Streamwise streaks induced by bedload diffusion
Summary (2 min read)
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.
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Citations
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Cites background from "Streamwise streaks induced by bedlo..."
...At the surface of a granular bed, streamwise grooves can be generated by counterrotating vortices when the flow is turbulent [17,18] or by the diffusion-like transport of grains from the troughs to the crests of the bed in a laminar regime [19]....
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References
2,182 citations
"Streamwise streaks induced by bedlo..." refers background in this paper
...Accordingly, the sediment flux resulting from their collective motion, called bedload transport, is usually expressed as a function of shear stress (Meyer-Peter & Müller 1948)....
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...Most authors relate bedload directly to the Shields parameter with a sediment transport law (Meyer-Peter & Müller 1948)....
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...A natural question to ask, then, will be whether the bedload instability grows as far as to split a broad channel into smaller ones, thus creating the precursor of a braided river (Stebbings 1963; Métivier, Lajeunesse & Devauchelle 2017)....
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...For randomly packed spheres, the packing fraction is approximately φ∼ 0.6 (Bernal & Mason 1960)....
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"Streamwise streaks induced by bedlo..." refers background in this paper
...Current ripples are iconic underwater bedforms found in streams, on beaches and sometimes in the sedimentary record (Allen 1982; Coleman & Melville 1994)....
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Frequently Asked Questions (23)
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.