CHAPTER 1-FELIX M. EXNER AND THE ORIGINS OF .ppt
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1、1,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 a
2、lso 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 i
3、n the Mississippi River, New Orleans, USA Image from LUMCON web page: http:/weather.lumcon.edu/weatherdata/audubon/map.html,2,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) sid
4、e 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.,Magnified view of previous image,Flow,3,DUNES IN THE RHINE DELTA, THE N
5、ETHERLANDS,Image courtesy A. Wilbers and A. Blom,4,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.,5,EXNERS QUESTION: WHY ARE DUNES ASYMMETRIC?,The parameters:
6、 x = streamwise distance L t = time T = bed elevation L qt = volume total sediment transportrate 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 1The notation in brackets denotes dimens
7、ions: 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.,6,THE EQUATIONS,St. Venant shallow water equations:,Exners equation of conservation of bed sediment:,Relation
8、 between sediment transport rate and flow hydraulics:,These are statements of conservation of water mass and momentum in a 1D river (constant width).,Exners seminal contribution: if more sediment enters a reach than leaves, the bed elevation in the reach increases.,The phenomenon of sediment transpo
9、rt was poorly known in Exners time. Exner guessed that a higher velocity caused a higher sediment transport rate.,7,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 formwhere c() denotes a
10、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),8,FELIX EXNER AND MORPHODYNAMICS,The field of morphodynamics consists of the class of proble
11、ms 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 deposit
12、ion 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 morphody
13、namics.,9,DERIVATION OF EXNERS 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
14、 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,or,or,qt,qt,10,1D SEDIMENT TRANSPORT MORPHODYNAMICS,While Exners 1D model successfully explains a) the tendency
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