1、1,CHAPTER 22: MORPHODYNAMICS OF RECIRCULATING AND FEED FLUMES,Laboratory flumes have proved to be valuable tools in the study of sediment transport and morphodynamics. Here the case of flumes with vertical, inerodible walls are considered. There are two basic types of such flumes:Recirculating flume
2、s andFeed flumesIn addition, there are several variant types, one of which is discussed in a succeeding slide.,Recirculating flume in the Netherlands used by A. Blom and M. Kleinhans to study vertical sediment sorting by dunes. Image courtesy A. Blom.,2,THE MOBILE-BED EQUILIBRIUM STATE,The flume con
3、sidered here is of the simplest type; it has a bed of erodible alluvium, constant width B and vertical, inerodible walls. The bed sediment is covered by water from wall to wall. One useful feature of such a flume is that if it is run long enough, it will eventually approach a mobile-bed equilibrium,
4、 as discussed in Chapter 14. When this state is reached, all quantities such as water discharge Q (or qw = Q/B), total volume bed material sediment transport rate Qt (or qt = Qt/B), bed slope S, flow depth H etc. become spatially constant in space (except in entrance and exit regions) and time. (The
5、 parameters in question are averaged over bedforms such as dunes and bars if they are present.),3,Friction relations:where kc is a composite bed roughness which may include the effect of bedforms (if present).,THE EQUILIBRIUM STATE: REVIEW OF MATERIAL FROM CHAPTER 14,The hydraulics of the equilibriu
6、m state are those of normal flow. Here the case of a plane bed (no bedforms) is considered as an example. The bed consists of uniform material with size D. The governing equations are (Chapter 5):,Water conservation:,Momentum conservation:,Generic transport relation of the form of Meyer-Peter and Ml
7、ler for total bed material load: where t and nt are dimensionless constants:,4,THE EQUILIBRIUM STATE: REVIEW contd.,In the case of the Chezy resistance relation, for example, the equations governing the normal state reduce to:,In the case of the Manning-Stickler resistance relation, the equations go
8、verning the normal state reduce with to:,Let D, kc and R be given. In either case above, there are two equations for four parameters at equilibrium; water discharge per unit width qw, volume sediment discharge per unit width qt, bed slope S and flow depth H. If any two of the set (qw, qt, S and H) a
9、re specified, the other two can be computed.,5,THE RECIRCULATING FLUME,In a recirculating flume all the water and all the sediment are recirculated through a pump. The total amounts of water and sediment in the system are conserved. In addition to the sediment itself, the operator is free to specify
10、 two parameters in operating the flume: the water discharge per unit width qw and the flow depth H. The water discharge (and thus the discharge per unit width qw) is set by the pump setting. (More properly, what are specified are the head-discharge relation of the pump and the setting of the valve o
11、n the return line, but in many recirculating systems flow discharge itself can be set with relative ease and accuracy.),The constant flow depth H reached at equilibrium is set by the total amount of water in the system, which is conserved. Increasing the total amount of water in the system increases
12、 the depth reached at final mobile-bed equilibrium.,Thus in a recirculating flume, equilibrium qw and H are set by the flume operator, and total volume sediment transport rate per unit width qt and bed slope S evolve to equilibrium accordingly.,6,THE FEED FLUME,In a feed flume all the water and all
13、the sediment are fed in at the upstream and allowed to wash out at the downstream end. Water is introduced (usually pumped) into the channel at the desired rate, and sediment is fed into the channel using e.g. a screw-type feeder at the desired rate. In addition to the sediment itself, the operator
14、is thus free to specify two parameters in the operation of the flume: the water discharge per unit width qw and the total volume sediment discharge per unit width qt reached at final equilibrium, which must be equal to the feed rate qtf.,Thus in a feed flume, equilibrium qw and qt are set, by the fl
15、ume operator, and equilibrium flow depth H and bed slope S evolve accordingly.,7,A HYBRID TYPE: THE SEDIMENT-RECIRCULATING, WATER-FEED FLUME,In the sediment-recirculating, water-feed flume the sediment and water are separated at the downstream end. Nearly all the water overflows from a collecting ta
16、nk. The sediment settles to the bottom of the collecting tank, and is recirculated with a small amount of water as a slurry. The water discharge per unit width qw is thus set by the operator (up to the small fraction of water discharge in the recirculation line). The total amount of sediment in the
17、flume is conserved. In addition, a downstream weir controls the downstream elevation of the bed. The combination of these two conditions constrains the bed slope S at mobile-bed equilibrium. Adding more sediment to the flume increases the equilibrium bed slope.,Thus in a sediment-recirculating, wate
18、r-feed flume, qw and S are set by the flume operator and qt and H evolve toward equilibrium accordingly.,8,The final mobile-bed equilibrium state of a flume is usually not precisely known in advance. Flow is thus commenced from some arbitrary initial state and allowed to approach equilibrium. This m
19、otivates the following two questions:How long should one wait in order to reach mobile-bed equilibrium?What is the path by which mobile-bed equilibrium is reached? It might be expected that the answer to these questions depends on the type of flume under consideration. Here two types of flumes are c
20、onsidered: a) a pure feed flume and b) a pure recirculating flume.In performing the analysis, the following simplifying assumptions (which can easily be relaxed) are made: The flow is always assumed to be subcritical in the sense that Fr 1. The channel is assumed to have a sufficiently large aspect
21、ration B/H that sidewall effects can be neglected. Bed resistance is approximated in terms of a constant resistance coefficientCf, so that the details of bedform mechanics are neglected. The sediment has uniform size D. The analysis presented here is based on Parker (2003).,MORPHODYNAMICS OF APPROAC
22、H TO EQUILIBRIUM,9,THE LEGEND OF SEDIMENT LUMPS IN RECIRCULATING FLUMES,In the world of sediment flumes, there is a persistent legend concerning recirculating flumes that is rarely documented in the literature. That is, these flumes are said to develop sediment “lumps” that recirculate round and rou
23、nd, either without dissipating or with only slow dissipation. The author of this e-book has heard this story from T. Maddock, V. Vanoni and N. Brooks. One of the authors graduate students encountered these lumps in a recirculating, meandering flume and showed them to the author (Hills, 1987).,10,THE
24、 LEGEND OF SEDIMENT LUMPS IN RECIRCULATING FLUMES,In the world of sediment flumes, there is a persistent legend concerning recirculating flumes that is rarely documented in the literature. That is, these flumes are said to develop sediment “lumps” that recirculate round and round, either without dis
25、sipating or with only slow dissipation. The author of this e-book has heard this story from T. Maddock, V. Vanoni and N. Brooks. One of the authors graduate students encountered these lumps in a recirculating, meandering flume and showed them to the author (Hills, 1987).,11,THE LEGEND OF SEDIMENT LU
26、MPS IN RECIRCULATING FLUMES,In the world of sediment flumes, there is a persistent legend concerning recirculating flumes that is rarely documented in the literature. That is, these flumes are said to develop sediment “lumps” that recirculate round and round, either without dissipating or with only
27、slow dissipation. The author of this e-book has heard this story from T. Maddock, V. Vanoni and N. Brooks. One of the authors graduate students encountered these lumps in a recirculating, meandering flume and showed them to the author (Hills, 1987).,12,THE LEGEND OF SEDIMENT LUMPS IN RECIRCULATING F
28、LUMES,In the world of sediment flumes, there is a persistent legend concerning recirculating flumes that is rarely documented in the literature. That is, these flumes are said to develop sediment “lumps” that recirculate round and round, either without dissipating or with only slow dissipation. The
29、author of this e-book has heard this story from T. Maddock, V. Vanoni and N. Brooks. One of the authors graduate students encountered these lumps in a recirculating, meandering flume and showed them to the author (Hills, 1987).,13,THE LEGEND OF SEDIMENT LUMPS IN RECIRCULATING FLUMES,In the world of
30、sediment flumes, there is a persistent legend concerning recirculating flumes that is rarely documented in the literature. That is, these flumes are said to develop sediment “lumps” that recirculate round and round, either without dissipating or with only slow dissipation. The author of this e-book
31、has heard this story from T. Maddock, V. Vanoni and N. Brooks. One of the authors graduate students encountered these lumps in a recirculating, meandering flume and showed them to the author (Hills, 1987).,14,x = streamwise coordinate t = time H = H(x, t) = flow depth U = U(x, t) = depth-averaged fl
32、owvelocity = (x, t) = bed elevation S = - /x = bed slope g = gravitational acceleration qt = volume bed material sediment transport rate per unitwidth qw = UH = water discharge per unitwidth b = boundary shear stress at bed L = flume length B = flume width D = sediment size p = porosity of bed depos
33、it of sediment,PARAMETERS,15,KEY APPROXIMATIONS AND ASSUMPTIONS,Flume has constant width B.Sediment is of uniform size D.H/B 1: flume is wide and sidewall effects can be neglected.Flume is sufficiently long so that entrance and exit regions can be neglected.Flow in the flume is always Froude- subcri
34、tical: Fr = U/(gH)1/2 1.qt/qw 1: volume transport rate of sediment is always much lower than that of water.Resistance coefficient Cf is approximatedas constant.,16,GOVERNING EQUATIONS: 1D FLOW,Flow mass balance,Flow momentum balance,Closure relation for shear stress: Cf = dimensionless bed friction
35、coefficient,Sediment mass balance,The condition qt/qw 1 allows the use of the quasi-steady approximation introduced in Chapter 13, according to which the time-dependent terms in the equations of flow mass and momentum balance can be neglected.,17,REDUCTION TO BACKWATER FORM,or thus,where,The equatio
36、ns of flow mass and momentum balance reduce to the standard backwater equation introduced in Chapter 5.,18,SEDIMENT TRANSPORT RELATION,D = grain size (uniform) s = sediment density = water density R = (s/ ) 1 1.65,Einstein number,Shields number,Sediment transport is characterized in terms of the sam
37、e generic sediment transport relation as used in Chapter 20, except that the parameter s is set equal to unity. Thus where,19,CONSTRAINTS ON A RECIRCULATING FLUME,The total amount of water in the flume is conserved. With constant width, constant storage in the return line and negligible storage in t
38、he entrance and exit regions (L sufficiently large), the constraint is (where C1 is a constant):At final equilibrium, when H = Ho, the constraint reduces to HoL = C1, according to which Ho is set by the total amount of water.,The total amount of sediment in the flume is conserved. Neglecting storage
39、 in the return line and the head box, the constraint is (where C2 is another constant):,Integrate the equation of sediment mass conservation,But from above,So a cyclic boundary condition is obtained:,Water discharge qw is set by the pump.,to get,20,CONSTRAINTS ON A FEED FLUME,Three constraints:,Wate
40、r discharge qw is set by the pump.,The upstream sediment discharge is set by the feeder. Where qtf is the sediment feed rate:,Let = + H denote water surface elevation. The downstream water surface elevation d is set by the tailgate:,The long-term equilibrium approached in a recirculating flume (with
41、out lumps) should be dynamically equivalent to that obtained in a sediment-feed flume (e.g. Parker and Wilcock, 1993).,21,The sediment transport relation reduces to the form:which applies whether or not mobile-bed equilibrium is reached.,MOBILE-BED EQUILIBRIUM,Let R, g, D, t, nt, c* and Cf be specif
42、ied. The equations in the boxes define three equations in five parameters Uo, Ho, qto, qw and So at mobile-bed equilibrium.,At mobile-equilibrium the equation of momentum balance reduce to the relation for normal flow introduced in Chapter 5:,The equation for water conservation reduces to:,22,MOBILE
43、-BED EQUILIBRIUM contd.,Recirculating flume:Water discharge/width qw is set by pump.Total amount of water Vw in system is conserved. Assuming constant storage in return line and neglecting entrance and exit storage, Vw = HLB depth Ho is set.Solve three equations for qto, U0, So.,Feed flume:Water dis
44、charge/width qw is set by pump.Sediment discharge/width qto is set by feeder.Solve three equations for uo, Ho, So.,The subscript “o” denotes mobile-bed equilibrium conditions.,23,APPROACH TO EQUILIBRIUM: NON-DIMENSIONAL FORMULATION,Dimensionless parameters describing the approach to equilibrium (den
45、oted by the tilde or downward cup) are formed using the values Ho, qto, So corresponding to normal equilibrium.,Bed elevation is decomposed into into a spatially averaged value and a deviation from this d(x, t), so that by definition,The above two parameters are made dimensionless as follows:;,From
46、the above relations,= the dimensionless flume number,where,24,APPROACH TO EQUILIBRIUM contd.,The sediment transport relation is :,where,The backwater relation is:,The relation for sediment conservation decomposes into two parts:,The dimensionless relations governing the approach to equilibrium are t
47、hus as follows; where Fro and denote the Froude and Shields numbers at mobile-bed equilibrium, respectively,25,The boundary condition on the backwater equation iswhere d is a constant downstream elevation set by a tailgate. This condition must hold at all flows, including the final mobile-bed equili
48、brium. Now the datum for elevation is set (arbitrarily but conveniently) to be equal to the bed elevation at the center of the flume (x = 0.5 L) at equilibrium, so that ao = 0 andBetween the above two relations and the nondimensionalizations, it is found that,APPROACH TO EQUILIBRIUM IN A FEED FLUME,
49、In a feed flume, the boundary condition on the sediment transport rate at the upstream end is , or in dimensionless variables,Thus the relations for sediment conservation of the previous slide reduce to,26,APPROACH TO EQUILIBRIUM IN A FEED FLUME: SUMMARY,The equations,must be solved with the sediment transport relation and boundary conditions:,and a suitable initial condition, e.g. where SI is an initial bed slope and aI is an initial value for flume-averaged bed elevation, , a = aI, d = SIL0.5 (x/L), or in dimensionless terms.,
