Emergence of large-scale coherent structures in a shallow separating
flow
Harmen Talstra, Wim S.J. Uijttewaal & Guus S. Stelling
Environmental Fluid Mechanics Section, Faculty of Civil Engineering and Geosciences, Delft University of
Technology, the Netherlands
ABSTRACT: In rivers, the phenomenon of flow separation past obstacles often gives rise to large-scale coherent structures. This study focuses on quasi-2d coherent structures associated with shallow flow separation
(vortex shedding). New insights about the physical mechanism governing vortex shedding have been obtained
from large-scale Particle Image Velocimetry (PIV) experiments on a Shallow Lateral Expansion (SLE) with a
variable inflow width. Analysis of the acquired data stresses the crucial role of the secondary circulation cell,
which is often present in shallow separation geometries. A sufficiently developed secondary circulation cell is
significantly contributing to vortex shedding. Moreover, the interaction between primary and secondary circulation cells causes a “scale jump” in the horizontal length scale of the shed vortices. An analysis of Reynolds’
stresses and the downstream development of conditional averaged eddies does clearly show this jump. Since
the scale jump is essentially due to interaction of discrete horizontal eddies, 3D Large Eddy Simulations
(LES) are performed in analogy with the measured PIV geometries. The obtained LES data will be used to develop a depth-averaged flow model that will reproduce the discrete horizontal eddy interaction in an accurate
way, thus enabling researchers and engineers to predict vortex shedding in river geometries.
1 INTRODUCTION
1.1 Background and objectives
The present study focuses on a combination of two
classes of turbulent flows. The first class is that of a
shallow free-surface flow: geometries with vertical
length scales much smaller than the horizontal scales
(typical aspect ratio 5% or less), in which most of
the large-scale turbulence may be considered as quasi two-dimensional. Both production and dissipation
of turbulent kinetic energy are greatly influenced by
the presence of bottom friction. The second class is
that of a separating flow: due to an adverse pressure
gradient the main flow separates from a wall, inducing a zone of flow recirculation and possibly the
presence of a street of vortices which are shed from
the separation point. Both classes come together in
many rivers and coastal flow geometries. Shallow
separating flows are abundant in nature as well as in
man-made environments.
In practice, large-scale coherent structures resulting from a separating shallow flow frequently turn
out to cause problems regarding e.g. navigation and
bed erosion, the latter combined with settling of sediment in undesirable places. These effects are wellknown from rivers normalized by long series of
groynes, or harbour entrances becoming less acces-
sible due to large-scale eddies and loss of navigable
depth. Since many shallow flows accommodate a variety of functions and interventions in flow systems
may have unexpected or unwanted effects, improving the understanding and modelling approach of
large-scale coherent structures is of practical relevance. In this study, therefore, an attempt is made to
visualize experimentally the emergence and behaviour of individual coherent structures shed from a
separation point. This experimental information is
being used to validate and inspire further development of numerical models of shallow separation
flow geometries.
1.2 General visualisation of shallow separation
Figure 1 shows the effect of separation on a shallow
channel flow past an obstacle, visualised by the injection of dye. In this case, the separation gives rise
to a primary and secondary recirculation cell (“bubble”). In literature, the maximum streamwise recirculation zone length (reattachment length) for this
type of lateral expansions is often estimated at approximately 8 times the obstacle size (see Babarutsi,
Nassiri & Chu, 1996). At the interface between main
flow and recirculation zone, lateral exchange of
momentum and dissolved matter takes place due to
the presence of a mixing layer and large-scale
Figure 2. Shallow Lateral Expansion (SLE) with time-averaged
streamlines, obtained by a preliminary HLES computation. Location of expected flow features: 1) primary bubble, 2) secondary bubble, 3) effect of intermittent opposite bubble, 4) mixing
layer, 5) induction of vortex shedding, 6) reattachment point
a “scale jump” in vortex sizes is visible. The location of this jump is important, because there is a
strong suggestion of positive interaction between the
large eddies and the primary bubble (“vortex merging”), while these eddies tend to remain small while
passing the secondary bubble slightly upstream. This
implies that “vortex shedding” is not a distinct
mechanism of eddy generation at the separation
point, but merely an internal flow feature associated
with interaction of existing eddies having the same
vorticity sign. This interpretation makes vortex
shedding here a typical quasi-2d turbulence phenomenon.
Figure 1. Large-scale structures in a shallow separating flow,
visualised by dye. Shown are: direction of the main flow (U),
primary bubble (1st), secondary bubble (2nd) and location of the
spatial “scale jump” in downstream development of the vortex
street
coherent structures. The presence of recirculations
makes the flow behaviour quite different from that
of a plane mixing layer due to a lateral velocity difference only.
In their experimental study on groyne fields,
Uijttewaal, Lehman & Van Mazijk (2001) point out
a qualitative difference between mixing layer vortices and “vortex shedding”. The latter phenomenon is
largest in scale and is associated with the presence of
a secondary recirculation cell, while the first phenomenon is associated with lateral shear. It is an
open question, however, whether there is indeed a
difference in physical mechanism for both types of
eddies.
From the experiments which are described in this
paper, it is visually observed that, indeed, a distinction can be made between the two mechanisms. As
soon as a sufficiently well-developed secondary
bubble appears, the initial mixing layer changes
character and much larger eddies seem to be shed
than before. This effect can be associated with the
term “vortex shedding”. It is observed, however, that
the largest eddy scales are emerging not from the
separation point, but from a point some distance
downstream – approximately at the interface between primary and secondary bubble. At this point,
2 DESCRIPTION OF PIV EXPERIMENTS
2.1 Experimental setup
In order to study the dynamics of individual largescale vortices, physical flow experiments have been
carried out in a wide and shallow free-surface flume,
using the measurement technique of Particle Image
Velocimetry (PIV). Because it is desired to study the
mere vortex shedding phenomenon with as least geometrical information as possible, the experimental
setup has been chosen to be a Shallow Lateral Expansion (SLE). This geometry involves only three
length scales: the inflow width, the outflow width
and a uniform depth. The SLE is well-known from
literature (e.g. Babarutsi, Ganoulis & Chu 1989), but
mostly only with respect to time-averaged quantities
rather than individual eddy dynamics. Moreover, it is
common practice to take an inflow width equal to
50% of the outflow width, thus effectively reducing
the number of length scales to two. In the present
study, the inflow width is explicitly chosen to be
varied to 3 different values. This yields 3 distinct
cases which are object of detailed PIV analysis; the
number of qualitative experiments carried out by
visual observation, however, is 18 because also
depth and discharge have been varied.
The experiments are carried out in a shallow
flume with a length of 20.00 m and an outflow width
b2 of 2.00 m. Bottom and wall conditions are hydraulically smooth. In all three PIV experiments, the
water depth h is 9.2 cm. The width b1 of the inflow
U0
y
sample 1
sample 2
(…)
sample n
x
Figure 4. Location of the series 1.50 x 1.50 m2 PIV samples
spectively 8, 6 and 4 samples, visualizing the greater
part of the vortex shedding areas in downstream direction (Figure 4).
2.3 Large eddy visualisation using vector potentials
Figure 3. Overview of the 3 PIV geometries (1:4, 2:4 and 3:4)
section is either 0.50, 1.00 or 1.50 m. The three cases are respectively referred to as the 1:4, 2:4 and 3:4
case. The discharge of the first 2 cases being 0.027
m3/s, the average depth-based Reynolds’ number is
Re = 13500. The discharge of the wide 3:4 case is
increased to 0.048 m3/s (Re = 24000) in order to
keep the entrance velocity sufficiently high and
comparable to the 2:4 case. On the other hand, the
1:4 case discharge is not decreased in order to prevent the recirculation zone from laminarizing. See
Figure 2 and 3 for an overview of the experimental
geometry and the expected time-averaged flow pattern.
2.2 Measurement tools
Floating black polypropylene tracer particles with a
diameter of 1.5 mm, contrasting with the white
flume bottom, are used to visualize the surface velocity field. With a sampling frequency of 10 Hz (2:4
and 3:4 cases) or 15 Hz (1:4 case), instantaneous velocity fields of size 1.50 x 1.50 m2 are obtained by a
1 Mpixel digital camera and the PIV algorithm of
Davis (version 6.2). Every set of 2 subsequent camera frames is cross-correlated by the algorithm, using
an interrogation window overlap of 75%. The results
are samples of 2d surface velocity fields on a rectangular co-located grid with a spacing of 2.4 cm in
both x- and y-direction. The sample sizes are respectively 7000 frames (2:4 and 3:4 case) and 10500
frames (1:4 case), both accounting for a measurement duration of 700 s.
Because the spatial domain of the camera image
is limited to 1.50 x 1.50 m2 by resolution requirements, series of samples have to be linearly fit together in order to cover the entire relevant flow area.
Therefore, each sample has an overlap of approximately 20% with neighbouring recording areas. The
1:4, 2:4 and 3:4 case data series are consisting of re-
A topic requiring special attention is the way in
which large-scale eddies within the PIV samples can
be detected and visualized. Although a simple look
at the velocity vector field often shows these large
eddies immediately, it is not always easy to define a
straightforward detection algorithm.
Bonnet et al. (1998), Scarano et al. (1999) and
Adrian et al. (2000) all give extensive overviews of
commonly used identification methods for coherent
structures, e.g. based on vorticity, swirling strength
or spatial correlations with a “mask eddy”. The
drawback of the latter method is the spatial inflexibility of the results and their slight dependence on
the mask eddy length scale which is chosen a priori;
the former methods have the disadvantage that, like
all gradient-based methods, they tend to enhance the
dominance of the small length scales within the velocity signal, rather than the large scales.
In this study, the vector potential of a velocity
field is used to detect large eddies. The use of vector
potentials is quite common within the context of
electromagnetism but not within fluid dynamics, in
spite of the great mathematical analogy between
both scientific fields. Yet, vector potentials are very
elegant in use because they allow large vortex scales
to be determined directly from instantaneous flow
kinematics. They very much resemble the concept of
2d stream functions, but are computed in a different
way in order to make a correction for the nonsolenoidality of a 2d plane within a 3d velocity field.
Vector potentials can be constructed by solving the
Poisson equation for each component of the vorticity, using homogeneous Neumann boundary conditions. See the Appendix for a full mathematical explanation of construction and relevance of vector
potentials.
Each local maximum or minimum of a vector potential function uniquely identifies a large eddy core
of positive, resp. negative vorticity sign. The shape
of the eddy is given by the surrounding isolines.
Within the instantaneous vector potential fields, the
permanently present primary and secondary recirculation cells are by far dominant; when the timeaveraged pattern is subtracted from the instantaneous
patterns, however, the result is the visualisation of a
straight streamwise line of large vortices being shed
from the separation point.
3 RESULTS
3.1 Visual observations: general flow features
In order to explore the parameter space of relevant
experimental cases, over 30 different flow cases
were observed merely visually, using dye and the
same black tracer particles as described before. Every flow case was running for about 2 hours and was
used to gain insight in the occurring flow features:
over-all flow patterns, bubble sizes and large eddy
length scales. Water level, discharge and inflowoutflow width ratio were varied in such a way that
the flow remained sufficiently turbulent and fully
subcritical. In the end, 18 flow cases were concluded
to be suitable for mutual comparison.
Table 1 shows the primary and secondary recirculation cell sizes, scaled by the lateral expansion
width d1, for the 2 different discharges (q) and 3
width ratios predefined in Section 2.1. The 18 water
depth values are varying for each case because of the
practical difficulty to get these exactly equal. The
primary bubble size is defined by the reattachment
length (L1) and the secondary bubble size by the xlocation of the secondary separation point along the
wall (L2), both taking the expansion’s lower left corner as the origin. Both water depth (h) and bubble
sizes are made dimensionless by the lateral expansion width.
First, the general impression is that the dimensionless reattachment length L1/d1 tends to decrease
for decreasing dimensionless depth, compared to the
maximum deep water limit known from literature.
This limit is estimated to lie between 7.0 and 8.0
(the latter number is proposed by Babarutsi, Ganoulis & Chu 1989). The values found in the 3:4
case seem to confirm this. It is however striking that
the values observed in the 1:4 and 2:4 cases exceed
this limit, even up to values 9.0 and over, although
the reattachment point is usually slightly oscillating.
This exceedance may be explained by the hydraulically smooth wall conditions of the present experimental setup, which often are not reached in practice. Also, the relatively large size and the high
Reynolds number of this setup may play a role of
importance (the experiments of e.g. Babarutsi et al.
were carried out on an approximately 3 times smaller scale and with larger values of roughness).
Second, the secondary bubble sizes L2/d1 in the
2:4 and 3:4 cases show a pattern opposite to that of
the reattachment length: as the relative shallowness
increases, the secondary separation point moves further downstream. This can be understood by the
primary bubble becoming weaker: as the primary reTable 1. Primary and secondary recirculation cell sizes for 18
different flow cases; visual observations
_______________________________________________
No. Geometry
Flow
parameters Recirculation
sizes
_____________
______________
q [m3/s] h/d1* [-] L1/d1* [-] L2/d1* [-]
_______________________________________________
1
1:4 case
0.027 0.052 7.2
2.2
2
1:4 case
0.027 0.064 7.6
2.2
3
1:4 case
0.027 0.075 8.4
3.6
4
1:4 case
0.048 0.073 8.8
2.2
5
1:4 case
0.048 0.084 8.8
2.6
6
1:4
case
0.048
0.107
8.8
3.2
_______________________________________________
7
2:4 case
0.027 0.059 8.4
2.7
8
2:4 case
0.027 0.078 8.4
2.7
9
2:4 case
0.027 0.108 10.2
1.8
10 2:4 case
0.048 0.071 9.6
2.7
11 2:4 case
0.048 0.092 9.6
2.4
12
2:4
case
0.048
0.124
10.2
1.8
_______________________________________________
13 3:4 case
0.027 0.090 6.0
2.4
14 3:4 case
0.027 0.162 6.6
2.4
15 3:4 case
0.027 0.226 6.6
1.8
16 3:4 case
0.048 0.118 6.6
1.8
17 3:4 case
0.048 0.194 7.2
1.5
18
3:4
case
0.048
0.256
7.2
1.5
_______________________________________________
* Length scales are made dimensionless by expansion width d1
circulation flow decreases in discharge and energy, it
will break away sooner from the solid wall because
of the local adverse pressure gradient which previously brought the secondary bubble into being.
Hence, with increasing shallowness, the primary
bubble is eaten up from two sides, sometimes even
to the point that the primary and secondary bubble
become comparable in size. The secondary zone of
the 1:4 case, on the contrary, is enhanced together
with the primary zone. It is observed that, in these
cases, the secondary bubble has grown too big to
maintain itself within such shallow conditions, and
hence breaks up in two or more intermittent structures. In some cases, even a small “tertiary bubble”
in the lower left corner is observed.
Third, because of its large relative shallowness
(small value of h/d1), the 1:4 case shows an intermittent bubble opposite to the primary recirculation.
The presence of this opposite bubble depends strongly on the passing by of large vortices shed from the
separation point. These vortices are able to entrain so
much main flow fluid into the mixing layer that the
main flow separates from the opposite continuous
wall. This feature is however intermittent and is located too close to the outflow boundary to be studied
in an accurate way. In the 2:4 and 3:4 cases, it is absent.
Finally, the most important flow feature to be described is the vortex shedding phenomenon. Flow
patterns can be described in good detail because of
the abundance of tracer particles. Striking is the stability of the near-field mixing layer, in spite of both
its small width and the presence of a very strong lateral shear due to the secondary bubble which is almost touching the main flow. The small coherent
structures emerging from the separation point seem
to be almost immune to their neighbourhood. As
soon as the mixing layer approaches the primary
bubble, however, the structures appear to be boosted
significantly, both in energy and spatial scale (“scale
jump”).
This can be explained from the positive interaction between vorticity cores in both mixing layer and
primary zone, i.e. the effect of “vortex merging”.
This effect is well-known from quasi twodimensional flow theory and can be associated with
the suppression of the vortex stretching term in the
vorticity equation. The boosted eddies are penetrating to a large extent in both main flow and primary
bubble, and are able to exchange a lot of mass and
momentum. (This, too, may be an explanation for
the remarkably large reattachment lengths that are
found in the 1:4 and 2:4 cases.) The zones in between two large shed vortices are observed to be areas of considerable shear. The flow is stretched in
streamwise direction here, as well as compressed in
spanwise direction. Also, secondary flow in vertical
direction is affecting these zones, in contrast with
the large vortex cores which appear to be predominantly two-dimensional.
In the 3:4 case the scale jump effect is much
weaker, if not absent. In this geometry the secondary
bubble zone is relatively deep (largest value of h/d1).
Therefore the bubble remains small and does not
touch the main flow. The influence of the primary
bubble, hence, is reaching all the way upstream to
the separation point, resulting in a smoother and less
dramatic mixing layer development. From that point
of view, it is not surprising that the reattachment
lengths found in the 3:4 case answer better to the expected maximum limit than the 1:4 and 2:4 cases.
In the 1:4 case, the scale jump effect is present
but weaker than in the 2:4 case, which contains the
most obvious jump. This is explained by a limited
influence of the secondary bubble in the 1:4 case,
because it tends to break up in separate structures as
mentioned above (which is visible in Figure 5a).
Based on the 18 flow cases described above, a
representative geometry is selected for detailed PIV
analysis. Only the inflow-outflow width ratio remains variable (hence 3 cases). Dimensions and other figures of this geometry have been described in
Section 2.1.
a)
b)
c)
Figure 5a-c. Time-averaged flow quantities of the 1:4 case: a)
time-averaged stream function  [m²/s], b) spanwise turbulent
kinetic energy u '² [m²/s²], c) horizontal Reynolds’ stresses
u ' v ' [m²/s²]
d)
e)
f)
Figure 5d-f. Time-averaged flow quantities of the 2:4 case: d)
time-averaged stream function  [m²/s], e) spanwise turbulent
kinetic energy u '² [m²/s²], f) horizontal Reynolds’ stresses
u ' v ' [m²/s²]
g)
h)
3.2 PIV data analysis: time statistics
All recorded PIV time series have a sampling period
of 700 s, which is equivalent to about 30-70 consecutive large eddies shed from the separation point. In
this section, over-all time statistics are shown for the
3 measured flow cases. For each case, neighbouring
measurement recording areas were fit together by
linear interpolation in the overlapping areas.
i)
Figure 5g-i. Time-averaged flow quantities of the 3:4 case: g)
time-averaged stream function  [m²/s], h) spanwise turbulent
kinetic energy u '² [m²/s²], i) horizontal Reynolds’ stresses
u ' v ' [m²/s²]
Figures 5a to 5i show the contour plots of timeaveraged flow quantities: stream function, spanwise
turbulent kinetic energy and horizontal Reynolds’
stresses of the 1:4, 2:4 and 3:4 cases respectively.
The stream functions show that, in the 3:4 case, the
secondary bubble is indeed less developed than in
the other 2 cases, resulting in a deviant vortex shedding behaviour. Please note the tendency of the 1:4
secondary bubble to break up in two parts.
Reynolds’ stress values can be seen as timeaveraged indicators for the presence of large-scale
coherent structures (velocity correlations). Therefore, it is telling that in the 1:4 and 2:4 cases, the
scale jump is visible in the Reynolds’ stress contours. In these cases, the negative peak of the Reynolds’ stress suddenly widens and deepens from a certain point downstream, while the 3:4 case does not
show this behaviour. Furthermore, it is striking that
the components of the turbulent kinetic energy show
a quite linear development, while the Reynolds’
stress contours are more discontinuous. This can be
understood by the fact that the velocity variance contains only temporal and local information, whereas
the velocity covariance is also pointing towards the
spatial structure of the flow field. Apparently, the
development of the shed vortices in terms of total
energy is pretty continuous, comparable to that of a
plane mixing layer; but downstream from the scale
jump location, the kinetic energy distribution does
shift towards larger length scales. This suggests that
the scale jump effect is spatial rather than temporal.
Figure 6 shows the development of the lateral
turbulent kinetic energy downstream from the separation point for each of the 3 cases. As expected, no
jump is visible here. Examining the downstream development of the associated temporal power density
spectra yields the same result. On the other hand,
one will expect to find such a jump in the spatial
power density spectrum development. Unfortunately,
the horizontal dimensions of the PIV camera samples are too limited to allow for reliable spatial spectra.
From the computed Reynolds’ stresses a turbulent
length scale (Prandtl mixing length) can be defined
using a simple mixing length approach. Figure 7
shows the downstream development of this mixing
length  for the three PIV cases. The 2:4 shows the
clearest jump. It is however not surprising that this
method yields eddy sizes which are almost an order
of magnitude too small, because we are using timeaveraged statistics of an intermittent phenomenon.
For both reasons mentioned above, spatial vortex
scale development can best be examined by conditional averaging of the intermittent shed vortices.
Results of this averaging operation are described in
the next section.
v ' ² / U 02
-3
7
6
x 10
Figure 6. Development of the lateral turbulent kinetic energy
downstream from the separation point, normalized by the
squared mean inflow velocity. Dashed line: 1:4 case, dash-dot
line: 2:4 case, dotted line: 3:4 case, continous: trend line. The
downstream distance is scaled by the lateral expansion width.
The 2:4 case vortex shedding is containing the largest amount
of turbulent kinetic energy relative to the energy of the mean
flow. The 1:4 and 3:4 case are quite comparable
/d1
0.025
0.02
0.015
0.01
0.005
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x/d1
Figure 7. Development of the Prandtl mixing length  downstream from the separation point, based on measured Reynolds’
stresses. Dashed line: 1:4 case, dash-dot line: 2:4 case, dotted
line: 3:4 case, continous: trend line downstream of the scale
jump location (approximately x/d1 = 0.8). Both the downstream
distance and the mixing length are scaled by the lateral expansion width. The 2:4 case shows the most pronounced scale
jump behaviour
3.3 PIV data analysis: conditional averages
The location of the conditional averages is the line
along which the relevant large eddies are found: the
straight streamwise “shadow line” downstream of
the separation point. For each point on this line
where PIV data are located, the cross section of the
flow is checked for the presence of a large eddy.
This is done by searching local minima in the vector
potential time series relative to the mean flow pattern (the latter is needed because otherwise only the
primary and secondary bubble would be found). After that, statistical operations can be performed on
y [m]
x 10
1.2
6
1
4
0.8
2
-3
cond / d1
1.5
1
0.6
0
0.4
U0
-2
0.5
0.2
-4
0
-6
0
0.5
1
1.5
x [m]
Figure 8. Typical example of a conditionally averaged shed
vortex (in the far field of the 2:4 case), visualised by a vector
potential field. The computed streamlines agree very well with
the associated vector field. The local minimum in the centre of
the picture identifies a clockwise large eddy with a diameter of
approximately 0.8 m
the conditional vortex samples. Figure 8 gives an illustration of the procedure in which the large eddies
are detected by means of vector potentials.
In principle, conditional averaging can be done
for every point within the measured flow domain.
Near the edges of the recording domains, however,
the conditional averaged flow fields show biases because of the difficulty to determine a local minimum
near a boundary. Therefore, the fitting of neighbouring data recording areas is problematic for conditional averaged quantities. The over-all patterns and
trends, however, remain sufficiently clear.
The conditionally averaged spanwise velocity
signal (again relative to the mean velocity) is suitable for determining local large eddy length and time
scales. Please note that vector potential quantities are
used only for performing the conditional averaging
process itself and for large eddy detection, but are
not suitable as an object of any further data statistics.
For determining eddy scales, it is better to return to
the velocity signal itself.
The spanwise velocity makes a zero crossing at
the location of conditionally averaged eddy cores.
An eddy length scale can be defined as the distance
between the centroid locations of the vortex velocity
profiles left and right of such a zero-crossing; this
scale can be computed using first order moments of
the signal. A more robust approach, however, is determining the dominant time scales of conditionally
averaged vortices and multiplying these by the propagation speed of the large eddy core, thus obtaining
an “eddy wave length” (the real eddy size is about
50% of this wave length). The latter approach is
much less sensitive to noise and boundary biases,
because the obtained PIV data allow for much better
temporal than spatial statistics.
0
0
0.5
1
1.5
2
2.5
3
x/d1
Figure 9. Development of the dominant large eddy scales cond
downstream from the separation point, based on conditional
averaging. Dashed line: 1:4 case, dash-dot line: 2:4 case, dotted
line: 3:4 case. The 2:4 case shows the largest eddy scales relative to the lateral expansion width d1, which is understandable
because of the large relative energy of the large-scale turbulence in this case (see also Figure 6)
Figures 9 shows the computed length scales of the
dominant eddies caused by vortex shedding, as a
function of x, for each of the 3 PIV cases. Both axes
are scaled by the lateral expansion width d1. Unlike
the scales shown in Figure 7, the conditionally averaged eddy sizes do have the correct order of magnitude (of the order of d1). It is obvious that the 1:4
and 2:4 cases do show a spatial scale jump downstream of the point x/d1 = 1.0, whereas the 3:4 case
eddy size development is more linear.
3.4 Discussion
In literature, as pointed out in Section 1.2, a distinction can be found between a shallow mixing layer on the one hand and the phenomenon of vortex
shedding on the other hand. The latter phenomenon
is seen as an extra effect additional to a mere mixing
layer, typical for shallow separating flows, while a
mere mixing layer is considered to be a more general
feature of shallow shear flows. With respect to this,
e.g. Jirka (2001) treats both types of shallow-flow
turbulence separately and speaks of two different
mechanisms: respectively the mechanisms of topographical forcing (due to flow separation) and internal transverse shear instabilities (due to a lateral
flow velocity difference). The reason why both
mechanisms differ, however, sometimes remains an
open question.
The observed phenomenon of the “scale jump”
due to a well-developed secondary recirculation cell
is a very helpful tool to explain why shallow flow
separation yields large vortices that have a character
different from mixing layer vortices. Both vortex
types are initially induced by lateral shear, but in a
shallow separation geometry, the presence of per-
manent recirculation cell vorticity greatly influences
the spatial eddy development. Hence, it can be concluded that in a separation geometry the influence of
solid wall friction (inducing recirculations) is never
negligible, while a plane mixing layer theoretically
does not have walls at al. With that respect, a shallow separation flow is always intrinsically a combination of free turbulence and wall turbulence,
whereas a mere mixing layer problem has a free turbulence character to a much larger extent.
The geometry-dependency of shallow separation
problems, however, puts also a limit to the amount
of conclusions that can be drawn from the current
experiments. In practice, many shallow separation
geometries do not contain a sufficiently developed
secondary bubble (for example due to the presence
of downstream obstacles like groynes), so that a less
pronounced scale jump effect should be expected
there. Also, the local wall and bottom roughness as
well as the presence of non-vertical sidewall slopes
(which are almost always there in the case of
groynes) will play a huge role in most river engineering problems in reality. It is not easy to say how especially the presence of wall slopes will influence
the scale jump behaviour. It is not unthinkable that,
in the case of groyne fields, scale jump locations will
lie so close to a separation point that large eddies
seem to be shed directly from the groyne tip.
Therefore, it will be worthwhile to repeat the current experiments replacing the vertical-wall lateral
expansions by several real-life groyne geometries,
maintaining the level of detail in which the largescale vortices are detected. However, introducing
wall slopes and more associated geometrical parameters will cause the experiments to be less generic in terms of length scales, which will make it more
difficult to draw general conclusions from them.
In terms of engineering applications, the current
experiments strongly suggest that any attempt to
prevent scale jumps from occurring will yield more
favourable river flow conditions, thus preventing the
well-known navigation and sedimentation problems
to a large extent. It is possible to do this by changing
shallow separation geometries in an effective way.
Especially changing river groyne shapes could be a
great asset to river engineers. If groyne shapes can be
modified in such a way that they will “shelter off”
most of the secondary bubble regions from the main
river, this may yield a considerable reduction of the
effective lateral shear. A very concise pilot experiment, executed within the same experimental setup
as described in this paper, does indeed show that a
strong reduction of the vortex shedding intensity is
feasible.
4 CONCLUSIONS AND OUTLOOK
Detailed PIV experiments have been performed
on a Shallow Lateral Expansion with a variable inflow width. The hypothesis seems to be confirmed
that the vortex shedding phenomenon is caused by
the presence of a sufficiently developed secondary
recirculation cell. A new element of information,
however, is that the emergence of large-scale shed
vortices is not caused by the flow separation itself,
but by interaction of initially induced mixing layer
vortices with the primary and secondary recirculation areas. This vortex interaction can be interpreted
as a typical feature of shallow (quasi twodimensional) flows. Also, the location of this large
vortex emergence (scale jump) does not coincide
with the separation point. The decomposition of
“mixing layer vortices” and “vortex shedding” opens
a perspective of effective manipulation of large eddies in real-world shallow flow situations by changing the geometry with respect to the secondary recirculation cell.
Further analysis of the vortex shedding phenomenon is currently done by performing a series of full
3D Large Eddy Simulations (LES), in full analogy
with the three examined PIV geometries. This type
of analysis seems to be rewarding because, like PIV
data, LES data allow for not only flow statistics but
also a very detailed look at the discrete horizontal
vortices that are responsible for the observed typical
vortex shedding behaviour in shallow flows. Moreover, conditional averaging of large horizontal eddies
is an option for LES data analysis too. Although the
general flow and turbulence features shown by the
already performed LES simulations look quite reasonable, a detailed analysis of the vortex shedding
process is not yet available. The obtained LES data
will be used to develop and inspire a more concise
shallow flow model, possibly fully depth-averaged,
yet able to reproduce horizontal eddy interaction in
rivers accurately. Meanwhile, examining the vortex
shedding process in more various shallow separating
flow geometries by detailed physical experiments thus verifying the proposed scale jump hypothesis –
will remain a subject of further research.
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APPENDIX: ABOUT VECTOR POTENTIALS
Given a 3d solenoidal velocity vector field:
u  (u , v, w) , with  u  0 and  u  
(i)
For such a vector field, a vector potential  exists
such that
u    ,
(ii)
for     0 by definition. Please note that  is
determined apart from an arbitrary gradient field  ,
because  '     is also satisfying (ii).
The vorticity  now can be written as
          2
(iii)
Because  has a degree of freedom, it can be chosen such that  is also solenoidal. In that case, expression (iii) for the vorticity reduces to a Poisson
equation:
  2
(iv)
The wonderful thing about the Laplacian operator in
this expression is that it operates on each vector
component separately. Therefore, if only one component of the vorticity is known (which is  z in this
study, for only u and v are known in the horizontal
PIV-plane), yet the full z-component of the vector
potential can be constructed. On the vector field
edges, it is sufficient to use homogeneous Neumann
boundary conditions for solving the Poisson equation.
Taking the curl of the constructed  z should
yield in turn the original velocity field (u,v). The latter, of course, does only make sense if the vorticity
component  z is the dominant component within 3d
context. This condition is obviously satisfied in case
of a shallow quasi-2d flow with large eddies in the
horizontal plane. In that case, taking the curl of the
vector potential is practically equivalent to:
u   z / y
and
v   z / x ,
(v)
which shows that  z very much resembles a 2d
stream function of (u,v), except for the fact that an
important correction has been made to circumvent
the nonsolenoidality of the (u,v)-plane.
In fact, computing a vector potential is a way to
integrate the associated velocity field, showing
large-scale rotation patterns in the end. On the contrary, computing a vorticity means taking a derivative of that velocity field, effectively showing smallscale rotation patterns. This can be understood by
considering a velocity signal being a sum of spatial
periodical signals (Fourier components). Taking derivatives of the total signal implies multiplying each
component i by its wave number ki, thus enhancing
the dominance of the high wave numbers (small
scales). Integrating the signal, however, will lead to
a division by ki, hence focussing on the low wave
numbers (largest scales). This explains why experimental vorticity data are often very noisy while, on
the other hand, vector potential data are quite
smooth and hence much easier to handle.
Each local maximum or minimum of a vector potential function identifies a vortex core of positive
respectively negative vorticity sign. The function
isolines are very well parallel to the original vector
field. It may be concluded that vector potentials are a
most suitable tool for identifying large eddies.