Coherent structures in flow over hydraulic engineering surfaces
RONALD J. ADRIAN, School for Engineering of Matter, Transport and Energy, Arizona State
University, Tempe, AZ 85287, USA
Email: rjadrian@asu.edu (Corresponding Author)
IVAN MARUSIC, Professor, Department of Mechanical Engineering, University of
Melbourne, Victoria 3010, Australia
Email: imarusic@unimelb.edu.au

Coherent structures in flow over hydraulic engineering surfaces

ABSTRACT
Wall-bounded turbulence manifests itself in a broad range of applications, not least of which in hydraulic
systems. Here we briefly review the significant advances over the past few decades in the fundamental
study of wall turbulence over smooth and rough surfaces, with an emphasis on coherent structures and
their role at high Reynolds numbers. We attempt to relate these findings to parallel efforts in the
hydraulic engineering community and discuss the implications of coherent structures in important
hydraulic phenomena.

Keywords: Coherent structures, wall turbulence, roughness, Reynolds number effects
1 Introduction
Flows over surfaces in hydraulic engineering are almost always intensely turbulent, owing to
the low viscosity of water and the characteristically large scales of length, 0, and mean flow
velocity, U. The archetypes for this class of flows are steady mean motions over smooth, flat
surfaces with large fetch, e.g. turbulent boundary layers or internal wall flows such as those in
pipes and channels.
Classically, understanding of these flows is based largely on average behavior of the important
aspects of the flow such as mean velocity and mean wall shear stress, w. The mean velocity
exhibits at least two different layers, an inner layer in which the wall shear stress, expressed in
terms of the friction velocity ut = t w / r , and the kinematic viscosity , are the important
external parameters; and an outer layer in which the depth of the flow 0 (equal to the boundary
layer thickness , the pipe radius R, or channel depth h) and the free stream velocity U ¥ or the
bulk velocity Ub determine the average behavior of the mean velocity profile. These layers share
a common part, the logarithmic layer, in which the mean velocity varies logarithmically with
distance from the wall, y . Coles’ logarithmic plus wake formulation (Coles 1956) gives the
mean velocity in the outer layer according to

U+ º

U ( y)
ut

= k -1 ln(y + ) + A + PW (y / d 0 ), y + > 30 ,

(1)

where von Karman’s constant, k @ 0.41and A @ 5 are empirical constants, and Coles’ wake
factor  is an empirical, non-dimensional parameter that depends upon the free stream pressure
gradient. The empirical fit W  sin2(y/0) describes the deviation of the mean velocity from the
logarithmic variation in the so-called wake region, and y+ = yu is the distance from the wall
in units of the viscous length scale, /u. The logarithmic variation dominates for y  0.150,
nominally.
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The mean velocity in the inner layer is described classically by von Karman's logarithmic law
above y+  30, and a viscously dominated buffer layer for 0  y+  30. (Modern investigations
suggest that the mean velocity does not vary logarithmically until higher values, y+  200in
boundary layers (Nagib et al. 2007) and 600 in pipes (Zagarola and Smits 1998), but for the
purposes of this discussion, it suffices to use y+ = 30 for reference.) Thus, the logarithmic layer
nominally exists between
30/R < y/0 < 0.15,

(2)

where
R  u

(3)

can be interpreted either as a turbulent Reynolds number or as the ratio of the layer depth to the
viscous length scale, known as the von Karman number.
Neo-classically, there has been considerable research effort to understand the behavior of the
flow statistics in terms of structural elements, variously called motions, coherent structures or
eddies (Townsend 1976; Cantwell 1981; Hussain 1986). Coherent motions are recurrent,
persistent motions that characterize the flow and play important roles in determining mean flow,
stress and other statistical properties. They may have rotational and irrotational parts. Eddies are
similar, but in the spirit of Townsend (1976) they are definitely rotational. Further discussion
can be found in Marusic and Adrian (2013), but for present purposes it suffices to think of
coherent structures as building blocks of flows that are recognizable, despite randomness, by
their common topological patterns, and that occur over and over again.
The quantitative validity of the logarithmic variation of the mean velocity and the scaling laws
that pertain to it have been questioned (Barenblatt 1993), especially for boundary layers
(George and Castillo1997), but there is now no doubt (Smits et al. 2011) that the logarithmic
law continues to be one of the cornerstones of wall turbulence, and that the physics of the
logarithmic region play a central role in the overall fluid mechanics of wall turbulence. This role
extends to such important issues as the proper boundary conditions for Reynolds averaged
Navier-Stokes equations and large eddy simulations, and to the asymptotically infinite Reynolds
number structure of the eddies of wall turbulence.
Despite the clear importance of the logarithmic layer at high Reynolds number and over a
variety of surfaces, surprisingly little of our knowledge about the structures of eddies within the
logarithmic layer is used in the treatment of hydraulic wall flows. For example, it is well known
that the logarithmic law can be derived by postulating that the mixing length grows in
proportion to y , and that it varies qualitatively as shown in Fig. 1(a). This proportionality in the
logarithmic layer is consistent with Townsend’s Attached Eddy Hypothesis, which states that
the eddies in wall turbulence have sizes that are proportional to their distance from the wall
(Fig. 1(b). But, very little else about the geometry of the eddies, their origin or their dynamics is
used in the classical hydraulic engineering literature.
The place of understanding coherent structures within the hydraulics research portfolio is
developing, and its ultimate applications remain to be established. Certainly, understanding how
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structures create motions that transport momentum, energy and scalars can be expected to
materially improve the ability to predict average behavior. Further, understanding the
component structures of a turbulent flow is also likely to provide a conceptual framework
within which observations of hydraulic phenomena can be assessed. Lastly, understanding the
coherent structures may make the design of hydraulic structures easier.
The purpose of this ‘vision paper’ is to summarize what is known about the structure of
coherent structures in wall turbulence, especially the high Reynolds number turbulence of
hydraulic flow applications, and to offer some ideas on the significance of the structures in
problem areas such as sedimentation, erosion and flow-structure interactions. Throughout, we
shall relate the coherent structures to the known regions of the mean velocity profile, as
discussed above.
2 Coherent structures on smooth walls
2.1 Near-wall structures
Before considering rough and irregular surfaces it is valuable to consider the large body of work
done on hydrodynamically smooth surfaces. Particularly, as theory (Townsend 1976, Jimenez
2004) indicates that for roughness length scales less than a few percent of the boundary layer
thickness, the logarithmic and fully outer regions are not affected by roughness, apart from
setting the inner boundary condition for the friction velocity, u.
The coherent structures that occur in the near-wall portion of the inner layer, have been
extensively reviewed by Kline (1978), Brodkey (1978), Cantwell (1981), Hussain (1986),
Robinson (1991), Adrian (2007) and others. Many characteristic elements have been recognized
and documented in the near-wall layer, including: low-speed streaks with spacing of 100
viscous wall units and the burst process (Kline et al. 1967), sweeps and ejections (Brodkey,
Wallace and Eckelmann 1974), quasi-streamwise vortices, Q2/Q4 events (Wallace et al. 1972,
Willmarth and Lu 1972) and associated VITA (Variable Integration Time Average) events
(Blackwelder and Kaplan 1976) and inclined shear layers (Kim 1987). Here Q2/Q4 refers to
events in the second and fourth quadrants of the u-v map, which thus contribute a positive
contribution to the Reynolds shear stress, -uv . It is noted here that we define u and v as the
fluctuating components of velocity in the streamwise and wall-normal directions respectively.
The bursting process in the near-wall region, in which low speed fluid is ejected abruptly away
from the wall, is considered to play an important role in the overall dynamics of the boundary
layer.
Different interpretations exist as to what type of coherent structures exist and what role they
play in the near-wall region, and many of these viewpoints are reviewed by Robinson (1991),
Panton (2001), Schoppa and Hussain (2006), Adrian (2007), Marusic et al. (2010) and Jimenez
(2012). Here, we emphasize the hairpin vortex as a simple coherent structure that explains many
of the features observed in the near-wall layer (Theodorsen 1952, Head and Bandyopadhyay
1981), or its more modern, and demonstrably more common variant, the asymmetric hairpin or
the cane vortex (Guezennec, Piomelli and Kim 1989, Robinson 1991, Carlier and Stanislas
2005). For brevity we shall not distinguish between symmetric and asymmetric hairpins, nor
4

will we distinguish between hairpins and horseshoes, since available evidence suggests that
these structures are variations of a common basic structure at different stages of evolution or in
different surrounding flow environments. In this regard, it may be also useful to group all such
eddies into the class of turbines propensii (referring to ‘inclined eddies’) to de-emphasize the
connotations of shape that are intrinsic to the term ‘hairpin’.
Theodorsen’s (1952) analysis considered perturbations of the spanwise vortex lines of the mean
flow that were stretched by the shear into intensified hairpin loops. Smith (1984) extended this
model and reported hydrogen bubble visualizations of hairpin loops at low Reynolds number.
While there is evidence for a formation mechanism like Theodorsen’s in homogenous shear
flow (Rogers and Moin 1987, Adrian and Moin 1987), it is clear that Theodorsen’s model
requires modification near a wall to include long quasi-streamwise vortices spaced about 50
viscous wall units apart and connected to the head of the hairpin by vortex necks inclined at
roughly 45° to the wall (Robinson 1991). With this simple model, the low speed streaks are
explained as the viscous sub-layer, low speed fluid that is induced to move up from the wall by
the quasi-streamwise vortices. A schematic illustrating these essential features of a hairpin
vortex is shown in Fig. 2. The second quadrant ejections are the low speed fluid that is caused to
move through the inclined loop of the hairpin by vortex induction from the legs and the head,
and the VITA event is the stagnation point flow that occurs when the Q2 flow through the
hairpin loop encounters a Q4 sweep of higher speed fluid moving toward the back of the
hairpin. This part of the flow constitutes the inclined shear layer. This picture is substantiated by
the direct experimental observations of Liu, Adrian and Hanratty (1991) who used PIV to
examine the structure of wall turbulence in the streamwise wall-normal plane of a fully
developed low Reynolds number channel flow. They found shear layers growing up from the
wall which were inclined at angles less than 45° from the wall. Regions containing high
Reynolds stress were associated with these near-wall shear layers. Typically, these shear layers
terminate in regions of rolled-up spanwise vorticity, which could be the heads of hairpin
vortices. In the near-wall hairpin model, ejections are associated with the passage of hairpin
vortices.
Perhaps the strongest experimental support for the existence of hairpin vortices in the
logarithmic layer was originally given by Head and Bandyopadhyay (1981), who studied highspeed, time-sequenced, images of smoke-filled boundary layers over a large Reynolds number
range. They concluded that the turbulent boundary layer consists of hairpin structures that are
inclined at a characteristic angle of 45° to the wall. Head and Bandyopadhyay (1981) also
proposed that the hairpins occur in groups whose heads describe an envelope inclined at 15-20°
with respect to the wall. The picture is similar to Smith’s (1984) interpretation of flow
visualizations in water, but instead of being based on data below y+=100, Head and
Bandyopadhyay (1981) appear to have based their construct on direct observations of ramp-like
patterns on the outer edge of the boundary layer (Bandyopadhyay 1981), plus more inferential
conclusions from data within the boundary layer. The observations of Head and
Bandyopadhyay (1981) lead Perry and Chong (1982), with later refinements by Perry, Henbest
and Chong (1986) and Perry and Marusic (1995), to develop a mechanistic model for boundary
layers based on Townsend’s (1976) attached eddy hypothesis where the statistically
representative attached eddies are hairpin vortices.
5

An important aspect of the attached eddy modelling work is that a logarithmic region requires a
range of scales to exist with the individual eddies scaling with their distance from the wall.
However, achieving such a range of scales requires a sufficiently high Reynolds number, which
makes measurements difficult due to the large dynamic range required. A major advance in this
regard came with the development of high resolution PIV. Adrian et al. (2000), were the first to
extensively use PIV to study the logarithmic and fully outer regions of boundary layers over a
range of Reynolds numbers. Their work was particularly important as the PIV measurements
provided images of the distribution of vorticity and the associated induced flow patterns without
invoking the inferences needed to interpret flow visualization patterns. The patterns revealed
that the logarithmic region is characterized by spatially coherent packets of hairpin vortices,
with a range of scales of packets coexisting. This scenario explained the observed inclined
regions of uniform momentum where the interfaces of these regions coincided with distinct
vortex core signatures. A sample instantaneous PIV result is shown in Fig. 3. The “attached”
hairpin packet scenario explains, or at least is consistent, with a number of observations made in
turbulent boundary layers. For example, it explains the observation that the spacing of the lowspeed streaks in the streamwise velocity fields increases across the logarithmic region with
distance from the wall (Tomkins and Adrian 2005; Ganapathisubrani et al. 2003, 2005).
Moreover, if one associates a burst with a packet of hairpins, this construct offers an
explanation both for the long extent of the near-wall low-speed streaks and for the occurrence of
multiple ejections per burst, which has been documented in a number of studies (Bogard and
Tiederman 1986, Luchik and Tiederman 1987, Tardu 1995). Thus, the original conception of a
turbulent burst being a violent eruption in time is replaced by a succession of ejections due to
the passage of a packet of hairpin vortices, the smallest hairpin creating the strongest ejection
velocity.
2.2 Large Scale Motions and Very Large-Scale Superstructures
Flow visualizations of boundary layers, an example of which is shown in Fig. 4, highlight that
in the outer layer, the edge of the turbulent zone has bulges that are about 2-3 long
(Kovasznay, Kibens and Blackwelder 1970) separated by deep crevasses between the back of
one bulge and the front of another (Cantwell 1981). The backs have stagnation points formed by
high-speed fluid sweeping downward, and the shear between the high-speed sweep and the
lower speed bulge creates an inclined, -scale shear layer. The bulges propagate at about 8085% of the free stream velocity.
Long streamwise lengths are also prominent in streamwise velocity energy spectra, as reported
by Balakumar and Adrian (2007). They showed that two large length scales emerge in pipe,
channel and boundary layer flows where one peak in energy is associated with large-scale
motions (LSM) of typical length 2-3, and a second longer wavelength peak is associated with
very-large scale motions (VLSM), or superstructures, on the order of 6  for boundary layers
(Hutchins and Marusic 2007). On the basis of the shapes of the streamwise power spectra and
the uv co-spectra, Balakumar and Adrian (2007) nominally placed the dividing line between
LSM and VLSM at 3. Using this demarcation, Balakumar and Adrian (2007) showed that the
LSM wavelength persists out to about y/ ~ 0.5 (consistent with the observed bulges in
6

visualizations), while the very large superstructure wavelengths do not extend beyond the
logarithmic region, ending at approximately y/ = 0.2.
While the reported lengths for the very-large superstructure events from spectra are
approximately 6 for boundary layers, this is considerably less than the observed values in pipe
and channel flows (Kim and Adrian 1999, Monty et al. 2007, 2009), suggesting that geometrical
confinement issues may play a role. However, what the actual lengths of the very-large
superstructures are remains an open question. Hutchins and Marusic (2007) used time-series
from a spanwise array of hot-wires (and sonic anemometers in the atmospheric surface layer) to
infer lengths well in excess of 10, and this is consistent with the high-speed PIV study of
Dennis and Nickels (2011). Sample results of instantaneous measurements from Hutchins and
Marusic (2007) and Dennis and Nickels (2010) are shown in Fig. 5.
The Dennis and Nickels (2011) results also shed invaluable information on the three-dimension
structure of the largest motions, and while not conclusive, strongly support the suggestion by
Kim and Adrian (1999) that the very-large superstructures are a result of a concatenation of
packets. Support for this also comes from atmospheric surface layer and laboratory
measurements as described in Hambleton et al. (2006) and Hutchins et al. (2012), as shown in
Fig. 6, where simultaneous x-y and x-z plane three-component velocity measurements reveal
signatures entirely consistent with the superstructure events consisting of an organized array of
packet structures. The lower schematics in Fig. 6 indicate comparisons with the Adrian et al.
(2000) packet paradigm with Biot-Savart calculations of an idealized packet of hairpin vortices
to infer what the corresponding spanwise velocity signatures would be in the relevant
orthogonal planes.
2.3 Interactions across scales
An important consequence of the large-scale and very-large superstructure motions in the outer
region (which includes the logarithmic region) is their role in interacting with the inner nearwall region, including their influence on the fluctuating wall-shear stress. There has been debate
over many decades as to whether the inner and outer regions do interact, or whether they can be
considered as independent, as assumed in all classical scaling approaches. Considerable
evidence now exists that outer scales are important for characterizing near-wall events. This
stems from a large number of studies that have documented a Reynolds number (or equivalently
an outer length scale) dependence in the near-wall region. These include the studies of Rao et al.
(1971), Blackwelder and Kovasznay (1972), Wark and Nagib (1991), Hunt and Morrison
(2000), DeGraaff and Eaton (2000), Metzger and Klewicki (2001), Abe et al. (2004), Hoyas and
Jimenez (2006), Hutchins and Marusic (2007), Orlu and Schlatter (2011) and others. Many of
the above studies support the viewpoint that some superposition of the large-scale motions is
experienced right to the wall. Hutchins and Marusic (2007b) went further and proposed that this
interaction also involved a modulation of the large scales on the near-wall small-scale motions.
Previous suggestions of modulation effects have also been made by Grinvald and Nikora (1988).
Mathis et al. (2009) studied the modulation effect extensively using data over a large range of
Reynolds number and showed that the degree of modulation increased with increasing Reynolds
number, and hence is a key aspect of high Reynolds number wall turbulence.
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Marusic, Mathis and Hutchins (2010) extended the observations of a superposition and
modulation of the large-scale outer motions in the near-wall region to a predictive model,
whereby a statistically representative fluctuating streamwise velocity signal near the wall could
be predicted given only a large-scale velocity signature from the logarithmic region of the flow.
The model was shown to work well over a large Reynolds number range for various statistics,
including higher-order moments. The formulation involves a universal signal and universal
parameters, which are determined from a once-off calibration experiment at an arbitrarily
chosen (but sufficiently high) Reynolds number. Marusic et al. (2011) further extended the
model to predict the fluctuating wall-shear stress given only a large-scale streamwise velocity
signal from the logarithmic region, and were able to reproduce the empirical result of
Alfredsson et al. (1988) and Orlu and Schlatter (2011) that showed that the standard deviation
of the inner-scaled fluctuating wall shear stress increases as a logarithmic function of Reynolds
number.

3 Effect of high R in hydraulic engineering
The significance of the logarithmic layer depends on the Reynolds number. At low Reynolds
number most of the change of the velocity from the wall to the free-stream occurs from the wall
to the top of the viscous-inertial buffer layer because the thickness of the logarithmic layer is
small, and there is relatively little change in velocity in the wake region. For example, in
turbulent channel flow at Reynolds number R = 180 (corresponding to Ubh/ = 2800) the mean
velocity at the edge of the buffer layer is approximately 75% of the centerline velocity, and the
velocity change across the logarithmic layer is very small. If one interprets the skin friction
coefficient as a quantity that specifies the free stream velocity corresponding to a given level of
wall shear stress, the foregoing consideration indicates that over half of the skin friction
coefficient is determined by the fluid mechanics of the buffer layer at low Reynolds number,
and hence that drag reduction strategies must concentrate on modifying the flow in the buffer
layer. This view is supported by the fact that the rate of production of turbulent kinetic energy
per unit volume, -uv ¶U / ¶y achieves a large maximum within the buffer layer, while it is
much smaller in the logarithmic layer, suggesting that the preponderance of the turbulence is
created in the buffer layer at low Reynolds number.
However, at high Reynolds numbers these conclusions must be altered substantially, simply
because the logarithmic layer becomes much thicker, and thereby becomes more important.
Consider for the sake of estimation equation (1). The velocity change from the wall to the top of
the buffer layer is 13.2 friction velocities, while the velocity change from the top of the buffer
layer to the top of the log layer (using y/0 =0.15) is 2.41ln-12.8. The ratio of the velocity rise
across the logarithmic layer to the velocity rise across the buffer layer is 0.183-0.97, implying
that the velocity change across the buffer layer vanishes as 5.5/ln0+ for large Reynolds
number. Thus, as Reynolds number becomes infinite, essentially all of the velocity change
occurs across the logarithmic layer, and hence all of the skin friction is associated with the
logarithmic layer.
8

Practically, this conclusion is too strong, because the logarithmic dominance increases very
slowly. For example, for ninety per-cent of the velocity change from the wall to the top of the
logarithmic layer to occur across the logarithmic layer, the Karman number must exceed 10 23,
far above the value achieved by any terrestrial flow. On the other hand, for typical Reynolds
number laboratory flows (say, = 2000) the velocity changes across the buffer layer,
logarithmic layer and wake region are nominally 50%, 25% and 25% of the free stream
velocity, respectively. Thus, the logarithmic layer does not dominate laboratory flows, but its
contribution is very substantial.
Similar conclusions can be drawn regarding the contribution that the logarithmic layer makes to
the total production of turbulent kinetic energy. For example, while the production per unit
volume does peak in the buffer layer, the volume of the logarithmic layer is much greater, so the
ratio of the production integrated over the logarithmic layer to the total production from within
the buffer layer grows as ln y+ as Reynolds number approaches infinity. They are equal at
approximately 0+ = 35,000.
Considerations like the foregoing plus others have led Smits et al. (2010) to conclude that a
reasonable criterion for wall turbulence to be considered high Reynolds number is0+ > 13,300
for boundary layers and 0+ > 50,000 for pipe flow. These values are achieved commonly in
hydraulic flows, so it is safe to assert that nearly all hydraulic flows are high Reynolds number
wall turbulence. (For example, the turbulent Reynolds number of a boundary layer in a water
flow with a free stream velocity of 2.5 m/s and a depth of 1 m is approximately 100,000.) This
simple rule implies that hydraulic wall turbulence
1. Possesses a clear range of logarithmic behavior in the mean velocity profile and a clear
range of k -5/3 behavior in the inertial sub-range of the power spectrum of the
streamwise velocity
2. Has larger production of turbulent kinetic energy in the logarithmic layer than in the
buffer layer
3. Possess a spectral peak at very long wavelengths that is distinct from the spectral peak
corresponding to the inner layer motions.
With regard to the coherent structures, high Reynolds number implies ample room for the
eddies to grow from their initially small scales at the wall to the depth of the flow. The range of
scales in the outer layer increases as /100, if we take 100 viscous wall units as the
representative height of the smallest first generation hairpin and  as the tallest coherent
structure. If attention is confined to the self-similar structures in the logarithmic layer the scale
ratio is approximately 0.15/100 = 150 at R = 100,000, making room for at least seven
doublings of the original height of the smallest hairpin (100  27 =12,800 < 15,000). This
implies seven or more different uniform momentum zones across the logarithmic layer.

4 Roughness effects on coherent structure
The surfaces bounding hydraulic flows are seldom smooth, and the height of the roughness
elements can easily exceed the thickness of the viscous buffer layer at the high Reynolds
9

numbers of hydraulic flows. Roughness elements disrupt the flow within the buffer layer, and
they may completely destroy it, replacing the effects of fluid viscosity with the effects of wall
roughness and replacing the viscous length scale with the roughness element length scale, k (of
course, a thin viscous sublayer is still attached to the surface of roughness elements but its very
small thickness makes it dynamically insignificant). A measure of the importance of the
roughness elements is the non-dimensional roughness element height k+=ku. Small values of
k + correspond to incomplete roughness, and large values correspond to complete or fully
developed roughness. While roughness may destroy the viscous buffer layer, it appears to have
much less effect on the logarithmic layer, other than shifting the effective slip velocity of the
logarithmic layer with respect to the wall (Townsend 1976). The logarithmic law in equation (1)
is thus replaced by
U + = k -1 ln y + + B(k + ) + PW (y / d 0 )

(4)

We shall refer to this phenomenon as robustness of the logarithmic layer. The persistence of the
logarithmic layer implies that the under-lying structures, like hairpins packets and related
turbines propensii also persist. Their form need not be identical to the structures over smooth
walls, but the evidence suggests that they are not very different (Hommema and Adrian 2003;
Guala, et al. 2012). We therefore adopt, as a working hypothesis for now, the idea that the
structures in the outer layer of turbulent flow over rough walls having roughness elements that
are smaller than the logarithmic layer are similar to those occurring in the outer layer of
turbulent flow over smooth walls.
If the roughness elements become a significant fraction of the logarithmic layer, they can
severely disrupt the self-similar structures, and the logarithmic layer is replaced by different
behavior. A hint as to how this may happen is contained in the companion paper to this paper
(Guala, et al. 2012) in which tall hemispherical roughness elements are placed sparsely on an
otherwise smooth surface. Measurements show two types of structures co-existing: hairpin
packets from the smooth surface, and hairpin packets from the individual hemispheres. The
essential difference between the two types is that the latter grow at a steeper angle than the
former and each of the latter packets is rooted to the hemisphere that generates it, much like
wake vortices shed from a stationary cylinder. This behavior hints at the effects that might be
expected from rivets on the surfaces of marine vessels or very large roughness elements in
streams and beds, such as large rocks.

5 Coherent structures and hydraulic phenomena
Turbulent transport plays a critical role in heat and mass transfer at the free surface, mixing and
dispersion, erosion and sedimentation, inlet conditions to hydraulic devices, interaction with
vegetation and, of course, resistance to flow. As such, insights into the coherent structures that
influence transport provide new ways of looking at each of these phenomena (Nezu, 2005,
Nikora et al. 2007, Nikora 2010, Grant and Marusic 2011).
5. 1. Coherent structures in canonical open channel flows

10

Here we consider flow in straight, wide channels of depth H with smooth walls, unless
otherwise stated. The most obvious coherent feature of open channel flow is the boil
phenomena (Nezu and Nakagawa 1993, Yalin 1992). These localized, intense upwellings occur
one after another in streaks along the streamwise direction with a spacing of approximately 2H
(Tamburrino and Gulliver 1999), which corresponds, to the large scale motions (bulges) in
turbulent boundary layers. The streaks of boils coincide with streaks of low speed flow,
upwelling and lateral spreading at the surface. They are separated by streaks of high-speed flow
lateral convergence and downwelling (Tamburrino and Gulliver 1999, 2007). From the
upwelling and downwelling long, streamwise-oriented rolling vortices apparently first inferred
by Velikanov (1958) (see Shvidchenko and Pender, 2001) and observed by many subsequent
workers (Klaven and Kopaliani 1973 and more recently Tamburrino and Gulliver (1999, 2007),
and Rodriguez and Garcia (2008) to cite a few).
The roll cells, also called large streamwise vortices (Gulliver and Halverson 1987) or long
longitudinal eddies (Imamoto and Ishigaki 1986), look like secondary flows in the plane
perpendicular to the streamwise flow (Nikora and Roy 2012). True secondary flows have nonzero long time averages, and they affect the distribution of mean velocity, turbulence intensities,
Reynolds shear stresses, and bed shear stress throughout the channel. If the channel is wide
enough, width > 5H, Nezu and Rodi (1986) observed that secondary flows are hard to see in the
long time averages, but they exist, nonetheless. PIV measurements of the cross-stream flow find
cellular secondary currents that vary in time regardless of the aspect ratio (Onitsuka and Nezu,
2001). This suggests that the long streamwise vortices meander in time as the aspect ratio
increases, causing their features to be lost in time average measurements. Tamburrino and
Gulliver (2007) observed that large-scale eddies having spanwise (lateral) widths of 1-1.5h
oscillate slowly in the mid channel, but fixed stationary secondary flows form in the vicinity of
the side walls. Nezu and Nakayama (1997) observe both secondary currents and time varying
cellular currents in the interaction between the mainstream and a flood plain. Correlation
measurements of the streamwise surface velocity made in many rivers indicate positive
correlation over 2-5H followed by negative correlation between 5-10H, and finite correlation,
either positive or negative over lengths extending to 10-20H (Sukhodolov et al. 2011). The
oscillating sign of the correlation in Sukhodolov et al. (2011) implies that the streaks either
waver or drift laterally so that a streamwise line of observation alternately crosses high speed
and low speed streaks.
A simple drawing summarizing these features is presented in Fig. 7. Note that the secondary
flows are steady and aligned with the side-walls, and the long streamwise vortices are unsteady
and inclined. While the cellular picture in Fig. 7 is appealing, the reality of open channel flows
is more complicated. Direct observations of multiple circulations perpendicular to the main
channel flow have been made by Nezu (2005), and their instantaneous streamlines clearly
fluctuate considerably from cell to cell. Further, the cells do not appear to extend down to the
bed. Consequently, the interior cells in Fig. 7 are too regular to represent the instantaneous flow,
and the reader should think of them as a conditional average of the roll cells given the location
of the center of the cell as it meanders.
The irregularity of real roll cells can be explained in part by their close association with
turbulent ‘bursts’ in the low speed zones. The term ‘burst’ will be used in the present discussion
11

in deference to common usage in the hydraulics literature. However, there is good evidence that
the concept of a burst as a rapid, perhaps even violent, ejection should be replaced by the
concept of a packet of hairpin vortices passing and creating a sequence of ejection events, each
associated with one of the hairpins. Since the packet evolves relatively slowly, the appearance
of rapid change is caused by the fast passage of the packet (Adrian et al. 2000). Observations
show that a burst can originate at the bed and cross the entire channel depth to impinge on the
surface and cause a boil (c.f. Shvidchenko and Pender 2001 for a summary of the observations).
The bursts reaching the surface have height H, length 2-5H and width 1-2H, virtually the same
as the large scale motions or bulges discussed earlier. In turbulent boundary layers the bulges
are likely to be the ultimate form assumed by the hairpin vortex packets upon reaching the edge
of the boundary layer. Consequently, Fig. 7 indicates hairpin vortex packets of various sizes,
with the largest (colored red) causing the surface boils. The smaller packets grow and merge
with others to ultimately form the largest packets. Particle image velocimeter measurements in
the streamwise verticle plane strongly support the similarity between internal packets in open
channel flow and turbulent boundary layers (Nezu and Sanjou 2005, Fig. 5).
While the association between the low speed streaks and the succession of bursts that creates
‘street’ of boils is well established, there is a very interesting issue of cause and effect.
Shvidchenko and Pender (2001) assert that bursts give rise to the long, streamwise oriented
rolling vortices. But, in their reply to this discussion Tamburinno and Gulliver note that the
rolling vortices may cause the ejections and the sweeps, rather than vice versa. A similar idea
has been developing independently in the turbulence community. The evidence presented earlier
for modulation of the small near-wall scales by the large outer scales supports this picture. The
authors’ view is that both mechanisms are plausible, and that it is likely that they operate
cooperatively. In this scenario, lateral motion of the cells towards the low speed streaks sweep
the smaller, growing hairpins and packets into the streaks (Toh and Itano 2005, Adrian 2007)
and create the alignment of the large-scale motions. That alignment creates the very large-scale
motions. Since the hairpins and packets are themselves elements of low momentum, their
congregation around the VLSM’s low speed streaks intensifies the momentum deficit. Schoppa
and Hussain (2002) have shown that low speed streaks are necessarily associated with quasistreamwise roll cells, so intensified low momentum would actually support formation of the roll
cells. In this way, a closed loop feedback cycle would exist in which the roll cells feed
themselves by sweeping low momentum hairpins and packets into the low speed streaks.
The close relationship between the meandering very large-scale motions of turbulence structure
research and the long cellular motions of open channel flow research is impossible to ignore. It
seems likely, in fact that they are one and the same. Sukhodolov et al., figure 5b, shows
correlation out to 5-10h in a compilation of time delayed streamwise correlation functions from
many rivers, and their figure 5a shows alternating high speed low speed zones extending up to
free surface. This is very similar to results for meandering VLSMs in pipes, channels and
turbulent boundary layers and the atmospheric boundary layer.

5.2 Structure in channels with significant roughness
Understanding of coherent structures in rough walled channels is limited, but generally
12

speaking the picture is similar to that for smooth walls. Several observations report structures
resembling large-scale motions that grow up from the wall and reach the surface (Roy et al.
2004, Hurther et al. 2007, Nikora et al. 2007). Surface lengths of 3-5H are reported, but
observed widths of 1H are somewhat smaller than the 1-1.5 H width of turbulent bulges. It is
well known that rough walls reduce the streamwise correlation length. Flow visualization of the
bursts from the bed (Roy et al. 2004, figure 16) show structures whose growth angle looks
similar to the ~15 angle of hairpin vortex packets, followed by structures that grow much more
rapidly, at least 45. The latter probably emanate from single roughness elements, and the rapid
growth angle offers the simplest explanation for the foreshortening of the streamwise length.
The companion paper by Guyala et al. (2012) offers some insight into the structures created by
sparse roughness elements.
5.3 Heat and mass transfer at the free surface
Free surface boils and other structures at the surface are hydraulic manifestations of coherent
structure rising to the surface. The interactions of the coherent structures with the free surface
are also important in the gas exchange at the surface, a major factor in evaluation of greenhouse
gas effects. The boils and the upwelling/downwelling streaks are the basis for surface renewal
theories, as discussed by Komori et al. (1982). In this regard, Calmet and Magnaudet (2003)
have shown the significance and utility of Hunt and Graham’s (1978) rapid distortion theory for
eddies approaching a surface, and this looks like a promising improvement on surface renewal
theory.
5.4 Mixing and dispersion
Mixing is perhaps one of the most important turbulent processes in problems involving dilution
of thermal and material effluents and density stratification in hydraulic flows. Since the
importance of coherent structures in the transport of momentum has been established
conclusively, it is clear that transport of heat and mass must also exhibit a strong dependence
upon coherent structures. The dispersion of heat and pollutants may be affected by the structure
of wall turbulence in shallow channel flows. Jirka (2001) studied wakes, jets and shear layer in
wide open channels and noted that three-dimensional turbulent bursts can affect these mainly
two-dimensional flows.
Dispersion of scalars is classically modeled as a random walk process that occurs on top of a
mean flow field (Sawford 2001, Balachandar and Eaton 2010). The random walk naturally
leads to concentration fields caused by dispersion from a point source that are Gaussian
functions of position. But in reality, the coherent structures in the flow produce a different
picture of the dispersion process. The anisotropy of the structures and their inhomogeneity are
factors that are difficult to incorporate realistically into Gaussian models, and the short-term
inhomogeneity that is associated with very large-scale superstructures, and their associated large
streaks, is almost never accounted for. If the surface were flat and wide, the long streaks would
meander with no preferred spanwise location, so that long time averages would indeed be
independent of the spanwise location. But over short times, the streaks tend to stay in one
location, causing substantial inhomogeneity. The presence of small-scale inhomogeneity such
as rocks, asperities, etc. could cause the streaks to stabilize, meaning that spanwise
13

inhomogeneity would be lost. In such cases, it is very important to model the realizations of the
coherent structures rather than their long time mean values.
5.5 Erosion and sedimentation
Erosion and sedimentation often lead to the formation patterns in solid boundaries such as dunes
and meanders in streams, and it is therefore not unreasonable to look for associations between
the formation of these patterns and the coherent patterns of flow in the fluid, at least in the
incipient or early stages of erosion when the bed form is essentially flat. Gyr and Schmid (1997)
have shown that at incipient erosion on a flat sandy bed, only the sweeps move the sand gains.
Erosion processes are also likely to feel the consequences of coherent structures because the low
probability, extreme events responsible for high local erosion rates are parts of the natural cycle
of flow. Roughness can also create fluctuation in the wall shear stress that are comparable to the
fluctuations caused by coherent structures in smooth walled flows (Cheng, 2006).
When sedimentation and erosion are strong enough to alter the bed form, the coherent structures
above the bed may be radically modified, especially by the process of flow separation. For
example, Kadot and Nezu (1999) show that the flow behind the crest of a dune is a turbulent
shear layer containing spanwise vortices. Nezu et al. (1988) and Nezu and Nakagawa (1989b)
found that the organized fluid motions and the associated sediment transport occurred
intermittently on a movable plane sand bed. After the sand ridges were formed, the roll cells
appeared stably across the whole channel cross section. As shown in the inset to Fig. 7 the sand
is eroded in the downwelling side of a cell and sedimented on the upwelling side, Roll cells are
also generated on beds with smooth and rough striping (Nakagawa et al. 1981; McLean 1981;
Studerus 1982). There is an extensive literature on the modification of turbulence statistics by
various bed form geometries c.f. Cellino and Graf (2001). A further comprehensive discussion
of coherent structures in sediment dynamics can be found in Garcia (2008).
6 Future challenges and prospects
Our present knowledge of coherent structures in flows over smooth flat surfaces is enough to
see how such structures could be of importance in hydraulic engineering. Efforts are needed to
exploit understanding of the structure to improve hydraulic engineering design in many areas.
Sedimentation, erosion, dispersion and entrance flows to hydraulic devices such as power
facilities, spillways, and barbs are importantly related to the large-scale and very large-scale
motions, and considerable advancement can be expected if we can adequately characterize and
predict these motions and possibly manipulate them in a controlled way. The interactions of the
large-scale motions with the near-wall region, and thus the bed shear stress, also need to be
studied and better exploited. Existing predictive models based on the outer region large-scale
motions (Marusic et al. 2010) need to be extended beyond smooth-wall flows and offer the
prospect of real predictive capability given only the large-flow field information. Such
information can be obtained by reasonably spatially-sparse, low-frequency measurements, or
preferably from numerical simulations, such as large-eddy simulations, where the large-flow
field information is resolved. Fully understanding the scaling behavior at high Reynolds
numbers also opens the way for refined scale up from models and better-informed designs.
14

At this point in time the various types of structure have been identified, but one cannot claim
that we fully understand their scaling or their functions. Investigations of the scaling of each
type of motion are needed. They may provide better definitions of the motions and improve
understanding of the their relative importance in different ranges of Reynolds number. The
interactions of the various motions have only begun to be understood, and much work,
especially dynamic experiments and theoretical analyses are needed to establish true cause and
effect in these interactions. For example, erosion by very large-scale motions may be caused by
direct action of the very large-scale motions, but it may also be the case that the very large
scales mainly organize and collect the smaller motions, and it is the latter that perform most of
the erosion. Understanding cause and effect is essential to management of fluid flows by design.
It would be truly disappointing if improved understanding of the structures in turbulent flows
and their roles in sedimentation, erosion and dispersion could not significantly improve the
accuracy and reliability of turbulence models of all kinds. Ultimately, incorporation of structural
properties into the models is one of the more important and more challenging tasks ahead of the
field. It is hoped that improved paradigms of turbulent flow will stimulate new and innovative
theoretical descriptions and computational modeling.
The very large Reynolds number inherent to hydraulic flows make them attractive for the study
of turbulent structure in the presence of a wide hierarchy of scales and important to turbulent
flow science. The wide range of scales across the logarithmic layer would be especially helpful
in this regard. The persistence of the logarithmic layer and attached eddies above rough surfaces
must be confirmed more fully, as this is an important piece of evidence concerning the robust
nature of structures in the outer region. Acquiring such information experimentally will require
resolving these flows with an unprecedentedly large dynamic range. However, rapid advances
in laser and digital camera technologies combined with evolving three dimensional velocimetry
techniques (Adrian and Westerweel 2011) make this a realistic proposition in the not too distant
future.

Acknowledgements
The authors gratefully acknowledge the support of NSF Grant CBET-0933848 (RJA) and the
Australian Research Council (IM).
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M
ixi
ng
le
ng
th

Zagarola M.V., Smits A.J. (1998) Meanflow
b scaling of turbulent pipe flow. J.
Fluid Mech. 373, 33–79.

a

y/h

y
0

y/h

1

22

Figure 1 (a) classical mixing length profile; (b) schematic illustration of Townsend's Attached
Eddy Hypothesis in which the attached eddies grow in size in proportion to their distance from
the wall.

Figure 2 Schematic of hairpin eddy attached to the wall; (b) signature of the hairpin eddy in the
streamwise/wall-normal plane (from Adrian et al. 2000).

23

Figure 3 PIV measurements of the velocity field and vorticity field (colored contours) in a
turbulent boundary layer flowing left to right. The ramp-like structures bounded by groups of
concentrated vorticies are evidence of hairpin vortex packets in which hairpins occur in a
streamwise alignment with smaller, upstream hairpins auto-generated by larger, downstream
hairpins. The velocity fields magnified in the upper inset figure posses the characteristics of
hairpins identified in Figure 2 (from Adrian et al. 2000)..

Figure 4 Flow visualization of a turbulent boundary layer. Flow is from left to right and the
visualization details are as described in Cantwell et al. (1978). Photo courtesy of Don Coles.

24

Figure 5 Very large-scale superstructure signatures: (a) From rake of hot wire traces from
Hutchins and Marusic (2007); u signal at y/ = 0.15 for R = 14400 (b) Same with only low
speed regions highlighted. (c). High-frame rate stereo-PIV measurements from Dennis and
Nickels (2011a,b) in a turbulent boundary layer at R = 4700, showing similar features to the
hot-wire rake measurements. Here the black isocontours show swirl strength, indicating the
corresponding location of vortical structures with the low-speed (blue) and high-speed (red)
regions. After Marusic and Adrian (2013).

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Figure 6 Top panel: Instantaneous velocity fluctuations in the streamwise-wall-normal (x-y)
plane and instantaneous streamwise velocity fluctuations in the streamwise/spanwise (x-y)
planes for data from laboratory PIV (Hambleton et al. 2006) and for the atmospheric surface
layer using an arrays of sonic anemometers (Hutchins et al. 2012). High positive w regions are
indicated by red while blue denotes highly negative w regions. High negative u regions are
indicated by dark gray while light gray shade denotes highly positive u regions. Bottom panel
shows the Biot Savart law calculations for an idealized packet of hairpin vortices with their
image vortices in the wall, as per the schematic of Adrian et al. (2000) shown on the left side.

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Seco
ndar
y circ

Low-

High
-s

spee

d st r
eak

peed
s

boil

upw

Ha i

rpin

vor
te

x pa

cke

t hi
era

rch

ellin

treak

, dow

nwel

ulatio

n

ling

g

y

sedimentation

Large-scale motions=
Largest HPVPs
erosion

Figure 7 Cartoon of coherent structures in open channel flows.

27