On sediment transport under dam-break flow

Transcription

1 J. Fluid Mech. (2002), vol. 473, pp c 2002 Cambridge Univerity Pre DOI: /S Printed in the United Kingdom 265 On ediment tranport under dam-break flow By DAVID PRITCHARD AND ANDREW J. HOGG Centre for Environmental and Geophyical Flow, School of Mathematic, Univerity of Britol, Univerity Walk, Britol BS8 1TW, UK (Received 5 Augut 2002 and in revied form 23 Augut 2002) We preent eact olution for upended ediment tranport under one-dimenional dam-break flow, both over a dry bed and into a mall depth of tail water. We eplicitly calculate the upended ediment concentration, including eroion and depoition, and invetigate the effect of varying the eroional and depoitional model employed. Thee olution offer inight into ediment tranport procee under flood or heet flow event, and we alo dicu their application a tet-bed olution for the validation of numerical model. 1. Introduction Dam-break flow, which repreent the udden releae of fluid from behind a vertical barrier, provide the implet available model for a number of important phenomena, uch a break-out flood, heet flow event and the formative tage of lahar or debri flow. An idealized configuration for dam-break flow i hown in figure 1: the barrier at ˆ = 0 divide fluid of different depth ĥ0 and ĥ2, until at time ˆt = 0 the barrier i intantaneouly removed, and fluid flood into the hallower region. The claical decription of thee flow obtained by Ritter (1892) and by Stoker (1957) are among the implet non-trivial olution of the hallow-water equation, and may alo be interpreted a a limiting form of contant-flu gravity current (Gratton & Vigo 1994). They are not of merely mathematical interet, a they have been confirmed eperimentally to provide a reaonably good decription of the flow once tranient aociated with the initial releae have died down (Stanby, Chegini & Barne 1998). Recent tudie by Capart & Young (1998) and by Fraccarollo & Capart (2002) have conidered dam-break flow a agent of ediment tranport. Both tudie invetigated flow over an erodible bed coniting of looe coare ediment which could be mobilized en mae a bedload: they found that the flow form an eroional bore, which advance digging a uniform cour pit beneath it and puhing the eroded ediment before it a a teep debri nout. The current paper may be regarded a complementary to thee tudie, and deal with the rather different edimentary regime in which the bed conit of fine or coheive material which, once a threhold hear tre i eceeded, i entrained into the water column and then tranported in upenion. Our interet i in characterizing the quantity of ediment which uch flow may mobilize and tranport. A econdary motivation for thi work i computational. The hallow-water equation are among the principal tool employed in numerical tudie of ediment tranport procee and morphodynamic, and important morphodynamical flow frequently involve feature uch a internal bore and moving horeline boundarie which preent a ignificant challenge to the accuracy and tability of numerical cheme. A variety of analytical tet-bed olution, of which dam-break flow are

2 266 D. Pritchard and A. J. Hogg ĥ ĥ 0 (a) Still water ( reervoir ) û(ˆ,tˆ) ˆ = Uˆtˆ ĥ Epanion wave ĥ 0 ˆ = ˆh (tˆ) Dry bed (b) ˆ Epanion wave Still water ( reervoir ) ĥ 1 ĥ 2 ˆ = Uˆtˆ ˆ = ˆf Uniform flow û = û 1 ˆ = Vˆtˆ Still water ( tail water ) ˆ Figure 1. Schematic of the (a) Ritter and (b) Stoker olution for dam-break flow on a horizontal bed. perhap the implet and mot popular, are ued to validate the hydrodynamic component of uch cheme. However, there i till a need for eact olution which may be ued to verify the accuracy of the correponding prediction of ediment tranport. In 2, we decribe the hallow-water model which we employ to decribe ediment tranport. In 3 we contruct and dicu eact olution, and in 4 we comment on the application of our reult. 2. The hallow-water equation The one-dimenional hallow-water equation for flow over a horizontal bed may be derived from the Euler equation under the aumption that the characteritic horizontal length cale of the flow i much greater than the characteritic vertical length cale. In their non-dimenional form, the equation for the conervation of fluid ma and momentum have the form h t + u (uh) =0, t + u u + h =0, (2.1) where h repreent the depth of water and u i a depth-averaged horizontal velocity. The variable here have been non-dimenionalized with repect to a length cale ĥ0 correponding to the initial height of the dam and a velocity cale Û =(ĝĥ0) 1/2, where ĝ i the acceleration due to gravity; they are given by h = ĥ/ĥ0, = ˆ/ĥ0, u = û/û, t = ˆtÛ/ĥ0. (2.2) Throughout, caret denote dimenional quantitie; dimenionle quantitie are unadorned. In dimenional form, the equation for the conervation of ediment tranported a a dilute well-mied upenion i given by ĉ ˆt + û ĉ ˆ = 1 ĥ ( ˆq e ˆq d ), (2.3)

3 On ediment tranport under dam-break flow 267 where ĉ repreent the ma concentration of upended ediment and ˆq e and ˆq d are, repectively, the ma eroion and ma depoition flue which repreent echange between the water column and the bed. In thi formulation, we have aumed that the diffuive tranport of particle in the horizontal direction i negligible compared to advective tranport. We will alo aume that the ma echange flue may be epreed in term of the vertically averaged quantity ĉ rather than of the concentration of upended ediment near the bed. Thi i formally valid when vertical miing in the water column i ufficiently intene to maintain a vertically uniform concentration profile, in other word in the limit of infinite friction velocity. For finite friction velocitie, the concentration profile i both vertically non-uniform and time-dependent, with a characteritic repone time which ha been invetigated for teady flow by Stanby & Awang (1998). However, the approimation which we make here i well-etablihed a the tandard method by which empirical depoition formulae may be incorporated into a depth-averaged model: like the depth-averaged model itelf, it provide an informative leading-order decription of the ytem. In general, the quantitie ˆq e and ˆq d are function of both the fluid velocity û and the upended ediment concentration ĉ. In thi paper, we conider the following form for the eroion rate ˆq e : (û2 n ˆm ˆq e (û) = e 1) for û û û 2 e e (2.4) 0 for û < û e. Thi model i commonly ued to decribe the eroion of ediment from a coheive bed (Teion et al. 1993) or from a bed of fine coheionle material (Dyer & Soulby 1998), where ome critical hear tre, correponding to a velocity û e, mut be eceeded in order to break up the bed and entrain particle. The quantity ˆm e i a dimenional ma eroion rate, while n i a dimenionle eponent, typically in the range n =1 to n =3.5. We note that the approach decribed in thi paper may readily be applied to other form of the epreion ˆq e (û). A number of epreion have been propoed to decribe the depoition of coheive ediment at low flow rate (Teion et al. 1993). However, a we will demontrate, we need only conider depoition when û>û e, and in thi regime there are two poible model: either ˆq d = ŵ ĉ, where ŵ repreent the ettling velocity of the larget floc which can urvive the near-bed turbulence; or turbulence i too trong for flocculated matter to ettle at all, and ˆq d = 0. It i often potulated that ˆq d =0 when velocitie are high enough to erode ediment, but Sanford & Halka (1993) have demontrated that thi aumption may not be appropriate to decribe edimentary procee outide the laboratory. For thi reaon, and in order to make our reult applicable to coheionle particle with contant ettling velocity ŵ, we will conider both form of ˆq d. We can now define a reference concentration Ĉ = ˆm e /ŵ (in the cae ˆq d =0,wemay et Ĉ = ˆm e /Û intead), and non-dimenionalize equation (2.3) by defining c = ĉ/ĉ, q e = ˆq e / ˆm e and q d = ˆq d / ˆm e. It i alo helpful to rewrite equation (2.3) in Lagrangian term. We conider the concentration of ediment c L (t) in a fluid parcel with poition L (t), depth h L (t) and velocity d L /dt = u L (t). In thi Lagrangian framework, equation (2.3) i then equivalent to dc L dt = E h L [q e (u L ) q d (c L )], (2.5)

4 268 D. Pritchard and A. J. Hogg where E = ŵ /Û i a non-dimenional bed echange rate repreenting the ratio of hydrodynamic to edimentary time cale. The degenerate initial condition for dambreak flow allow u to eliminate E from the problem by the further recaling E 1 and t E 1 t; in ubequent ection, thi i preumed to have been done. 3. Solution for ediment tranport Over a horizontal bed, the dam-break olution have the form of imple wave (Stoker 1957). The mot elementary cae i Ritter olution for flow over a dry horizontal bed, h 2 = 0. In thi cae (figure 1a), the flow field ha the form of an epanion wave, analagou to a rarefaction wave in ga dynamic. When the region >0 initially contain till fluid of depth h 2 > 0, the imple wave olution involve a rightward-propagating bore a well a an epanion wave (ee figure 1b) Ritter olution for flow over a dry bed Hydrodynamic olution The epanion-wave olution due to Ritter (1892) ha the form h(, t) = 1 ( 2 ) 2 2 (, u(, t) = 1+ ) (3.1) 9 t 3 t in the region 1 /t 2. We may readily decribe thi flow in Lagrangian term. We conider a fluid parcel at = L (t): it velocity i given by d L /dt = 2(1 + 3 L/t), and thu L (t) =2t + Bt 2/3 for ome contant B which label the parcel. Thi lead immediately to u L (t) = d L = dt Bt 1/3, h L (t) =h( L (t),t)= 1 9 B2 t 2/3. (3.2) When conidering ediment tranport, it i natural to relate the labelling quantity B to t e, the time at which the velocity of the fluid element attain the critical velocity for eroion. We find that B(t e )= 3(u 2 e 2)te 1/3, where u e = û e /Û i the dimenionle threhold velocity for eroion. We note that we require u e < 2 in order for eroion to occur, and thu that B<0 for all fluid parcel, o each parcel accelerate aymptotically toward a maimum velocity u = 2. Hence with the initial condition c =0 everywhere, non-zero concentration can only occur in region where u>u e : thi jutifie our previou tatement that we need only conider q d in thi velocity range Contructing olution for the concentration field Under the flow defined by equation (3.2), equation (2.5) become dc L dt = 9t2/3 B 2 [q e (u L (t)) ɛc L (t)] for t>t e, (3.3) where ɛ =0forq d = 0 and ɛ =1forq d = c. The initial condition i given by c L (t e )=0. Thi equation may be integrated to give t [ 9z 2/3 q e (u L (z)) c L (t) =A(t) dz, where A(t) = ep 27ɛ ] t e A(z) B 2 5B 2 t5/3. (3.4) The integral cannot in general be epreed in term of elementary function, but i eaily evaluated uing tandard package uch a maple.

5 On ediment tranport under dam-break flow (a) 30 (b) 2 20 c c (c) 0.16 (d) Figure 2. Solution for ediment tranport with q d = c and u e = 1. Succeive plot at t =0.2 to 2.0. Concentration field c for (a) n = 1 and (b) n = 3; ediment load = ch for (c) n = 1 and (d) n = 3. Solid line how eact olution, dahed line how olution with c = c eq. Finally, by ubtituting for B in term of t e, and then for t e (, t), given by t e (, t) = 1 [ ( )] 3 2 2t, (3.5) t u e we obtain the Eulerian concentration field c(, t). Before dicuing the olution, it i helpful to define the quantity c eq = q e (u(, t)). In the cae q d = c, thi quantity repreent an intantaneou equilibrium between eroion and depoition in a fluid parcel, and it i traightforward to how that in any given parcel c L (t) c eq (u L (t)) a t. (In fact, the convergence i rather rapid, epecially in the hallower part of the flow, and thi provide a ueful mean of evaluating c(, t) cloe to the front of the flow = h (t), where plotting program may have difficulty handling the eponential in equation (3.4).) Since c eq (t) i clearly bounded above by the horeline concentration c h = q e (2), we obtain the reult that when depoition occur, concentration mut be bounded by c = c h, even in very hallow water cloe to the front. Figure 2 illutrate thi point Eample of the olution Figure 2 illutrate the olution with continuou depoition for two value of n. The principal feature of the plot of concentration c i the maimum at the front, where flow i fatet and ha been eroional for longet. However, the ediment load = ch i highet in omewhat deeper water, and in fact tend trongly to zero near the front. A n i increaed, the horeline concentration increae dramatically and the plot of c become more concave, becaue of the enhanced eroion at higher

6 270 D. Pritchard and A. J. Hogg 4 (a) (b) Figure 3. Solution for ediment tranport with q d = 0 and u e = 1. Succeive plot at t =0.2 to 2.0, howing ediment load = ch for (a) n = 1 and (b) n =3. velocitie. However, the overall ediment load increae much more lowly, becaue of the reduced eroion at low velocitie. Becaue of the concavity of c(, t), the maimum of occur rather cloer to the front for higher n, and ince velocitie are higher there, the maimum ediment flu q = cuh, which decribe the tranport potential of the flow, i further enhanced for higher n. Figure 3 illutrate the olution when depoition i neglected entirely. In thi cae, the plot of c are dominated by the unbounded maimum which develop at the front, and o we how only the plot of total ediment load = ch. Thi quantity increae monotonically toward it value at the front = h. By conidering the ediment tranport equation evaluated at = h, we obtain the reult [ (ch) t + (chu) ] h = d h dt u + h h = d h dt + 2 3t h = q e (2), (3.6) and ince q e (2) i a contant, and there i initially no ediment in upenion, we obtain h = 3 5 tq e(2). (3.7) The unboundedne of c at the front i unphyical, ince when concentration become ufficiently high, the preence of upended matter ignificantly affect the rheology of the flow, and the hydrodynamic mut be modified accordingly (Iveron 1997). Thi may, of coure, occur alo for the olution in which depoition i included, in which cae it will depend on the value of Ĉ a well a on the criterion for the upenion to be regarded a dilute. However, a comparion between figure 3 and figure 2 ugget that including depoition mut ignificantly alter any etimate which we are able to make of the time taken for uch a flow to bulk up in thi manner Net cour A final point which thee olution illutrate i the importance of depoition in controlling the net cour beneath the flow. We have carried out thee calculation auming that the total quantity of ediment eroded i ufficiently mall that the bed geometry i not ignificantly altered. If the depth of cour, which i proportional to the amount of material eroded per unit length, can increae unboundedly, then thi aumption will become invalid after ufficient time ha paed. We can quite readily etimate the total amount of ediment eroded. When depoition occur, the concentration field i bounded above by c eq (u(, t)), and o the total

7 On ediment tranport under dam-break flow 271 quantity of ediment in upenion, S(t), i bounded by S(t) 2t 0 c eq (u(, t))h(, t)d = t 2 0 c eq (u(ξ))h(ξ)dξ, where ξ = t. (3.8) Since the length of the region from which material ha been eroded alo increae linearly in time, we may conclude that when depoition i allowed for, the depth of cour beneath the flow may not increae unboundedly, and o the olution obtained here may remain formally conitent. When depoition i neglected, we have no upper bound for the concentration field. However, we may write 2t ds dt = q e (u(, t)) dt = t q e (u(ξ)) dξ, (3.9) 0 0 o S(t) t 2, for any form of q e. Hence the maimum depth of cour mut increae at leat linearly with time, and o when depoition i neglected, olution for eroive dam-break flow which neglect the effect of cour on the bed mut become inconitent at large time Stoker olution and the effect of tail water Stoker (1957) etenion to Ritter dam-break olution decribe the propagation of a dam-break wave into a region of till tail water of caled depth h 2 < 1. The olution i illutrated in figure 1(b): it conit of a bore propagating into the tail water with velocity V, behind which i a region of uniform flow with u = u 1 and h = h 1, which i itelf linked by Ritter epanion wave olution to the quiecent reervoir behind. Given the downtream depth h 2 = ĥ2/ĥ0, the quantitie V, h 1 and u 1 may eaily be deduced from the conervation of ma and momentum acro the bore, and from the condition that velocity and urface elevation are continuou at = f, where the epanion wave meet the region of uniform flow. We may epre V, h 1 and h 2 a function of u 1, and then invert the olution numerically to obtain u 1, and hence V and h 1, a function of h 2. (For further detail of the hydrodynamic olution, ee Stoker 1957.) A feature of the olution i that it varie trongly at h 2 = 0, with u /4 h 1/4 2, V 2 2 7/4 h 1/4 2, h 1 2 3/2 h 1/2 2 for h 2 1, (3.10) o even a mall tail water depth ha a dramatic effect on u 1 (ee figure 5a below). Thi i of interet becaue of the poible preence of tail water in phyical contet uch a a breakout flood entering a river valley, and alo becaue a popular mean of handling the horeline condition for beach and dam-break problem i to impoe a mall notional water depth everywhere in the numerical domain, o that the horeline become effectively a bore. It i therefore important to invetigate the effect thi may have on ediment tranport. We may epre the flow field in Lagrangian form a we did for the Ritter olution, and contruct olution for ediment tranport in eactly the ame manner a before (ee equation (3.3) and (3.4)). The algebraic detail of thi contruction are rather lengthy and are not informative, and we therefore omit them here. However, certain feature of the reulting ediment tranport are worth calling attention to. Figure 4 illutrate a typical olution, with the rather mall tail water depth h 2 =0.01. The main feature of the plot i the region of near-contant ediment concentration which coincide with the region of uniform flow. A fluid parcel may 2

8 272 D. Pritchard and A. J. Hogg 0.4 (a) (b) c Figure 4. Solution for ediment tranport under Stoker dam-break olution with q d = c, h 2 =0.01, u e = 1 and n = 1. Succeive plot at t =0.2 to 2.0: (a) concentration field c; (b) ediment load = ch (compare with the Ritter olution, figure 2a and 2c). enter thi region either from the epanion wave behind or through the bore from the tail water ahead; having entered, it remain in the region of uniform flow, and the concentration adjut aymptotically to it equilibrium value in thi region, c L (t) =c eq (u 1 ) Ke t/h 1, (3.11) where K i a contant of integration which depend on the initial poition of the fluid parcel. Figure 4(b) illutrate the effect the preence of tail water ha on the total ediment load, and thu on the capacity of the flow to erode and tranport ediment. Comparing figure 2(c) and 4(b), we oberve that the difference between the ediment load for h 2 = 0 and h 2 =0.01 i quite ubtantial: the maimum ediment load i reduced from =0.087 to =0.064, and the ediment-laden region etend only to = Vt rather than to =2t. Figure 5 provide the eplanation for thi: it illutrate how the very rapid decreae in u 1 a h 2 increae from zero drive a trong decreae in c eq (u 1 ), and hence in both the ediment load and the ediment flu in the uniform region. The effect i amplified for higher n by the tronger dependence of c eq on u. Thee reult indicate that coniderable caution i required when numerically modelling ediment tranport under flow with a moving horeline, ince if even a mall bore develop it can lead to unepectedly large change in the ediment tranport rate. A well a indicating the poibility of uch error, the olution preented here offer a mean to identify them by more thoroughly teting numerical cheme. 4. Concluion We have preented eact olution which decribe upended ediment tranport under dam-break flow. For dam-break flow over a dry bed, the concentration field increae monotonically toward the front, while the total ediment load ha a maimum omewhat behind it. Converely, for flow into quiecent tail water, a region of near-uniform ediment concentration rapidly develop behind the bore, correponding to a local balance between the eroion and depoition of ediment. Furthermore, the maimum concentration and maimum ediment load in the flow are both ubtantially reduced compared to the dry bed olution, for even a relatively mall tail water depth. Thi ugget that, while the dry bed olution i the implet prototype for many phyically relevant flow, the inight provided by the Stoker olution i important in interpreting the dry bed reult.

9 On ediment tranport under dam-break flow (a) 1.0 (b) u h c (c) 0.3 (d) q h 2 h 2 Figure 5. The effect of a mall tail water depth h 2 on hydrodynamic and upended ediment behind the bore. (a) Maimum velocity u 1 (olid line) and depth h 1 (dahed line; caled by a factor of 10 for clarity) in the uniform region. (b), (c) and (d) Scaled equilibrium concentration c = c eq (u 1 )/c eq (2); equilibrium ediment load = h 1 c eq (u 1 ); and equilibrium ediment flue q = h 1 u 1 c eq (u 1 ), for n =1 (olid), n = 2 (dahed), n = 3 (dotted). A well a indicating the enitivity of ediment tranport to the preence of tail water, our olution illutrate how the quantity of ediment mobilized depend on the model employed for the eroion and depoition of upended ediment, and the implication of thi for the bulking up of the upenion into a debri flow. A econdary role for thee olution i a tet-bed cae, which are eaily reproduced and may be ued to validate numerical cheme for predicting ediment tranport within hallow-water model. Throughout thi tudy, we have employed the implet available decription of ediment ditribution in the water column. One etenion which might yield intereting reult i to conider in more detail the vertical ditribution of upended ediment (Stanby & Awang 1998); another i to invetigate how dam-break flow might fluidize a coheive bed and tranport material a a fluid mud layer, analagou to the non-coheive bedload conidered by Capart & Young (1998) and by Fraccarollo & Capart (2002). Both etenion lie beyond the cope of thi paper. However, they are natural development of the eiting work in thi area, and could provide further inight into problem uch a the early tage of lahar formation. The method ued to contruct the eact olution preented here i eaily applied to other problem in which the flow field may be epreed in Lagrangian term. An immediate etenion of the olution decribed here i to the dam-break flow on a lope which ha been ued a a model for the wah that reult from a breaking wave on a plane beach (Shen & Meyer 1963), and work on thi problem i currently underway. A imilar approach ha recently been applied to decribe ediment tranport under tidal flow on mudflat (Pritchard 2001).

10 274 D. Pritchard and A. J. Hogg D. P. and A. J. H. acknowledge the financial aitance of EPSRC and HR Wallingford Ltd, and would like to thank two anonymou referee for their helpful uggetion. REFERENCES Capart, H. & Young, D. L Formation of a jump by the dam-break wave over a granular bed. J. Fluid Mech. 372, Dyer, K. R. & Soulby, R. L Sand tranport on the continental helf. Annu. Rev. Fluid Mech. 20, Fraccarollo, L. & Capart, H Riemann wave decription of eroional dam-break flow. J. Fluid Mech. 461, Gratton, J. & Vigo, C Self-imilar gravity current with variable inflow reviited: plane current. J. Fluid Mech. 258, Iveron, R. M The phyic of debri flow. Rev. Geophy. 35, Pritchard, D Some problem in two-phae flow. PhD thei, Univerity of Britol. Ritter, A Die Fortpflanzung der Waerwellen. Z. Verein. Deutch. Ing. 36, Sanford, L. P. & Halka, J. P Aeing the paradigm of mutually ecluive eroion and depoition of mud, with eample from upper Cheapeake Bay. Mar. Geol. 114, Shen, M. C. & Meyer, R. E Climb of a bore on a beach. Part 3. Run-up. J. Fluid Mech. 16, Stanby, P. K. & Awang, M. A. O Repone time analyi for upended ediment tranport. J. Hydr. Re. 36, Stanby, P. K., Chegini, A. & Barne, T. C. D The initial tage of dam-break flow. J. Fluid Mech. 374, Stoker, J. J Water Wave: The Mathematical Theory with Application. Intercience. Teion, C., Ockendon, M., Le Hir, P., Kranenburg, C. & Hamm, L Coheive ediment tranport procee. Coatal Engng 21,