the area are semi diurnal with a tidal range of 1.6 m on average. The tidal inlet bed is sandy and covered with compound bedforms (Lefebvre et al., 2011a): large ebb-oriented bedforms (wavelengths of several hundred metres and heights of several metres) on which smaller bedforms (wavelengths of 3–5 m and heights of 0.15–0.3 m) are superimposed. These secondary bedforms reverse directions and migrate in the direction of the tidal currents while the primary bedforms stay ebb-oriented throughout the tidal cycle. Repetitive ship-based surveys were conducted over two 700 mlong transect lines (Transect North and Transect South) crossing 3 and 4 primary bedforms (Fig. 2) with the RV Senckenberg on 17 October 2009 during a full tidal cycle (Lefebvre et al., 2013a). Flow velocity magnitudes and directions were measured using an acoustic Doppler current profiler (ADCP) operating at 1200 kHz

A. Lefebvre et al. / Continental Shelf Research 91 (2014) 57–69

59

depth (m)

-10

Transect North

A -15

B bedform 1

-20

0

100

bedform 3

bedform 2 200

300

400

bedform 4 500

600

700

depth (m)

-10

Transect South

Lander

C -15

D bedform 1

-20

0

100

200

bedform 3

bedform 2 300

400

500

600

700

distance along the transect (m) Fig. 2. Location of the field work area and bathymetry along Transects North and South showing the positions of the crest and troughs of the bedforms (filled circles and diamonds respectively); along Transect South, the triangles indicate the position of the ADVs.

and the seabed bathymetry was recorded using a high-resolution multibeam echosounder (MBES) system operating at 455 kHz. The measurements of seabed bathymetry and flow velocity were performed simultaneously while the vessel was moving against the main tidal current in order to maintain a straight course at a constant and relatively low vessel speed. A total of 16 repetitive runs were carried out over each transect line. The tidal range during the survey was 1.7 m, and maximum depth- and transectaveraged flow velocities were 0.9 m s
1 during the flood and 1.1 m s
1 during the ebb. The bathymetric data were gridded with a cell size of 0.5  0.5 m2 and bed elevation profiles were extracted from the gridded bathymetry along the transect lines. The average bathymetry of the 16 recorded transects for each transect line was used to characterise the bedform: the troughs and crests of the large bedforms were determined following Lefebvre et al. (2011b) as the lowest and highest elevation along each bedform and used to calculate bedform dimensions (Table 1). Two Nortek Vector Acoustic Doppler Velocimeters (ADVs) were fixed on a seabed observatory (Lander) deployed along Transect South close to the crest of bedform 2 (Fig. 2). The ADVs were fixed at 1.05 and 0.5 m above the bed and measured three dimensional flow velocities at 32 Hz 0.15 m below the transducer heads during 5 min-long bursts every 15 min. These data were used to calculate burst-averaged horizontal and vertical velocities and Turbulent Kinetic Energy (TKE): TKE ¼ ðu02 þv02 þw02 Þ=2 where u and v are horizontal velocities, w is vertical velocity, the bar denotes timeaveraged quantities and the prime stands for velocity fluctuations.

2.2. Numerical model Delft3D (Deltares, 2011) is a process-based open-source integrated flow and transport modelling system. In Delft3D-FLOW the 3D non-linear shallow water equations derived from the three dimensional Navier–Stokes equations for incompressible free surface flow are solved. In order to capture non-hydrostatic flow

Table 1 Bedform dimensions along the transect lines; Hb ¼ bedform height and Lb ¼ bedform length; h ¼ water depth. Mean angle (α) and maximum angle over 5 m (α) over the gentle and steep sides. Maximum and minimum Hb/h values are given for the minimum and maximum water depths during the tidal cycle. Hb (m)

Transect North Bedform 3.9 1 Bedform 2.6 2 Bedform 4.9 3 Bedform 4 4 Transect South Bedform 8.2 1 Bedform 3.4 2 Bedform 5.3 3

Lb (m)

Hb/h

Gentle side

Steep side

Minimum Maximum α (deg)

α (deg)

α (deg)

α (deg)

80

0.25

0.28

3.3

12.5

7.7

17.9

215

0.17

0.19

1.9

8.0

9.1

17.6

138

0.31

0.35

2.3

12.0

14.7

21.9

188

0.26

0.29

1.3

4.8

6.9

24.1

284

0.52

0.58

1.5

12.3

14

21.5

151

0.22

0.25

2.3

8.6

9.1

25.1

156

0.34

0.38

1.7

7.1

12.7

21.3

phenomena such as flow recirculation on the lee of bedforms, the non-hydrostatic pressure is computed by using a pressure correction technique: for every time step, a hydrostatic step is first performed to obtain an estimate of the velocities and water levels; a second step, taking into account the effect of the non-hydrostatic pressure is then carried out and the velocities and water levels are corrected, such that continuity is fulfilled (Deltares, 2011). In an earlier study (Lefebvre et al., 2014), the non-hydrostatic Delft3D modelling system had been verified in a setup, calibration and validation of the numerical model on horizontal velocities, TKE and water level data from the laboratory flume experiments of McLean et al. (1999). It proved to be able to correctly reproduce flow separation over idealised, angle-of-repose bedforms under

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A. Lefebvre et al. / Continental Shelf Research 91 (2014) 57–69

Table 2 Summary of the parameters used to set up the numerical model. Boundary at entrance

Logarithmic velocity profile

Boundary at exit Bed resistance Horizontal grid Vertical grid

Water level Roughness length z0 0.5 m Non-equidistant layers with a size of 0. 2 m between the lowest trough and the highest crest increasing gradually to 1 m in the water column Transect North: 52 layers Transect South: 58 layers 0.15 s k–ε

Time step Vertical turbulence model

flow crest

SEP

flow se

para

ze ro v

HFSZ

eloc

tion

lin

e

ity li

ne

RET

MID trough

LFSZ crest

crest FSZ SEP

unidirectional flow conditions; in particular the shape and length of the FSZ and wake region were correctly modelled (Lefebvre et al., 2014). Here the model is further verified against field data before being used to characterise the form-influenced flow field and roughness over natural bedforms during a tidal cycle. All simulations were performed on a two-dimensional vertical (2DV) plane Cartesian model grid over a fixed bed (i.e. no sediment transport). The multibeam echo sounder data were used to construct the 700 m-long model bathymetry along the Transects. The settings used to set up the model are summarised in Table 2. 2.3. Flow separation zone (FSZ) and wake region characterisations As the spatial resolution of the ship-based ADCP field data is insufficient to accurately determine the position and extent of the FSZ (Lefebvre et al., 2013a) the present study characterises the FSZ and wake region based on numerical simulation experiments. The method used to determine the presence and shape of the FSZ is adapted from Paarlberg et al. (2007) and Lefebvre et al. (2014). The FSZ is sought on the lee of each bedform for each simulation (Fig. 3). First, the profiles with negative velocity are found (horizontal velocities are set to be positive in the main flow direction) and the height of zero velocity point is calculated along each of those velocity profiles. Using all the zero velocity points, the zero velocity line is determined and parameterised. It is observed to be composed of two segments: (1) upper zero velocity line: over the slip face, i.e., from the first profile with a negative velocity until the profile where the angle of the bed becomes gentler than 121 (MID, see Fig. 3a), the zero velocity line is a straight line and is fitted with a 1st order polynomial; and (2) lower zero velocity line: over the trough and stoss side (angle of the bed profile 4
121 until last profile with negative velocity) the lower zero velocity line is fitted with a 3rd order polynomial (Lefebvre et al., 2013b). The intersection between the upper segment of the zero velocity line and the bed defines the separation point (SEP, beginning of the FSZ). The height of the separation points is estimated along each profile by calculating the height at which the integral of the velocity between the bed and that point is zero (Paarlberg et al., 2007). The flow separation line is calculated by fitting a 3rd order polynomial through the flow separation points. The reattachment point (RET) is defined as the point where the flow separation line crosses the bed. The FSZ length (LFSZ) is calculated as the horizontal distance between the separation point and the reattachment point. In case of a complicated morphology near the crest, the flow separates at the edge of the slip face and not close to the crest (highest point along the bedform, Fig. 3b); in this case, the FSZ length is determined from the highest point between the crest and the separation point (crest FSZ). The height of the FSZ (HFSZ) is calculated as the height

HFSZ RET

LFSZ

LFSZ HFSZ trough FSZ

trough Fig. 3. Schematic of the FSZ and definition of the parameters (see text for details).

between the separation point and the trough. In case the FSZ does not extend to the bedform trough, the lowest point between the separation and reattachment point (trough FSZ, Fig. 3c) is used to determine the height of the FSZ. The length of the FSZ is normalised by either the height of the flow separation zone: L0 FSZ ¼ LFSZ/ HFSZ or by the bedform height LHb0 FSZ ¼ LFSZ/Hb. The former gives an indication of the FSZ proportions and the latter refers to the more conventional way of normalising the LFSZ. The wake region is defined following Lefebvre et al. (2014) using the TKE calculated by the k–ε turbulence closure model. For each simulation, the point of maximum turbulent kinetic energy TKEmax is found. The wake region (Fig. 1) is defined as the region where TKE is at least 70% of TKEmax. The wake length Lwake here is defined as the horizontal distance between the bedform crest and the wake maximal horizontal extent; the wake height Hwake is the vertical distance between the bedform trough and the wake maximal vertical extent.

3. Model verification The numerical model has been validated with lab measurements of horizontal flow velocity and turbulence over regular bedforms for laboratory scaled bedforms only (Lefebvre et al., 2014).Thus before using the numerical model to assess the influence of varying flow velocity, water depth and morphology

A. Lefebvre et al. / Continental Shelf Research 91 (2014) 57–69

on the form-influenced flow field, model results were compared to ADCP and ADV field data. No additional model tuning of parameters and model calibration has been performed. However, the choice of a representative bed roughness length is discussed below as the simulated roughness length has a prominent influence on velocity profiles, turbulence and ultimately the length of the FSZ (Lefebvre et al., 2014). The median grain diameter (d50) along Transect South was measured to be on average 640 μm (unpublished data). The grain related roughness length thus may be estimated by z0 ¼ d50/ 12¼ 0.00005 m (Soulsby, 1997). As small (wavelengths of 3–5 m and heights of 0.15–0.3 m) secondary bedforms are present on the back of the bedforms analysed here, this grain roughness is assumed to affect the velocity profiles only in the lower 20 cm of the velocity profiles (Chriss and Caldwell, 1982); above this height, the roughness is likely to be dominated by the influence of the secondary bedforms. Lefebvre et al. (2013a) calculated z0 ¼ 0.01 m on average along the transect lines attributed to the influence of the secondary bedforms. As the vertical model resolution is 0.20 m, the effect of the secondary bedforms is assumed to be of sub-grid order, and consequently needs to be parameterised. Therefore, roughness lengths ranging from 0.00005 to 0.02 m were tested to reflect on this assumption. The model was set-up with the parameters given in Table 2 and the bathymetry recorded over Transect South (over which the ADVs were located). Open boundary conditions at the lateral upstream end of the model domain were defined as harmonic flow velocity forcing derived from a tidal analysis of ADCP data for the three main tidal constituents (M2, M4 and M8). The water level boundary at the domain exit was defined as harmonic surface elevation, derived from a tidal analysis of water level data recorded by the ADVs. For each set up, two simulations were carried out: one during the flood with the entrance boundary situated on the left and the exit boundary on the right and one during the ebb with reversed boundaries. In order to assess the effect of bed roughness, the absolute difference (AD) between the modelled and measured velocities was calculated for each ADCP and ADV measurements as AD¼ |yi
xi| where xi represents an ADCP or ADV measurement and yi the corresponding simulated parameter (horizontal or vertical velocity or TKE). The mean absolute difference (MAD) was calculated as the mean of all AD for all ADCP transects and as timeaveraged mean for each ADV. Values of MAD of 0 would imply a

61

perfect fit. However as the measurement errors were not taken into account into the values of absolute difference, a perfect fit cannot be expected; the MAD was mainly used to estimate the influence of roughness length on the model simulations in order to assess which roughness length gave closest agreement to the measurement. Overall, similar patterns of horizontal velocity are recognised on the model results and ADCP measurements with acceleration of flow above the crest and deceleration above the trough (Figs. 4 and 5). The vertical velocity patterns are also reasonably well reproduced by the model, with a strong upward flow just before the crest, especially recognisable during the flood (Fig. 4). However the ADCP vertical velocities are clearly corrupted towards the surface (high velocities). Therefore, the MAD between vertical velocities measured with the ADCP and simulated by the model in the upper 5 m of the water column was discarding. The average MAD values for all roughness values tested are 0.12 and 0.04 m s
1 for horizontal and vertical velocities respectively (Table 3). At the position of the ADVs, the influence of roughness on horizontal velocities and TKE is relatively small during the flood and much stronger during the ebb (Fig. 6). This is because the ADVs were positioned on the bedform ebb lee side which is the bedform stoss side during the flood (Fig. 2) and the influence of roughness length on velocity profiles is strongest over the lee side than over the stoss side (Lefebvre et al., 2014). The horizontal velocities recorded by the ADV are reasonably well simulated by the numerical model (MAD of 0.11 and 0.06 m s
1 for z0 ¼0.01 m for the upper and lower ADVs, Table 3). The TKE calculated from ADV data is relatively low during the flood (maximum TKE of 0.01 m2 s
2) and showed higher values during the ebb (maximum TKE of 0.04 m2 s
2). The same pattern is seen from the simulated TKE, especially at the position of the upper ADV. During the flood, the TKE values calculated from ADV data are relatively well simulated by the model (MAD of 0.001 m2 s
2 for both the upper and lower ADV for z0 ¼0.01 m) whereas the difference between measured and calculated TKE is somewhat higher during the ebb (MAD of 0.008 and 0.006 m2 s
2 for upper and lower ADV for z0 ¼0.01 m). Considering that the patterns of horizontal and vertical velocities along the transect line as recorded by the ADCP and simulated by the model are similar (Figs. 4 and 5) and that the magnitude and variations of horizontal velocity and TKE at the ADV position are well reproduced by the model (Fig. 6), the

Fig. 4. Horizontal (a and b) and vertical (c and d) velocities simulated by the model and measured by the ADCP at the time of maximum flood velocity.

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A. Lefebvre et al. / Continental Shelf Research 91 (2014) 57–69

Fig. 5. Horizontal (a and b) and vertical (c and d) velocities simulated by the model and measured by the ADCP at the time of maximum ebb velocity.

Table 3 Summary of MAD calculated between the model results and the field measurements. ADV L (lower) was sampling 0.35 m above the bed and ADV U (upper) was sampling 0.9 m above the bed. u¼ the horizontal velocity; w¼ vertical velocity; TKE ¼ turbulent kinetic energy. z0 (m)
1

0.00005

0.001

0.005

0.01

0.02

ADCP

u (m s ) w (m s
1)

0.12 0.04

0.12 0.04

0.12 0.04

0.12 0.04

0.12 0.04

ADV U

u (m s
1) TKE (m2 s
2)

0.21 0.009

0.16 0.007

0.13 0.006

0.11 0.005

0.09 0.004

ADV L

u (m s
1) TKE (m2 s
2)

0.10 0.006

0.04 0.005

0.05 0.004

0.06 0.004

0.12 0.003

numerical model is considered a suitable tool to investigate certain flow properties of the interaction of bedforms and tidal flows. Overall, the best fit between the model and field data is achieved using a roughness length of z0 ¼ 0.01 m (Table 3) which corresponds to the roughness of the secondary bedforms (Lefebvre et al., 2013a) rather than grain roughness (estimated to z0 ¼ 0.0005 m). This is likely due to the fact that the first layer above the bed of the model has a size of about the size of the grain boundary layer. Above this first layer, the influence of the secondary bedforms is thought to be dominant, and therefore the roughness length of the secondary bedforms is to be used for the model. Therefore z0 ¼0.01 m is used in the following model experiments.

4. Model experiments 4.1. Experimental setup Three experiments (Table 4) were carried out using the numerical model in order to determine the influence of varying velocity, water levels and morphology on the form-influenced flow field over the Transects North and South. In the first experiment, the mean average bed level of the 16 bathymetric transects measured with the MBES was used to define the model bathymetry along the 700 m-long domain; the water level (exit boundary) was kept constant; and the velocity (entrance boundary) was defined as harmonic varying velocity profiles. Two separate simulations were carried out for ebb and flood conditions so

that the velocity was always prescribed at the entrance boundary. In the second experiment, the bed level and velocity boundary were kept as in experiment 1 and the water level boundary (exit boundary) was defined to harmonic variation. Finally, the third experiment used the 16 bed levels calculated from the MBES data. For each bed level, a simulation was carried out using a steady velocity and water level corresponding to the velocity and water level calculated from the corresponding ADCP transect data. These simulations therefore used steady velocities and water levels and depicted the change in bed elevation due to natural sediment transport during the tidal cycle. During the field campaign, two transect lines were investigated: Transect North and Transect South (Fig. 2). The bedforms along the two transect lines differ in morphology (Table 1). Therefore, simulations were carried out over the two transects in order to investigate FSZ and wake above bedforms with as many distinct morphologies as available.

4.2. Overall difference between ebb and flood All three experiments along the two transect lines show overall similar patterns: during the ebb a FSZ with a well-developed wake region is present over the steep lee side of all bedforms (Fig. 7). The average length of the FSZ during the ebb varies between 5.7 HFSZ and 6.7 HFSZ with an average value of 6.2 HFSZ. When compared to the bedform height, the average length of the FSZ varies between 3.6 Hb and 5.8 Hb with an average value of 5.1 Hb. The FSZ generally initiates at, or shortly behind the crest and the reattachment point is situated close to the bedform trough (Fig. 7). The wake extends to a length of 8.4–14 Hb from the crest (8.8–19.1 HFSZ) and a height of 1–1.2 Hb above the bedform trough (1.2–1.8 HFSZ). The wake has an overall oval shape starting close to the separation point and extending downstream over the next bedform stoss side, without reaching down to the reattachment point or to the next bedform crest. The shapes of the FSZ and wake region are relatively similar for all bedforms except for the bedform 1 over Transect North (Fig. 7). There, the lee side shows a hump and the FSZ stops just before that hump. This results in a relatively small FSZ compared to the bedform height (LFSZ ¼ 3.6 Hb) whereas the FSZ length to height ratio is similar to that of the FSZ over the other bedforms (LFSZ ¼6.4 HFSZ). The wake region over this bedform also differs from the wake regions over the other bedforms. Although the FSZ stops at the hump, the wake extends

A. Lefebvre et al. / Continental Shelf Research 91 (2014) 57–69

63

observed during the ebb with an average value of Lwake ¼26 HFSZ. The wakes extend to a height of 1.1 Hb above the bedform trough.

z = 0.00005 m 0

z0 = 0.001 m

4.3. Influence of velocity & water level

z = 0.01 m 0

z0 = 0.02 m ADV measurements

ebb

flood

1

u (m s-1 )

0.8 0.6 0.4 0.2 0 09:00

12:00

15:00

18:00

21:00

12:00

15:00 time

18:00

21:00

2 -2

TKE (m s )

0.05 0.04 0.03 0.02 0.01 0 09:00

Fig. 6. Horizontal velocities (a) and TKE (b) calculated from the upper ADV data and from model results using varying roughness length (z0).

Table 4 Summary of the simulations carried out with details of the experiment number (Exp. number), the number of simulations per transect (for Transects North and South) for each of these experiments (number of sim. per transect), the velocity and water level boundary and the bed level used. Exp. number

Number of sim. per transect

Velocity

Water level

Bed level

1 2 3

2 2 16

Tidally-varying Tidally-varying Steady

Mean Tidally-varying Steady

Mean Mean Varying

Experiment 1 was designed to assess the effect of tidallyvarying velocity magnitude on the form-influenced flow field, i. e., whether the FSZ length and shape of the wake region vary during a tidal phase. For velocity magnitudes less than 0.1 m s
1 (until around 10 min after slack water), no flow separation zone is forming (Fig. 9a). At the beginning of the tidal phase, as the flow is accelerating, the initially small FSZ rapidly grows to reach its stable size as the average velocity magnitude reaches 0.4 m s
1 (30– 50 min after slack water). Thereafter, the size of the FSZ varies by less than 0.3% until the velocity magnitude decreases to 0.4 m s
1 (30–50 min before the following slack water). Approaching slack water, when the flow is decelerating and reversing, the flow separation zone increases due to flow reversal. This is because the flow begins changing direction towards the reattachment point where the velocities are the smallest. As the FSZ is defined from profiles containing negative velocities, i.e., velocities having a direction opposite to the main velocity direction along the domain, those profiles where the flow has already begun turning near the reattachment point are considered to belong the FSZ, resulting in a large FSZ. When the average velocity magnitude is less than 0.1 m s
1 (10–20 min before slack water) no distinct flow separation zone can be detected. The extent of the wake shows a similar pattern to that of LFSZ/HFSZ, with a small wake developing as the FSZ appears. The extent of the wake is stable for velocity magnitudes exceeding 0.5 m s
1, in the period 40–60 min after and before slack water. Experiment 2 was performed to test the influence of changing water levels on the FSZ. During the tidal cycle simulated here, the tidal range is 1.7 m and the relative bedform height (bedform height/mean water depth) varies depending on the bedform from 0.17 to 0.52 at high water and 0.19–0.58 at low water with relative height varying by 0.03 on average during the tidal cycle (Table 1). The results of Experiment 2 show that varying water levels have only a small influence on the FSZ; taking into account only the FSZ which are not affected by the change in velocity, i.e., a stable FSZ when the velocity magnitude is greater than 0.4 m s
1, the size of the FSZ varied by less than 0.5% during a tidal phase. L0 FSZ decreases with increasing water depth (Fig. 9b). Scatter in the data shows that each FSZ reacts differently to changes in water level and prevents establishing a strong correlation between water depth and normalised LFSZ/HFSZ (R2 ¼0.32). Varying water levels do not have any significant influence on the wake with the wake dimensions from Experiment 2 being different from those of Experiment 1 by less than 0.5%. 4.4. Influence of morphology

further downstream; therefore, it has a large length compared to the FSZ height (Lwake ¼18 HFSZ). During the flood, no FSZ develops behind bedforms which have a flat crest (e.g., bedforms 2 and 4 of Transect North, Fig. 8). Over bedforms with a steeper part towards the crest (e.g., bedforms 1 and 3 of Transect North, Fig. 8), a small FSZ develops. The length to height ratio of these FSZ is higher than for the ebb FSZ with an average value of LFSZ ¼7.7 HFSZ. However, they have a small length compared to the bedform height with an average value of LFSZ ¼ 1.7 Hb, which shows that these small FSZ are related to the steep bed towards the crest rather than to the whole bedform. A wake region is associated with each FSZ. The length of the wake is small compared to the bedform height (Lwake ¼ 5.4 Hb on average) but much longer compared to the FSZ height than the wake region

The third experiment, during which the high-resolution bathymetric data were used to simulate the naturally-varying morphology during the tidal cycle, was designed to test the effect of morphological changes on the FSZ. The high-resolution MBES data collected during the field campaign provided bathymetric data that show the changes in bathymetry due to sediment transport during a tidal cycle. The location of the troughs of the bedforms did not change significantly during the tidal cycle, whereas the crests were clearly altered by sediment movement, as described in detail by Ernstsen et al. (2006) along similar large bedforms in the neighbouring Grådyb tidal inlet channel. The results from Experiment 3 show that varying morphology has a stronger influence on FSZ length than varying velocity or water levels (example in Fig. 10). During the flood, both the presence and size of the FSZ

64

A. Lefebvre et al. / Continental Shelf Research 91 (2014) 57–69

Fig. 7. Horizontal velocity and TKE during maximum ebb velocity over Transect North, the arrow indicates the flow direction; (bottom) detail of the zero-velocity line (dashdotted line), flow separation line (solid line) and wake region (dashed line) over the 4 bedforms.

Fig. 8. Horizontal velocity and TKE during maximum ebb velocity over Transect North, the arrow indicates the flow direction; (bottom) detail of the zero-velocity line (dashdotted line), flow separation line (solid line) and wake region (dashed line) over the 4 bedforms.

vary due to changes in bed morphology. For instance, no FSZ develops over bedform 3 of Transect South from the results of Experiments 1 and 2 (which used the averaged bathymetry from the 16 bathymetric transects). Over the same bedform, a FSZ was present in half of the simulations of Experiment 3 and its length varied from 5.2 to 11.1 HFSZ depending on the bathymetric data

used. During the ebb, a FSZ is present over all bedforms for all bed configurations; variations of the FSZ length and wake region extent are observed between each simulation (e.g., Fig. 10). Relationships were sought between bedform geometry and FSZ dimensions over all bedforms having a FSZ, i.e., ebb and flood together. It was expected for the length of the FSZ to be influenced

A. Lefebvre et al. / Continental Shelf Research 91 (2014) 57–69

stable FSZ

g

Transect South Transect North

de

ce flo lera w tin

ing at ler low ce f de

normalised LFSZ / HFSZ

1.5

flood ebb

stable FSZ

adapting no adapting FSZ FSZ FSZ

8 6

-0.5

ac

ce l f l o erat w i

ng

ing rat ele acc flow

EBB -1

LFSZ/ Hb

1

0.5

65

0

2

FLOOD 0.5

4

1

0

velocity (ms- 1 )

0

5

10

15

1.05

8

linear regression

small primary bedform FSZ FSZ

no FSZ

6 LFSZ/ Hb

normalised LFSZ / HFSZ

αl (°)

1

4 ebb FSZ

flood FSZ

2 0.95

14

14.5 15 water depth (m)

15.5

Fig. 9. LFSZ/HFSZ normalised by the average LFSZ/HFSZ of the stable FSZ of each bedform along each transect as a function of (a) flow velocity (during the ebb velocities are negative) and (b) water depth (only the stable FSZ are displayed).

0

0

5

10

15 αl (°)

20

25

30

Fig. 11. Length of the FSZ normalised by Hb as a function of (a) the mean angle of the bedform lee side (αl ) and (b) the maximum angle of the lee side over 5 m (αl).

LFSZ / HFSZ

7 Table 5 Summary of the characteristics of the normalised length of the FSZ (LFSZ/Hb) in dependence to the mean and maximum angle of the bedform side (α and α) and of the length and height of the wake (Lwake and Hwake) as a function of the length of the FSZ.

6 5 15:00

18:00

21:00

18:00 time

21:00

LFSZ / HFSZ

7 6 5 15:00

Fig. 10. Length of the FSZ normalised by HFSZ during the ebb over bedform 1 of Transect South (a) and Transect North (b) calculated with results from Experiment 1 (varying velocity, solid line), Experiment 2 (varying velocity and water levels, bold dashed line) and Experiment 3 (varying velocity, water levels and morphology, þ ).

by the slope of the bed between the crest and the separation point as suggested by Paarlberg et al. (2007) and by the slope between the trough and the reattachment point as observed by Lefebvre et al. (2014) over idealised angle-of-repose bedforms. However, no relationships are found between LFSZ/HFSZ or LFSZ/Hb and the angle of the bed between the crest and the separation point, or between the separation point and the MID point, or the MID point and the trough, or the trough and the reattachment point (R2 o0.5 in all cases). A relationship is found between LFSZ/Hb and mean lee angle (R2 ¼ 0.91, n ¼112, Fig. 11a), showing that an overall increase in mean lee angle leads to an increase of LFSZ/Hb. However, a more meaningful relationship exists between the maximum slope (here

LFSZ/Hb ¼ 2.2 log(α) – 0.1 LFSZ/Hb ¼ 0.3 α – 2.4 α o 101, no FSZ 101o α o 151, LFSZ/Hb ¼1.7 (flood-cap FSZ) α 4151, LFSZ ¼ /Hb ¼ 5.3 (fully developed FSZ) Lwake ¼ 1.6 LFSZ Hwake ¼ 0.2 LFSZ

along 5 m, i.e., the maximum slope over a 10-points runningaverage smoothed bathymetry for each lee side) and LFSZ/Hb (R2 ¼0.86, n ¼112, Fig. 11b) which gives some insight into when it is likely to have a FSZ depending on the slope of the steepest part of the lee side (Table 5). When the average maximum slope is less than 101, no FSZ develops; when it is between 10 and 151, only a small FSZ develops (flood FSZ related to the steep part of the bed near the crest); and only when the slope exceeds 151 does a primary bedform FSZ (ebb FSZ) develop with LFSZ ¼ 5.3 Hb on average. The length and height of the wake are also affected by changes in morphology more than by changes in velocity and water levels. In fact, they are both related to the length of the FSZ (Fig. 12) with Lwake ¼ 1.6 LFSZ (R2 ¼ 0.74, n ¼83) and Hwake ¼ 0.2 LFSZ (R2 ¼ 0.90, n ¼56 during the ebb). 4.5. Roughness In order to estimate the influence of the different roughness elements (grains, secondary and primary bedforms), roughness lengths were calculated using the Law of Wall following the

A. Lefebvre et al. / Continental Shelf Research 91 (2014) 57–69

flood ebb Transect South Transect North Length wake (m)

80 60 40 20 0

0

10

20

30 LFSZ (m)

40

50

60

0

10

20

30 LFSZ (m)

40

50

60

Height wake (m)

15 10 5 0

Fig. 12. (a) Length of the wake and (b) height of the wake as a function of the FSZ length.

methodology described in Lefebvre et al. (2013a) which uses the Law of the Wall. The bed morphology influences the overlying flow in the boundary layer, where the time-averaged current velocity profile ideally displays a logarithmic distribution above the bed and is commonly described by the von Kármán-Prandtl Law of the Wall: uðzÞ ¼

un

κ

ln

z z0

ð1Þ

where u is the time-averaged current velocity at the height z above the bed, κ is the von Kármán constant (0.41), un is the shear velocity and z0 (height z at which the current velocity is zero) is the roughness length. In case several roughness elements are present, the increase of velocity as described by the Law of the Wall becomes composed of several log-linear segments, each one related to the friction induced by one scale of roughness and representing a hierarchy of boundary layers (Chriss and Caldwell, 1982; Lefebvre et al., 2013a; Smith and McLean, 1977). Using this method, the roughness of each roughness element (e.g. grain and bedforms) can be calculated; the roughness of the highest segment comprising the total roughness of all the roughness elements. Applying this method, Lefebvre et al. (2013a) characterised the roughness of the primary and secondary bedforms from the spatially-averaged ADCP data over Transects South and North. They observed that during the flood tide, the velocity profiles were composed of one log-linear segment extending from a relative height (z/h where z is height above the bed and h is total water depth) of 0.06–0.6 and having an average roughness length at times of maximum velocity of 0.02 m attributed to the secondary bedforms. During the ebb, the spatially-averaged velocity profiles were composed of two segments: the lower segment, extending from z/h of 0.07–0.25 had z0 ¼ 0.01 m being controlled by the secondary bedforms and the upper one, extending from z/h between 0.25 and 0.6 gave z0 ¼0.50 m which was related to the

resistance of the primary bedforms. It was suggested that the primary bedform roughness developed during the ebb only due to the presence of a FSZ during the ebb and its absence during the flood. From all the simulations of Experiment 3, spatially-averaged velocity profiles (excluding the area of flow reversal) are calculated. These simulations were forced with a steady velocity and water level and are therefore not influenced by flow unsteadiness (see Lefebvre et al. (2013a) for discussion of the effect of flow unsteadiness on the velocity profiles). Log linear segments are sought on the spatially-averaged velocity profiles. As for the measured profiles in Lefebvre et al. (2013a), differences are observed between the ebb and the flood velocity profiles; moreover differences are recognised between the results of Lefebvre et al. (2013a) and the results calculated from the numerical simulations. The flood velocity profiles are generally composed of 2 segments converging at a relative height of 0.09 (Fig. 13a). The roughness length of the lower segment (on average z0 ¼0.024 m, Table 6) is interpreted to be related to grain roughness, which in the present case represents the input roughness length. The roughness length of the upper flood segment is most likely controlled by the roughness of the secondary bedforms (on average z0 ¼0.093 m, Table 6), including the influence of the small FSZ detected over the steep part of the primary bedforms. It was not expected to capture roughness created by the secondary bedforms from the model simulations due to the relatively coarse grid resolution compared to the secondary bedform size (average length of 3.3 m and height of 0.16 m). However it was noted that some velocity profiles with negative velocities are observed on the lee of some large secondary bedforms. Furthermore, higher turbulence than the background turbulence is seen in areas where there are many secondary bedforms (e.g., over bedform 2 of Transect North, Figs. 7 and 8). Since bedform roughness is thought to be mainly generated within the form-influenced flow field by the FSZ and associated turbulence, it is reasonable to assume that the secondary bedforms create roughness and therefore a segmentation of the velocity profiles during the flood. The small FSZ observed over the steepest part of the primary bedforms during the flood is also likely to contribute to the secondary bedform roughness. Those FSZ are small compared to the bedform height and are therefore more likely to be associated with the secondary bedform roughness than with the primary bedform roughness. To be more precise, they should be considered as the FSZ of the large

ebb

flood

0.18

z/h

66

10-1 0.09

10-2 0.2

0.4 velocity

10-1 0.09

0.6

0.8

(ms- 1 )

10-2

0.5 velocity

1

1.5

(ms- 1 )

Fig. 13. Example of spatially-averaged velocity profiles ( þ) and fitted log-linear segments (lines) over transect North at the time of maximum velocity during (a) the flood and (b) the ebb tide.

A. Lefebvre et al. / Continental Shelf Research 91 (2014) 57–69

Table 6 Summary of the calculated roughness lengths (z0 in m, number in bracket is the standard deviation calculated for 8 profiles) of each segment of the spatiallyaveraged velocity profiles from model results in Exp3. Transect South Flood

Transect North ebb

Flood

ebb

Grain roughness z/h o0.09 0.023 (0.001) 0.018 (0.000)

0.025 (0.001)

0.017 (0.001)

Secondary bedform roughness z/h 40.09 0.09 o z/h o 0.23 0.100 (0.005) 0.016 (0.003)

z/h 40.09 0.085 (0.004)

0.09 o z/h o 0.18 0.055 (0.013)

Primary bedform roughness z/h 40.23 0.271 (0.016)

z/h 40.18 0.129 (0.010)

secondary bedforms that develop towards the primary bedform crest rather than the FSZ of the primary bedforms. During the ebb, the spatially-averaged velocity profiles are composed of 3 segments (Fig. 13b). The lower segment (z/ho 0.09), having an average roughness length of 0.017 m (Table 6) is interpreted to be related to grain roughness, as during the flood. The middle segment (0.09 oz/ho0.23 and 0.09 oz/ho 0.18 for Transects South and North, respectively) is expected to be controlled by the roughness of the secondary bedforms. Finally, the upper segment is interpreted to be related to the primary bedform roughness. It was expected, and is observed, that the primary bedform roughness over Transect South would be larger than over Transect North because the primary bedforms over Transect South are larger than over Transect North (5.6 and 3.9 m on average, respectively). Furthermore, the height of the segmentation was suggested to be related to the bedform height (Lefebvre et al., 2013a) and therefore, it is in accordance that the segmentation between the middle and higher segment happens at a larger relative height for velocity profiles from Transect South than for those from Transect North (i.e., 0.23 vs. 0.18).

5. Discussion Let us first consider the ebb FSZ which are characteristic of the primary bedforms. The length of the FSZ is generally accepted to be between 4 and 6 Hb after the range initially suggested by Engel (1981) and further confirmed by other studies (Bennett and Best, 1995; Fernandez et al., 2006; Lefebvre et al., 2014; Nelson et al., 1993; Paarlberg et al., 2007; Stoesser et al., 2008). The average FSZ length from this study is 5.1 Hb, which falls into the range 4–6 Hb. However, the studies previously cited were all conducted over simple bedform shapes (typically a sinusoidal stoss side and a straight steep lee side) with a steep lee side (301) where the flow separates at the crest. The present study deals with natural compound bedforms; and therefore, the crests are not as “sharp” and the lee sides are not as steep as in the case of artificial ones. The flow separates directly at the crest for 29% of the FSZ observed over the bedforms of the ebb simulations of Experiment 3 (varying morphology). For all the other characterised FSZ, the flow separates on average at 0.5 Hb (1.9 m) downstream of the crest. As a result, the FSZ length compared to the FSZ height (on average LFSZ ¼6.2 HFSZ) is higher than the value (LFSZ ¼ 5.2 HFSZ) proposed by Paarlberg et al. (2007), who also included bedforms with a brink point in their study. Overall, it appears that, over natural bedforms, using a value of 4–6 Hb to find the distance between the separation and the reattachment point is suitable; however, the flow is likely to separate downstream of the crest. Ideally, the exact point where the flow separates compared to the crest

67

should be measured in the field in order to confirm our results; however it is very difficult to precisely determine the position of the separation and reattachment points from field data. The FSZ that develops over the flood lee side is small compared to the primary bedform height; it is shown that they are related to the large secondary bedforms often situated at the crest of the primary bedforms rather than to the primary bedforms themselves. The relative length of the flood FSZ is larger than those of the ebb FSZ (7.7 HFSZ and 6.2 HFSZ respectively). Mazumder and Sarkar (2013) observed that the FSZ over an upward facing step is influenced by the angle of the stoss side. Therefore, the flood FSZ could be influenced by the relatively steep flood stoss side (111 on average), compared to the ebb FSZ that experienced a gentle stoss side (21 on average). However, the stoss sides investigated by Mazumder and Sarkar (2013) were of 501 and 901, much larger than those tested in this study. It is therefore not evident whether the findings by Mazumder and Sarkar (2013) are applicable to the present study. Currently we have no explanation as to why LFSZ / HFSZ of the flood are larger than those of the ebb FSZ. Once the FSZ is established (i.e. when the velocity magnitude is larger than 0.4 m s
1) variations of velocity and water level have very little influence on the FSZ within the tested range of velocity and water depth. This agrees with results from Engel (1981) and Paarlberg et al. (2007) who, based on results from a large dataset of lab experiments, did not find any relationship between length of FSZ and flow velocity or water depth. On the other hand Lefebvre et al. (2014) observed an influence of bedform relative height on L0 FSZ. However, the range of relative height tested by Lefebvre et al. (2014) (0.05 o Hb/h o0.4) was much larger than those tested here (variations of maximum 0.04 Hb/h over a single bedform during the tidal cycle, Table 1). Therefore, it can be concluded that in a micro-tidal environment (tidal range o2 m), the change in water depth during a tidal cycle is too small to influence the length of the FSZ or the extent of the wake region. It is also interesting to note that a FSZ appears only 10 min after slack water; it then takes around 30 min to grow to its stable size. Thereafter and throughout the tidal phase until 30 to 50 min before the following slack water, LFSZ/HFSZ do not vary due to changes in velocity magnitude. This implies that the region where there is a potential reverse sediment transport is stable throughout the tidal cycle. The main variations of the FSZ size are caused by changes in the bed morphology. However, no correlations are found between the slope of the bed at the bedform crest or towards the reattachment point and LFSZ/HFSZ. In fact, no relationships could be established between the slope of the bed and LFSZ/HFSZ. This may be due to the fact that the bedforms all had varying morphologies bringing scatter in the data. For instance, bedform 1 of Transect North exhibits a specific morphology (having a hump on the steep side) which results in a particular form-influenced flow field during the ebb (Fig. 7). The reattachment point is situated just at the beginning of the hump. Downstream of the hump, the lee side has an average slope of 101 with a maximum over 5 m of 151. Therefore, it was expected for another FSZ to develop after the hump (i.e., maximum slope over 5 m greater than threshold for FSZ of 101). However, no reverse flow is observed, indicating that the flow after the hump is most likely still influenced by the FSZ over the first part of the lee side (before the hump). Although the flow does not separate again, the wake is “extended” after the hump by the steep slope of the bed and a much longer wake compared to the FSZ length is observed above this bedform than over the other bedforms. Therefore, the hump “stops” the FSZ but the turbulence still gets enhanced by the steep slope of the bed after the hump. This suggests some complicated interactions between the bed morphology and the form-influenced flow field. A study investigating specifically the relationship between the form-influenced flow field and varying morphology (for example testing the

68

A. Lefebvre et al. / Continental Shelf Research 91 (2014) 57–69

influence of the number and position of brink points) could help in clarifying the control of bedform morphology on the FSZ and wake shape. The numerical experiments suggest that a FSZ develops for maximum lee side angles in excess of 101 (over a distance of at least 5 m). Although this value is smaller than those suggested by Kostaschuk and Villard (1996) and Best and Kostaschuk (2002) (191 and 141 respectively), it agrees well with the assumption of Paarlberg et al. (2009) that flow separation is permanent in time for a slope of 101 or more. However, it is crucial to make the difference between the average lee slope (i.e., the slope between the highest and lowest bed positions along the bedforms) and the maximum bed slope, here taken over 5 m. For example, Holmes and Garcia (2008) reported flow separation happening over bedforms with lee side angles of 81 which would, according to our results, be too low an angle for flow separation. However, they did not report on the steepest slope over the lee side which is likely to be more than 10 or 151 and may explain the presence of a FSZ. Kostaschuk (2000) investigated flow over symmetric dunes in Canoe Pass, Canada. He reported that although the dunes were symmetrical, they exhibited steep lower lee side slopes which were likely to be steep enough to create flow separation. When assessing whether flow is expected to separate based on morphological parameters, it is therefore essential to consider the steepest part of the lee side rather than the average lee side slope or the bedform aspect ratio (bedform height/length). The present simulations were carried out using the Delft3D modelling system, which solves the Reynolds-averaged Navier Stokes (RANS) equations. The k–ε model turbulence closure model used with the non-hydrostatic module of Delft3D-FLOW allows the simulation of a time-averaged form-influenced flow field. Although this gave some insight as to how the time-averaged FSZ shape and wake extent vary during a tidal cycle, other processes, such as coherent flow structure dynamics or the intermittency of the flow separation zone, cannot be described by the present model. Yet the transport of suspended sediment over bedforms is essentially linked to coherent flow structures (Kostaschuk, 2000; Venditti and Bennett, 2000) created by turbulent processes originating along the shear layer generated by flow separation close to the bedform crest (Omidyeganeh and Piomelli, 2011) or by intermittent flow separation over the bedform lee (Best and Kostaschuk, 2002). Of particular interest is the period when the flow separation zone is observed to develop, at the time of strong flow acceleration, in the present study around 30–90 min after slack water, as it is a critical moment for the occurrence of clouds of suspended sediment entrained by boils (Kwoll et al., 2013). The present simulations showed that at that time, the time-averaged FSZ and wake region are developing to reach their stable size. However, a numerical model showing the details of turbulent processes at that time would allow a better description and understanding of the origins and dynamics of the initiation of the flow separation zone and how this affects the transport of suspended sediment. Large eddy-simulation (LES) models, which compute the contribution of the large turbulence structures and parameterise only the effect of the smallest scales of turbulence, would allow this as they offer the possibility to investigate in details turbulent processes such as coherent flow structures (Omidyeganeh and Piomelli, 2011; Stoesser et al., 2008). However, LES simulations are expensive in terms of computational time and it is at present unfeasible to model flow over bedforms at field scale during the whole, or even part of, a tidal cycle (Piomelli and Omidyeganeh, 2013). Overall, the results from the numerical model support the findings of Lefebvre et al. (2013a) that the total roughness over Transects North and South is much higher during the ebb than during the flood due to the influence of the asymmetric primary bedforms. Flow separation zones associated with the primary bedforms develop only during the ebb over the steep lee side,

which induces a strong turbulent field, visualised by the wake region. Associated with the flow field influenced by the primary bedforms, a segmentation of the velocity profiles and a high total roughness is induced. Lefebvre et al. (2013a) observed only one segment during the flood and two segments during the ebb. This is because they applied the Law of the Wall on ADCP data in which the lower 6% of the water column (z/ho 0.06) were not reliable and could not be taken into account for analysis. Therefore, grain roughness could not be characterised due to a lack of velocity measurements near the bed. In the present study, velocity data are available close to the bed and therefore, grain roughness (here close to the input roughness length) could be measured. Although a high roughness and the presence of a primary bedform FSZ are concomitant, it does not prove that the high roughness is created directly by the FSZ presence. Best and Kostaschuk (2002) showed that a shear layer was present over symmetric bedforms that did not develop a permanent FSZ. The bedforms investigated in the present study are strongly asymmetrical with a steep ebb lee side over which a FSZ always develops and a gentle flood lee side over which no FSZ associated with the primary bedforms develops. However, it is possible that if the flood lee side had been on average steeper, a shear layer and associated turbulence such as those observed by Best and Kostaschuk (2002) may develop and a higher roughness may be created. Therefore, a systematic study investigating simultaneously roughness and varying bedform shape would help in determining if the roughness is really created by the presence of a FSZ or by simply the presence of a steep lee side even if a FSZ does not develop. Regardless of the exact processes that create a higher roughness during the ebb compared to the flood, the results of the present study support the hypothesis that bedform asymmetry in relation to flow direction (i.e., whether the bedform steep side coincides with its lee) is a crucial component of bedform roughness in tidal environments (Buijsman and Ridderinkhof, 2007; Davis and Flemming, 1991; Hoitink et al., 2009; Lefebvre et al., 2011b). Therefore, in case asymmetric bedforms are present, the total roughness experienced by the flow may vary strongly from one tidal phase to the next. This may have important implications when carrying out numerical modelling at large scale where bedforms are represented by only one grid cell. Based on the results from field measurements of Lefebvre et al. (2013a) and high-resolution modelling of the present study, total roughness potentially has to be varied between the ebb and the flood. However, this hypothesis and the sensitivity of a numerical model to variation of roughness depending on e.g., grid size compared to bedform size still have to be tested.

6. Conclusions A numerical model was applied to investigate the changes in the form-influenced flow field during a tidal cycle over two transects along large asymmetric bedforms. The model results were verified with field measurements of vertical and horizontal velocities along the transect lines and point measurements of velocity and TKE throughout the tidal cycle. The influence of roughness lengths was shown. It was concluded that velocities and turbulence patterns along the transect lines were reproduced well by the model when using a roughness length of 0.01 m. The numerical model was then used to carry out three experiments in order to test the influence of tidally-varying flow velocity, water levels and bathymetry on the form-influenced flow field. Several conclusions can be drawn from the model simulations: – A well-developed flow separation zone and associated wake region are observed on the steep lee sides of the bedforms

A. Lefebvre et al. / Continental Shelf Research 91 (2014) 57–69

–

–

–

– –

during the ebb; during the flood, no flow separation zone develops over the flat-crested bedforms, only a small flow separation zone and wake region occur locally over the steepest part of the bed near the crest of some bedforms. For a given bed configuration, constant water levels, and tidally-varying velocity, a flow separation zone forms only when velocity magnitude is greater than 0.1 m s
1 (10 min after slack water); with the accelerating tide, the flow separation zone grows to reach its stable size for velocity magnitude over 0.4 m s
1 (30–50 min after slack water), thereafter, changes in flow velocity do not influence the size of the flow separation zone or the extent of the wake until flow magnitude decreases to 0.4 m s
1 (30–50 min before slack water) at which point the flow separation zone increases due to flow reversal. For a given bed configuration, variations in water level due to the tidal wave (tidal range of 1.7 m) do not influence the size of the flow separation zone or the extent of the wake. The influence of naturally-varying morphology was tested through carrying out simulations using high-resolution bathymetry collected during the tidal cycle. No correlations are found between the length of the flow separation zone and the angles of the bed at the crest or over the stoss side; however, the maximum slope (here calculated over 5 m) of the lee side is concluded to be a good criterion to assess the presence of a flow separation zone: for a maximum slope of less than 101, no flow separation zone develops; a small flow separation zone associated with the steepest part of the bed close to the crest rather than to the entire bedform develops for slope of 10–151; for maximum slopes greater than 151, a full size flow separation zone develops. The length and height of the wake region are related to the length of the flow separation zone. A greater total hydraulic roughness is calculated during the ebb than during the flood. It is expected to be due to the influence of the large flow separation zone developing over the steep lee side of the primary bedforms during the ebb and not during the flood; the small flow separation zones observed during the flood are thought to contribute to the roughness of the secondary bedforms and not to the roughness of the primary bedforms due to their small size compared to the primary bedforms.

Acknowledgements This study was funded through DFG-Research Center/Cluster of Excellence “The Ocean in the Earth System” and the Danish Council for Independent Research/Natural Sciences under the Project “Process-based understanding and prediction of morphodynamics in a natural coastal system in response to climate change” (Steno Grant 10-081102). Alice Lefebvre is also appreciative of the support provided by GLOMAR, Bremen International Graduate School for Marine Sciences. Yann Ferret is thanked for his enthusiasm in the early stage of the work. References Ashley, G.M., 1990. Classification of large-scale subaqueous bedforms: a new look at an old problem. J. Sediment. Res. 60, 160–172. Bartholdy, J., Bartholomae, A., Flemming, B.W., 2002. Grain-size control of large compound flow-transverse bedforms in a tidal inlet of the Danish Wadden Sea. Mar. Geol. 188, 391–413. Bennett, S.J., Best, J.L., 1995. Mean flow and turbulence structure over fixed, twodimensional dunes: implications for sediment transport and bedform stability. Sedimentology 42, 491–513. Best, J., 2005. The fluid dynamics of river dunes: a review and some future research directions. J. Geophys. Res. 110, 21.

69

Best, J., Kostaschuk, R., 2002. An experimental study of turbulent flow over a lowangle dune. J. Geophys. Res. 107, 3135. Buijsman, M.C., Ridderinkhof, H., 2007. Long-term ferry-ADCP observations of tidal currents in the Marsdiep inlet. J. Sea Res. 57, 237–256. Chriss, T.M., Caldwell, D.R., 1982. Evidence for the influence of form drag on bottom boundary layer flow. J. Geophys. Res. 87, 4148–4154. Dalrymple, R.W., Rhodes, R.N., 1995. Estuarine dunes and bars. In: Perillo, G.M.E. (Ed.), Developments in Sedimentology. Elsevier, pp. 359–422 (Chapter 13). Davis, R.A.J., Flemming, B.W., 1991. Time-series study of mesoscale tidal bedforms, Martens Plate, Wadden Sea, Germany. In: Smith, D., Reinson, G., Zaitlin, B., Rahmani, R. (Eds.), Clastic Tidal Sedimentology. Canadian Society of Petroleum Geologists, Memoir 16, Calgary (Canada), pp. 275–282. Deltares, 2011. User Manual Delft3D-FLOW. Deltares, Delft, The Netherlands. El Kheiashy, K., McCorquodale, J., Georgiou, I., Meselhe, E., 2010. Three dimensional hydrodynamic modelling over bed forms in open channels. Int. J. Sediment Res. 25, 431–440. Engel, P., 1981. Length of flow separation over dunes. J. Hydraul. Div. 107, 1133–1143. Ernstsen, V.B., Noormets, R., Winter, C., Hebbeln, D., Bartholomä, A., Flemming, B. W., Bartholdy, J., 2006. Quantification of dune dynamics during a tidal cycle in an inlet channel of the Danish Wadden Sea. Geo-Mar. Lett. 26, 151–163. Fernandez, R., Best, J., López, F., 2006. Mean flow, turbulence structure, and bed form superimposition across the ripple-dune transition. Water Resour. Res. 42, W05406. Hoitink, A.J.F., Buschman, F.A., Vermeulen, B., 2009. Continuous measurements of discharge from a horizontal acoustic Doppler current profiler in a tidal river. Water Resour. Res. 45, W11406. Holmes Jr., R.R., Garcia, M.H., 2008. Flow over bedforms in a large sand-bed river: a field investigation. J. Hydraul. Res. 46, 322–333. Kostaschuk, R., 2000. A field study of turbulence and sediment dynamics over subaqueous dunes with flow separation. Sedimentology 47, 519–531. Kostaschuk, R., Villard, P., 1996. Flow and sediment transport over large subaqueous dunes: Fraser River, Canada. Sedimentology 43, 849–863. Kwoll, E., Winter, C., Becker, M., 2013. Intermittent suspension and transport of fine sediment over natural tidal bedforms. In: Venditti, J.G., Best, J., Church, M., Hardy, R.J. (Eds.), Coherent Structures in Flows at the Earth's Surface. WileyBlackwell, London. Lefebvre, A., Ernstsen, V.B., Winter, C., 2011a. Bedform characterization through 2D spectral analysis. J. Coast. Res. SI64, 781–785. Lefebvre, A., Ernstsen, V.B., Winter, C., 2011b. Influence of compound bedforms on hydraulic roughness in a tidal environment. Ocean Dyn. 61, 2201–2210. Lefebvre, A., Ernstsen, V.B., Winter, C., 2013a. Estimation of roughness lengths and flow separation over compound bedforms in a natural-tidal inlet. Cont. Shelf Res. 61–62, 98–111. Lefebvre, A., Ferret, Y., Paarlberg, A., Ernstsen, V., Winter, C., 2013b. Variation of flow separation over large bedforms during a tidal cycle. In: Garlan, T., Van Lancker, V. (Eds.), Marine and River Dune Dynamics. MARID IV, Bruges, Belgium, pp. 169–175. Lefebvre, A., Paarlberg, A., Winter, C., 2014. Flow separation and shear stress over angle of repose bedforms: a numerical investigation. Water Resour. Res. 50, 986–1005. Mazumder, B.S., Sarkar, K., 2013. Turbulent flow characteristics and drag over 2-D forward-facing dune shaped structures with two different stoss-side slopes. Environ. Fluid Mech., 1–29. McLean, S.R., Wolfe, S.R., Nelson, J.M., 1999. Spatially averaged flow over a wavy boundary revisited. J. Geophys. Res. 104, 15743–15753. Nelson, J.M., McLean, S.R., Wolfe, S.R., 1993. Mean flow and turbulence fields over two-dimensional bed forms. Water Resour. Res. 29, 3935–3953. Nezu, I., Nakagawa, H., 1993. Turbulence in Open-Channel Flows. A.A. Balkema, Rotterdam. Omidyeganeh, M., Piomelli, U., 2011. Large-eddy simulation of two-dimensional dunes in a steady, unidirectional flow. J. Turbul., N42. Paarlberg, A.J., Dohmen-Janssen, C.M., Hulscher, S.J.M.H., Termes, P., 2007. A parameterization of flow separation over subaqueous dunes. Water Resour. Res. 43, W12417. Paarlberg, A.J., Dohmen-Janssen, C.M., Hulscher, S.J.M.H., Termes, P., 2009. Modelling river dune evolution using a parameterization of flow separation. J. Geophys. Res. 114, F01014. http://dx.doi.org/10.1029/2007JF000910. Piomelli, U., Omidyeganeh, M., 2013. Large-eddy simulations in dune-dynamics research. In: Van Lancker, V., Garlan, T. (Eds.), MARID 2013: Fourth International Conference on Marine and River Dune Dynamics. VLIZ Special Publication 65, Bruges, Belgium, pp. 15–22. Smith, J.D., McLean, S.R., 1977. Spatially averaged flow over a wavy surface. J. Geophys. Res. 84, 1735–1746. Soulsby, R.L., 1997. Dynamics of Marine Sands: A Manual for Practical Applications. Thomas Telford, London. Stoesser, T., Braun, C., García-Villalba, M., Rodi, W., 2008. Turbulence structures in flow over two-dimensional dunes. J. Hydraul. Eng. 134, 42–55. Vanoni, V.A., Hwang, L., 1967. Relation between bed forms and friction in streams. J. Hydraul. Div., 121–144. Venditti, J.G., 2013. Bedforms in sand-bedded rivers. In: Shroder, J., Wohl, E. (Eds.), Treatise on Geomorphology. Academic Press, San Diego, CA, pp. 137–162. Venditti, J.G., Bennett, S.J., 2000. Spectral analysis of turbulent flow and suspended sediment transport over fixed dunes. J. Geophys. Res. 105, 22035–22047.