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CHAPTER 1:

FELIX M. EXNER AND THE ORIGINS OF MORPHODYNAMICS

Felix Maria Exner was an Austrian researcher who was active in the early part of the 20th Century. His main area of interest was meteorology. At some point he became interested in the formation of dunes in rivers (Exner, 1920, 1925; see also Leliavsky, 1966). In the course of his research on the subject, he derived and employed one version of the various statements of conservation of bed sediment that are now referred to as “Exner equations.” In addition, he made an important early contribution to 1D nonlinear wave dynamics.

Dunes in the Mississippi River, New Orleans, USA

Image from LUMCON web page:

http://weather.lumcon.edu/weatherdata/audubon/map.html


DUNE ASYMMETRY

Dunes are rhythmic bedforms that are often seen on the beds of rivers. The upstream (stoss) side of a dune has a gentle slope, whereas the downstream (lee) side has a steep slope. This characteristic asymmetry allows the determination of flow direction.

In the image on the right, the flow direction is from top to bottom. Dunes migrate in the same direction as the flow, i.e. downstream.

Flow

Magnified view of previous image


DUNES IN THE RHINE DELTA, THE NETHERLANDS

Image courtesy A. Wilbers and A. Blom


DUNES IN THE LABORATORY

Flow

Flow

Dunes in a channel at St. Anthony Falls Laboratory, University of Minnesota, USA

Dunes in a channel at Tsukuba University, Japan.

Image courtesy H. Ikeda.


EXNER’S QUESTION:

WHY ARE DUNES ASYMMETRIC?

  • The parameters:

  • x = streamwise distance [L]

  • t = time [T]

  • = bed elevation [L]

    qt = volume total sediment transport

    rate per unit stream width [L2/T]

    p = bed porosity [1]

    g = acceleration of gravity [L/T2]

    H = flow depth [L]

    U = depth-averaged flow velocity [L/T]

    Cf = bed friction coefficient [1]

    The notation in brackets denotes dimensions: M denotes mass, L denotes length and T denotes time. The bed friction coefficient is defined such that b = CfU2, where b denotes bed shear stress [M/L/T2] and  denotes water density [M/L3].


THE EQUATIONS

St. Venant shallow water equations:

These are statements of conservation of water mass and momentum in a 1D river (constant width).

Exner’s equation of conservation of bed sediment:

Exner’s seminal contribution: if more sediment enters a reach than leaves, the bed elevation in the reach increases.

Relation between sediment transport rate and flow hydraulics:

The phenomenon of sediment transport was poorly known in Exner’s time. Exner guessed that a higher velocity caused a higher sediment transport rate.


THE RESULTS

The details of the analysis are not considered here. The essential result is a nonlinear equation for the evolution of bed elevation  of the general form

where c() denotes a positive wave speed of the bed that is an increasing function of bed elevation. The result is that any symmetric bedform sharpens in time to the asymmetry characteristic of dunes.

Image from Leliavsky (1966)


FELIX EXNER AND MORPHODYNAMICS

The field of morphodynamicsconsists of the class of problems for which the flow over a bed interacts strongly with the shape of the bed, both of which evolve in time.

That is, the flow and the bed talk to each other. The flow field over the bed determines a pattern of variation of sediment transport rate. This variation changes the bed by erosion or deposition of sediment. The changed bed now induces a changed flow field.

Felix Exner was the first researcher to state a morphodynamic problem in quantitative terms. The term “morphodynamics” itself evolved many decades afterward. This notwithstanding, Exner deserves recognition as the founder of morphodynamics.


DERIVATION OF EXNER’S EQUATION OF SEDIMENT CONSERVATON

Recall that qt denotes the volume sediment transport rate per unit width and p denotes bed porosity (fraction of bed volume that is pores rather than sediment). The mass sediment transport rate per unit width is then sqt, where s is the material density of sediment. Mass conservation within the control volume with a unit width requires that:

/t (sediment mass in bed) = mass sediment inflow rate – mass sediment outflow rate

qt

or

qt

or


1D SEDIMENT TRANSPORT MORPHODYNAMICS

While Exner’s 1D model successfully explains a) the tendency of dunes to migrate downstream and b) their tendency to become asymmetric, it does not explain the origin of dunes. Such an explanation requires a 2D analysis, and as such is not considered further here. The interested reader can refer to e.g. Engelund (1970), Smith (1970) and Fredsoe (1974).

Dunes are one of many examples of fascinating morphodynamic problems requiring 2D or 3D approaches. For example, the pattern of meandering and sediment sorting in the river on the right cannot be explained with a 1D approach.

So why is this book limited to 1D morphodynamics?

  • There are many interesting 1D problems.

  • A surprising number of 2D problems can be reduced and explained using a 1D formulation.

  • 1D morphodynamics is the gateway to 2D and 3D morphodynamics.

Ooi River, Japan.

Image courtesy H. Ikeda


RIVER STRAIGHTENING: A PROBLEM IN 1D MORPHODYNAMICS

Consider a meandering river.

In 1D approaches, the relevant streamwise coordinate is measured as an arc length down the channel centerline. The long profile of the river is as illustrated below. The fluctuations are induced by the meanders themselves.


RIVER STRAIGHTENING (contd.)

Now suppose the bends between points A and B are cut off due to river straightening.

The elevation drop between A and B is the same as before straightening (at least initially), but the distance between A and B is much shorter. As a result bed slope is elevated between A and B compared to the reaches upstream and downstream.


RIVER STRAIGHTENING (contd.)

Now let’s exaggerate the change in bed slope for the sake of illustration.

The bed slope is higher, the flow swifter and the sediment transport rate higher over the steepened reach.

Exner equation of sediment conservation over a reach with length x (set porosity p = 0 for illustration).

Bed elevation of the reach increases in time if more sediment enters than

leaves.


RIVER STRAIGHTENING (contd.)

The sediment transport rate is higher over the steepened reach.

The reach UA must degrade (bed elevation must decrease in time), as there is more sediment output than input.

The reach between B and D must aggrade (bed elevation must increase in time), as there is more sediment input than output.


RIVER STRAIGHTENING (contd.)

Initial long profile after straightening.

Evolving long profile.

Final equilibrium long profile: qt/x = 0 thus /t = 0

The upstream part of the reach must degrade and the downstream part must aggrade until a new equilibrium profile is established.

The above 1D picture is a simplified version of reality: too much aggradation downstream will cause the river to avulse (jump channel). Nevertheless, it captures the essentials of the problem.


AN EXAMPLE: THE EAST PRAIRIE RIVER, ALBERTA, CANADA

River straightening was popular in the 1960’s and 1970’s. The goal was flood protection: the steeper slope of the straightened reach allowed higher flow velocities, and thus conveyance of the same discharge at a lower depth. This short-term benefit was often outweighed by the long-term pattern of aggradation and degradation induced by straightening. The East Prairie River provides such an example. See Parker and Andres (1976) for more details.

Degrading upstream reach

Aggrading downstream reach

Straightened central reach

East Prairie River, Alberta, Canada

Images courtesy D. Andres


THE STRAIGHTENED WEST AND EAST PRAIRIE RIVERS, ALBERTA, CANADA

West Prairie River

East Prairie River

straightened

Straightened central reach

meandering

Image from NASA

https://zulu.ssc.nasa.gov/mrsid/mrsid.pl


REFERENCES FOR CHAPTER 1 CANADA

Engelund, F., 1970, Instability of erodible beds, J. Fluid Mech., 42(2), 225-244.

Exner, F. M., 1920, Zur Physik der Dunen, Sitzber. Akad. Wiss Wien, Part IIa, Bd. 129 (in German).

Exner, F. M., 1925, Uber die Wechselwirkung zwischen Wasser und Geschiebe in Flussen, Sitzber. Akad. Wiss Wien, Part IIa, Bd. 134 (in German).

Fredsoe, J., 1974, On the development of dunes in erodible channels, J. Fluid Mech., 64(1), 1-16.

Leliavsky, S., 1966, An Introduction to Fluvial Hydraulics, Dover, New York, 257 p.

Parker, G. and Andres, D., 1976, Detrimental effects of river channelization, Proceedings, ASCE Rivers '76 Conference, 1248-1266.

Smith, J. D., 1970, Stability of a sand wave subjected to shear flow of low Froude number, J. Geophys. Res., 75(30), 5928-5940.


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