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Streamwise streaks induced by bedload diusion
Anaïs Abramian, Olivier Devauchelle, Eric Lajeunesse
To cite this version:
Anaïs Abramian, Olivier Devauchelle, Eric Lajeunesse. Streamwise streaks induced by bedload dif-
fusion. Journal of Fluid Mechanics, Cambridge University Press (CUP), 2019, 863, pp.601-619.
�10.1017/jfm.2018.1024�. �hal-02001689�
This draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics
1
Streamwise streaks induced by bedload
diffusion
Ana¨ıs Abramian
1
†, Olivier Devauchelle
1
and Eric Lajeunesse
1
1
Institut de Physique du Globe de Paris, France
(Received xx; revised xx; accepted xx)
A fluid flowing over a granular bed can move its superficial grains, and eventually
deform it by erosion and deposition. This coupling generates a beautiful variety of
patterns such as ripples, bars and streamwise streaks. Here, we investigate the latter,
sometimes called “sand ridges” or “sand ribbons”. We perturb a sediment bed with
sinusoidal streaks, the crests of which are aligned with the flow. We find that, when
their wavelength is much larger than the flow depth, bedload diffusion brings mobile
grains from troughs, where they are more numerous, to crests. Surprisingly, gravity can
only counter this destabilising mechanism when sediment transport is intense enough.
Relaxing the long-wavelength approximation, we find that the cross-stream diffusion of
momentum mitigates the influence of the bed perturbation on the flow, and even reverses
it for short wavelengths. Viscosity thus opposes the diffusion of entrained grains to select
the most unstable wavelength. This instability might turn single-thread alluvial rivers
into braided channels.
Key words: Bedload transport, bedforms, pattern formation, river morphology, granular
diffusion.
1. Introduction
When water flows over a granular bed with enough strength, it dislodges some of the
superficial grains and entrains them downstream (Shields 1936; Einstein 1937; Bagnold
1973). As long as the flow-induced force is comparable to their weight, the entrained
grains remain close to the bed surface, where they travel with the flow, until they
eventually settle down. In steady state, the balance between entrainment and settling
sets the number of travelling grains, which thus depends primarily on the flow-induced
shear stress (Charru et al. 2004a; Lajeunesse et al. 2010). Accordingly, the sediment flux
resulting from their collective motion, called bedload transport, is usually expressed as
a function of shear stress (Meyer-Peter & M¨uller 1948).
Bedload transport is often heterogeneous—it scours away the bed somewhere, and
deposits the entrained material elsewhere (Exner 1925). The flow then adjusts to the
deformed bed, and alters the distribution of erosion and deposition. This fluid-structure
interaction generates bedforms through various instabilities (Seminara 2010; Charru et al.
2013).
Current ripples are iconic underwater bedforms found in streams, on beaches and
sometimes in the sedimentary record (Allen 1982; Coleman & Melville 1994). They result
from the inertia of the flow, which concentrates shear stress just upstream of their crest
† Email address for correspondence: anaisabramian@gmail.com
2 A. Abramian, O. Devauchelle, and E. Lajeunesse
y
x
z
S
Flow
D
2h
Figure 1. Sediment bed perturbed by longitudinal streaks. A layer of fluid (blue) flows over
a granular bed (brown). The reference frame is inclined with respect to gravity (downstream
slope S). The vertical grey line with a diamond marker symbolises a plumb line.
(Kennedy 1963; Charru 2006; Charru et al. 2013). Nascent ripples make the most of this
mechanism by orienting their crest across the flow. At least initially, they do not involve
any cross-stream sediment flux. By contrast, the oblique crest of alternate bars diverts
the water flow to induce the cross-stream bedload flux that makes them unstable (Parker
1976; Colombini et al. 1987; Devauchelle et al. 2010; Andreotti et al. 2012).
Although less common, streamwise streaks materialise cross-stream bedload more
neatly—their crest remains aligned with the stream as they grow (Karcz 1967; Colombini
& Parker 1995; McLelland et al. 1999). To initiate a cross-stream flux of sediment, these
bedforms use a subtle peculiarity of turbulence. When streamwise ridges perturb their
boundary, turbulent flows generates transverse, counter-rotating vortices (Colombini
1993; Vanderwel & Ganapathisubramani 2015). Over a granular bed, these slow sec-
ondary currents transport sediment across the primary flow to accumulate it in upwelling
areas, thus reinforcing the ridges that brought them about. A similar phenomenon occurs
when grains of different sizes make up the bed, the heterogeneous roughness of which
then plays the role of ridges (McLelland et al. 1999; Willingham et al. 2014).
In the above examples, the sediment grains travel along the force that entrains them.
This is certainly true on average, but bedload particles slide and roll over a rough bed,
which makes their trajectory seesaw across the stream (Nikora et al. 2002; Furbish et al.
2012). The experiments of Seizilles et al. (2014) show that these fluctuations cause
the grains to disperse laterally, like random walkers. Collectively, they diffuse across
the bedload layer towards areas of lesser transport, thus moving across the stream in
the absence of transverse flow. We speculate that this Fickian diffusion could create
streamwise streaks without secondary currents, provided bedload is less intense on the
bedforms’ crests.
To test this scenario, we investigate the stability of a flat sediment bed sheared by a
laminar flow. We begin with the shallow-water approximation (section 2). Extending our
analysis to two dimensions, we then find that viscosity, which diffuses momentum across
the stream, selects the size of the most unstable mode (section 3). We then consider a
stream covered with a rigid lid, which might facilitate measurements in a laboratory
experiment (section 4). Finally, we look for this instability in previous experiments
(section 5).
Streamwise streaks 3
2. Bedload instability
2.1. Base state
We consider an infinitely wide, flat granular bed sheared by a free-surface, laminar
flow (figure 1). A small slope S drives the fluid along x, the streamwise direction, but we
will neglect its effect on the weight of a grain later on. We further assume that the size
of a grain, d
s
, is much smaller than the flow depth D. In steady state, the shear stress
the fluid exerts on the bed, τ, is the projection of its weight on the streamwise direction:
τ = ρgDS (2.1)
where ρ is the density of the fluid, and g the acceleration of gravity.
To entrain a grain, the fluid needs to pull it with enough strength to overcome its
weight. Mathematically, this happens when the ratio of these two forces, θ, exceeds a
threshold value, θ
t
. Shields (1936) defined this ratio as
θ =
τ
(ρ
s
− ρ)gd
s
, (2.2)
where ρ
s
is the density of a grain.
Based on laboratory observations, Charru et al. (2004a) suggested that the number of
grains the fluid dislodges from the bed, per unit surface and time, is proportional to the
distance to threshold, θ −θ
t
. The bedload layer is fed by this constant input. Conversely,
it loses a fraction of its population through settling. When moving grains are too sparse
to interact, the settling rate is proportional to the number of moving grains per unit
area, n (Aussillous et al. 2016). At equilibrium, the density of moving grains thus reads
n =
α
n
d
2
s
(θ − θ
t
) (2.3)
where α
n
is, like θ
t
, a dimensionless, empirical parameter (Lajeunesse et al. 2010). For
illustration, θ
t
∼ 0.1 and α
n
∼ 0.01 are typical values for these parameters in a laminar
flow (Seizilles et al. 2014). Equation (2.3) is valid only above threshold, that is, when
θ > θ
t
; below threshold, the bedload layer is empty (n = 0).
After equation (2.3), a flow near threshold can only entrain a sparse bedload layer.
Then, the velocity of the travelling grains is that of the fluid near the bed, which is
proportional to shear stress in a laminar flow (Seizilles et al. 2014). As a consequence,
the average velocity in the bedload layer is proportional to Stokes’ settling velocity, which
reads
V
s
=
(ρ
s
− ρ)gd
2
s
18η
(2.4)
where η is the viscosity of the fluid. The sediment flux, q
s
, results from the collective
motion of the bedload grains:
q
s
= α
v
nV
s
, (2.5)
where α
v
is a dimensionless coefficient. In a laminar flow, Seizilles et al. (2014) found
α
v
∼ 0.4.
Most authors relate bedload directly to the Shields parameter with a sediment trans-
port law (Meyer-Peter & M¨uller 1948). Combining equations (2.3) and (2.5), we find that
bedload transport is proportional to the distance to threshold. The specific expression
of this law, however, is still debated, and is likely to depend on the particle Reynolds
number (Ouriemi et al. 2009). Here, we choose a simple law that compares reasonably
with near-threshold, laminar experiments (Charru et al. 2004a; Seizilles et al. 2014).
Equations (2.1) to (2.5) represent a uniform base state, both in the downstream
4 A. Abramian, O. Devauchelle, and E. Lajeunesse
Bed
Free surface
z
y
x
a
Transverse coordinate, y
Sediment flux, q
x
b
Figure 2. Mechanism of the bedload instability. a: Bed elevation (brown) and free surface flow
(blue). b: Distribution of the corresponding downstream sediment flux. Red arrows indicate
bedload diffusion.
direction x and in the cross-stream direction y. In the following, we add a perturbation
to it to introduce bedload diffusion.
2.2. Bedload diffusion
Heterogeneity drives diffusion. To introduce some of it in our system, we now carve
streamwise streaks into the granular bed, in the form of a sinusoidal perturbation of
amplitude h and wavelength λ (figure 1). The fluid flow and the sediment bed remain
invariant along x, and our system is now two-dimensional.
To illustrate the mechanism of bedload instability, we consider, in this section, that
the amplitude of the perturbation is much smaller, and its wavelength much longer, than
the flow depth. With these assumptions, we expect the shallow-water approximation to
yield a reasonable estimate of the shear stress τ , and therefore of the Shields parameter
θ. Both are then proportional to the local flow depth, D − h, and therefore of lesser
intensity at the crest of the perturbation. Mathematically,
θ =
ρDS
(ρ
s
− ρ)d
s
1 −
h
D
. (2.6)
According to equation (2.3), the bedload layer is thus denser in the troughs than on the
crests. Its density reads
n = n
0
−
α
n
θ
0
d
2
s
h
D
(2.7)
where n
0
and θ
0
are the density of moving grains and the Shields parameter in the base
state, respectively. Like equation (2.3), from which it is derived, the above equation only
holds above threshold, that is, when
h
D
6
θ
0
− θ
t
θ
0
. (2.8)
This condition sets the maximum amplitude the perturbation h can reach before the
following analysis breaks down.
Following Seizilles et al. (2014), we now treat the bedload grains as independent random
walkers. As they travel downstream at the average velocity α
v
V
s
, their cross-stream
velocity fluctuates around zero. We represent this process by a series of random sideways
steps, the amplitude of which is a fraction of the grain size. Statistically, the accumulation
