Streamwise streaks induced by bedload diffusion

TL;DR: In this paper, the authors perturb a sediment bed with sinusoidal streaks, the crests of which are aligned with the flow, and 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.

Abstract: 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.

Summary

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).
• The authors then consider a stream covered with a rigid lid, which might facilitate measurements in a laboratory experiment (section 4).
• The authors now introduce the transport number, Tr, as the ratio of the two terms of the diffusivity in equation (2.12): Tr = γD `d θ0 − θt θ0 . (2.13) Combining this definition with the bedload transport law (equation (2.5)) recasts the transport number into a dimensionless sediment flux: Tr = γDd2s αvαn`dVs qs0 θ0 . (2.14) Literally,.
• When the wavelength of the perturbation is much longer than the flow depth, the shallowwater approximation ensures that the latter condition is fulfilled.
• The authors linear stability analysis then holds for imperceptible streaks only.

4. Rigid lid

• The free surface of the flow makes the tracking of sediment grains difficult.
• The authors are thus left with equation (2.17) again, that is, the same dispersion relation as in section 2.
• Specifically, the authors replace the free-boundary condition with a no-slip one.
• This result encourages us to look for the bedload instability in a closed channel, where their observation should be easier than in an open one—although, to their knowledge they have never been reported in any laboratory experiment.
• This is a dramatic change: bedload diffusion and gravity now conspire to wear away even a long-wavelength perturbation.

5. Laboratory channels

• Bedload experiments in laminar channels or pipes are not uncommon (Charru et al.
• When the transport number is below one, the order of the most unstable mode increases with the aspect ratio of the channel.
• Such a transient may not last long enough for the instability to grow.
• Remembering that the roughness of the above estimates matches that of their simplified boundary conditions, the authors may now map the explored regions of the parameter space .
• For the most part, they lie in the stable domain, although the experiments by Aussillous et al. (2013) and Seizilles et al. (2014) marginally overlap the stable domain on the first mode.

6. Conclusion

• The linear stability analysis the authors have presented in this paper identifies a new instability associated with bedload transport, caused by the cross-stream diffusion of the travelling grains.
• This instability produces bed streaks aligned with the flow, in the absence of any secondary currents.
• The authors could not find any report of this instability appearing in a laboratory experiment.
• This, of course, raises the question of its actual existence, but the authors suspect that the basic reason for its absence from the literature is that the aspect ratio of laboratory channels is usually too small to accommodate its growth.
• O.D. was partially funded by the Émergence(s) programme of the Mairie de Paris.

Figures

Figure 5. Stability map of the bedload instability, for a free surface flow. a: Dispersion relation. Darker areas correspond to faster growth rate. Red line: most unstable wave number for a given value of the transport number. b: Growth rate of the most unstable wavenumber. c: Maximum growth rate as a function of the transport number.

Figure 3. Dispersion relation of the bedload instability, in a free-surface flow. Dashed lines: shallow-water approximation (equation (2.17)). Solid lines: two-dimensional Stokes flow (equation (3.9)).

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.

Table 1. Estimated parameter values for laminar bedload-transport experiments.

Figure 6. Saturation of the bedload instability. Red line: amplitude of the bed perturbation at the transition to the non-linear regime, hmax, as a function of the transport number. Inset : transition criterium, the amplitude of the perturbation of the Shields parameter reaches θ0− θt.

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.

Figure 8. Stability map for the bedload instability in a finite-width channel. Blue scale indicates growth rate of the fastest-growing mode. Red boxes show the range of laboratory experiments. a: open channel. b: fixed lid

Full text

HAL Id: hal-02001689
https://hal.archives-ouvertes.fr/hal-02001689
Submitted on 31 Jan 2019
HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entic research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diusion de documents
scientiques de niveau recherche, publiés ou non,
émanant des établissements d’enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.
Streamwise streaks induced by bedload diusion
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

Frequently asked questions

Q1. What are the contributions mentioned in the paper "Streamwise streaks induced by bedload diffusion" ?

Abramian et al. this paper studied the effect of shear stress on sediment motes in a stream. 

Q2. What are the future works mentioned in the paper "Streamwise streaks induced by bedload diffusion" ?

The authors speculate that this might not be fortuitous, as experimenters often wish the sediment bed to be invariant in the cross-stream direction. At this point, the authors can only imagine an ideal set-up, specially designed to observe the bedload instability. Once such a framework is set, it will become a matter of numerical routines to explore it beyond linearity. 

Q3. What is the effect of Stokes flow on streamwise streaks?

Stokes flowBedload diffusion reinforces streamwise streaks provided (i) bedload transport is weak, and (ii) the flow exerts a lower shear stress on the crests than in the troughs. 

Q4. What is the effect of the flow-induced force on the grains?

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. 

Q5. What is the shear stress the fluid exerts on the bed?

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. 

Q6. What is the important fact about the instability?

The bedload instability is sensitive to boundary conditions; it persists in a rectangular pipe, but disappears when the flow is driven by a travelling lid. 

Q7. What is the effect of the hmax on the stability of the flow?

Once the perturbation has outgrown hmax, the instability enters a nonlinear regime, to which the present analysis grants no access. 

Q8. What is the effect of the secondary currents on the bedforms?

Over a granular bed, these slow secondary currents transport sediment across the primary flow to accumulate it in upwelling areas, thus reinforcing the ridges that brought them about. 

Q9. What is the equation for the slope-induced flux?

The bedload density equation (2.7), the cross-stream flux equations (2.9) and (2.10), and the the Exner equation form a closed system, which the slope-induced flux makes nonlinear. 

Q10. What is the effect of the bedload particles on the stream?

they diffuse across the bedload layer towards areas of lesser transport, thus moving across the stream in the absence of transverse flow. 

Q11. Why does the instability grow near the threshold of sediment transport?

Because it can only grow near the threshold of sediment transport, the unstable perturbation quickly enters a nonlinear regime which the authors havenot investigated. 

Q12. What is the mechanism of a granular bed?

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). 

Q13. What is the effect of the free surface of the flow on the tracking of sediment grains?

Although most bedload experiments involve open channels, the free surface of the flow makes the tracking of sediment grains difficult. 

Q14. What is the effect of the stability theory on the bedload?

while the stabilising mechanism vanishes, the unstable coupling between the flow and the bed persists—the bedload instability then thrives. 

Q15. What is the parameter for a laminar flow?

In addition, the parameter γ, which accounts for the gravityinduced flux of sediment, has never been measured in a laminar flow (section 2.3). 

Q16. What is the balance between entrainment and settling?

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). 

Q17. How many parallel streaks should be allowed to grow?

The pipe should be wider than the wavelength of the perturbation; an aspect ratio of approximately 20, for instance, should allow two parallel streaks to grow. 

Q18. What is the reason for the absence of the instability in the literature?

of course, raises the question of its actual existence, but the authors suspect that the basic reason for its absence from the literature is that the aspect ratio of laboratory channels is usually too small to accommodate its growth. 

Q19. What is the way to understand the morphology of the sediments?

A reliable theory of bedload diffusion, tested against laboratory experiments, would therefore help us understand their morphology. 

Q20. How do the authors determine the bedload instability?

To identify the bedload instability unambiguously, one need only reduce the bedload flux until the wavelength of the instability reaches its minimum, which should be the above value. 

Q21. What is the shear stress at the crest of the channel?

Re ( ĥ exp ( ikyD))) , (4.6)thus confirming that, regardless of the wavenumber k, the shear stress is always stronger at the crest. 

Q22. What is the recent analysis of the bedload instability?

The linear stability analysis the authors have presented in this paper identifies a new instability associated with bedload transport, caused by the cross-stream diffusion of the travelling grains. 

Q23. What is the qd of the bedload?

the accumulationof these steps generates a diffusive flux, qd, towards the less populated areas of the bedload layer.