Introduction
Problem definition
Estuaries are tidal basins with some freshwater inflow that are long
relative to their inlet width. Reversing tidal flow is driven by the tidal
water level fluctuations at the seaward boundary. In nature, estuaries have
embayed or seaward widening planforms with coastal inlets and are partly
filled with intricate patterns of shoals, tidal sand bars, mudflats and
tidal marshes. The large-scale planform shape and bar-channel patterns within
evolved through biogeomorphological processes and are partly determined by
inherited initial conditions and changing boundary conditions
. Certain phenomena are unique to estuaries,
such as mutually evasive ebb- or flood-dominated channels separated by shoals
. These shoals hinder shipping and at the same
time may be important habitats. However, gaining understanding of their
behaviour is challenging because modelling sediment transport processes in
three-dimensional reversing flow remains overly sensitive to sediment
transport parameters , and field observations of
morphological development spanning decades to centuries are unavailable
. The third complementary method for research
is controlled laboratory experiments , which are rare for estuarine phenomena in
contrast to the large number of river experiments.
Only two sets of experiments simulating estuarine morphodynamics are
accessible in the literature: conducted a
large number of experiments in basins of various shapes, and
conducted two experiments in an exponentially widening
estuary. In both cases the flow was driven by periodic sea level
fluctuations,
but sediment mobility was too low compared to natural systems. Recently, an
alternative experimental method was discovered that caused sufficiently strong reversing flow
for sediment transport similarity in tidal inlets by tilting the entire flume
periodically. With relatively small set-ups this was shown to result in
dynamic channel and shoal patterns that are similar to those in tidal inlet
systems in nature. However, the experimental tilting principle has not yet
been applied to estuaries that are much longer than tidal inlets and therefore
require much better control on the tidal wave dynamics than short basins
. Gentle tilting drives the flow in a fundamentally
different way than tides do in nature, raising the question of to what degree
this method leads to similar spatial flow and sediment motion patterns.
Moreover, why the Reynolds method does not lead to sufficient sediment mobility
remained unresolved for over a century. Here we compare for the first time
the flow in idealized experimental estuaries measured in both the classic
Reynolds set-up and the novel tilting set-up and extend the comparison by
modelling from the smallest experiments to large estuaries.
Targets for tidal landscape experiments
Tidal flows in natural estuaries can be complex, but for the purpose of
pioneering laboratory experiments we focus firstly on the most fundamental
properties. In nature, flow is mostly driven by a primary tide causing
periodic sea level fluctuation, which propagates as a wave through the estuary
mouth. This is modified by other tidal components, river flow and
circulation in deep estuaries with salinity stratification
. The
length L of an estuary is typically up to half a tidal wavelength Lt,
which is estimated as Lt=Tgh with primary tidal period T and
shallow water wave celerity gh, where h is water depth and g is
gravitational acceleration. The tidal amplitude a is usually less than half
the water depth . The resulting flow
velocity u depends on tidal period and cross-sectional area A of the
inlet and on the tidal prism that depends on the planform geometry of the
estuary e.g.. Typically estuaries get narrower and
shallower in the landward direction as freely erodible substrates adapt to
spatial gradients in flow velocity so that flow velocity in many estuaries
does not vary more than an order of magnitude with distance from the inlet
. The aerial extent and elevation of channels, shoals,
mudflats and salt marshes modifies the magnitude, timing and duration of the
ebb-directed flow and the flood-directed flow, particularly if these vary
along the estuary and if the channels are dredged .
Driving flow in nature and in experiments with the requirement of
sediment mobility similarity. Given the same sediments in experiments as in
natural systems, the shear stress in experiments must be the same as in
nature. With much smaller water depths this requires much larger gradients,
which is straightforward for river experiments. In tidal experiments these
gradients are impossible to obtain in flumes by sea level fluctuation, but
quite feasible to obtain by tilting the flume periodically.
In turn, sediment mobility and transport are driven by the flow to cause
morphological change. Here, mobility is expressed as the Shields number
θ=τ/g(ρs-ρ)D, where τ is the bed shear stress by
the flow, g=9.8 ms-2 is gravitational acceleration, ρ and ρs
are the density of water and sediment, respectively, and D is a
representative particle size of the bed sediment. The critical Shields number
for the onset of sediment motion is about θc≈0.04. The bed
shear stress is calculated as τ=ρfu|u| or τ=ρgu|u|/C2, where u is depth-averaged flow velocity driven by the energy
gradient, f is a dimensionless bottom drag coefficient and C is the
Chézy coefficient with f=g/C2 by definition. The characteristic
timescale of large-scale morphological change is much larger than the tidal
period . This has an important
consequence for modelling and experimentation: the time-dependent phase
differences of flow velocity as a function of distance from the mouth are not
of first-order importance for the morphodynamics as long as spatial
variations in velocity, residual currents and the resulting sediment mobility
are present. This conclusion can also be drawn from linear theory for tidal
bar properties, for which a rigid lid flow assumption was sufficient, meaning
that water surface fluctuations are only of secondary importance
. The reversing tidal flow is therefore much
more important for morphology than the tidal wave behaviour.
The prime challenge for morphodynamic tidal experiments is that the reversing
flow should cause sufficient bed shear stress for periodically reversing
sediment motion . As the bed sediment calibre cannot be
scaled with the same ratio nL as the dimensions of the system, the energy
gradient S of the laboratory system must be increased such that the
mobility θ remains the same as in the prototype. For medium sands this
slope is typically S=0.01 mm-1, accounting for both particle weight and the
large bed roughness in experiments . This required
energy slope for mobile sediment is feasible in river experiments, but
not in estuary experiments driven by periodic sea level fluctuation for the
following reasons (Fig. ). Consider an experimental tidal
system with a depth of h=2 cm. Here the tidal water level amplitude can be
at most about a=1 cm, meaning that a/h is as large as 0.5. Given a typical
aspect ratio of the estuary mouth of W/h>100, this means that the width of
the experiment should be about W=1 m. However, with a minimum slope of
0.01 mm-1, the distance from the mouth with sufficient gradient to move
sediment is effectively only about 1 m given the maximum water surface
amplitude. In laboratory-sized systems this creates a short tidal basin
rather than the long estuary we aim for.
This, in turn, leads to a number of other scale problems. The first is a
problem with tidal period. For the 2 cm depth and 1 m basin length, the
required tidal period is about 4–9 s, which is very fast for water to
accelerate and for pumps to deal with. The second problem is that this wave
causes very low flow velocities of O(10-3) ms-1, which is far below that
required for sediment motion. When the tidal amplitude is enlarged with
a/h>0.5, a new problem arises: the flow causes a net export of sediment
results on the seaward sloping bed so that the tidal system excavates until
it is in static equilibrium, as probably happened in a number of the
experiments of .
An expensive solution would perhaps be to make impractical experimental
set-ups of O(102) m long, which renders morphological timescales
impractically long and requires very large pumping capacity. The tilting
flume principle, on the other hand, has been qualitatively shown to attain
the required sediment mobility, but the principle is counterintuitive: the
real world does not tilt periodically. This may be the reason that this
principle was not invented in the past 130 years. The fundamentally different
driving mechanism for the flow raises the question of to what degree the tidal
flow is similar to that in nature and what consequences this may have for
the morphological development of estuaries on the experimental scale.
Objectives and approach
In our preliminary work in small flumes so far, a comparison between periodic
sea level and periodic tilting was done only qualitatively. Measurements are
needed of flow velocity fields and flow depth for a more in-depth analysis of
the laboratory flow behaviour compared to natural tidal systems, and a larger
facility with a higher tilting frequency is needed for better scaling of
basin size relative to tidal wavelength. Furthermore, a numerical model
reproducing the main dynamics of the experiments is needed to assess whether
the tidal flows in the Metronome are similar to those in nature and, if so,
to scale up from the smallest laboratory experiments to the largest natural
tidal systems on the planet in order to uncover possible scale problems.
The first objective of this paper is to compare tidal flows generated in the
tilting flume and in the Reynolds flume, focussing on the magnitude of
reversing flow velocity and sediment mobility along the estuary. To this end
we present flow measurements in a large tilting flume facility. Specifically,
experiments were designed to directly compare tidal wave behaviour, flow
velocity magnitude and tidal asymmetry driven by periodic tilting or sea level
fluctuation in idealized straight tidal channels with rough beds and the
largest possible tidal amplitudes. To exclude complex morphodynamic
feedbacks, this study is limited to idealized channels without bars and
shoals and with fixed rough beds or natural sand beds with conditions below
the beginning of sediment motion. The second objective is to assess how the
dimensions and dynamics of experiments are to be scaled up to prototype
systems and at what cost in terms of scale problems and distortions. To this
end we adapted a one-dimensional model of the shallow water equations to
include bed tilting and verified whether the most important tidal behaviour
is reproduced. We then compared modelled flow in systems with length scales
ranging from a small laboratory set-up to a large natural estuary and with
flow driven by both methods, characterized by morphologically relevant
dimensionless variables.
The Metronome tidal facility. Note PhD candidates for scale. The
flume floor below the sand is covered in artificial grass (see text). Note
the vertically mounted actuators that drive the flow. This pilot experiment
started as a 0.2 m × 0.03 m straight initial channel and ran for about 12 h
with a slope amplitude of 0.005 mm-1 and a period of 30 s and
100 Lh-1
river inflow.
Geometry of the Metronome facility. The inner basin measures 20.00 m
in length, 3.00 m in width and 0.40 m in depth, and the maximum tilting
amplitude is 0.5 m at the end tank, resulting in a tilting slope amplitude of
0.05 mm-1. Both flume ends have end tanks with a 0.3 m long stilling basin
functioning as a sediment trap and pumped water inflow separated by an
automated weir from the outside 0.2 m long overflow basin with a 2 mm mesh to
capture PIV particles. Motion is controlled by four 20 kN actuators for
tilting and two small actuators for each end tank weir. Cameras C1–7 are
mounted 3.7 m above the flume floor.
Application of the vegetation friction relation
to the artificial vegetation in the Metronome. (a) Flow velocity profile
(red) between the bed and the water surface (blue). Height of vegetation
indicated in green. (b) Ratio between water surface velocity usurf and
depth-averaged velocity udepth-avg as a function of total flow depth.
This is independent of slope. Given the insensitivity to water depth
variations, a constant value of 1.95 is assumed for comparison of measured
and modelled flow velocities.
Design of the Metronome facility
Between 2014 and 2015 we constructed the Metronome, a 20 m long flume designed
for periodic tilting to create tidal systems
(http://www.uu.nl/metronome; Fig. ; building
plans in the Supplement). The basic components are a steel basin that
tilts over the short central axis, motion control, water recirculation with a
constant head condition at the seaward end and optical imaging
(Fig. ).
The principle can be reproduced by simple means, namely with any basin, stream
table or flume that can be tilted over its axis, a consumer-grade garden pond
pump and a camera. The periodic tilting can be driven by an actuator, an
excentre mechanism or an adjustable stroke mechanism with a gearbox and motor
to drive sinusoidal motion with a period of tens of seconds and a slope
amplitude up to 0.02. For the purposes of future reference for ongoing
biomorphological experiments and for replication in other laboratories, the
specific design of the Metronome is briefly described below.
The steel basin has inner dimensions of 20.00 m long by 3.00 m wide and
0.40 m deep. The flume has two end tanks for water supply, water level
control and outflow and for sediment trapping. The tilting axis is directly
below the steel floor to minimise longitudinal motion and the entire flume
set-up is symmetrical about this axis. The basin was constructed from 4 mm
stainless steel plates cut and folded such that the sidewalls are suitable
for a gantry to screed the bed and set up measurement equipment and are a
structural part of the basin to minimise bending. Further stiffness was
accomplished by using a ribbed structure and steel beams along and across the
flume. Finite-element modelling on the design showed that the maximum
expected bending of the flume was at most 2 mm under extreme loads in
emergency conditions. This model was also used to select the required range
and power of the actuators and motion control and to estimate the loading and
required reinforcement of the floor. The steel basin was curved slightly
upwards during production such that it is straight under typical water and
sediment loading when supported by the tilting axis.
The end tanks were designed to function as constant head tanks, with sediment
traps at the inside of movable weirs. Water is supplied by four garden pond
pumps with a maximum discharge of 4.7 Ls-1 or 9 m head each in a 12 m3 sump
tank, which is an inflatable swimming pool in the basement of the building.
The 3 m wide weirs in both end tanks are broad-crested with a length of
0.06 m and rounded edges due to the folding of the steel plate from which it
was constructed. Small actuators control the motion of the weirs. This set-up
means that the flow is critical on the broad-crested weirs so that the water
depth hc at the weir depends on the specific discharge q=uh over it as
hc=q2/g1/3. Consequently, the water surface elevation at
the seaward boundary is not exactly as set by the weir height but is modified
slightly depending on the tidal prism. The effect of this will be taken into
account in the interpretation of the results herein. In future live-bed
experiments, the water depth can be corrected by motion of the weir at an
appropriate amplitude and phase shift relative to the tilting motion
depending on the tidal prism, such that the water level at the shoreline of
the live bed remains approximately constant.
The four actuators to tilt the flume operate in pairs with motion mirrored at
the tilting axis. The maximum force is 20 kN, but in the downward direction had
to be limited as the reinforced concrete floor supports the downward force
well,
but the upward pulling force not very well. The motion and forces are
monitored and internal safety controls prevent values above this that might
be damaging. The motion at periods and amplitudes as used in this paper is
typically 0.01 mm accurate. The actuators keep repeatable positions at all
times, also during rest, such that the flume does not deform. We found that
the flume was best set horizontal through manual measurement with a leveller
and 0.5 mm graded rulers on the sand screed riding on the cart and applying
offset positions to all four tilting actuators.
Landscape experiments often show channels clinging to sidewalls and, when
insufficient sand is used, channels that erode down to the flume floor where
erosion is enhanced because of the smooth surface. Using groynes or ribs does
not solve this because these force their own patterns on the flow and
morphology. A surface of uniform roughness is needed with a roughness scale
larger than the viscous sublayer thickness, much smaller than the smallest
bedforms , and with a gradual transition from an
alluviated sand bed to a fixed rough surface. We therefore covered the
Metronome floor and sidewalls with small-scale uniform roughness:
artificial grass about 15±1 mm (1σ) high that is spatially uniform in
stem density and glued to the floor in places and further kept down by a few
millimetres
of sand. The grass is so stiff that it did not bend noticeably in the
strongest experimental flows tested. The glue was applied such that water
cannot flow under the grass. An alternative would be sandpaper or any other
rough surface, but a practical advantage of the artificial grass is easier
sand removal and flume floor protection against shovels. We used the grass
roughness in the fixed-bed experiments and buried it under sand in the
sand-bed experiments. The sand has a D10 of 0.33 mm, a D50 of
0.57 mm and a D90 of 1.2 mm, which has a larger roughness length than
the viscous sublayer. We will assume the same sediment properties for all
modelled estuaries independent of length scale.
Boundary conditions applied in all experiments: auxiliary
fixed-slope experiments to determine the roughness of the artificial grass
bed, periodic tilting experiments with one or two open boundaries and
periodic sea level variations. Experiments with sand beds were conducted with a
shallow sea of 2 m length to reduce boundary effects and were closed on the
upstream boundary.
Bed
Period
Tilt slope amplitude
Sea level amplitude
Boundaries
Rationale
s
×10-3 mm-1
×10-3 m
Grass
0.9
0
Both open
Steady flow: control
Grass
2.3
0
Both open
Same, faster flow
Grass
60
0
10
x=0 m closed
Same, longer wave
Grass
30
0
20
x=0 m closed
Reynolds method
Grass
30
9.1
0
Both open
Reach within estuary
Grass
15
9.1
0
Both open
Same, shorter tidal excursion length
Grass
30
4.5
0
Both open
Reduced tidal energy (not shown)
Grass
30
9.1
0
x=0 m closed
Basin with reflective landward boundary
Grass
15
9.1
0
x=0 m closed
Same, short tidal excursion length
Sand
40
3.6
0
x=0 m closed
Tilting, natural roughness
Sand
40
0
3.5
x=0 m closed
Reynolds method, natural roughness
Experimental set-up and materials
Geometry and flow conditions
We conducted experiments with various initial and boundary conditions
(Table ). Most importantly, we applied periodic
tilting and periodic sea level fluctuation for comparison. Both experimental
approaches were applied on a sand bed and on a rough, artificial grass bed.
The majority of experiments were conducted on the artificial grass bed
because this allowed for the most freedom in conditions that would have led to
significant sediment motion on the sand bed. We tested two different boundary
conditions for the tilting experiments with the grass bed: one open sea boundary
and one reflective boundary to represent an estuary with the landward
boundary closed and two open boundaries to represent a reach within a long
estuary. We conducted auxiliary experiments with a constant flume gradient to
test the flow resistance formulation for the artificial grass bed. The
sand-bed experiments were conducted to assess the effects of the typical
roughness in live-bed experiments on the flow and had one open boundary in
all cases. The sand bed was pre-soaked.
The precise geometry of an estuary strongly determines tidal flow patterns
along the river. Given the aim in this paper, we chose the simplest
geometries and boundary conditions possible: straight channels and periodic
motion (Table ). An alternative could have been to
create exponentially convergent estuaries in which the friction loss in the
tidal wave is compensated for by the landward narrowing such that the flow
velocity amplitude is approximately constant along the estuary .
However, this requires careful matching of the convergence length with the
tidal conditions and effective friction, which we did not know in advance.
Moreover, we do not know yet whether this shape is applicable in the tilting
flume. For the artificial grass-bed experiments, a straight channel 0.7 m
wide was sectioned off from the remainder of the flume by 0.1 m diameter
cotton hoses filled with sand pressed down into the grass. For the sand-bed
experiments, a channel 0.6 m wide and 0.03 m deep was carved in a 0.065 m
thick sand bed over the first 18 m of the flume, leaving a basin 2 m long
and 3 m wide uncovered. This “sea” allows ebb delta formation in future
live-bed experiments. The side effect is that the mass of water available for
inflow and outflow of the channel is unhindered by the limited capacity of
the pumps and the critical flow condition on the weir, making this set-up
insensitive to adverse seaward boundary effects.
The average water depths were set at about h=0.028 m in all grass
experiments by applying the same (average) heights of the weirs to submerge
the vegetation-like roughness at all times. The Reynolds-type experiments
were done with a period of 30 s and a seawater surface amplitude of 0.02 m,
which is the same period as in many of our other experiments with an extreme
tidal amplitude, and a less energetic condition with a 60 s period and a
0.01 m amplitude that is closer to conditions in experiments reported in
the literature. The most basic tilting experiment has two open boundaries with
constant elevations of both overflow weirs, approximating constant head
conditions. For this condition two experiments were run with tilting periods
of 30 s, which is similar to other experiments in the literature, and 15 s to
investigate the possibility of reducing the tidal excursion length estimated
as Le≈uT/2. In most grass-bed experiments a tilting slope amplitude
(maximum slope during sinusoidal tilting) of 9.1×10-2 mm-1 was
applied. The second set-up has one boundary closed (here at 0 m) and the other
open, representing a tidal basin of finite length. Here again 15 and
30 s
periods were applied. As a control experiment to test the friction relation,
the steady flow was measured at constant slopes of 0.91×10-3 mm-1
and 2.3×10-3 mm-1 and the same water depth as the other experiments.
Conditions in the sand-bed experiment were set such that the channel did not
overflow and the sediment hardly moved, which was attained at a mean water
depth of h=0.018 m and a 40 s period, the typical period for live-bed
experiments to be reported in future papers. The Reynolds experiment on sand
was conducted with a 3.5×10-3 m water surface amplitude to prevent
sediment motion. The tilting experiment was conducted with a tilting slope
amplitude of 3.6×10-2 mm-1, for which we observed no significant
sediment motion.
Imaging, measurements and data reduction
Flow was measured by water depth measurements and large-scale surface
particle imaging velocimetry (PIV) the same method
as. The PIV was conducted by spreading white
floating particles on the water surface of the flume with repeated photography
and image processing to obtain the motion of the particles as detailed below.
Water depth was measured in the grass-bed experiments at various locations
along the flume with rulers with 0.5 mm grading supported by small
thin-legged tripods. This is rather inaccurate because of irregularities in
bed elevation and because of the meniscus of the water surface on the rulers.
These data were detrended with still water measurements. In the sand-bed
experiments conducted later we measured water surface elevation relative to
still water with an ultrasonic device at a sound frequency of 150 kHz mounted on
the tilting flume. The distance of about 0.2 m from the bed with
a temperature-corrected distance measurement resulted in a footprint of about
0.03 m and a vertical accuracy of about 1 mm. Measurements were collected for
three tidal cycles at a 10 Hz sampling frequency in phase with the tilting and
phase-averaged by fitting with a spline at 1 s intervals for presentation.
For the PIV, seven industrial cameras were mounted 3.7 m above the floor of
the flume, approximately above the centerline at equal distances. However,
camera alignment was hampered by the roof supports in the temporary lab
location so that axis positions and directions differ between cameras and are
not perpendicular to the flume floor. This caused the geometry of the optical
system relative to the flume to be suboptimal, resulting in higher tilt
angles and a few pixels of mismatch between adjacent cameras. This does not
affect the conclusions of this paper because the velocity is spatially
averaged along the flume and the focus is on general characteristics and
behaviour. The cameras are CMOS MAKO colour cameras with a resolution of 2048
by 2048 pixels. The cameras have a lens with fixed focal length of 12.5 mm.
The footprint is about 3.15 m so that a pixel on average covers about
1.5–2 mm. Hardware and software are designed to allow simultaneous
25 Hz
imaging for the purpose of PIV. The trigger for the cameras is taken from the
tilting motor controller at exactly defined moments in the tidal cycle. For
the PIV this trigger starts a 25 Hz pulse train from a frequency generator in
order to have accurate, computer-clock-independent timing.
The flume is illuminated at about 600 lux with daylight-coloured TL aimed at
a white diffusive ceiling at about 4.5 m above the flume floor, designed for
imaging and future vegetation growth. This allows for low exposure
durations, but we later found that the ceiling reflected on the water surface
to hinder imaging in live-bed experiments. By the time we conducted the
additional sand-bed experiments a diffusive white sheet was suspended below
the ceiling and lamps in the shape of a tent. This improved lighting although
it reduced light intensity, but this did not affect the PIV imaging.
The procedure for data collection was as follows. White floating particles of
2–3 mm diameter were seeded on the water surface along the flume and newly
supplied by operators at both boundaries where necessary. After about five
tidal cycles the flow was considered in equilibrium. In 16 phases of the
tide, 10 images were collected at 25 Hz simultaneously by all cameras. Water
levels were measured before seeding the PIV particles. Control experiments
with constant slope were conducted in the same manner but with lower slopes
because of the rapid evacuation of floating particles.
Raw images were first debayered to obtain RGB colour images, from which only
the green layer was taken for analysis. Background images were subtracted
that were obtained for the same tidal phase without floating particles. These
images were then rectified using the Caltech camera calibration toolbox in
MATLAB (http://www.vision.caltech.edu/bouguetj/calib_doc/; version
15 October 2004) after obtaining camera calibrations.
Flow velocities were calculated for every pair of consecutive images using
the MPIV toolbox in MATLAB (,
http://www.oceanwave.jp/softwares/mpiv/). The focus of this paper is on
width-averaged flow in a uniform channel so that the conventional
cross-correlation algorithm for PIV suffices. This means that the peak
cross-correlation is used as mean particle displacement in a given window.
This was run with a window size of 100 pixels with 50 % overlap. Subsequently
the vector fields were scaled by the footprint of the cameras, which was
calculated from the geometry of the flume, the average height of the cameras and
the camera resolution, and the instantaneous tilting angle. As a result flow
velocity vectors were determined on a regular grid at about 77 mm of spacing.
Erroneous vectors resulted from windows that were partially filled with flume
wall, spots empty of particles, mismatched particles and reflections on the
water surface. After filtering out the 1 % most extreme values that are
assumed to be errors, width-averaged velocities were obtained along the flume
for each cross section within 0.36 s at 16 phases in the tidal cycle.
Numerical flow model
Model formulation
We use a one-dimensional model that has been demonstrated to reproduce the
most important tidal dynamics . We assume a
rectangular channel of constant width and depth and solve the shallow water
equations for friction-dominated conditions; a condition that we will check
later. Here we modify the model to tilt the bed periodically so that it can
be applied to the tilting flume.
Continuity is conserved as
w∂η+zb∂t=∂uw(η+zb)∂x,
where w = width, h = depth, u = flow velocity, water depth h=η+zb
with the water surface located at level z=η, bed level at z=-zb,
t = time and x = streamwise coordinate. Here, cross-sectional area A=hW
and discharge Q=uhW in our rectangular channel. The left-hand side
represents the time rate of change of the wetted cross-sectional area, given
constant width entirely due to water level changes, and the right-hand side
represents volume flux convergence along the channel. To tilt the system
periodically, zb is imposed as a function of time and therefore changes
on
the tidal timescale in contrast to most studies in which it evolves on a much
longer morphodynamic timescale.
The momentum balance equation is given as
∂Q∂t+gA∂η∂x+gQ|Q|PC2A2=0,
where the terms from left to right represent local acceleration,
along-channel pressure gradient and bottom friction. Furthermore,
C = Chézy roughness coefficient and P=2h+W is the wetted perimeter for
the rectangular cross section.
We excluded advection because this term is an order of magnitude smaller than
the inertia, friction and pressure gradient terms. Inertia will scale as
[U] / [T], where [U] is the typical velocity scale and [T] is the timescale
over which the velocity changes, typically a quarter of the tilting period.
Advection scales as [U2] / [Lx], where Lx is a typical length scale.
For the Metronome this will be on the order of half the length of the basin.
Hence, advection with respect to inertia scales as [UT]/[2Lx], which is
half the tidal excursion length divided by the basin length. For typical
conditions in the Metronome this ratio is smaller than 0.1–0.2. In order to
keep this scale the same in smaller flumes, the tilting period needs to be
reduced linearly with the flume length.
The set of equations are discretized on a staggered grid with n flux points
and n-1 bed elevation points and solved by an explicit numerical scheme
that is second order in both time and space. The condition that Courant
numbers for surface wave celerity and flow velocity are below unity was
checked for every model run. Typical model settings are time step
dt=0.05 s
and spatial step dx=0.05 m for a domain of L=20 m with T=40 s and
h=0.025 m. We assume a constant width in all cases.
Three sets of boundary conditions simulate three different experimental
set-ups. In a Reynolds set-up, the bed zb is static, the landward boundary is
closed and the seaward water level is a function of time, i.e. η=asin(2πt/T) at x=L. As in nature, this enforces the pressure gradient
at the seaward boundary only, neglecting upstream rivers. The other two
set-ups are tilting basins with one or two boundaries open where zb is a
function of time; for example, at x=0 m, zb=z0+asin(2πt/T), with
z0 being the position of the bed at zero tilt. This means that the
pressure gradient is forced to be equal along the flume and that water
depth can change because both bed level and water surface are functions of
time and space. Two tilting scenarios applied: one with the landward boundary
closed and the seaward boundary open with a fixed water depth and free flux
(abbreviated henceforth as “tilt1”) and one with both the landward and
seaward boundary open (“tilt2”).
Hydraulic resistance
The artificial grass cover of the Metronome floor causes hydraulic resistance
similar to that of submerged unbending vegetation. This flow resistance is
calculated with the relation found by their Eq. 74.
Furthermore, the surface flow velocity is derived from the model calculations
in order to compare with the PIV data.
The Chézy roughness coefficient for submerged vegetation is calculated as
the combined effect of bottom roughness, throughflow resistance and overflow
resistance (Fig. a):
C=1Cb2+cDNsDsHs2g-1+gκlnhHs,
where Ns = number of stems, here measured at 50 000 m-2, Ds = stem
diameter, here measured at 0.4 mm, Hs = vegetation height, here 14 mm, and
κ=0.4 is Karman's constant. The first term represents the bed friction
below the vegetation; the second term represents the friction for flow
through the vegetation, and the third term represents friction for flow over
the vegetation. The drag coefficient cD of vegetation is here made
dependent on the Reynolds number, Re=uh/ν with ν=1×10-6,
because during flow reversal it may drop below typical turbulent flow
values. The drag coefficient
is dynamically calculated with a Coleman-type
constitutive relation,
cD=1+30Re+15Re0.6,
so that for high Re, cD≈1. We assume a minimum Re=30 so that the
maximum cD≈4, which occurs for velocities below about 0.001 ms-1. The
roughness of the bottom of the vegetated layer is calculated as
Cb=1810log12hks,
where ks = is the Nikuradse roughness length, here taken to be equal to the
90th percentile of the particle size distribution. In the sand-bed
experiments a constant C=25 m1/2s-1 was assumed. The dimensionless
friction factor is calculated from the Chézy coefficient as f=g/C2.
To be of use for the present purpose, the flow velocity at the water surface
is needed for a comparison of model results with PIV-derived data. Corrections
usually reported in the literature assume a logarithmic flow velocity profile,
but in the present case a layer of water is “skimming” over the vegetation so
that the partitioning of flow between lower and higher layers differs. A
correction factor was therefore calculated by using the combination of equations in
for a range of water depths above the submergence height
of the vegetation. The ratio of surface velocity and depth-averaged velocity
was found to be insensitive to total water depth (Fig. b),
meaning that water depth variations during the tidal cycle do not change the
ratio between depth-averaged and surface velocity more than 5–10 %.
Here flows with emergent vegetation were avoided because the present method
of PIV is impossible to use under these conditions. Furthermore, the flow is
not well described for the transition between barely submerged vegetation to
emerged vegetation at which the aforementioned ratio rapidly drops to unity, so
these conditions are also avoided. In the remainder of this paper the
modelled velocities are corrected with a constant multiplication factor of
1.95 for the grass-bed experiments and 1.60 for the sand-bed experiments,
which leads to an estimated error smaller than ±5 % for the lowest and
highest water levels, respectively.
Unidirectional flow data from PIV for the constant low slope and
high slope experiments. (a) Flow velocity at the water surface along the
flume. Drawn lines are analytically calculated surface flow velocities.
(b) Approximate water depths measured for both experiments. Drawn lines are
average values used in flow calculation. Still water depth measurements show
variation in bed level due to irregularities in artificial grass height and
sand layer thickness at the bottom of the grass.
Flow data from PIV and modelled flow with 30 s period tilting at
0.009 mm-1 slope amplitude with both boundaries open. Flow velocity is defined
as positive in the ebb direction with x=0 m being the upstream boundary.
(a) Flow velocity at the water surface along the flume for selected phases of
the tidal cycle. (b) Flow velocity at the water surface in one tidal cycle
for selected positions along the flume measured from x=0 m, indicated in
the legend. (c) Water level as a function of phase in the tidal cycle for
selected positions along the flume. Measured water levels have correct
amplitude and phase but possibly erroneous vertical offsets. Data are plotted
as symbols and model results are plotted as drawn lines.
Flow data from PIV and modelled flow with 15 s period tilting at
0.009 mm-1 slope amplitude with both boundaries open. (a) Flow velocity at the
water surface along the flume for selected phases of the tidal cycle.
(b) Flow velocity at the water surface in one tidal cycle for selected positions
along the flume measured from x=0 m, indicated in the legend. (c) Water level
as a function of phase in the tidal cycle for selected positions along the
flume. Measured water levels have correct amplitude and phase but possibly
erroneous vertical offsets. Data are plotted as symbols and model results are
plotted as drawn lines.
Experimental results and comparison to model results
A comparison of all experiments shows that flow velocities in the tilting flume
are much larger than in the Reynolds set-up. High velocities occur nearly
simultaneously along the flume as expected because it is driven by the gradient
of the entire flume rather than the gradient caused by a tidal wave initiated
at the seaward boundary. These results are consistent with the numerical
model. The model scenarios of initial conditions and boundary conditions are
the same as in the experiments. Below the results are described and compared.
Tilting flume experiments with two open boundaries
The tilting with two open boundaries shows nearly symmetrical reversing flow
(Figs. , ). Spatial
patterns in flow velocity along the flume appear consistent between tidal
phases and with the unidirectional flow experiments and are caused by camera
alignment and irregularities on the flume bed. These are further ignored. The
flow velocity lags behind the periodic tilting by about 2–3 s in both
the 30 and 15 s period tilting (Figs. b and
b). Measured water level fluctuates
periodically near the boundaries, especially at the 20 m boundary
(Fig. c). The faster tilting (15 s) experiment has
a lower velocity amplitude that also occurs nearly simultaneously along the
flume. On the other hand, the slower tilting has a higher velocity amplitude
in the middle of the flume. The slower tilting also shows more deformation in
the velocity signal than the faster tilting (compare
Figs. b and b).
Tidal amplitude (a) and phase (b) in the velocity signal of the
model runs compared to the experiments for the tilting flume. The principal
tide T1 is the full tidal period of 30 or 15 s, and the first overtide
T2 and second overtide T3 are the higher harmonics. Dashed lines indicate the
perfect fit plus or minus an error of 0.01 ms-1 to indicate the uncertainty
range of the data.
Flow data from PIV and modelled flow with 30 s period tilting at
0.009 mm-1 slope amplitude with upstream boundary closed. (a) Flow velocity at
the water surface along the flume for selected phases of the tidal cycle.
(b) Flow velocity at the water surface in one tidal cycle for selected positions
along the flume measured from x=0 m, indicated in the legend. (c) Water level
as a function of phase in the tidal cycle for selected positions along the
flume. Measured water levels have correct amplitude and phase but possibly
erroneous vertical offsets. Data are plotted as symbols and model results are
plotted as drawn lines.
Flow data from PIV and modelled flow with 15 s period tilting at
0.009 mm-1 slope amplitude with upstream boundary closed. (a) Flow velocity at
the water surface along the flume for selected phases of the tidal cycle.
(b) Flow velocity at the water surface in one tidal cycle for selected positions
along the flume measured from x=0 m, indicated in the legend. (c) Water level
as a function of phase in the tidal cycle for selected positions along the
flume. Measured water levels have correct amplitude and phase but possibly
erroneous vertical offsets. Data are plotted as symbols and model results are
plotted as drawn lines.
The model results show a fairly simple periodic flow that is nearly uniform
along the flume, with a very minor reduction of flow velocity at the
boundaries (Fig. a). Likewise, the modelled water
levels are nearly static (Fig. c). Modelled flow
velocities fit the observations fairly well when local accelerations due to
bed irregularity and discontinuities due to camera positioning are ignored.
However, a wave forms at both boundaries in all tilting experiments that
leads
to velocity peaks coinciding with water level peaks.
The time lag differs between the model and the flume
(Fig. b) so that the highest velocities of the
tidal cycle are approximately modelled correctly, but there is a mismatch
between the model and the observations near the slack. The measured flows begin to
decelerate sooner after the peak and accelerate slower after the slack, while
the modelled flow has a more rapid reversal of flow.
We compared the amplitudes and phases of tidal components in the observed and
modelled velocity signals in the middle of the flume
(Fig. ). For clarity, the full tilting period of the
flume is called “principal tide” or T1 rather than M2. The comparison shows
that the tidal velocity signal is dominated by the tilting period. The
“second overtide” (T3 rather than M6) is about 2 % of the velocity amplitude
due to friction and the “first overtide” (T2 rather than M4) is even lower
due to the negligible water level fluctuations. For the latter the deviation
between modelled and observed velocity is also the largest but this cannot be
considered significant given an uncertainty in the velocity data of a few
percent. The phase lag of T3 is surprisingly opposite in the model compared
to the observations. However, the phase lags are much smaller for the
principal tide. Possible causes are discussed later.
Tilting flume experiments with one open boundary
The observed and modelled flows in experiments with one boundary closed are
fairly similar to those with two open boundaries with two major differences
(Figs. , ). First, the
flow velocity reduces to zero at the closed boundary over a distance of about
1–2 m for the flood current (towards the closed boundary) in both
experiments and increases to its maximum value over a distance of about 5 m
for the ebb current in the 30 s experiment and about 3 m in the
15 s
experiment. This asymmetry between ebb and flood currents is caused by the
fact that water depth increases during the flood stage and decreases during
the ebb stage.
The second difference with the open boundary experiments is the effect of
the reflection of the tidal wave on the closed boundary. This leads to water
depth and velocity fluctuations close to the boundary
(Fig. ). As a result a water surface wave with a
velocity peak travels seaward over an 8–10 m distance whilst the tilting
slope peaks and reverses to dampen out at the peak flood velocity. The
primary effect of this wave superimposed on the tilting is a reduction of
velocity near the upstream boundary. In the middle and downstream reaches of
the flume, the observed and modelled flow shows negligible differences with
the cases of two open boundaries. We visually observed that the wave formed a
bore several millimetres high in the experiments.
The modelled and observed water level amplitudes at the upstream boundary
agree fairly well (Fig. c). The absolute level
differs, but this is meaningless in the experiments because the data were
detrended.
The harmonic analyses show that the runs with one boundary closed plot close
to the runs with both boundaries open for all tidal components except for
the T2, which is 2 orders of magnitude smaller than the T1
(Fig. ). This means that the flow in the middle of the
flume is not affected by the upstream boundary being closed, in agreement
with the observations made above.
Flow data from PIV and modelled flow with 60 s period sea level
fluctuation at 0.01 m of amplitude with the landward boundary closed. Positive
flow velocity is in the ebb direction. (a) Flow velocity at the water surface
along the flume for selected phases of the tidal cycle. (b) Flow velocity at
the water surface in one tidal cycle for selected positions along the flume
measured from x=0 m, indicated in the legend. (c) Water level as a function of
phase in the tidal cycle for selected positions along the flume, indicated in
the legend. Measured water levels have correct amplitude and phase but
possibly erroneous vertical offsets. Data are plotted as symbols and model
results are plotted as drawn lines.
Flow data in an 18 m sand-bed channel with a 2 m long by 3 m wide
sea from PIV and modelled flow with a 40 s period sea level fluctuation of
0.007 m amplitude with the landward boundary closed. Note that the vertical
axis range is half that of the tilting experiment. (a) Flow velocity at the
water surface along the flume for selected phases of the tidal cycle.
(b)
Flow velocity at the water surface in one tidal cycle for selected positions
along the flume measured from x=0 m, indicated in the legend. (c) Water level
measured by acoustics as a function of phase in the tidal cycle for selected
positions along the flume. Data are plotted as symbols and model results are
plotted as drawn lines.
Flow data in an 18 m sand-bed channel with a 2 m long by 3 m wide
sea from PIV and modelled flow with 40 s period tilting at 0.004 mm-1 slope
amplitude with the landward boundary closed. (a) Flow velocity at the water
surface along the flume for selected phases of the tidal cycle. (b) Flow
velocity at the water surface in one tidal cycle for selected positions along
the flume measured from x=0 m, indicated in the legend. (c) Water level
measured by acoustics as a function of phase in the tidal cycle for selected
positions along the flume. Data are plotted as symbols and model results are
plotted as drawn lines.
Reynolds-type experiments
Flow in the Reynolds set-up with periodic sea level fluctuations is weak
(Figs. ). The 30 s experiment with the amplitude
exceeding half a water depth showed effects of drying and flooding,
invalidating this experiment for the present purposes. In general the
strongest flows are generated at the sea level boundary, decaying rapidly
towards the closed boundary. The data show that velocity halves within the
first 3 m in both experiments. Furthermore, a local minimum velocity occurs
in the middle of the flume and a slight increase in flow velocity at one-quarter of the length with opposite phase to that at the mouth. The numerical
model roughly reproduces this pattern but predicts higher velocities in the
upstream half of the flume than observed.
However, this experiment shows a velocity limitation. Even though a 0.01 m
sea level amplitude was imposed, the observed sea level amplitude at 0.1 m from
the upstream boundary is only half this value. This may be due to the pump
capacity limitation at the seaward boundary. For this reason we ran the model
with half the design amplitude, which resulted in fairly close correspondence
of flow velocity in the most seaward few metres. Furthermore, higher modelled
water level amplitudes did not result in equally higher flow velocities
because of the non-linear effects of friction in shallower flow.
Sand-bed experiments
The sand-bed experiments with the tilting and Reynolds set-ups behave largely
the same as the grass-bed experiments
(Figs. , ). The flow velocity
amplitude is much larger in the tilting experiment than in the Reynolds
experiment despite the modest tilting slope amplitude. Despite the perfectly
symmetrical tilting motion, the ebb and flood phases are asymmetrical: flood
velocities occur at higher water levels than the same ebb velocities in the
first few metres from the closed boundary despite the perfectly symmetrical
tilting motion. On the other hand, the velocity amplitude in the Reynolds
experiment decays rapidly in the landward direction, but the sea level amplitude
is already 30–40 % of the water depth and cannot be increased much.
The sand-bed experiments have a complex geometry with a narrow, shallow
channel connected to a wide and deep sea. This leads to two-dimensionality in
the flow pattern that the one-dimensional model cannot cover well, such as
the high peak in modelled flow velocity at the transition from sea to
channel, which is gentler in the experiment due to convergence and divergence
in the sea. Also, the large spread in the flow velocities at x=18.6 m in
Figs. b and b is due to the
two-dimensional variation. The spatial and temporal patterns in the velocity
data are qualitatively similar to the model results with magnitudes of flow
velocity within about 20 %. However, despite the water available in the sea
for rapid inflow and outflow of the channel and the narrowed flume, the
inflow velocity again appeared to be limited. When half the sea level
amplitude was imposed in the model, as in the grass-bed experiments, we
obtained a velocity and water level amplitude similar to that in the
experiments.
The water surface amplitude and phase reasonably modelled in the Reynolds
set-up (Fig. c) were somewhat overestimated at the
landward boundary and imperfectly predicted in the tilting set-up, again
particularly at the upstream boundary (Fig. c). Possible
reasons are irregularities in the sand bed. As in the grass-covered
experiments, bores a few millimetres high form
(Figs. c and c). A small ebb
bore initiates near the upstream boundary and a larger flood bore initiates
at the seaward boundary. Furthermore, the measured velocity amplitude is
reduced
faster in the landward direction than the modelled velocity in the Reynolds
experiments. This is surprising because with a sea present in the tilting
flume, we did not expect the flux from the seaward boundary to be limited by
the pumps, so we expected the measured flow to resemble the model better.
The sand-bed experiments were designed as the initial condition for live-bed
experiments to be done later and are in that sense closer to future
morphological experiments than the grass-bed experiments. However, the sudden
transition from sea to channel renders the data less straightforward to
interpret. Nevertheless, the general correspondence in behaviour between the
grass-bed and sand-bed experiments and the model runs shows consistent
behaviour of the tilting flume in comparison to the Reynolds set-up, which
allows for general conclusions.
The tilting leads to two unexpected effects that need to be cancelled by
the periodic motion of the overflow weir. Flow depth over the weir is controlled
by the specific discharge given that the Froude number remains constant. This
means that compensation is required, approximately in phase with the tilting
and depending on discharge to maintain constant sea level. To have space for
the development of an ebb delta, a “sea” needs to be installed over a length
of a few metres as in the pilot experiment in Fig. .
However, the tilting would cause fast outflow into the sea during ebb, which
leads to water level change at the coastline. This can be prevented by
opposite-phase correction of the downstream weir to maintain constant
sea level at the coastline rather than at the weir.
Control experiments with constant slope
Measured flow in the constant slope experiments is on average uniform as
expected (Fig. ). However, there are spatial variations of up to
20 % that are consistent between the two experiments for flow velocity and
for water depth. Some of the variation occurs at the transitions between
camera images, which can be explained by deviations in camera orientation.
However, a larger part is also seen in the water depth measurements, including
the still water depth, and can therefore be attributed to irregularities in
the elevation of the artificial grass and the thickness of the sand bed. For
example, the increased velocity at 16–18 m coincides with shallower flow and
the lowest velocities occur at 6–7 and 12–13 m.
The highest water depths occur at the upstream boundary (0 m) and the
downstream boundary (20 m), perhaps because here the grass was glued to the
flume floor and the sand was not spread out as well. The water depth is lowest
for the highest flow velocity as expected because only slope was changed.
The predicted flow velocity based on measured average water depth and imposed
slope is approximately correct, which we take as sufficient evidence that the
measured artificial vegetation characteristics lead to the correct predicted
friction coefficient in the model. The effective Chézy coefficient is
about 11 m/s for these conditions, which is typical for shallow flume
experiments with a rough bed.
Tidal asymmetry
We used the model to explore tidal asymmetry and the magnitudes of overtides
(extending the range of models shown in Fig. ). Tidal
asymmetry is often used to indicate sedimentation tendencies. This was here
calculated as a function of the tilting slope amplitude. Tidal analysis for a
range of tilting slopes shows a straightforward increase in velocity
amplitude for the main component with deviatory behaviour only within the
first metre of the upstream boundary. The higher harmonics, however, do not
increase monotonously with tilting amplitude but the velocity amplitudes are
at least an order of magnitude smaller than the principal tide.
Sensitivity of flow to depth variation in the basin (drawn lines)
compared with the ideal depth (dashed lines) for 30 s period model runs.
Inset in panel (c) shows modified bed elevation. Runs with original depth are the same
as in Fig. . (a, c) Flow velocity at the water
surface along the flume for selected phases of the tidal cycle. Legend as in
Fig. . (b, d) Water level and flow velocity in one
tidal cycle showing hysteresis. (a, b) Tilting at 0.09 m of amplitude with
upstream boundary closed. (c, d) Reynolds method with sea level fluctuation at
0.01 m of amplitude with the landward boundary closed.
The model runs further show that near the closed landward boundary, flood
velocities are higher and ebb duration is longer, meaning that the head of
the estuary fills rapidly with water and empties slowly. This would lead to
sedimentation as expected with principal tide and without river inflow. The
inlet has approximately symmetrical tides but is slightly ebb dominant.
Halfway into the flume and in the upstream direction, the currents are ebb dominated
and the ebb duration is also longer than the flood, which is mostly due to a
minor second overtide contribution. The behaviour towards and at the
upstream boundary is sensitive to the tilting amplitude: above the large
gradient of 0.02 mm-1 (results not shown) the currents become flood dominant
and the flood duration exceeds the ebb duration. However, the strongest
responses all occur in the upstream few metres of the flume near the closed
boundary. Here the experiments show much smaller water surface fluctuations,
perhaps because the upstream boundary is less reflective and more subject to
friction than in the model. Furthermore, in a live-bed experiment,
sedimentation would rapidly modify the morphology, all of which would reduce
these asymmetries.
Effects of bed imperfections on the measured flows
The data indicate that irregularities in the grass and sand bed in the flume
affected the tidal flows. We tested this by conducting a model run with depth variation
along the flume. The sensitivity of the flow to water depth variations is
rather large (Fig. ): a gradually increased depth with a
maximum of 5 mm, less than 20 % of the original depth, causes large
and unexpected spatial variations in flow velocity and depth. In particular,
the increased depth causes increased ebb velocities at the seaward boundary
during some phases and decreased flood velocities. The flow velocity
patterns with modified bed elevation are more non-uniform, even in the middle
of the flume (Fig. a), and resemble those observed in the
experiments. This makes it likely that the irregularities of the bed in the
experiments caused at least some of the deviations between the model and
experimental data, in addition to potential bias in the data due to imperfect
camera positioning and calibration.
The model generally reproduces tidal dynamics in the experiments. Two main
differences emerged between the model and the data that need to be taken into
account in interpretations. Firstly, the water level amplitude is smaller in
the experiments than in the model, while flow velocity amplitude is larger in
the experiments. Secondly, bores form in the experiments during both flood
and ebb phases. In both the model and the experiments the tidal flow at the sea level
boundary transitions from currents without water level fluctuations to water
level fluctuations without current fluctuations at the closed landward
boundary.
Hypotheses for differences between measured and modelled velocities
There are minor differences between the modelled and measured velocity in the
tilting set-up during slack, suggesting a phase difference. A possible reason
is that the water depth varies by about 5 mm in the experiments over the
tidal cycle, which changes inertia, whereas the modelled water depths show no
significant temporal variation. In the flume there are stilling basins from
which water flows in at nearly zero velocity to rapidly accelerate into the
flume. In the model, on the other hand, there is no velocity gradient at the
boundaries. An alternative explanation is the effect of the critical flow
over the weir and the capacity of the pumps. During inflow, the water depth
at and near the boundary is reduced as the pump capacity is constant because
less water flows out of the flume. This reduces the inflow velocity. This
effect could in future be removed by increasing the pump capacity or
decreasing the effective width of the channel.
During outflow, the water depth at and near the boundary increases as the
broad-crested weir forces flow to be critical. This adverse behaviour could
in future be removed by compensation of the weir elevation. On the other
hand, inflow velocity appeared to be limited in the sand-bed experiments, too,
which had a considerable water volume in the “sea” that should have buffered
inflow limitations. This suggests that the pumps are not limiting the inflow
after all. We speculate that the inflow from the stilling basin (and sediment
trap) over a sharp edge onto the grass-covered flume floor causes flow
losses.
Properties of Reynolds and tilting estuaries across a range of
scales from experiments to natural estuaries. Linear friction is defined as
r=8gu/(3πhC2) with the given velocity rather than the unity value that
is usually assumed. The ratio between friction and inertia is calculated as
rT/2π. Velocities for the model scenarios were taken from the model runs
at relative length 0.8. Experiments with the Reynolds set-up were taken from
and and with the tilting set-up from
. Data for the Dovey (UK) are from ,
for the Thames from and for Westerschelde from .
Case
Metronome
Kleinhans
Fig. 2
Metronome
Tambroni 1
Reynolds tank A
prototype
Dovey
Thames
Westerschelde
model
experiment
experiment
model
experiment
experiment
model
nature
nature
nature
Configuration
tilt
tilt2
tilt1
Reynolds
Reynolds
Reynolds
Reynolds
Reynolds
Reynolds
Reynolds
Length L (m)
20
3.5
20
20
24.14
3.62
20 000
20 000
95 000
200 000
Width W (m)
1.5
1.3
1.5
1.5
0.3
1.18
1500
800
4300
6000
Depth h (m)
0.03
0.004
0.025
0.03
0.082
0.05
30
5
8.5
15
Amplitude a (m)
0.1
0.018
0.05
0.01
0.05
0.05
10
2
2
1.75
Period T (s)
40
72
30
40
180
53
40 000
44 712
44 712
44 712
Period T (h)
11
12.42
12.42
12.42
Chézy (m0.5s-1)
11
11
11
11
11
11
35
35
50
55
u (ms-1)
0.2
0.1
0.25
0.07
0.24
1
1.2
1
1.5
θ (–)
1
0.25
0.5
0.11
0.33
10
1.99
1.26
2.56
Fr (–)
0.37
0.5
0.5
0.13
0.27
0.06
0.17
0.11
0.12
Re (–)
6000
400
6300
2100
19 700
30 000 000
6 000 000
8 500 000
22 500 000
Excursion Le (m)
4
3.6
3.75
1.4
21.6
20 000
26 827
22 356
33 534
Wavelength Lt (m)
22
14
15
22
161
37
686 000
313 000
408 000
542 000
Friction r (–)
0.46
1.72
0.69
0.16
0.2
0.0002
0.0016
0.00039
0.00028
Aspect W/h
50
325
60
50
4
24
50
160
506
400
Amplitude a/h
0.33
0.61
1
0.33
0.4
0.24
0.12
Excursion Le/L
0.2
1.03
0.19
0.07
0.89
1
1.34
0.24
0.17
Wavelength Lt/L
1.1
4
0.8
1.1
6.7
10.2
34.3
15.7
4.3
2.7
Lt/Le
5.5
3.9
4
15.7
7.5
34
12
18
16
Friction / inertia
2.9
19.7
3.3
1
5.7
1.3
11.4
2.8
2
advection / inertia
0.2
1
0.2
0.1
0.9
1
1.3
0.2
0.2
Results of scaling by analysis and modelling
The shallow flow equations used here are well known to reproduce tidal
dynamics in idealized tidal basins on a prototype scale
and were shown above to reproduce tidal dynamics reasonably
well in tidal basins on the experimental scale for both the Reynolds set-up
and the tilting set-up. In this section we apply the model to length scales
ranging from small experiments to the largest estuaries on Earth in order to
explore scale effects and develop an understanding of how to upscale future results
of live-bed experiments to natural scales. Here, scale is defined as nL=Lprototype/LMetronome. We proceed in the opposite
direction of usual scaling analysis: given the Metronome we explore what
systems in nature are similar to it in important properties. The consequences
for scaling, expressed in characteristic dimensionless numbers for tidal
systems, are discussed on the basis of modelled velocities and compared to
data from real systems and some experiments (Table ).
Furthermore, experiments in smaller set-ups than the Metronome are discussed.
Application of the numerical model to scale up to prototype systems
As the most critical test, we assumed no distortion but simply multiplied
length, width, depth, tidal amplitude and tidal period with scale factors 0.1
to 10 000, covering small 2 m long flumes to 200 km long estuaries. The
results are hypothetical dimensions and dynamic properties of estuaries on
the full natural scale that are geometrically the same as in the experiments
(Table ). The modelling is conducted to investigate
whether the tidal flow is similar as well. Comparison between the
hypothetical and real natural estuaries in the next section on the basis of
modelled and observed velocities will show whether the assumed scaling is
realistic.
Sediment mobility calculated for tidal flows driven by periodic
tilting (drawn lines) and by the Reynolds method with periodic sea surface
fluctuation (dashed lines). Legend as in Fig. .
Grey area indicates immobile sediment.
Scaling from Metronome to natural estuaries by the numerical model
as nh=nT=nL for all settings, a constant a for the tilting set-up
(dashed lines for tilt1 and dotted lines for tilt2) and a constant a/h for
the Reynolds method (drawn lines). Legend indicates the length of the basin in
metres,
and the relative downstream distance is from 0 : landward to 1 : inlet.
The main scaling requirement for morphodynamic similarity between experiments
and reality is that sediment mobility is the same regardless of the length
scale. Here the grain-related Shields number is calculated from the
depth-averaged flow velocity and skin friction as θ=ρu2/[Cb2(ρs-ρ)D]. Furthermore, the relative excursion length Le/L
should be similar and flow should be subcritical (Fr<1), which is not an
arbitrary requirement in small experiments . The
Reynolds set-up is limited by the relative tidal amplitude a/h. The tilting
set-up drives tidal flow by the pressure gradient along the flume, meaning
that a different scaling is required: the pressure gradient depends on water
depth, which is linearly scaled with length, and gradient, which should
therefore remain constant and be independent of the scale number. Tides are
generated in the Reynolds set-up with a=0.01 m at the seaward boundary,
scaled with nL, and a=0.1 m for the tilting amplitude that is
kept constant across scales. Other model settings on the experimental scale
are W=1.5 m (no convergence) and h=0.03 m, while C=11 m1/2s-1 is
scaled by nL1/6. Note, however, that in live-bed experiments these are
dependent variables.
Modelled velocities and sediment mobility in the tilt2 runs remain
approximately the same across all scales because the scaling is entirely
linear in nL except for C and for the tilting amplitude
(Fig. ). Ebb or flood dominance along the system is
negligible. This means that with two open boundaries the flume simulates a
reach within an estuary with a nearly static water surface and periodic flow
velocity that is similar to the rigid lid assumption in tidal bar theories
. The tilt1 runs are similar except for the
surface amplitude increase and velocity amplitude decrease towards the
landward closed boundary. Further model tests (not shown) suggest that
convergent planform shapes could compensate for the friction loss. The Reynolds
runs have a landward decaying velocity amplitude with flood dominance in flow
velocity and a longer ebb duration landward of the mouth region. Moreover,
the velocity amplitude of the Reynolds set-up is rapidly reduced on experimental
scales, while it becomes independent of scale for the largest cases. For the
Reynolds set-up on the prototype scale, sediment is in motion along nearly the
entire estuary, whereas on the experimental scale the mouth region has barely
mobile sediment. On the smallest scales the tidal wave fits a number of times
in the basin whereas the largest scales have short basin properties.
Increasing the tidal amplitude in the experiments much beyond the present
a/h=0.33 is not an option, and neither is reducing the tidal period because
then the tidal wavelength would become smaller than the flume length, while
estuaries typically have lengths less than half the tidal wavelength.
The ratio of peak flood and peak ebb flow velocity indicates whether tidal
basins are respectively importing or exporting sand. The ratio of flood and
ebb duration, on the other hand, indicates the tendency for mud sedimentation
at slack tide. Here, flood duration is defined as the period that flow was
landward and the ebb duration as the period that flow was seaward. The
tilting set-ups have nearly symmetrical tides, whereas the Reynolds set-up has
flood-dominant conditions except near the mouth (Fig. ).
Consequences for morphodynamic experiments and possibilities to control tidal
asymmetry are discussed later.
Comparison of dimensionless numbers across a range of scales in idealized and real systems
The question is now how the models on different scales and in experiments from
the literature and the tilting set-up compare to natural estuaries. Dimensional
and dimensionless variables are given in Table .
Comparison between the experimental and natural estuaries shows the expected
differences in flow depth, flow velocity, sediment mobility and roughness.
For the prototype scale, the assumed roughness in the model is similar to that in
real estuaries, but the depth is considerably larger. This could suggest that
the chosen depth in the model on the experimental scale is large relative to
the length and width, but this remains to be investigated in live-bed
experiments in which the roughness is probably smaller, as the sand-bed
experiments had a Chézy value 2 times larger than the grass-floor
experiments. Smaller depth and the concurrent smaller water surface amplitude
would reduce sediment mobility in the Reynolds set-up even further. The Froude
number is below unity on the experimental scale, but it is much larger than in
nature. A smaller depth would increase the Froude number, but Fr=1 would
not be exceeded on a mobile bed . The width is rather
arbitrarily chosen here and leads to smaller channel aspect ratios than
observed in nature, which agrees with the suggestion that the modelled depth
is rather large. However, in nature the channel width is a property
determined by antecedent geology, tidal prism and the strength of the
banks. In turn, the width determines bar dimensions and bar pattern,
suggesting an interplay between self-formed bank properties and channel
dimensions similar to that in rivers. Investigating this further requires
morphodynamic experiments with salt marsh and riparian vegetation.
The different wave behaviour between scales is caused by the assumed linear
dependence of Lt/L on scale: tidal wave celerity depends on
h1/2,
while time is scaled linearly in our scenario. The tidal wave propagates in
the Reynolds set-up but is closer to standing in the tilting set-up. The depth
in the 200 km long model scenario is 300 m, which is unrealistically deep,
but again note that our chosen depth at nL=1 is not the result of
morphological experiments but merely a first estimate that can be adjusted on
the basis of morphodynamic experiments. The Reynolds set-up is flood dominant
landward of the mouth, whereas the tilting shows no significant asymmetry.
Friction dominates over inertia except in the Reynolds experiments. This
means that a linear upscaling of the tilting Metronome works for estuaries
the size of the Dovey estuary and perhaps larger estuaries
(Table ). However, in smaller experiments, such as those
of (Table ) and the 2 m scale in
the model runs (Fig. ), the friction dominates much more
over inertia, while advection is also more important than in the larger flume
(Table ). Since all these numbers depend on the flow
velocity, tidal period and length of the flume, they show that a flume longer
than at least 10 m is required to obtain acceptably similar conditions
to natural systems. While useful exploratory work can be done in a 2–3 m
tilting flume, similarity in tidal behaviour, flow conditions and sediment
mobility is less satisfactory. All this is ultimately the consequence of
having to use relatively coarse sediment to prevent cohesion and adverse
hydraulically smooth boundary effects.
The horizontal dimensions of a simulated estuary can be expressed as tidal
wavelength relative to basin length and relative tidal excursion length.
Here, the tidal excursion length is the distance that a parcel of water
travels during half a tidal cycle. In a natural estuary the tidal excursion
length is O(104) m, several times shorter than the estuary, while the
tidal wavelength is an order of magnitude larger and several times longer
than the estuary. The same is the case for the experiments, which was an
important scaling requirement. However, both the tidal excursion length and
the tidal wavelength are large relative to the flume length, meaning that
these simulate only part of the estuary length. Reducing the tidal tilting
period would reduce tidal excursion length and tidal wavelength linearly.
Discussion: implications for morphodynamic tidal experiments
The key result of the experiments and numerical modelling is that the
periodic flow velocities in a tilting flume set-up are roughly uniform along
the tilting flume. In contrast, periodic fluctuation of the sea level, as in
the
Reynolds set-up, causes much lower velocities that decay rapidly in the
landward direction and are too small to move sand. The effect on sediment
transport would be that sediment is immobile along most of the Reynolds set-up
and mobile in the tilting set-up (Fig. ). The
ongoing morphological experiments (Fig. ) are also
conducted with this sediment. The peak values of the Shields number in the
tilting experiments with a grass bed are 0.2–0.3, and in the sand-bed
experiments with lower tilting slope amplitude, mobility was kept
deliberately at about the threshold for motion. Much larger Shields values
can be obtained by using higher tilting slopes. This means that the typical
mobilities of natural systems, which are O(1), are within reach with the
Metronome. On the other hand, to approach such conditions in the Reynolds
set-up, a flume of at least 100 m length is needed with a pumping capacity of
several m3s-1. At a length of about 200 m the flow conditions of the
Reynolds set-up and the tilting set-up are similar (Fig. ),
while for smaller flumes only the tilting set-up maintains sufficient sediment
mobility.
The width and depth of self-formed estuaries depends on the strength of the
banks, and in turn the bar pattern depends on the channel aspect ratio.
Experimental creation of intertidal mudflats and salt marsh requires
a slightly cohesive mud simulant and vegetation with a flow resistance and
rooting depth appropriate on the experimental scale and in practice
constrained by the typical minimum sizes of fast-sprouting vascular plants
. This means that fine sediments, perhaps of lower
density than sand, and seeds will need to be suspended up onto the bars in
tidal experiments. However, a major complication in past meandering river
experiments was the rapid drop in sediment mobility and turbulence on the
shallower parts of the bed. Indeed, the Reynolds numbers in the smaller
experiments are close to the transition from turbulent to laminar flow
(Table ) and are far below it in the smallest
experiment for width-averaged flow conditions. As both flow velocity and
water depth decrease onto bars, while the roughness length remains the same,
the Reynolds number decreases quadratically with depth. Therefore flow on
bars is likely laminar in metre-scale experiments. This problem is
considerably reduced for the 20 m flume, but not entirely removed. The 20 m
flume will also be more appropriate for the use of live plant seedlings to
simulate vegetation, for which hydraulic resistance is Reynolds-number dependent
and the rooting length relative to channel depth is more similar than in
experiments in 2 m flumes.
Flood dominance accomplished in the tilting flume by adding an
overtide to the tilting with an amplitude of 20 % of the principal tide with
a phase delay of π/2. Ebb flow is positive. The transport is cumulative
in time to show the flood dominance in two tidal cycles.
The Reynolds set-up is slightly flood dominated compared to the tilting set-up
(Fig. ), suggesting that Reynolds experiments would
import sediment but the tilting set-up would not. These results led us to
hypothesize that tidal asymmetry can be imposed at any degree through asymmetric
tilting by adding an overtide, which is confirmed by the model
(Fig. ). This is, for example, similar to having an
M4 tidal component at the seaward boundary as occurs in nature in shallow
seas such as the North Sea. To show whether the higher peak flood velocity or
the longer ebb duration dominates net transport, we calculated sediment
transport from qs=α(θ-θc)1.65 and
cumulated over two tidal periods. In this example, an overtide tilting
magnitude of 20 % of the principal tide and a phase delay of π/2 is
predicted to give strong flood-dominated transport. The transport without the
imposed overtide is not exactly symmetrical because of the secondary overtide
generated in the flume (Fig. ).
It is technically straightforward to tilt the Metronome with higher harmonics
and tidal asymmetry (Fig. ), to add a constant
discharge at one closed boundary and to impose any initial planform shape
and depth along the system. This opens up possibilities to drive ebb- or
flood-dominant transport in the flume, which is the cause of sediment import,
export and equilibrium in natural systems , and simulate, for example, the infilling of flood-dominated
estuaries. The broad similarity between conditions in the 15, 30, 40 and
60 s
experiments and further model tests (not shown) indicates that a range of
combinations of tidal wavelengths, tidal excursion lengths and sediment
mobility can be attained in the Metronome to design preferred scales. We
expect that self-formed morphology will not cause such strong spatial
variations in flow velocity as observed in the present experiments and
models because such flow divergence in a friction-dominated flow would cause
spatial gradients in sediment transport that modify the morphology to eventually reduce
the spatial variations in flow velocity .
It is clear from the present results that obtaining sufficiently mobile
sediment over the length of an experimental estuary is impossible in the
Reynolds set-up at practical laboratory sizes. On the other hand, by tilting
we have a high degree of control over current velocities and water levels and
tidal asymmetry. With this, the question of whether flooding or tilting is
better suited for morphodynamic experiments with tidal systems is partially
answered in that the tilting method is clearly more suited to obtain
periodically reversing sediment transport at any required mobility and tidal
asymmetry similar to those in natural systems. Moreover, this technology is
potentially widely available because of the simplicity of periodic tilting
and the smallest metre-scale flume size at which interesting results are
obtained despite serious scale problems of shallow flow . The Metronome set-up opens up the possibility to conduct
experiments on estuary development and biogeomorphodynamics following similar
principles as for rivers, including bar formation and interactions with
self-forming floodplains with cohesive sediment and vegetation
.
Conclusions
The prime requirement for scale experiments with tidal systems is to obtain
reversing currents that cause high sediment mobility along the entire system,
while tidal wave behaviour is of secondary importance. However, it is
impossible to scale the sediment size by the same factor as system length and
width. Here we show that the degree of similarity of tidal flows and sediment
mobility in experiments and in nature depends strongly on the size of the
flume and the method of generating tidal flows.
The method of Reynolds with periodically fluctuating sea level cannot lead to
sufficient bed shear stress for bidirectional sediment transport except with
low-density sediments and in impractical set-ups larger than hundreds of
metres. The reason is that the tidal wave rapidly dampens out in the landward
direction due to friction, which is higher in experiments than in nature and
much higher in flumes of a few metres long. This cannot be compensated for by
higher tidal amplitude because it should not exceed half the water depth. A
practical problem is that these experiments require considerable pumping
capacity even to reach the limited mobility in the inlet.
A periodically tilting flume of 20 m length causes reversing flows with
sufficient strength in both the flood and ebb direction to transport sand. A
sinusoidal tilting pattern with two open boundaries causes an approximately
sinusoidal flow velocity pattern along the entire flume with uniform width
and depth, whilst the water level hardly fluctuates. This means that the rigid lid
condition is approximated with two open boundaries. When one boundary is
closed, reflection of the tidal wave causes large depth fluctuations and
enhanced ebb currents near the closed boundary, whilst the flow velocity
along most of the flume is almost the same as in the experiments with two
open boundaries. The flow remains subcritical but the Froude number is much
larger than in natural estuaries. In nature this would affect tidal wave
propagation and resulting flow velocity, but in the tilting flume the tidal
wave is independently imposed by the tilting. The flow is turbulent in the
present experimental conditions, but the Reynolds number would drop rapidly
to laminar conditions for shallower flows above bars, meaning that larger
flumes are better for suspended sediment transport onto bars and mudflats
and for interactions with live seedling vegetation.
Numerical modelling for a range of scales shows that similar sediment
mobility, relative tidal excursion length and relative tidal wavelength can
be attained in tilting flumes tens of metres long. The tidal flow is
friction dominated, as in natural systems that are 3 orders of magnitude
larger, while advection is of minor importance. However, for the smallest
possible experimental estuaries of a few metres in length, the friction is much
higher than in nature, while the flow becomes laminar above bars, although the
required sediment mobility may be attained, which is useful for exploratory
experimentation.
The tilting flume set-up allows for independent control over tidal period and
tidal asymmetry. This, in turn, allows for experimental control over the
simulated length of the tidal basin, tidal excursion length and tendency to
import or export sediment without compromising the sediment mobility. The
implication is that the Metronome tidal facility opens up new possibilities
for tidal morphodynamics research that are complementary to numerical
modelling and field observations.