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Wave––current interaction at an angle 1: experimentPradeep C. Fernando a , Junke Guo b & Pengzhi Lin ca Department of Civil Engineering, National University of Singapore, 10 Kent RidgeCrescent, Singapore, 117576 E-mail: pradchamjerda@yahoo.com.sgb Department of Civil Engineering, University of Nebraska-Lincoln, 1110 S 67th St,Omaha, NE, 86182, USA E-mail: jguo2@unl.educ State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University,Chengdu, Sichuan, 610065, China

Available online: 08 Jul 2011

To cite this article: Pradeep C. Fernando, Junke Guo & Pengzhi Lin (2011): Wave––current interaction at an angle 1:experiment, Journal of Hydraulic Research, 49:4, 424-436

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Research paper

Wave–current interaction at an angle 1: experiment

PRADEEP C. FERNANDO, Department of Civil Engineering, National University of Singapore, 10 Kent RidgeCrescent, Singapore 117576.Email: pradchamjerda@yahoo.com.sg

JUNKE GUO (IAHR Member), Department of Civil Engineering, University of Nebraska-Lincoln, 1110 S 67th St,Omaha, NE 86182, USA.

Email: jguo2@unl.edu

PENGZHI LIN (IAHR Member), State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan

University, Chengdu, Sichuan 610065, China.

Email: cvelinpz @scu.edu.cn (author for correspondence)

ABSTRACTThis research presents an experimental study in a spatial wave basin for nonlinear interaction of regular waves with a perpendicular shear current over amovable bed. Detailed measurements were collected for current velocity profiles under different conditions, from which the bed shear stress and appar-ent roughness heights were determined. These combined with the measured velocities were used to validate theoretical models for wave–current inter-action at an angle. It is found that for small wave heights all models agree well with the measured mean current velocities in the combined flow, while forlarger wave heights only two models are reasonably close to the measurements for current velocities. For very large waves none of the models ade-quately describes the near-surface current velocities, which deviate significantly from the log-law. For the bed shear stress and the apparent roughness,only one model describes the measured data well.

Keywords: Bed roughness, bed shear stress, theoretical model, velocity distribution, wave–current interaction

1 Introduction

Waves in coastal environments generally co-exist with currents,

where the bed shear stresses and flow fields differ significantly

from these under pure waves or pure currents (Grant and

Madsen 1979, Kemp and Simons 1982, 1983, You et al. 1991,

Huang and Mei 2003). For example, under a wave–current

condition more sediments can be picked up from the bottom

and transported to other places than under a current or waves

alone.

During the past four decades researchers contributed to the

understanding of the mechanism of wave–current interactions.

The first orthogonal wave–current experiment was reported by

Bijker (1967). This experiment was conducted in a 27 m long

and 17 m wide wave basin, where the combined wave–current

flow tests were carried out over both fixed and movable beds.

Detailed velocity profiling was not attempted, however. The

bed shear stress was found by means of the energy gradient

and the bedload transport was measured using a sediment trap.

Based on these measurements, Bijker (1967) proposed empirical

equations for bed shear stress and bedload transport for the

combined wave–current flow. The first theoretical model was

proposed by Grant and Madsen (1979) who described the

wave–current interactions at an arbitrary angle with a time-

invariant eddy viscosity that linearly varies within and outside

a wave–current boundary layer, concluding that wave presence

increases the bottom roughness. This approach was further

developed by Tanaka and Shuto (1984), Christoffersen and

Jonsson (1985), Grant and Madsen (1986), and Myrhaug and

Slaattelid (1990). Fredsøe (1984) discussed this topic using a

momentum defect method. Jose et al. (2003) used a numerical

turbulent-closure model to determine the bottom shear stress in

the wave–current interaction and proposed parameterization

for time-series shear stress. These models were tested with

Journal of Hydraulic Research Vol. 49, No. 4 (2011), pp. 424–436

doi:10.1080/00221686.2010.547036

# 2011 International Association for Hydro-Environment Engineering and Research

Revision received 6 December 2010/Open for discussion until 29 February 2012.

ISSN 0022-1686 print/ISSN 1814-2079 onlinehttp://www.informaworld.com

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flume experiments under the acting angles of 0o or 180o, but they

were not validated for arbitrary acting angles, because neither the

experimental data were detailed enough (van Rijn and Havinga

1995, Khelifa and Ouellet 2000, Andersen and Faraci 2003)

nor the flow conditions compatible with the model assumptions

(Visser 1986, Arnskov et al. 1993).

Experimental studies on wave–current interactions were con-

ducted in wave flumes with fixed beds (Bakker and van Doorn

1978, Kemp and Simons 1982, 1983, Mathisen and Madsen

1996a, 1996b, 1999, Fredsøe et al. 1999, Monismith et al.

2007) and movable beds (Bijker 1967, Nieuwjaar and van der

Kaaij 1987, Nap and van Kampen 1988, van Rijn et al. 1993,

Marin 1999). These indicated that (i) the apparent roughness

increases with increasing wave height, (ii) sediment suspension

by pure waves is much higher than that by pure currents, and

wave-related transport is much less than current-related

transport, and (iii) combining waves and a current significantly

increases the sediment transport capacity. These experiments,

however, do not reflect the real situation on a continental

shelf, where waves and currents normally interact at a certain

angle.

Several experiments of wave–current interactions at an angle

f in wave basins with fixed beds are available (Visser 1986, f ¼

908, Sleath 1990, f ¼ 908, Arnskov et al. 1993, f ¼ 728, 908,1088, Musumeci et al. 2006, f ¼ 908). Few experiments were

also reported on wave–current flows at angles over movable

beds (van Rijn and Havinga 1995, f ¼ 608, 908, 1208, Khelifa

and Ouellet 2000, f ¼ 608, 908, Andersen and Faraci 2003, f

¼ 908, Madsen et al. 2008, f ¼ 908). These mainly concentrate

on bed ripple formation without detailed quantitative velocity

profile measurements for a wide range of combined wave and

current conditions, which are needed for the validation of theor-

etical and numerical models.

Herein a series of well-controlled measurements on an orthog-

onal wave–current flow over a movable bed in a wave basin for

different wave and current conditions was conducted. Emphasis

was made to study the wave height effect on the change of

current profiles. The objectives of the experiments are: (i) to

study the ripple development under various combined wave–

current flow conditions, (ii) to analyse the corresponding current

velocity profiles, bed shear stresses, physical and apparent rough-

ness, (iii) to carry out validations of the existing theoretical models

for wave–current interactions at an angle, and (iv) to develop a new

theoretical model to be detailed in a separate research.

2 Experimental set-up

The experiment was conducted in a 24 m long, 10 m wide, and

0.9 m deep wave tank at the Hydraulics Laboratory, National

University of Singapore. Figure 1 shows the set-up for wave–

current perpendicular interaction over movable beds.

The current was generated in a channel 7.2 m long and 2 m

wide, recirculating across the basin as shown in Fig. 1.

To minimize the effects of channel inlet, three measures were

taken as follows: (i) the inflow tank was constructed to its

maximum possible size of 1.4 m × 10 m and the PVC inlet

pipes were submerged to minimize turbulence effects, (ii) a hon-

eycomb filter was placed at the entrance to the current channel to

improve the current flow, and (iii) two baffle nets were placed

at the two inlet ends to minimize the impact of water jets.

After the current passed through the honeycomb filter, the flow

was further directed by a 2 m wide and 1.89 m long channel.

The tailgate height was adjusted to obtain a steady water depth

of h ¼ 0.35 m inside the basin.

Regular waves were generated by programmable wave

paddles. Seven capacitance-type wave gauges were used to

check the uniformity of the wave height in the basin. Four

wave gauges were placed in the zone without currents and three

were fixed on the carriage to monitor wave heights in the inter-

action area. Plywood walls were coated with resins to minimize

the side wall flow resistance for the propagating waves. The inci-

dent wave energy was effectively dissipated by the 1:4 porous

slope made of 25 mm mean-sized crushed stones. The reflection

coefficient from the slope was controlled to be less than 6%.

Furthermore, three layers of sandbags were randomly placed

along the side walls of the basin to dissipate diffracted waves.

A 10 cm thick quartz sand of d10 ¼ 0.087 mm, d50 ¼

0.220 mm and d90 ¼ 0.282 mm layer was deployed on the

bottom, and two ends of the sand bed across the basin were

supported by a 1:13 plywood sloping structure to support

smooth wave transformation from the concrete bed to the

movable sand bed.

In the experiment, 20 runs of different wave and current con-

ditions were conducted (Table 1). For all, the constant water

depth of h ¼ 35 cm was maintained and the constant wave

period T ¼ 1.5 s was used. The wave height varied from 4.40

to 18.64 cm while two depth-averaged mean current velocities

of U ¼ 10.5 cm/s and 13.5 cm/s, based on the total current dis-

charge divided by the cross-sectional area, were applied. An

Acoustic Doppler Velocimeter (ADV) was used for three-dimen-

sional velocity measurements at 25 Hz.

For each run, the combined wave–current flow was initially

conducted over the plane sand bed (undisturbed without

ripples). The ripple height and length were measured at 15, 45

and 105 min to check whether the bed ripples were fully devel-

oped. It was found that after 45 min, the ripple geometry

remained almost unchanged. Therefore, measurements of vel-

ocity and bed geometry were taken after 90 min when the bed

ripples were fully developed.

3 Data analysis and results

3.1 Flow development

The laboratory study of orthogonal wave–current interaction is a

difficult task. It requires the working area, where current and

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waves overlap, to be large enough so that both current and waves

are fully developed and adapted to each other. Herein, this area

measured 2.0 m × 4.11 m (Fig. 1), while previous similar exper-

iments used various sizes, e.g. Sleath (1990) 1.2 m × 0.81 m,

Musumeci et al. (2006) 2.5 m × 4 m, Khelifa and Ouellet

(2000) 2.8 m × 3.7 m, Madsen et al. (2008) 2.7 m × 6.6 m,

Andersen and Faraci (2003) 3.5 m × 5.5 m, van Rijn and

Havinga (1995) 4 m × 26.5 m, Arnskov et al. (1993) 8 m ×23 m, and Visser (1986) 6.5 m × 26.5 m, in which the first

length is the width of the current channel and the second the

length.

To verify that the wave field was fully-developed the horizon-

tal and vertical velocities under waves were measured and com-

pared to Stokes’ wave theory, resulting in reasonable agreement.

To prove that the current was fully-developed, both theoretical

computation and laboratory measurement were used. Based on

Schlichting (1960), the thickness dc of a developing current

boundary layer is

ln30dc

kn

( )= 0.57 2.9 + 0.69 ln

y

kn

( )[ ]1.25

(1)

where y ¼ 4.0 m is the distance from the honeycomb filter to the

measurement point A (Fig. 1) and Nikuradse’s equivalent sand

roughness height kn ¼ 0.00687 m was based on the measured

physical roughness height, resulting in dc ¼ 0.212 m. In the

data analysis, only data below 15 cm above the bed were used

to ensure that these were within the fully-developed current

boundary layer. The test run of pure current, which gave a

Figure 1 Experimental set-up: (a) plan view, (b) 3-D view of basin

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good agreement between the measured velocity and the log-law

profile, also supported this theoretical finding.

To verify the wave and current adapting to each other, the

current velocity at a certain elevation and the wave height

along and across the current channel in the working area

(Fig. 1) were measured for all 20 runs. It was found that the

current velocity remained nearly unchanged within this area.

Similarly, the uniformity of wave height distribution inside the

area was also checked. Reasonable uniformity of wave field

was obtained with a standard deviation within 10% of the

mean wave height.

3.2 Ripple development

Visual observations were first made for ripple development

under orthogonal wave–current flows with various wave

heights and current speeds. It was found that when the current

was relatively weak as considered herein, the formation of

ripples mainly depended on wave height. For small amplitude

waves of H , 8 cm, nearly two-dimensional serpentine ripples

were formed, whereas for large waves with H . 8 cm, three-

dimensional honeycomb ripples were developed (Fig. 2).

Table 1 presents the measured bed ripple parameters. It was

found that both ripple height and length increase with wave

height. For serpentine ripples, the averaged ripple height is

0.61 + 0.02 cm and the length in wave propagation direction

3.99 + 0.13 cm. For honeycomb ripples, these values increase

to 0.70 + 0.02 cm and 4.77 + 0.23 cm, respectively. These

values of ripple height and length are in the close range of the

earlier experiments with similar sediment size and experimental

conditions. For example, Khelifa and Ouellet (2000) reported an

average ripple height of 0.79 cm and length of 3.85 cm (test code

90-26 to 90-27). Madsen et al. (2008) observed an average ripple

height of 0.54 cm and length of 4.98 cm, van Rijn and Havinga

(1995) found an averaged ripple height of 0.80 cm and ripple

length of 9.00 cm (test code T7 0 90 to T14 30 90).

The second task was to estimate the hydraulic roughness of

the ripple bed after it was fully developed. This was achieved

by stopping the combined wave–current flows, and then

running a pure current over the developed bed. A small current

speed of, say U ¼ 10.5 cm/s, was sought causing neither sedi-

ment suspension nor bed load movement. The ADV was used

to take the point measurements at vertical intervals of about

1.5 cm from the bottom to 15 cm above it, at a spacing of

about 3.0 cm above 15 cm. The velocity measuring height

origin was the undisturbed sand bed. The velocity profile in a

pure current above a rough bed is described by the log-law as

u(z) = u∗k

lnz

z0(2)

where u ¼ mean velocity at distance z from the bottom, u∗ ¼

friction velocity, k ¼ 0.4 as von Karman constant, z ¼ distance

to the bottom (undisturbed sand bed), and z0 ¼ hydraulic rough-

ness. By fitting the measured mean velocity data to Eq. (2) using

linear regression, the hydraulic roughness z0 and the associated

Table 1 Test conditions and bed ripple geometry measurements

Run Measuring position (Fig. 1) H (cm) U (cm/s)

Bed ripple geometry

D (cm) l (cm) Pattern z0 (cm)

1 D 4.4 10.5 0.58 3.83

Serpentine 0.0229 (+0.0147)

2 D 5.63 13.5 0.57 3.77

3 A 6.01 10.5 0.62 3.85

4 B 7.18 10.5 0.61 4.13

5 C 7.19 13.5 0.62 4.15

6 D 7.28 10.5 0.63 4.22

7 B 7.72 13.5 0.61 3.96

8 D 8.45 13.5 0.66 4.27

Honeycomb 0.0264 (+0.0081)

9 A 8.62 13.5 0.66 4.31

10 A 10.51 10.5 0.71 4.73

11 C 11.12 13.5 0.69 4.55

12 B 11.28 10.5 0.69 4.98

13 A 11.67 13.5 0.67 4.48

14 B 13.4 13.5 0.68 4.68

15 E 14.89 13.5 0.69 4.93

16 A 15.07 13.5 0.72 4.81

17 E 15.36 10.5 0.72 4.76

18 A 17.49 10.5 0.71 5.14

19 B 18.64 10.5 0.74 5.18

20 B 18.64 13.5 0.72 5.13

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friction velocity u∗ were found (Table 1). Considering that ripple

height and length do not change significantly for either the ser-

pentine or honeycomb ripple types, a single averaged value of

z0 was defined. Further, kn ¼ 30z0 ¼ 0.792 cm for honeycomb

ripples, i.e. about 1.13 times the average ripple height, in agree-

ment with van Rijn and Havinga (1995) who found kn ¼ 0.5 to

1.5 times the bed ripple height.

3.3 Near-bed velocity profile, apparent roughness, and bedshear stress

Under combined wave–current flows, the current profile is

modified by the presence of waves due to the increase of effec-

tive bed shear stress. Previous theoretical work of Grant and

Madsen (1986), laboratory work of Kemp and Simons (1982)

and field measurement by Cacchione et al. (1987) have

suggested that the resulting near-bed current profile still

follows the log law as

u(z) = u∗c,wc

kln

z

z0a(3)

except that both friction velocity u∗c,wc and hydraulic roughness

z0a become larger. The hydraulic roughness z0a in wave–current

flows is often called the apparent roughness.

The point velocities at different levels were measured using

the ADV for the orthogonal wave–current flows. The ADV

elevation was changed with a 1.5 cm interval up to 15 cm

above the bottom and thereafter at 3.0 cm intervals up to the

elevation lying about 5 cm below the wave trough level. At

each elevation, the velocity data were recorded for 2 minutes

to capture 80 wave cycles. The mean current speed was then

derived from the time history record. The mean current speeds

are listed in the Appendix. Selected runs for U ¼ 10.5 cm/s

are plotted in Fig. 3, together with the corresponding pure

current flows. Note that in all runs the current profiles under

waves deviate significantly from the corresponding pure

current profiles. The near-bed current velocities are reduced by

wave presence, while the upper layer velocities are increased.

These effects were more pronounced for relatively weak currents

and strong waves (Fig. 3f). A reason for this effect, at least under

the rough flow regime, could be the generation of strong turbu-

lence within the thin wave bottom boundary layer (Rosales

et al. 2008), consistent with previous experiments on a rough

bed for parallel wave–current flows (Kemp and Simons 1982,

1983, Brevik and Aas 1980, van Rijn et al. 1993, Bakker and

van Doorn 1978, Supharatid et al. 1992).

With the mean velocity data the linear regression method was

used to find the best fit to the data, from which both z0a and u∗c,wc

were obtained (Table 2). Note that the lowest measurement point

is 0.8 cm above the bottom, i.e. outside of the wave boundary

layer whose thickness was estimated (Christoffersen and

Jonsson 1985) to be smaller than 0.5 cm. Since the near-free

surface current profile deviates significantly from the logarithmic

curve under wave presence, only the measurements up to 15 cm

above the bed were used in this analysis. It was found that the

apparent roughness z0a is generally an order of magnitude

larger than the hydraulic roughness z0. The bed shear stress

also increases accordingly due to the increase of near-bed

mean velocity gradient. For U ¼ 10.5 cm/s under pure current

over rippled bed, averaged pure current friction velocities were

0.647 cm/s and 0.648 cm/s for serpentine and honeycomb

patterns, respectively. The actual values of z0a and u∗c,wc

depend on both current speed and wave height.

3.4 Near-free surface velocity profile

Studies in wave flumes of Bakker and van Doorn (1978), Kemp

and Simons (1982, 1983), and Swan et al. (2001) indicate that the

near-surface current velocity is reduced by following waves, but

is increased by opposing waves. Under perpendicular wave–

current interaction, the effect of wave steepness on the near-

surface velocity profiles was studied. The presence of small

amplitude waves slightly increases the near-surface velocity

(Fig. 3a,b). As the wave amplitude increases, the near free

surface velocity increases while the velocity gradient nearby

decreases. For large waves (H/h . 0.45), the mean velocity

gradient near the surface is significantly reduced and the

maximum current velocity appears at the middle water depths

(Fig. 3f).

Similar experimental observations under wave–current flows

were reported by Nieuwjaar and van der Kaij (1987), Nap and

van Kampen (1988) and van Rijn et al. (1993). It is generally

believed that besides the bed effect, the wave-induced stress

(radiation stress) should be considered when analysing the

near-surface velocity profile (You 1996, Yang et al. 2006). The

wave-induced stress effect can be significant if wave nonlinearity

becomes strong (Lin 2008). Swan et al. (2001) observed that the

current profile in wave–current flows depends on both wave

steepness and current vorticity distribution. However, so far

there has been no satisfactory theoretical explanation for this

phenomenon, especially for wave–current interactions at an

angle.

Figure 2 Ripple development under orthogonal wave–current flows(a) serpentine type for H , 8 cm, (b) honeycomb type for H . 8 cm;ruler in wave propagation direction

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4 Validation of theoretical models

Existing wave current models can be mainly categorized into

three types, namely time-invariant eddy-viscosity models,

mixing length models, and numerical models. Eddy viscosity

models are the most popular because of their simplicity of defin-

ing the relationship between velocity field and shear stress.

Herein, the models by Christoffersen and Jonsson (1985,

briefed as CJ85), Grant and Madsen (1986, briefed as GM86),

and Myrhaug and Slaattelid (1990, briefed as MS90) were

selected for validation with the measured data for 908 wave–

current interactions. In addition to eddy-viscosity models, the

model by Fredsøe (1984, briefed as FR84), was also selected

as this is the only using the momentum defect approach. These

models were not validated for interaction angles other than 08and 1808.

4.1 Review of theoretical models

CJ85 is based on a two-layer time-invariant eddy viscosity

model, which has two forms: Model I for “large” roughness

and Model II for small. The parameter J¼ u∗c,wc/knva was

used to determine the range of model applicability, where

u∗c,wc, kn, va are current-associated bed friction velocity, Nikur-

adse roughness height and absolute angular frequency, respect-

ively. For all experimental runs J , 3.47 and therefore CJ85

Model I was used. Its eddy viscosity was assumed to be constant

within the wave–current boundary layer, while having a para-

bolic distribution above the boundary layer. Model I assumes

that: (i) the current has small Froude numbers that vary slowly

in space, (ii) the bed is locally horizontal, and (iii) the dissipation

of wave energy outside the wave–current boundary layer is neg-

ligible. The model is applicable only in the rough turbulent flow

regime, and was validated with laboratory measurements by

Figure 3 Comparison of mean current velocity distribution in presence (filled triangle) and absence (open diamond) of orthogonal waves for T ¼ 1.5 sand U ¼ 10.5 cm/s

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Table 2 Comparison of measurements and model predictions on apparent roughness and current associated bed shear stress, 95% confidence limits in brackets

Run N R (u∗c,wc)exp (cm/s) (z0a)exp (cm)

CJ85 FR84 GM86 MS90

u∗c,wc (cm/s) z0a (cm) u∗c,wc (cm/s) z0a (cm) u∗c,wc (cm/s) z0a (cm) u∗c,wc (cm/s) z0a (cm)

1 9 0.994 1.118 (+0.101) 0.234 (20.040, +0.049) 0.886 0.112 0.781 0.060 0.941 0.108 0.825 0.057

2 10 0.996 1.388 (+0.095) 0.273 (20.035, +0.041) 1.138 0.112 1.001 0.059 1.145 0.117 1.004 0.062

3 10 0.992 0.960 (+0.094) 0.150 (20.028, +0.033) 0.912 0.129 0.829 0.081 0.972 0.152 0.833 0.076

4 9 0.998 1.061 (+0.047) 0.241 (20.021, +0.023) 0.927 0.139 0.859 0.097 0.992 0.184 0.839 0.089

5 10 0.995 1.288 (+0.104) 0.174 (20.026, +0.030) 1.165 0.125 1.053 0.076 1.218 0.148 1.053 0.076

6 10 0.997 1.453 (+0.077) 0.697 (20.071, +0.080) 0.929 0.140 0.863 0.099 0.976 0.191 0.826 0.092

7 10 0.995 1.441 (+0.115) 0.307 (20.045, +0.052) 1.173 0.129 1.067 0.082 1.248 0.158 1.074 0.080

8 9 0.991 1.484 (+0.175) 0.303 (20.064, +0.082) 1.218 0.153 1.112 0.100 1.307 0.190 1.114 0.095

9 10 0.994 1.331 (+0.119) 0.222 (20.037, +0.043) 1.220 0.154 1.115 0.101 1.302 0.195 1.108 0.098

10 9 0.991 1.182 (+0.137) 0.274 (20.060. +0.078) 0.989 0.184 0.915 0.130 1.159 0.279 0.955 0.132

11 9 0.990 1.386 (+0.172) 0.198 (20.044, +0.055) 1.252 0.172 1.175 0.130 1.456 0.239 1.223 0.118

12 10 0.996 1.313 (+0.087) 0.380 (20.047, +0.055) 0.996 0.190 0.970 0.170 1.218 0.291 1.007 0.137

13 10 0.996 1.452 (+0.096) 0.321 (20.041, +0.046) 1.258 0.176 1.186 0.135 1.410 0.263 1.179 0.128

14 9 0.990 1.526 (+0.187) 0.335 (20.073, +0.093) 1.276 0.187 1.220 0.154 1.484 0.297 1.228 0.143

15 10 0.970 1.358 (+0.276) 0.190 (20.064, +0.097) 1.289 0.195 1.250 0.171 1.605 0.314 1.329 0.152

16 9 0.993 1.549 (+0.156) 0.233 (20.044, +0.054) 1.291 0.196 1.253 0.173 1.664 0.307 1.380 0.149

17 8 0.990 1.285 (+0.183) 0.444 (20.106, +0.139) 1.027 0.216 1.026 0.214 1.240 0.413 0.999 0.188

18 9 0.996 1.328 (+0.094) 0.292 (20.040, +0.047) 1.040 0.227 1.049 0.235 1.467 0.419 1.182 0.195

19 9 0.996 1.568 (+0.124) 0.519 (20.079, +0.095) 1.047 0.233 1.062 0.247 1.494 0.446 1.200 0.206

20 8 0.990 1.949 (+0.271) 0.404 (20.098, +0.130) 1.320 0.215 1.301 0.203 1.867 0.360 1.535 0.175

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Bakker and van Doorn (1978), and Kemp and Simons (1982,

1983) for parallel wave–current flows (f ¼ 08 or 1808).GM86 model is the improved model of Grant and Madsen

(1979) based on a two-layer eddy viscosity concept by assuming

two linear eddy viscosities within and outside the wave–current

boundary layer. The original model used the shear velocity u∗c,wc

associated with the maximum bed shear velocity of wave–

current combined flow (u∗wcm,wc). In the GM86 model, both

u∗c,wc and u∗wcm,wc were used, defined from the bottom shear

stress due to current and maximum shear stress of combined

flow, respectively. The GM86 model is applicable only for

rough turbulent regime with u0/vz0 . 300. Its wave–current

boundary layer thickness is between (1 and 2)ku∗wcm,wc/v. For

the present experiments, the value of wave–current boundary

layer thickness was taken as 1.75ku∗wcm,wc/v.

MS90 utilized the analogy between wave boundary layer flow

and planetary boundary layer flow by using similarity theory. It

was also validated with the laboratory measurements by

Bakker and van Doorn (1978) and Kemp and Simons (1982,

1983) for parallel wave–current flows (f ¼ 08 or 1808). The

same eddy-viscosity distribution within and outside the wave

current boundary layer similar to GM86 model was considered.

Unlike other models, the MS90 model applies for any flow

regime of rough, smooth or transitional by using the relevant

equation set for each regime. The MS90 model constants are

determined from rough turbulent pure oscillatory flow data.

The FR84 model describes wave–current interactions of arbi-

trary angles using the depth-integrated momentum equation

assuming the log-laws both inside and outside the wave boundary

layer. Unlike CJ85, GM86 and MS90, the wave–current bound-

ary layer thickness is treated as time-variant. Furthermore, the

wave–current associated friction velocity is also time-dependent.

The upper limit of the boundary layer is taken as dwc + kn/30

where the instantaneous velocity is treated to be the vector sum

of the potential flow and the mean current velocities. The

apparent roughness height is determined by matching the

current velocities (intersection of two layers) at the upper

limit of boundary layer. As the boundary layer thickness is

time-dependent, the mean value of boundary layer at vt ¼ p/2

and vt ¼ 3p/2 is taken. This model was also validated with

the data of Bakker and van Doorn (1978) for parallel wave–

current flows.

All four models (CJ85, GM86, MS90, FR84) analytically

describe the current velocity profile, the bed shear stresses, the

wave–current boundary layer thickness and the apparent rough-

ness height in the combined wave–current interaction at an arbi-

trary angle. The solution of the GM86 and MS90 models are

based on the reference height velocity while for CJ85 and

FR84 the computations are based on the depth-averaged

current speed. These models are applicable only for the rough

turbulent flow regime except for MS90. The main deficiency

of these models is the physically unrealistic discontinuity associ-

ated with eddy viscosity. All models process an iterative calcu-

lation procedure in finding the combined flow parameters.

4.2 Comparisons with experimental data

General input for models

The experimental data of the current profile, bed shear stress, and

apparent roughness height for orthogonal wave–current flows

are compared with the model predictions of CJ85, GM86,

MS90 and FR84. For GM86 and MS90 the current velocity at

the reference height of 0.3h was employed, while for CJ85 and

FR84 the depth-averaged current velocity was used. The exper-

imentally estimated hydraulic roughnesses z0 of Table 1 were

used as input to the models. The wave orbital velocity u0 was

calculated theoretically from experimental wave height, period

and water depth using the wave theory. The parameters in

CJ85 are r ¼ 0.45 and b ¼ 0.0747. For MS90, the model

parameters are B ¼ 1.28 and c ¼ 0.30.

Mean current profile

The comparisons of mean current profiles are shown in Fig. 4.

Note that for small wave heights (H/h , 0.25), all models

predict the mean current velocity well, as shown by Runs 3, 4,

5 and 9. For larger wave heights (0.25 , H/h , 0.45), GM86

and MS90 agree better with the present experimental data than

CJ85 and FR84, as shown by Runs 11, 12 and 13. For large

wave heights (H/h . 0.45) none of the models predicts the

mean current profile near free surface well, and the measured

data curled towards the free surface, deviating from the logarith-

mic profile, as shown by Runs 16 and 19. Similar observations

were made by Kemp and Simons (1982), van Rijn and

Havinga (1995), and Musumeci et al. (2006). The present data

further reveal that such an effect is more pronounced as the

wave height increases. The reason of the deviation from the log-

arithmic profile is as yet unclear. Nielsen (1992) and You (1996)

attributed it to wave-induced Reynolds stresses, while Yang et al.(2006) believed it is due to net vertical velocity. Future exper-

iments for the mean stress and the variation of mean water

level are needed to understand this fact.

Since the current velocity may deviate from the log-law near

the free surface, the model computations based on the depth-

averaged current velocity (CJ85 and FR84 models) assuming

the logarithmic profile up to the free surface may not adequately

simulate the actual wave–current interaction. This may be the

reason for the poorer performance for larger waves. However,

both the GM86 and MS90 models are based on the current vel-

ocity at a reference point. As this velocity in combined flows

satisfies the log-law up to intermediate depths, the selection of

a reference point close to the bed resulted in a better prediction

as shown herein.

Bed shear stress and apparent roughness

Table 2 summarizes the comparisons of the current associated

bed shear stress and apparent roughness height. It also shows

the 95% confidence limit on experimental estimations. Devi-

ations of experimental estimations to theoretical predictions are

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Figure 4 Comparison of (open circle) measured mean current velocities under orthogonal wave–current flows with model predictions by (dotted line)FR84, (thin line) CJ85, (dashed line) GM86, and (dash-dotted line) MS90 for various wave and current conditions

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shown in Table 3. Here, the absolute value of difference between

theory and experiment is added for each run and divided by the

number of runs 20. While all four models predict the friction

velocity reasonably, CJ85, FR84 and MS90 underestimate the

apparent roughness. In general, GM86 gives the best overall

predictions for the bed shear stress and apparent roughness

(Table 3), as well as the mean current velocity profile.

5 Conclusions

Laboratory experiments were conducted in a spatial wave basin

to generate orthogonal wave–current flows over a movable bed.

The relatively weak currents of 10.5 cm/s and 13.5 cm/s depth-

averaged flow velocity and strong waves of wave heights ranging

from 4.40 cm to 18.64 cm were generated under a still water

depth of 35 cm. The ripple patterns then mainly depend on the

wave height. If the wave height is less than 8 cm, the ripples

are nearly two dimensional serpentine with a hydraulic rough-

ness height equal to 0.0229 cm. If the wave height is larger

than 8 cm, the ripples are three-dimensional honeycomb, of

hydraulic roughness height equal to 0.0264 cm.

The current flow was significantly modified by the presence

of orthogonal waves, which reduce the near-bed current vel-

ocities by increasing the associated bed shear stress and apparent

roughness. Further results include: (i) For small waves of wave

height to water depth ratios less than 0.25, all of the selected

theoretical models for wave–current interactions at an arbitrary

angle agree well with experimental measurements for the mean

current velocity profiles. Moreover, the predictions of the

current-associated bed shear stress and apparent roughness

height given by all of the theories agree reasonably well with

the experiments, (ii) For larger waves of wave height to water

depth ratio between 0.25 and 0.45, models based on the reference

point agree well with the measured current velocities, yet one

model predicts the bed shear stress and the apparent roughness

height well. The other three generally underestimate the bed

shear and apparent roughness, and (iii) For very large waves of

wave height to water depth ratio larger than 0.45 the

near-surface current velocity deviates strongly from the log

law and it cannot be properly predicted by any of the theoretical

models.

This experimental study is expected to enrich the wave–

current database, which can be used to develop and validate

new numerical or theoretical wave–current interaction models.

Further studies, both experimental and theoretical, are needed

to better understand the detailed interaction mechanisms for

large waves with strong wave nonlinearity. The effect of inter-

action angle on the performance of various theoretical models

is also to be investigated to apply these to realistic coastal

problems.

Acknowledgements

This work was partly funded by research grants from the

National Science Foundation of China (50525926 and

51061130547), Ministry of Science and Technology of China

(2007CB714150), and the National University of Singapore

(R-264-000-182-112). The authors would like to thank Prof.

O.S. Madsen, MIT, and Prof. Cheong H.F. & Prof. Chan E.S.

at NUS for useful discussions on the experimental set-up.

Table 3 Deviations of theoretical predictions from experiment estimations

Deviation from experimental estimation∑i=20

i=1(u∗c,wc)exp − (u∗c,wc)theory

∣∣ ∣∣/20 (cm/s)∑i=20

i=1(z0a)exp − (z0a)theory

∣∣ ∣∣/20 (cm)

CJ85 0.253 0.141

FR84 0.317 0.174

GM86 0.121 0.093

MS90 0.276 0.187

Appendix

Data of current velocity profiles in combined flow (all at 298C water temperature).

z (cm) u (cm/s) z (cm) u (cm/s) z (cm) u (cm/s) z (cm) u (cm/s) z (cm) u (cm/s)

Run 1 Run 2 Run 3 Run 4 Run 5

1.15 4.59 1.10 5.11 1.20 4.92 0.80 3.22 1.00 5.57

2.60 6.17 2.80 7.61 2.80 6.73 2.30 4.92 2.57 8.29

4.17 7.58 4.28 9.33 4.50 8.43 3.60 7.09 4.16 10.70

5.74 8.86 5.86 10.72 6.20 9.15 5.30 8.15 5.80 11.60

7.20 9.82 7.41 11.67 8.00 9.93 6.80 8.76 7.40 12.09

(Continued)

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Appendix. Continued

z (cm) u (cm/s) z (cm) u (cm/s) z (cm) u (cm/s) z (cm) u (cm/s) z (cm) u (cm/s)

8.90 10.36 8.86 12.20 9.86 10.15 8.30 9.64 8.60 12.63

10.35 10.74 10.50 12.90 11.50 10.17 9.77 9.85 10.10 12.87

11.77 11.05 11.90 13.00 12.30 10.56 11.31 10.09 11.50 13.65

13.30 11.16 13.05 13.26 13.61 10.69 12.90 10.50 13.02 13.60

15.09 11.54 14.36 13.74 14.70 10.82 14.40 10.93 14.50 14.20

18.21 11.78 17.34 14.03 17.59 11.33 17.15 11.43 17.40 14.09

21.04 11.90 20.40 14.44 20.54 11.66 20.10 11.38 20.37 14.93

24.01 12.25 23.44 14.63 23.65 12.06 23.14 11.63 23.44 15.66

27.06 12.31 26.61 15.32 26.70 12.34 26.00 12.20 26.51 15.92

Run 6 Run 7 Run 8 Run 9 Run 10

1.20 2.02 1.15 4.99 1.08 5.28 1.30 6.25 1.00 5.23

2.70 4.75 2.56 7.34 2.58 7.12 2.80 8.10 2.66 6.62

4.52 6.75 4.07 9.43 4.08 8.55 4.40 9.54 4.03 7.73

6.02 7.92 5.64 10.16 5.58 10.92 5.80 10.60 5.63 9.42

7.52 8.86 7.06 11.26 7.08 11.21 7.20 11.96 8.04 10.08

9.02 9.25 8.49 12.15 8.58 12.60 8.44 12.30 9.54 10.23

10.52 9.79 9.83 12.90 10.08 12.98 10.13 12.86 11.04 10.94

12.02 10.71 11.10 12.47 11.58 13.78 11.64 13.12 12.54 11.22

13.52 10.47 12.60 13.51 13.08 14.10 13.10 13.47 13.42 11.63

15.02 11.08 14.20 13.89 14.21 14.29 14.44 13.99 14.92 11.68

18.02 11.64 17.25 13.89 17.21 14.78 17.50 14.24 17.30 12.06

21.00 11.79 20.30 15.18 20.21 15.11 20.47 14.66 20.33 12.33

24.00 12.07 23.36 15.41 23.40 15.08 23.51 15.04 23.33 12.36

27.00 11.98 26.39 15.20 26.20 14.98 26.51 15.21 26.13 12.32

Run 11 Run 12 Run 13 Run 14 Run 15

0.89 5.56 1.45 4.45 1.37 5.38 1.78 6.08 0.90 6.23

2.45 6.19 2.95 6.44 2.87 7.87 3.28 8.23 3.20 8.16

3.60 9.24 4.30 8.03 4.50 9.26 5.00 9.53 4.63 9.99

5.27 10.99 5.95 9.09 5.73 10.72 4.00 10.06 6.30 11.63

6.41 12.67 7.45 9.85 7.23 11.04 5.50 11.16 7.80 12.67

7.88 12.84 8.87 10.40 8.80 12.25 8.30 12.41 8.70 13.29

9.69 13.48 10.37 10.89 10.47 12.99 9.80 12.97 10.40 14.05

11.15 14.00 11.87 11.61 12.54 13.28 11.47 13.29 11.48 13.93

12.15 14.34 13.08 11.55 13.64 13.45 13.20 14.03 12.70 14.80

13.41 14.70 14.58 11.60 14.80 13.85 14.80 14.07 14.60 14.59

16.66 15.37 17.55 12.00 18.21 14.15 17.43 14.64 17.58 15.43

19.64 15.75 20.55 12.11 20.90 14.27 20.40 15.38 20.45 15.09

22.54 15.76 23.84 12.25 24.15 15.29 23.49 15.31 23.71 15.38

Run 16 Run 17 Run 18 Run 19 Run 20

1.10 7.68 1.30 3.94 1.05 5.70 1.30 2.59 1.03 6.67

2.60 9.48 2.62 5.53 2.50 7.20 2.80 6.42 2.53 8.56

4.10 10.59 4.46 7.00 4.00 8.45 4.30 8.14 4.03 11.09

5.60 12.43 6.22 8.20 5.50 9.71 5.80 9.74 5.53 13.59

7.10 13.63 7.70 8.99 7.00 10.79 7.30 10.56 7.03 13.99

8.60 14.12 8.38 10.28 8.50 11.38 8.80 11.30 8.53 14.92

10.10 14.54 10.87 10.14 10.00 11.64 10.30 11.73 10.03 15.54

11.60 15.17 11.51 10.54 11.50 12.19 11.80 12.04 11.53 16.33

13.10 15.51 12.67 11.57 13.00 12.46 13.30 12.52 13.03 16.57

14.60 15.93 14.40 11.82 14.50 12.99 14.80 13.09 14.53 16.52

17.60 16.25 17.35 12.14 17.50 12.88 17.80 13.37 17.53 16.82

20.60 16.32 20.47 12.16 20.50 12.99 20.80 13.16 20.53 17.00

23.60 16.28 22.75 12.36 23.50 13.16 22.80 13.06 22.03 16.57

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Notation

B, c ¼ model constants of Myrhaug and Slaattelids’

theory

d50 ¼ diameter of median sediment particle size

h ¼ water depth

H ¼ wave height

N ¼ number of points for linear regression

R ¼ linear regression correlation coefficient

kn ¼ Nikuradse’s equivalent sand roughness height

r, b ¼ model constants of Christoffersen and Jonsson

model

T ¼ wave period

u ¼ current velocity

uc,wc ¼ current velocity in the combined flow

u0 ¼ wave orbital velocity

u∗ ¼ friction velocity

u∗c ¼ pure current friction velocity

u∗c,wc ¼ current friction velocity in the combined flow

u∗wcm,wc ¼ friction velocity associated with maximum bed

shear stress of wave–current combined flow

U ¼ magnitude of depth averaged current velocity

z ¼ height above bottom for velocity prediction/

measurement

z0 ¼ hydraulic roughness height

z0a ¼ apparent roughness height

k ¼ von Karman constant

f ¼ wave–current interaction angle

v ¼ wave angular frequency

va ¼ absolute angular frequency

dwc ¼ thickness of wave–current boundary layer

D ¼ bed ripple height

l ¼ bed ripple length

Subscriptsexp ¼ estimated by experiment

theory ¼ predicted by theory

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