Phenomenology of Meandering of a Straight River

E3S Web of Conferences 40, 05004 (2018) https://doi.org/10.1051/e3sconf/20184005004 River Flow 2018

Phenomenology of meandering of a straight river

Sk Zeeshan Ali* and Subhasish Dey Department of Civil Engineering, Indian Institute of Technology Kharagpur, Kharagpur 721302, India

Abstract. In this study, at first, we analyse the linear stability of a straight river. We find that the natural perturbation modes maintain an equilibrium state by confining themselves to a threshold wavenumber band. The effects of river aspect ratio, Shields number and relative roughness number on the wavenumber band are studied. Then, we present a phenomenological concept to probe the initiation of meandering of a straight river, which is governed by the counter-rotational motion of neighbouring large-scale eddies in succession to create the processes of alternating erosion and deposition of sediment grains of the riverbed. This concept is deemed to have adequately explained by a mathematical framework stemming from the turbulence phenomenology to obtain a quantitative insight. It is revealed that at the initiation of meandering of a river, the longitudinal riverbed slope obeys a universal scaling law with the river width, flow discharge and sediment grain size forming the riverbed. This universal scaling law is validated by the experimental data obtained from the natural and model rivers.

1. Introduction Meandering rivers are ubiquitous in earth-like planetary surfaces. Since past, numerous attempts were made to probe the cause of meandering of a river. Several concepts including earth’s revolution [1], riverbed instability [2–6], helicoidal flow [7], excess flow energy [8] and macro-turbulent eddies [9] were proposed so far. Despite these concepts, the exact mechanism of the meandering of a straight river remains a vexing phenomenon. Lane [10] proposed that for nearly straight to meandering rivers, the longitudinal riverbed slope S can be expressed as a function of flow discharge Q in the form of S = aQb, where a and b are empirical constants. Then, Henderson [11] introduced the effects of sediment grain size to refine Lane’s [10] empirical formula. However, such relationships exclusively stand on empirical foundation and thus, they are dimensionally inhomogeneous inviting uncertainties. Interestingly, Yalin [9] introduced the concept of macro-turbulent eddies to explain the meandering of a river. He argued that the longitudinal length scale of macro-turbulent eddies is approximately equal to the longitudinal length of alternate bars in a straight river. He considered the length scale of the macro-turbulent eddies as six times the river width. Although, this concept provided a qualitative idea of the meandering of a

* Corresponding author: [email protected]

© The Authors, published by EDP Sciences. This is an open access article distributed under the terms of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). E3S Web of Conferences 40, 05004 (2018) https://doi.org/10.1051/e3sconf/20184005004 River Flow 2018

river a reise hsial mehanism and a uantifiation of this henomenon is still unnon n fat the initiation of meandering as found to e fundamentall deendent on the formation of stead or slol migrating alternate ars in a river – esite several mehanisms of the river meandering dnamis reorted in earlier studies – a reise henomenologial frameor of the initiation of meandering is still unnon n suseuent setions a linear stailit analsis of a straight river is first erformed hen the henomenolog of the initiation of meandering of a straight river is desried inall onluding remars are made

2. Linear stability analysis

2.1. Mathematical analysis

z H y

D x B

eferene level

Fig. 1. eth of a straight river and the oordinate sstem

igure deits the shemati of a straight river ith a onstant idth of B floing over an erodile granular ed onfined to to arallel riverans ith referene to the artesian oordinate sstem x y here x is the streamise distane and y is the lateral distane from the river entreline the dethaveraged veloit omonents in x y are U V resetivel urther D and H denote the flo deth and the free surfae elevation from a non referene level resetivel and z reresents the vertial oordinate from the referene level he ed shear stress omonents and the volumetri sediment flu omonents in x y are denoted Tx Ty and Qx Qy resetivel he flo ontinuit euation is

 DUˆˆ  DV ˆˆ xyˆˆ

he flo momentum euations are

Uˆ  UH ˆˆTˆ UVˆˆ  Ax  xˆ  yx ˆˆ Dˆ

ˆ ˆˆ ˆ V  VH T y UVˆˆ  A  xˆ  yy ˆˆ Dˆ

he sediment flu ontinuit euation is

2 E3S Web of Conferences 40, 05004 (2018) https://doi.org/10.1051/e3sconf/20184005004 River Flow 2018 river a reise hsial mehanism and a uantifiation of this henomenon is still    x y unnon n fat the initiation of meandering as found to e fundamentall deendent on FHˆˆ D   tˆ xyˆˆ the formation of stead or slol migrating alternate ars in a river – esite several mehanisms of the river meandering dnamis reorted in earlier studies – a ˆ ˆ ˆ ˆ reise henomenologial frameor of the initiation of meandering is still unnon xˆ yˆ x yB D DDm U V U VUm H H F Dm F n suseuent setions a linear stailit analsis of a straight river is first erformed UmgDm g hen the henomenolog of the initiation of meandering of a straight river is desried ˆ ˆ A BDm T x T y Tx Tyf U m f inall onluding remars are made tˆ tUmQrB t Qr gd – UmDm  g – ff g 2. Linear stability analysis d  x y Qx Qygd subscript “m” 2.1. Mathematical analysis Tˆ Tˆ Tˆ Tˆ f Uˆ Vˆ Uˆ z x y x y Vˆ f – f z H y   ˆ U k s D   x B ˆ f  u D Rf 

ˆ u* k s ksDm ks R UD

 ks eferene level ks d         Fig. 1. eth of a straight river and the oordinate sstem x y x y   –  – igure deits the shemati of a straight river ith a onstant idth of B floing over rA  yˆ F Hˆ – Dˆ  an erodile granular ed onfined to to arallel riverans ith referene to the  r artesian oordinate sstem x y here x is the streamise distane and y is the lateral –   Vˆ Uˆ Vˆ distane from the river entreline the dethaveraged veloit omonents in x y are U    –   V resetivel urther D and H denote the flo deth and the free surfae elevation from c c  a non referene level resetivel and z reresents the vertial oordinate from the c referene level he ed shear stress omonents and the volumetri sediment flu Uˆ Vˆ ˆ omonents in x y are denoted Tx Ty and Qx Qy resetivel H he flo ontinuit euation is ˆ ˆ ˆ ˆ D H m U V H D kxˆ –  t  O –

 kˆ – DUˆˆ  DV ˆˆ ˆˆ ˆ ˆ ˆ ˆ xy T x  T x f U c D c kxˆ –  t    ˆ ˆ  ˆ    – he flo momentum euations are m U c D c kx – t c – mfm f c ˆ – fmf D – mfmf c mmc c mmc ˆ ˆˆ ˆ ˆ U  UH T x   D – O UVˆˆ  A  m xˆ  yx ˆˆ Dˆ

 F a  a aa  aaa aM ˆ ˆˆ ˆ V  VH T y UVˆˆ  A   m xˆ  yy ˆˆ Dˆ

he sediment flu ontinuit euation is

3 E3S Web of Conferences 40, 05004 (2018) https://doi.org/10.1051/e3sconf/20184005004 River Flow 2018

   aa aa aaa  aM  aM  a a  a aa  a  M   a  aa

ˆ ˆ ˆ r a c i k  Afm a a a i k a c – a  i k a ˆ i k c a –rF A

2.2. Results and discussion

s t rpic iustrti t ss sitis ui ris r t s f – – g rspcti iti t sr s ubr R* is csir t crctri t ruic ru ri t tt t citi –i iicts pti rt rt i t citi –i scribs pti c rt iur –c prsts t –i –i s ucti isi ubr kˆ cput r r irt us rir spct rti A is ubr  rti russ ubr  i t utis –i –i r stui r irt spct rtis A b pi t rti russ ubr  t is ubr  s cstt r i kˆ it kˆ t –i s c it crs i A ctrst r i kˆ it kˆ t –i rs it A ritis –i it kˆ sust tt r i kˆ it kˆ t ruc citti is st ipt A t tr r i kˆ it kˆ t ruc citti uic cs it crs i A i b t utis –i –i r i r irt is ubrs  b pi t spct rti A t rti russ ubr  s cstt r spciic  t strit rir rsps rpi t t tr prturbtis bcus t ritis –i it kˆ r pti rt r c cui i t iciit kˆ  r i kˆ it kˆ t –i cs it  t ctrr r i kˆ it kˆ t –i icrss it  r kˆ pprcs uit t –i r rr  rpi rs it i ruc ritis –i it kˆ s tt r i kˆ it kˆ t ruc citti is prctic irit  r r i kˆ it kˆ t ruc citti icrss it  i c t utis –i –i r prst r irt rti russ ubrs  b pi t spct rti A t is ubr  s cstt r i kˆ it kˆ t –i iiiss it  ritis –i it kˆ iict tt r i kˆ it kˆ t ruc citti is ipt  urtrr r i kˆ it kˆ t ruc citti s rucs it crs  t ctrr r i kˆ it kˆ t ruc citti icrss it  trsti i i –c r s i ubrs kˆ tr ists ubr b kˆ r ic bt –i –i r uit s t us s – –i –i uc ubr b c b tut s trs

4 E3S Web of Conferences 40, 05004 (2018) https://doi.org/10.1051/e3sconf/20184005004 River Flow 2018

   aa aa aaa  aM  aM  a a  a aa  a  M  a aa    ˆ ˆ ˆ r a c i k  Afm a a a i k a c – a  i k a ˆ i k c a –rF A

2.2. Results and discussion

urtrr r i kˆ it kˆ t ruc citti s rucs ˆ it crs  t ctrr r i kˆ it kˆ t ruc Fig. 2. – – k citti icrss it  A   ˆ trsti i i –c r s i ubrs k tr ists ubr b kˆ r ic bt –i –i r uit s t us s – –i –i uc ubr b c b tut s trs

5 E3S Web of Conferences 40, 05004 (2018) https://doi.org/10.1051/e3sconf/20184005004 River Flow 2018

3. Conceptual framework

3.1. Theory

E b

Um B

E a c

Lmw

Fig. 3.

B Dm S Um Q d the secondary currents of Prandtl’s second kind E E E Uw gDm Lmw

6 E3S Web of Conferences 40, 05004 (2018) https://doi.org/10.1051/e3sconf/20184005004 River Flow 2018

ae to traerse the distance eteen the alternate ars is the sae as that for the ean flo to coer a straiht ath eteen the alternate ars undaentally the ean flo elocity has a close link ith the turulent enery sectru lyin the turulence – henoenoloy the ean elocity can e eressed as Um agDmS dDm here a is a coefficient he tie taken y the flo to traerse a straiht ath eteen the 3. Conceptual framework alternate ars is LmwUm n the other hand the tie taken y the ae to traerse this 2 distance is L mw B Uw he Uw is eressed as Uw agDm here a is a 3.1. Theory – coefficient he flo dischare is ien as Q BDmUm aBDmgDmS dDm herefore euatin the aoe tie eriods results in

 S BQ dg

E b 3.2. Results and discussion Um B

E a c

Lmw

Fig. 3.

B Dm S Um Q d

Fig. 4. ariation of streaise riered sloe S ith flo dischare Q the secondary currents of Prandtl’s second kind E uation roides a scalin la of the initiation of eanderin of a straiht rier t is aarent that for constant alues of the raitational acceleration rier idth and sedient rain sie the threshold slope is proportional to the “–2/9”th oer of the flo dischare iure dislays the functional reresentation of SQ and the eeriental data collected fro seeral natural and odel riers featured y aroiately straiht to hihly E E eanders he ean sloe of the lotted data and that oeys a “–2/9” scaling law is shon y the straiht line ote that the oerall scatter of the eeriental data is oin to the ariaility of rier idths and rain sies Preiously ane roosed the folloin forula S –Q– Therefore, the “–2/9” (= – scalin la deried fro this study satisfactorily corresonds to that roosed y ane 4. Conclusion linear staility analysis of a straiht rier shos that there eists a threshold aenuer and for hich the natural erturation odes aintain an euiliriu state oer a road rane of rier asect ratios hields nuers and relatie rouhness nuers he initiation U gD w m of eanderin of a rier is oerned y the turulent flo ith a counterrotational otion L mw of the neihourin larescale eddies in succession to enerate the rocesses of alternate erosion and deosition of sedient rains his concet is elained y a atheatical odel stein fro the ersectie of the turulence henoenoloy to roide a

7 E3S Web of Conferences 40, 05004 (2018) https://doi.org/10.1051/e3sconf/20184005004 River Flow 2018

antitatie insight into the eanering of a straight rier t is fon that the longitinal riere slope, rier with, flow ischarge an eian grain sie are connecte a nie scaling law which has a satisfactor agreeent with the eperiental ata

References instein, ie atrwissenschaften 14, – (92 2 riein, nite tates aterwas perient tation, icsrg (9 allaner, li ech 36, – (99 ngeln, ogaar, li ech 57, 29–2 (9 rese, li ech 84, 9–2 (9 lonea, einara, li ech 157, 9– (9 Tanner, eophs es 65, 99–99 (9 T ang, rol 13, 2–2 (9 9 alin, Mechanics of sediment transport (ergaon, 9 ane, nite tates r ngineer iision, issori ier iision, orps of ngineers, aha, erasa (9 enerson, Trans oc i ng 128, – (9 2 lesen, onications on hralics, 9–, elft niersit of Technolog (9 lesen, h thesis, elft niersit of Technolog (9 r, rosato, Tassi, ater esor 93, 2– (2 olei, einara, li ech 438, –2 (2 rosato, osselan, ater esor es 45, 2 (29 sahi, hii, elson, arer, eoph es 118, 22–2229 (2 einara, Tino, ater esorces onograph, , es ea an arer (2 9 e aintVenant, Comptes Rendus de l’Academic des Sciences, vol 73, Paris, pp –, 2–2 ( 2 ner, itngserichte er aeie er issenschaften 134, –2 (92 2 oleroo, hite, roc oc on 161, – (9 22 ngeln, ansen, cta oltech can 35, – (9 2 ea, ral i 108, 9–99 (92 2 Talon, trisa, an ierlo, ral es 33, 9– (99 2 e, Fluvial hydrodynamics: hydrodynamic and sediment transport phenomena (pringer, 2 2 ao, ener, eng, ral ng 132, 9–99 (2 2 lara, ein, ral ng 137, – (2 2 erner, Trans eophs nion 32, 9–92 (9 29 ioia, oarelli, hs e ett 88, (22 eopol, olan, rofessional paper 22, nite tates eological re, ashington, (9