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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 preise physial mehanism and a quantifiation of this henomenon is still unknown In fat the initiation of meandering as found to be fundamentally deendent on the formation of steady or slowly migrating alternate bars in a river [12–14]. esite several mehanisms of the river meandering dynamis reorted in earlier studies [15–18], a preise phenomenologial framewor of the initiation of meandering is still unnown. In subseuent setions, a linear staility analysis of a straight river is first erformed. Then the henomenology of the initiation of meandering of a straight river is desried inally, onluding remarks are made 2. Linear stability analysis 2.1. Mathematical analysis z z H y D x 2B Referene level Fig. 1. Sketh of a straight river and the oordinate sstem. igure 1 deits the shemati of a straight river ith a onstant width of B, flowing over an erodile granular ed, onfined to two arallel riverbanks With referene to the Cartesian oordinate system x y), here x is the streamwise distane and y is the lateral distane from the river entreline, the deth-averaged veloity omponents in x y) are (U V), resetively. Further, D and H denote the flow deth and the free surfae elevation from a nown referene level respetively, and z reresents the vertial oordinate from the referene level The bed shear stress omponents and the volumetri sediment flux components in x y are denoted y (Tx Ty) and (Qx Qy), resetively. The flo ontinuity euation is [19] (DUˆˆ ) ( DV ˆˆ )0 (1) xyˆˆ The flo momentum euations are [19] ˆ ˆˆTˆ ˆˆU UH x UV A 0 (2) xˆ yx ˆˆ Dˆ ˆ ˆ ˆˆT ˆˆV VH y UV A 0 (3) xˆ yy ˆˆ Dˆ The sediment flu continuit equation is [20] 2 E3S Web of Conferences 40, 05004 (2018) https://doi.org/10.1051/e3sconf/20184005004 River Flow 2018 river a preise physial mehanism and a quantifiation of this henomenon is still 2 x y unknown In fat the initiation of meandering as found to be fundamentally deendent on ()FHˆˆ D 0 (4) tˆ xyˆˆ the formation of steady or slowly migrating alternate bars in a river [12–14]. esite several mehanisms of the river meandering dynamis reorted in earlier studies [15–18], a ˆ ˆ ˆ ˆ 2 2 preise phenomenologial framewor of the initiation of meandering is still unnown. wher ( xˆ yˆ ) (x y)/B D D/Dm (U V ) (U V)/Um, H H /(F Dm) F [ 1/2 In subseuent setions, a linear staility analysis of a straight river is first erformed. Um/(gDm) is flow Froud number h unperturbed tate g is gravitation Then the henomenology of the initiation of meandering of a straight river is desried ˆ ˆ 2 acceleration, A [ B/Dm is e river aspec atio, ( T x T y ) (Tx Ty)/(f U m ) f is h mass inally, onluding remarks are made 3 1/2 density f luid, tˆ tUmQr/B t is time, Qr gd ) /[(1 – 0)UmDm] [ g – f)/f] is submerged elativ density f edimen grains, g is mass ensity f edimen 2. Linear stability analysis grains, d is median of edimen rains, 0 is porosity f edimen (x y) 3 1/2 (Qx Qy)/(gd ) and subscript “m” refers th uantities linked wit he unperturbed 2.1. Mathematical analysis uniform ow. 2 Th bed hear tress mponents ( Tˆ Tˆ ) ar given ( Tˆ Tˆ ) (f/8) Uˆ Vˆ )(Uˆ z x y x y 2 1/2 Vˆ ) , her f is Darcy–Weisbach riction factor. The f is pressed [21] z H y 1.1 ˆ 1 1 U k s 2.51 D 0.86 ln (5) x 2B 1/2 1/2 ˆ 1/2 f 8 u * 14.8D Rf ˆ wher u* is h shear velocity, k s ks/Dm ks is th bed oughness eigh R ( UD/) the Reynolds numbe n is coefficien of inemati viscosity f fluid. he ks is Referene level considered ks 2.5d [22]. The ( ) ar given ( ) (co , i ), her is ang created by he Fig. 1. Sketh of a straight river and the oordinate sstem. x y x y grain trajectory with h streamwise direction. The i expressed as in–1{sin – 1/2 2 igure 1 deits the shemati of a straight river ith a onstant width of B, flowing over [r/(A) ](/ yˆ )(F Hˆ – Dˆ )} [23], her is th ang between shear tress vector an erodile granular ed, onfined to two arallel riverbanks With referene to the and h streamwise direction is e hields umber and r is efficien ( 0.5) [24] Cartesian oordinate system x y), here x is the streamwise distane and y is the lateral –1 2 2 1/2 The i given a =sin [ Vˆ /( Uˆ Vˆ ) ]. Considering e edload anspor the distane from the river entreline, the deth-averaged veloity omponents in x y) are (U prevailing mod of sedimen ransport, the is pressed – )1.5 [25], her V), resetively. Further, D and H denote the flow deth and the free surfae elevation from c c is h threshold Shields umber. The is determined rom e mpir formu a nown referene level respetively, and z reresents the vertial oordinate from the c proposed y ao al. [26]. referene level The bed shear stress omponents and the volumetri sediment flux Uˆ Vˆ ˆ components in x y are denoted y (Tx Ty) and (Qx Qy), resetively. To perform th tability analysis, th rimitiv ariab are pressed ( H , 1/2 The flo ontinuity euation is [19] ˆ ˆ ˆ ˆ D ) 1 H m 1) [(U V H D )exp kxˆ – t )] [6], her i O(1), = –1) ˆ k is e ondimension wavenumber d –i is e mplex uantity. Therefore, (DUˆˆ ) ( DV ˆˆ )0 (1) ˆˆ ˆ ˆ ˆ ˆ xy expansions f T x and produce T x f/8){1 [(U c1 D c2)exp kxˆ – t )] ˆ ˆ ˆ –1 The flo momentum euations are [19] and m{1 [(U c3 D c4)exp kx – t )]}, ere c1 = – ( m/fm)( f/ )] , c2 ˆ –1 (1/fm)(f/ D – (m/fm)(f/)] , c3 m/m)(/)c1 and c4 m/m)(/)c2 ˆ ˆˆTˆ ˆ ˆˆU UH x (1/m / D ). Considering h ects, h governing uations [Eqs. 1)–(4 O( UV A 0 (2) xˆ yx ˆˆ Dˆ can expanded d ter om algebra, w find: 22 i {F [( a a ) aa ] [ aaa aM]} ˆ ˆˆ ˆ 11 14 22 31 13 22 31 11 V VH T y UVˆˆ A 0 (3) m xˆ yy ˆˆ Dˆ The sediment flu continuit equation is [20] 3 E3S Web of Conferences 40, 05004 (2018) https://doi.org/10.1051/e3sconf/20184005004 River Flow 2018 22aa14 41 1 2aa13 41 (aaa13 22 31 aM 11 ) aM44 a45 ()a11 a 14 aa 22 31 a 43 M a11 aa22 11 (6) ˆ ˆ ˆ whr a11 c1 i k Afm/8 a13 = a31 = a34 i k a14 (c2 – 1) a22 i k a ˆ 2 1/2 i k c3 and a = –rF /(A) .
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