Water Wave Induced Boundary Layer Flows Above a Ripple Bed
Philip L.-F. Liu, Khaled A. Al-Banaa, and Edwin A. Cowen
School of Civil and Environmental Engineering, Cornell University
Ithaca, NY 14853
By using the Digital Particle Tracking Velocimetry (DPTV), laboratory experiments have been
performed to investigate the surface gravity wave induced boundary-layer flow field over a fixed ripple bed. The
spatial and temporal variations of the velocity field have been recorded. Two sets of wave parameters, yielding
two different Keulegan-Carpenter and Reynolds numbers, were used. The phase-dependant mean velocity and
vorticity fields were presented in details, illustrating the vortex generation and ejection processes. By using the
bootstrap percentile technique, the random errors from the velocity measurements were calculated. The root-sumsquare (RSS) was used to estimate the random errors in the vorticity measurements. The wave-induced streaming
profile was determined from the experimental data and its implications were discussed. The evolutions of the
Reynolds stress and turbulent kinetic energy were also presented. An estimate of the turbulence eddy viscosity
was then calculated inside the turbulent boundary layer. The physical processes are also modeled and calculated
numerically. The model is based on the Reynolds Averaged Navier-Stokes (RANS) equations with a
k − ε turbulence equation. In this extended abstract the setup and procedure of the experiments is discussed.
However, only a small portion of experimental data is presented. The full discussion of experimental and
numerical results will be given in the workshop.
INTRODUCTION
Sand ripples are frequently found on the seafloor. The appearance of sand ripples affects the sediment
transport rate and cause additional energy dissipation and enhances mixing in the vicinity of the ripples. The flow
motions, including oscillatory and steady components, directly influence the formation and the stability of sand
ripples. In recent years, many researchers have investigated analytically the relationships between boundary layer
flow patterns and various physical parameters, such as the ripple slope, ε , Reynolds number, Re, and KeuleganCarpenter number, α  Although these investigations have provided significant physical insights, they are limited
to two-dimensional laminar flows. To consider the effects of turbulence, turbulence closure models are necessary
to solve the oscillatory turbulent boundary layer and only the numerical solutions have been obtain. The
development of appropriate turbulence closure model for oscillatory boundary layer flows is still a challenging
research topic. To accomplish this task, accurate experimental data are essential. Furthermore the numerical
models or analytical solutions need to be validated by experimental studies.
Most of the velocity measurements above ripple beds have been obtained by a point measurement
technique, which is either a hot-film velocimeter or a laser-Doppler velocimeter. Because of the nature of the point
measurement techniques, the spatial resolution of the reported data is commonly low. Consequently, these data
could not provide accurate information for the vorticity field and other physical variables associated with the
turbulent fluctuations, such as the Reynolds stress and eddy viscosity. Recently, the Particle Image Velocimetry
(PIV), which is a whole-field measurement technique, has been successfully employed in measuring the vorticity
field in the vicinity of an object subject to either a steady current or oscillatory flows. Chang & Liu (1998 and
1999) have also reported PIV measurements of velocity and vorticity field under breaking waves.
The objective of this paper is to report an experimental study of the flow field above a fixed ripple bed,
using the digital particle tracking velocimetry (DPTV) (Cowen & Monismith 1997). This technique uses crosscorrelation based PIV as an estimate of the velocity field allowing individual particles to be tracked accurately at
traditional PIV seeding densities. The result is an order of magnitude more independent velocity vectors as well
as improved accuracy relative to traditional correlation based PIV. This allows us to investigate the generation
and transport of vorticity and turbulence under various wave conditions.
EXPERIMENTAL SET-UP AND PROCEDURE
The experiments were conducted in the wave tank in the DeFrees Hydraulics Laboratory at Cornell
University. The wave tank is 32m long, 0.6m wide and 0.9m deep. It is constructed of plate glass sidewalls with a
painted steel bottom. The middle test section has been fitted with a two-dimensional rippled bed constructed of
painted wood with the mean position between the ripple crest and trough aligned vertically with the steel bottom.
The ripple crests are perpendicular to the centerline of the wave tank. The ripple bed consists of 60 ripples, each
ripple is 1.2 cm in height and 3 cm in length. The removable section of the wave tank is 1.82m long and 0.60m
wide. The wave tank is equipped with a piston-type wave generator, which is driven by a hydraulic servo-system
and is capable of generating both regular and irregular waves. The downstream section of the tank is fitted with a
1:20 sloping beach and horsehair (a packing material) to minimize the reflected wave energy. The reflection
coefficient was less than 8%.
In the following table wave parameters and associated dimensionless parameters are listed for this study.
Table 1 Physical variables and dimensionless parameters
Quantity, Symbol (Unit)
Wave period, T (sec)
Wave length, λ (m)
Water depth, h (m)
Wave height, H (m)
Horizontal particle displacement, A (m)
Ripple wavelength, L (m)
Ripple amplitude, η (m)
Case A
2.1
2.85
0.20
0.0244
0.03
0.03
0.006
Case B
2.19
3.12
0.22
0.0440
0.045
0.03
0.006
Maximum velocity above the boundary layer, Uo
(m/s)
0.0897
0.1290
Boundary layer thickness, δ =
Dimensionless parameters
0.82
0.84
Reynolds no. Re = ω A / ν
2700
5800
Ripple slope, ε = η / L
0.2
0.2
Wavelength ratio, β = L / λ
0.01
0.01
Keulegan-Carpenter no., α = A / L
1.0
1.5
2ν / ω mm)
2
The primary difference between these two experiments is the Keulegan-Carpenter (K-C) number, α . In Case A
the K-C number is unity and the maximum horizontal fluid particle displacement outside of the boundary layer is
the same as the ripple wavelength. On the other hand, in case B the K-C number is increased to 1.5 and therefore
fluid particles travel 1.5 times of the ripple wavelength over one half of the wave period. Moreover, the Reynolds
number in Case B is more than twice of that for Case A.
The boundary layer flow above the ripple bed was measured using digital particle tracking velocimetry
(DPTV) (Cowen & Monismith 1997). Images were collected with a Silicon Mountain Designs (SMD) 1M60
digital CCD camera (1024 x 1024 pixel, 12-bit, 30 Hz). The camera was fitted with a 35mm extension tube and a
Nikkor 105mm f/2.5 lens. The PTV illumination source was a Spectra-Physics PIV-200 twin-pulsed Nd:YAG
system (140 mJ/pulse, 10 Hz on each laser head). A cylindrical lens (f = -50 mm) and a spherical lens (f = 1.0 m)
were used to form a light sheet. This arrangement yielded an approximately 1 mm thick light sheet centered on a
ripple trough (between the 30th and the 31st ripple) and located 13 cm off the tank centerline. The camera image
area was centered in this light sheet and the optical arrangement yielded a viewing area of 4.04 cm x 4.04 cm.
The camera/laser/wave-maker system was controlled with National Instruments Labview. A single
burst of 21 image pairs for case A and 22 image pairs for case B were collected at 10Hz directly to host computer
RAM. The time between illuminating the images in each pair was 4ms. This process was repeated 333 times. The
DPTV algorithm tracked 7.7 million valid velocity vectors across all the phases. These vectors were ensemble
averaged at the different phases into non-overlapping 1.26 mm x 0.98 mm bins across the 333 waves at
approximately 1027 active measurement locations. An average of 358 vectors were included in the ensemble
average in each bin. Notably near the boundaries the vector count in the bins drops but remains greater than 200
vectors per bin in essentially all bins except those adjacent to the boundaries.
The wave-maker was turned on for two hours before measurements were initiated. The flow field was
assumed to be at a quasi-steady state. The minimum spin-up time was determined by monitoring the phase shift
between the wave-maker driving signal and the water surface elevation measured at the test location with a
resistance type wave gage. After two hours the phase shift became a constant.
The uncertainty analysis was carried out to determine the random errors associated in this experiments.
The bootstrap technique was used to estimate the 95% confidence interval for the mean velocities, turbulent
intensities and Reynolds stresses, which will be presented and discussed in the following sections. The Root-SumSquare (RSS) technique was also used to estimate the random errors in the mean vorticity measurements.
EXPERIMENTAL RESULTS AND DISCUSSIONS
In this section, the experimental data for the phase-averaged velocity will be presented and discussed.
The results for vorticity, turbulent intensity, Reynolds stresses, and turbulent eddy viscosity will be presented will
be presented in the workshop. In figure 1 the time history of the free surface displacements above the
measurement area for case A is shown. The circles and the associated number represent the phases at which
experimental data will be analyzed and presented.
To analyze the experimental data we first examine the
phase-averaged velocity. In the present experiments the
phase-averaged velocity results from sampling 21 times
per wave period for case A (see figure 1).
To plot the spatial variation of velocity
vectors, the spatial domain is divided into nonoverlapping bins of 1.26mm x 0.98mm. At a given
phase all the velocity vectors inside a bin across 333
consecutive waves are averaged to give the phaseaveraged velocity vector for the bin. If there were no
data dropouts, the total number of velocity vectors
should exceed 333. However, near the solid boundary
and along the edge of the field of view (FOV),
Figure 1 Time histories of free surface displacements at the
data dropouts occur. Nevertheless, for most of the flow
measurement location for case A and B. The solid-circles
domain, there are more than 200 velocity vectors within represent the wave phases where data are analyzed.
each bin.
For case A the spatial variations of the phase-averaged velocity field at different phase are displayed in
figure 2. In each figure the velocity vectors are plotted over the measurement area. Furthermore, the vertical
profiles of the horizontal velocity component at four vertical cross sections are also shown. The 95% uncertainty
intervals for the velocity data are displayed on the velocity profiles. At the phase ω t = 0.26π (figure 2a), the
flow field is accelerating as the wave crest is approaching the measurement area (see figure 1). The velocity field
above and away from the ripple bed appears to be uniform and unidirectional. However, close to the ripple bed the
velocity profiles in the vicinity of the ripple crest are very different from those near the ripple trough.
Approaching the ripple crest from above (x = 0.6 cm in figure 2a), the horizontal velocity increases first so as to
satisfy the continuity and it vanishes on the boundary. On the other hand, in the neighborhood of ripple trough (x
= 1.84 cm in figure 2a), the velocity decreases monotonically. Consequently, fluid particles experience
deceleration on the lee side (down slope) of the ripple crest and a reverse pressure gradient also exists. These
features suggest the possibility of flow separation, which indeed occurs and is shown in figure 2b ( ω t = 0.35π ).
It is interesting to point out that the flow separation occurs before the wave crest reaches the measurement area
(see figure 1), which indicates a phase shift between the oscillatory boundary layer and the free surface profile.
Moreover, the velocity reaches its maximum above the ripple bed before the wave crest passes the measuring area
and has a value greater than the maximum velocity outside the boundary layer in the unidirectional region, i.e.,
< u > max / U o = 1.33. Because of the flow separation, a lee vortex is developed and is convected towards the ripple
trough. As the wave crest passes the measurement area, the ambient velocity above the ripple bed decreases and
the lee vortex continues the forward movement as shown in figure 2c at phase ω t = 0.55π . However, at
ω t = 0.83π (figure 2d), although the wave field is still moving forward (in the positive x-direction), the vortex
has started to move backwards. As the ambient velocity decelerates further, the vortex begins to rise from the
ripple crest where the vortex was originally generated ( ω t = 1.02π , figure 2e). As the wave trough approaches
the measurement area, the magnitude of the negative (backward) velocity near the ripple bed grows stronger
indicating a possibility of vortex ejection. At ω t = 1.21π flow separation has occurred on the weather side of the
second ripple crest (figure 2f). Thus, the generation, convection, and dissipation of a vortex are repeated every
one-half of wave period. The ejected vortex is transported across the ripples during the next half period. This
movement can be seen more clearly when the vorticity field is examined.
References
Chang, K.-A. & Liu, P. L.-F. 1998 “Velocity, acceleration and vorticity under a breaking wave,” Phys. Fluids, 10,
327-327.
Chang, K.-A. & Liu, P. L.-F. 1999 “Experimental investigation of breaking waves in intermediate water,” Phys.
Fluids, 11, 3390-3400
Cowen, E. A. & Monismith, S. G. 1997 “A hydrid digital particle tracking velocimetry technique,” Exp. Fluids,
22, 199-211.
Figure 2 (a-f) The spatial variations of the phase-averaged velocity field at different wave phase.
4
4
3.5
3.5
3
3
2.5
2.5
)
m
c(
Z
)
m
c(
Z
2
2
1.5
1.5
1
1
0.5
a
0
10 (cm/s)
0.5
1
1.5
2
2.5
3
0.5
0
3.5
g
10 (cm/s)
0.5
1
1.5
X (cm)
X = 1.84(cm)
X = 2.47(cm)
X = 1.22(cm)
X = 0.60(cm)
2.5
3
X = 1.84(cm)
4
4
4
4
4
4
4
4
3
3
3
3
3
3
3
2
2
2
2
2
2
1
1
1
1
1
0
< u > (cm/s)
0
-20
20
0
< u > (cm/s)
0
-20
20
0
< u > (cm/s)
20
)
m 2
c(
Z 1
0
-20
0
< u > (cm/s)
0
-20
20
4
4
3.5
3.5
3
3
2.5
2.5
)
m
c(
Z
)
m
c(
Z
2
20
0
-20
0
< u > (cm/s)
20
1
0
-20
0
< u > (cm/s)
20
0
-20
20
1
0.5
b
0.5
10 (cm/s)
0
0
0.5
1
1.5
2
2.5
3
3.5
X = 0.60(cm)
X = 1.22(cm)
X = 1.84(cm)
4
4
4
3
3
3
3
2
2
2
1
1
1
0
-20
0
-20
0
-20
)
m 2
c(
Z 1
0
-20
0
< u > (cm/s)
20
0
< u > (cm/s)
20
0
< u > (cm/s)
20
10 (cm/s)
0.5
1
X = 0.60(cm)
X = 2.47(cm)
4
k
)
m
c(
Z
0
< u > (cm/s)
1.5
2
2.5
3
3.5
X (cm)
X (cm)
X = 1.84(cm)
X = 2.47(cm)
4
4
4
3
3
3
3
2
2
2
2
1
1
1
0
-20
20
X = 1.22(cm)
4
4
4
3.5
3.5
3
3
0
< u > (cm/s)
20
0
-20
0
< u > (cm/s)
20
1
0
-20
0
< u > (cm/s)
20
0
-20
0
< u > (cm/s)
20
2.5
2.5
)
m
c(
Z
2
2
1.5
1.5
1
1
d
10 (cm/s)
0.5
1
1.5
2
2.5
3
0.5
0
3.5
X (cm)
X = 1.22(cm)
X = 0.60(cm)
k
10 (cm/s)
0.5
1
1.5
2
2.5
3
X = 2.47(cm)
X = 0.60(cm)
X = 1.22(cm)
X = 1.84(cm)
X = 2.47(cm)
4
4
4
4
4
4
4
3
3
3
3
3
3
3
3
2
2
2
2
2
2
1
1
1
1
1
1
0
-20
0
-20
0
-20
0
-20
0
-20
)
m 2
c(
Z 1
0
< u > (cm/s)
20
0
< u > (cm/s)
20
3.5
X (cm)
X = 1.84(cm)
4
0
-20
0
< u > (cm/s)
1.5
1
0
0
< u > (cm/s)
2
1.5
0.5
3.5
X = 2.47(cm)
3
)
m 2
c(
Z 1
0
-20
)
m
c(
Z
2
X (cm)
X = 1.22(cm)
X = 0.60(cm)
0
< u > (cm/s)
20
)
m 2
c(
Z 1
0
< u > (cm/s)
20
0
-20
0
< u > (cm/s)
20
0
< u > (cm/s)
20
0
< u > (cm/s)
20
0
-20
0
< u > (cm/s)
20