Sediment transport mechanisms 1. Bed-load transport

Transcription

1 10 Sediment tranprt mechanim 1. Bed-lad tranprt 10.1 Intrductin When the bed hear tre exceed a critical value, ediment are tranprted in the frm f bed-lad and upended lad. Fr bed-lad tranprt, the baic mde f particle mtin are rlling mtin, liding mtin and altatin mtin (Fig. 10.1). In thi chapter, frmulatin t predict the bed lad tranprt rate are preented. Figure 10.2 hw a natural tream ubjected t igni cant bed-lad tranprt. Nte 1. Saltatin refer t the tranprt f ediment particle in a erie f irregular jump and bunce alng the bed (Fig. 10.1(a)). 2. In thi ectin, predictin f bed-lad tranprt are develped fr plane bed. Bed frm mtin and bed frm e ect n bed-lad tranprt are nt cnidered in thi ectin (ee Chapter 12). De nitin The ediment tranprt rate may be meaured by weight (unit: N/), by ma (unit: kg/ ) r by vlume (unit: m /). In practice the ediment tranprt rate i ften expreed per unit width and i meaured either by ma r by vlume. Thee are related by: _m ˆ 10:1 where _m i the ma ediment w rate per unit width, i the vlumetric ediment dicharge per unit width and i the peci c ma f ediment Empirical crrelatin f bed-lad tranprt rate Intrductin Bed lad tranprt ccur when the bed hear tre exceed a critical value ( c.in

2 196 Sediment tranprt mechanim 1. Bed-lad tranprt (a) Fig Bed-lad mtin. (a) Sketch f altatin mtin. (b) De nitin ketch f the bed-lad layer. (b) dimeninle term, the cnditin fr bed-lad mtin i: > c Bed-lad tranprt 10:2 where i the Shield parameter (i.e. ˆ = ÿ 1 gd ) and ( c i the critical Shield parameter fr initiatin f bed lad tranprt (Fig. 8.4) Empirical bed lad tranprt predictin Many reearcher attempted t predict the rate f bed lad tranprt. The rt ucceful develpment wa prped by P.F.D. du By in Althugh hi mdel f ediment tranprt wa incmplete, the prped relatinhip fr bedlad tranprt rate (Table 10.1) prved t be in gd agreement with a large amunt f experimental meaurement. Subequently, numeru reearcher prped empirical and emi-empirical crrelatin. Sme are lited in Table Graf (1971) and van Rijn (199) dicued their applicability. The mt ntriu crrelatin are the Meyer-Peter and Eintein frmulae.

3 10.2 Empirical crrelatin f bed-lad tranprt rate 197 (a) (b) Fig Bed-lad tranprt in natural tream. (a) Suth Krrumbyn Creek NSW, Autralia lking uptream (25 April 1997). Stream bed 2 km uptream f the Krrumbyn Creek weir, fully-ilted tday (Appendix A2.1). (b) Cedar Creek QLD, Autralia lking dwntream (9 December 1992) (curtey f Mr J. Hacker). Nte the care material left after the d.

4 198 Sediment tranprt mechanim 1. Bed-lad tranprt Table 10.1 Empirical and emi-empirical crrelatin f bed lad tranprt Reference Frmulatin Range Remark (1) (2) () (4) By (1879) ˆ ÿ c wa called the characteritic ediment ce cient. Schklitch (190) Shield (196) Eintein (1942) Meyer-Peter (1949,1951) Eintein (1950) Schklitch (1950) Nielen (1992) ˆ 0:54 Schklitch (1914) ÿ g Labratry experiment with unifrm grain f variu kind f and and prcelain. / d ÿ=4 Straub (195) 0:125 < d < 4 mm Baed upn labratry data. ˆ 0 in k q ÿ q c 0:05 < d < 7:02 mm Baed upn labratry q c ˆ 1: ÿ2 d in ÿ4= experiment. in ˆ 10 ÿ c 1:06 < < 4:25 g ÿ 1 d 1:56 < d < 2:47 mm ˆ 2:15 exp ÿ0:91 ÿ 1 gd _m 2= in ÿ 9:57 g ÿ 1 10=9 ˆ d ÿ 0:462 ÿ 1 g _m 2 2= d p <0.4 1:25 < < 4:25 0:15 < d < 28:6mm Labratry experiment. Weak ediment tranprt frmula fr and mixture. Nte: d d 5 t d 45. 1:25 < < 4:2 Labratry experiment. Unifrm grain ize ditributin. ˆ ÿ 1 gd Labratry 4 experiment. Particle =2 mixture. ÿ 0:188 ÿ 1 gd Nte: d d 50. q Deign chart p < 10 Labratry q ÿ 1 gd ÿ 1 gd ˆ f experiment. 1:25 < < 4:25 Fr and mixture. 0:15 < d < 28:6mm Nte: d d 5 t d 45. _m ˆ 2500 in =2 q ÿ q c Baed upn labratry q c ˆ 0:26 ÿ 1 5= d =2 40 in ÿ7=6 experiment and eld meaurement (Danube and Aare river). ˆ r 12 ÿ 0:05 ÿ 1 gd ÿ 1 gd 1:25 < < 4:22 0:69 < d < 28:7mm Re-analyi f labratry data. Nte: _m ˆ ma water w rate per unit width; _m ˆ ma ediment w rate per unit width; q ˆ vlumetric water dicharge; ˆ vlumetric ediment dicharge per unit width; c ˆ critical bed hear tre fr initiatin f bed lad. Nte P.F.D. du By ( ) wa a French hydraulic engineer. In 1879, he prped a bed lad tranprt mdel, auming that ediment particle mve in liding layer (By 1879).

5 10. Bed-lad calculatin 199 Fig. 10. Bed-lad tranprt rate: cmparin between Meyer-Peter frmula, Eintein calculatin and labratry data (Meyer-Peter et al. 194, Gilbert 1914, Chien 1954). Dicuin The crrelatin f Meyer-Peter (1949, 1951) ha been ppular in Eurpe. It i cnidered mt apprpriate fr wide channel (i.e. large width t depth rati) and care material. Eintein' (1942, 1950) frmulatin derived frm phyical mdel f grain altatin, and they have been widely ued in America. Bth the Meyer-Peter and Eintein crrelatin give cle reult (e.g. Graf 1971, p. 150), uually within the accuracy f the data (Fig. 10.). It mut be nted that empirical crrelatin huld nt be ued utide f their dmain f validity. Fr example, Engelund and Hanen (1972) indicated explicitly that Eintein' (1950) bed-lad tranprt frmula di er igni cantly frm experimental data fr large amunt f bed lad (i.e. = > 10). 10. Bed-lad calculatin Preentatin Bed-lad tranprt i clely aciated with inter-granular frce. It take place in a thin regin f uid cle t the bed (metime called the bed-lad layer r altatin layer) (Fig. 10.1, 10.4). Viual bervatin ugget that the bed-lad particle mve within a regin f le than 10 t 20 particle-diameter height. During the bed-lad mtin, the mving grain are ubjected t hydrdynamic frce, gravity frce and inter-granular frce. Cnverely the (ubmerged) weight f the bed lad i tranferred a a nrmal tre t the (immbile) bed grain. The

6 200 Sediment tranprt mechanim 1. Bed-lad tranprt Fig Sketch f bed-lad mtin at equilibrium. nrmal tre e exerted by the bed lad n the immbile bed particle i called the e ective tre and it i prprtinal t: e / ÿ 1 g c C 10: where i the bed-lad layer thickne, C i the vlumetric cncentratin f ediment in the bed-lad layer and i the lngitudinal bed lpe. The nrmal tre increae the frictinal trength f the ediment bed and the bundary hear tre applied t the tp layer f the immbile grain becme: ˆ c e tan 10:4 where ( c i the critical bed hear tre fr initiatin f bed lad and i the angle f repe. Nte 1. The cncept f e ective tre and aciated bed hear tre (a preented abve) derive frm the wrk f Bagnld (1956, 1966). 2. Fr ediment particle, the angle f repe range uually frm 268 t 428 and hence 0:5 < tan < 0:9. Fr and, it i cmmn t che: tan 0: Bed lad tranprt rate The bed lad tranprt rate per unit width may be de ned a: ˆ C V 10:5 where V i the average ediment velcity in the bed-lad layer (Fig. 10.1(b)). Phyically the tranprt rate i related t the characteritic f the bed-lad layer: it mean ediment cncentratin C, it thickne which i equivalent t the average

7 10. Bed-lad calculatin 201 altatin height meaured nrmal t the bed (Fig and 10.4) and the average peed V f ediment mving alng the plane bed. Nte 1. A teady ediment tranprt in the bed-lad layer i metime called a (nupenin) heet- w. 2. Nte that the vlumetric ediment cncentratin ha a maximum value. Fr runded grain, the maximum ediment cncentratin i Dicuin Several reearcher have prped frmulae t etimate the characteritic f the bedlad layer (Table 10.2). Figure 10.5 preent a cmparin between tw frmulae. Overall the reult are nt very cnitent. In practice there i till great uncertainty n the predictin f bed lad tranprt. Nte that the crrelatin f van Rijn (1984a) are prbably mre accurate t etimate the altatin prpertie (i.e. C, /d and V /V ) (within their range f validity). Table 10.2 Bed-lad tranprt rate calculatin Reference Bed-lad layer characteritic Remark (1) (2) () Fernandez-Luque and van Beek (1976)! V ˆ 9:2 1ÿ 0:7 c V Nielen (1992) C ˆ 0:65 d ˆ 2:5 ÿ c Van Rijn (1984a,199) V V ˆ 4:8 C ˆ 0:117 2 d ÿ 1 g! 1= ÿ 1 c! 1= 0:7 r ÿ 1 c! 1= 2 ÿ 8 c! 1= ÿ 1 g ˆ 0: d d 2 V ÿ 1 g ˆ 9 2:6 lg V 10 d C ˆ 0:117 d 2 ÿ 1 g ˆ 0: d d 2 V V ˆ 7 ÿ 1 ÿ 1 g c! 1= 0:7 r ÿ 1 c Nte: V ˆ hear velcity; c ˆ critical Shield parameter fr initiatin f bed lad. Labratry data 1:4 4:58 0:9 d :mm 0:08 d 0:12 m Simpli ed mdel. Fr < 2 and d ˆ d 50. c Baed n labratry data 0:2 d 2mm d > 0:1m Fr < 0:9 d ˆ d 50. Baed n labratry data 0:2 d 2mm d > 0:1m Fr < 0:9

8 202 Sediment tranprt mechanim 1. Bed-lad tranprt Fig Dimeninle bed-lad tranprt rate /(d V ) a a functin f the dimeninle Shield parameter / c (Table 10.2). Nte 1. The calculatin detailed in Table 10.2 apply t at channel (i.e. in <0:001 t 0.01) and in abence f bed frm (i.e. plane bed nly). 2. Fr teep channel everal authr hwed a trng increae f bed-lad tranprt rate. It i believed that the lngitudinal bed lpe a ect the tranprt rate becaue the threhld cnditin (i.e. initiatin f bed lad) are a ected by the bed lpe, the ediment mtin i changed with teep bed lpe and the velcity ditributin near the bed i mdi ed. Dicuin The predictin f bed-lad tranprt rate i nt an accurate predictin. One reearcher (van Rijn 1984a) tated explicitly that: the verall inaccuracy [...] mayntbele than a factr Applicatin Applicatin N. 1 The bed-lad tranprt rate mut be etimated fr the Danube river (Central Eurpe) at a particular cr-ectin. The knwn hydraulic data are: w rate f abut 50 m /, w depth f 4.27 m, bed lpe being abut The channel bed i a ediment mixture with a median grain ize f m and the channel width i abut 4 m. Predict the ediment-lad rate uing the Meyer-Peter crrelatin, the Eintein frmula, and equatin (10.5) uing bth Nielen and van Rijn ce cient.

9 Firt calculatin Auming a wide channel, the mean hear tre and hear velcity equal repectively: ˆ gd in ˆ 998:2 9:81 4:27 0:0011 ˆ 46:0Pa p V ˆ gd in ˆ 0:215 m/ The Shield parameter equal: ˆ 46:0 ˆ ÿ 1 gd 998:2 1:65 9:81 0:012 ˆ 0:27 auming ˆ 2:65 (quartz particle). And the particle Reynld number i: Re ˆ Vd ˆ 0:215 0:012 ˆ :007 10ÿ6 Fr thee w cnditin, Re, the Shield diagram predict ediment mtin: ˆ 0:27 > c 0:05 Nte that V =w i mall and the w cnditin are near the initiatin f upenin. In the rt apprximatin, the upended ediment tranprt will be neglected. Apprach N. 1: Meyer-Peter crrelatin Fr the hydraulic w cnditin, the dimeninle parameter ued fr the Meyer- Peter frmula i: ÿ 1 gd 998:2 1:65 9:81 0:012 ˆ ˆ 4: Applicatin f the Meyer-Peter frmula (fr a ediment mixture) lead t: 10.4 Applicatin 20 Hence: ˆ 0:005 m 2 /. ˆ 0:66 Apprach N. 2: Eintein functin Fr the hydraulic w cnditin, the dimeninle parameter ued fr the Eintein frmula i: ÿ 1 gd 5 In the abence f infrmatin n the grain ize ditributin, it will be aumed: d 5 d 50. It yield: ÿ 1 gd 5 4:215 Fr a ediment mixture, the Eintein frmula give: Hence: ˆ 0:0045 m 2 /. ˆ 0:85

10 204 Sediment tranprt mechanim 1. Bed-lad tranprt Apprach N. : bed-lad calculatin (equatin (10.5)) The bed lad tranprt rate per unit width equal: ˆ C V Uing Nielen' (1992) impli ed mdel, it yield: 10:5 ˆ 0:65 2:5 ÿ c d 4:8V ˆ 0:65 2:5 0:27 ÿ 0:05 0:012 4:8 0:215 ˆ 0:008 m 2 = With the crrelatin f van Rijn (1984a), the altatin prpertie are: C ˆ 0:00145 d ˆ 1:6 V ˆ 7 V and the ediment tranprt rate i: ˆ 0:0008 m 2 /. Summary Meyer-Peter Eintein Eq. (10.5) (Nielen) Eq. (10.5) (van Rijn) Q (m /) (?) (m 2 /) (?) Ma ediment rate (kg/) (?) Fur frmulae were applied t predict the ediment tranprt rate by bed-lad. Three frmulae give reanably cle reult. Let u review the variu frmulae. Graf (1971) cmmented that the Meyer-Peter frmula `huld be ued carefully at [...] high ma w rate', emphaizing that mt experiment with large w rate ued by Meyer-Peter et al. (194) were perfrmed with light ediment particle (i.e. lignite breeze, ˆ 1:25). Graf tated that ne advantage f the Meyer-Peter frmula i that `it ha been teted with large grain'. The Eintein frmula ha been etablihed with mre varied experimental data than the Meyer-Peter frmula. The preent applicatin i within the range f validity f the data (i.e. = ˆ 0:85 10). Equatin (10.5) give reanably gd verall reult uing Nielen' (1992) impli ed mdel. In the preent applicatin, the grain ize (0.012 m) i large cmpared with the larget grain ize ued by van Rijn (1984a) t validate hi frmulae (i.e. d 0:002 m). Hence it i undertandable that equatin (10.5) with van Rijn' frmulae i inaccurate (and nt applicable). Fr the preent applicatin, it might be recmmended t cnider the Meyer-Peter frmula, which wa develped and teted in Eurpe. Nte All bed-lad frmulae wuld predict the maximum bed-lad tranprt rate fr a tream in equilibrium. Thi tranprt capacity may r may nt be equal t the

11 10.4 Applicatin 205 actual bed-lad if the channel i ubjected t degradatin r aggradatin (ee Chapter 12) Applicatin N. 2 A wide tream ha a depth f 0.6 m and the bed lpe i The bed cnit f a mixture f heavy particle ˆ 2980 kg/m ) with a median particle ize d 50 ˆ 950 mm. Cmpute the bed-lad tranprt rate uing the frmulae f Meyer-Peter and Eintein, and equatin (10.5) fr unifrm equilibrium w cnditin. Firt calculatin Auming a wide channel, the mean hear tre and hear velcity equal repectively: ˆ gd in ˆ 998:2 9:80 0:6 0:0008 ˆ 4:70 Pa p V ˆ gd in ˆ 0:069 m/ The Shield parameter equal: ˆ 4:70 ˆ ÿ 1 gd 998:2 1:98 9:80 0:00095 ˆ 0:255 The particle Reynld number i: Re ˆ Vd 0:069 0:00095 ˆ 1: ÿ6 ˆ 65:1 Fr thee w cnditin, Re, the Shield diagram predict ediment mtin: ˆ 0:255 > c 0:05 Nte that V =w i le than 0.7. In the rt apprximatin, the upended ediment tranprt i negligible. Apprach N. 1: Meyer-Peter crrelatin Fr the hydraulic w cnditin, the dimeninle parameter ued fr the Meyer- Peter frmula i: ÿ 1 gd 998:2 1:98 9:80 0:00095 ˆ ˆ :91 4:7 Applicatin f the Meyer-Peter frmula (fr a ediment mixture) lead t: Hence: ˆ 9:82 10 ÿ5 m 2 /. ˆ 0:76 Apprach N. 2: Eintein functin Fr the hydraulic w cnditin, the Eintein frmula i baed n the d 5 grain ize. In the abence f infrmatin n the grain ize ditributin, it will be aumed: d 5 d 50.

12 206 Sediment tranprt mechanim 1. Bed-lad tranprt Fr a ediment mixture, the Eintein frmula give: Hence: ˆ 1:29 10 ÿ4 m 2 /. 1 Apprach N. : bed-lad calculatin (equatin (10.5)) The bed-lad tranprt rate per unit width equal: ˆ C V Uing Nielen' (1992) impli ed mdel, it yield: 10:5 ˆ 0:65 2:5 ÿ c d 4:8V ˆ 0:65 2:5 0:255 ÿ 0:05 0: :8 0:069 ˆ 1:05 10 ÿ4 m 2 = With the crrelatin f van Rijn (1984a), the altatin prpertie are: C ˆ 0:019 d ˆ 5:848 V ˆ 7 V and the ediment tranprt rate i: ˆ 0:5 10 ÿ4 m 2 /. Summary Meyer-Peter Eintein Eq. (10.5) (Nielen) Eq. (10.5) (van Rijn) (m 2 /) 0:98 10 ÿ5 1:2 10 ÿ4 1:05 10 ÿ4 0:5 10 ÿ4 Ma ediment rate (kg//m) All the frmulae give cnitent reult (within the accuracy f the calculatin!). Fr mall-ize particle, (i.e. d 50 < 2 mm), the frmulae f van Rijn are recmmended becaue they were validated with ver 500 labratry and eld data. Nte, hwever, that `the verall inaccuracy f the predicted [bed-lad] tranprt rate may nt be le than a factr 2' (van Rijn 1984a, p. 145) Applicatin N. The Nrth Frk Tutle river w n the nrth-wet lpe f Munt St. Helen (USA), which wa devatated in May 1980 by a vlcanic eruptin. Since 1980 the river ha carried a large vlume f ediment. Meaurement were perfrmed n 28 March 1989 at the H tadt Creek bridge. At that lcatin the river i 18 m wide. Hydraulic meaurement indicated that the w depth wa 0.8 m, the depth-averaged velcity wa.06 m/ and the bed lpe wa in ˆ 0:0077. The channel bed i a ediment mixture with a median grain ize f 15 mm and d 84 ˆ 55 mm.

13 Predict the ediment-lad rate uing the Meyer-Peter crrelatin, the Eintein frmula, and equatin (10.5) uing bth Nielen and van Rijn ce cient. Firt calculatin Auming a wide channel (d ˆ 0:8 m 18 m), the mean hear tre and hear velcity equal repectively: ˆ gd in ˆ 998:2 9:81 0:8 0:0077 ˆ 62:6Pa p V ˆ gd in ˆ 0:25 m/ The Shield parameter equal: ˆ ˆ 0:258 ÿ 1 gd auming ˆ 2:65 (quartz particle) and uing d ˆ d 50. And the particle Reynld number i: Re ˆ Vd ˆ 725 Fr thee w cnditin, Re, the Shield diagram predict ediment mtin: ˆ 0:258 > c 0:05 Nte that V =w i mall (V =w 0:5) and the w cnditin are near the initiatin f upenin. In the rt apprximatin, the upended ediment lad will be neglected. Apprach N. 1: Meyer-Peter crrelatin Fr the hydraulic w cnditin, the dimeninle parameter ued fr the Meyer- Peter frmula i: ÿ 1 gd ˆ :87 uing d ˆ d 50. Applicatin f the Meyer-Peter frmula (fr a ediment mixture) wuld lead t: Hence: ˆ 0:0057 m 2 /. ˆ 0:78 Apprach N. 2: Eintein functin Fr the hydraulic w cnditin, the dimeninle parameter ued fr the Eintein frmula i: ÿ 1 gd 5 In the abence f infrmatin n the grain ize ditributin, we aume: d 5 d 50. Fr a ediment mixture, the Eintein frmula give: Applicatin 207

14 208 Sediment tranprt mechanim 1. Bed-lad tranprt But nte that the w cnditin are utide f the range f validity f the frmula. That i, the Eintein frmula huld nt be ued. Apprach N. : bed-lad calculatin (equatin (10.5)) The bed-lad tranprt rate per unit width equal: ˆ C V 10:5 Uing Nielen' (1992) impli ed mdel, it yield: ˆ 0:65 2:5 ÿ c d 4:8V ˆ 0:0061 m 2 = With the crrelatin f van Rijn (1984a), the altatin prpertie are: C ˆ 0:00129 d ˆ 8:97 V ˆ 7 V And the ediment tranprt rate i: ˆ 0:001 m 2 /. Summary Meyer-Peter Eintein Eq. (10.5) (Nielen) Eq. (10.5) (van Rijn) Data Q (m /) 0.10 N/A (m 2 /) N/A Ma ediment rate (kg/) 274 N/A Pitlick (1992) decribed in-depth the eld tudy perfrmed at the H tadt Creek bridge n the Nrth Frk Tutle river. On 28 March 1989, the main bervatin were: d ˆ 0:8 m; V ˆ :06 m=; in ˆ 0:0077; f ˆ 0:054; ˆ 6 N=m 2 ; C ˆ 0:01; _m ˆ 11:4kg=m= The channel bed wa frmed in dune (up t 0.16 m high). Dicuin Firt let u nte that tw methd f calculatin are incrrect: the Eintein frmula and equatin (10.5) uing van Rijn' crrelatin. The w cnditin and ediment characteritic are utide f the range f applicability f Eintein' frmula a = > 10. In additin the grain ize (0.015 m) i larger than the larget grain ize ued by van Rijn (1984a) t validate hi frmulae (i.e. d 0:002 m). Secndly it i wrth nting that the Meyer-Peter frmula and equatin (10.5) uing Nielen' crrelatin give reanable predictin. Thi lat applicatin i an intereting cae: it i well dcumented. The river w i characterized by heavy bed-lad tranprt and the bed-frm are a igni cant feature f the channel bed.

15 10.5 Exercie Exercie Numerical lutin t me f thee exercie are available frm the Web at Cnidering a 20 m wide channel, the bed lpe i and the berved w depth i.2 m. The channel bed i andy (d ˆ 0:008 m). Calculate: (a) mean velcity, (b) mean bundary hear tre, (c) hear velcity, (d) Shield parameter and (e) ccurrence f bed-lad mtin. If bed-lad mtin ccur, calculate: (f ) bed-lad layer ediment cncentratin, (g) bed-lad layer thickne, (h) average ediment velcity in bed-lad layer and (i) bed-lad tranprt rate. (Aume that the equivalent rughne height f the channel bed equal the median grain ize. Ue the Nielen impli ed mdel.) Cnidering a wide channel, the dicharge i 20 m 2 /, the berved w depth i 4.47 m and the bed lpe i The channel bed cnit f a and mixture (d 50 ˆ 0:011 m). Calculate the bed-lad tranprt rate uing: (a) Meyer-Peter crrelatin, (b) Eintein functin, (c) Nielen impli ed mdel and (d) van Rijn crrelatin. (Aume that the equivalent rughne height f the channel bed equal the median grain ize.) A 25 m wide channel ha a bed lpe f The bed cnit f a mixture f light particle ( ˆ 250 kg/m ) with a median particle ize d 50 ˆ 1:15 mm. The w rate i 7.9 m /. Calculate the bed-lad tranprt rate at unifrm equilibrium w cnditin uing the frmulae f Meyer-Peter, Eintein, Nielen and van Rijn. (Aume that the equivalent rughne height f the channel bed equal the median grain ize.)