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Multiple time scales of alluvial rivers carrying suspended sediment and their implications for mathematical modeling
Advances in Water Resources 30 (2007) 715–729 www.elsevier.com/locate/advwatres
Multiple time scales of alluvial rivers carrying suspended sediment and their implications for mathematical modeling Zhixian Cao *, Yitian Li, Zhiyuan Yue State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan, Hubei 430072, China Received 13 January 2006; received in revised form 27 May 2006; accepted 6 June 2006 Available online 17 August 2006
Abstract Flow, sediment transport and bed deformation in alluvial rivers normally exhibit multiple time scales. Enhanced knowledge of the time scales can facilitate better approaches to the understanding of the fluvial processes. Yet prior studies of the time scales are based upon the concept of sediment transport capacity at low concentrations, which however is not generally applicable. This paper presents new formulations of the time scales of fluvial flow, suspended sediment transport and bed deformation, under the framework of shallow water hydrodynamics, non-capacity sediment transport and the theory of characteristics for the hyperbolic governing equations. The time scale of bed deformation in relation to that of flow depth is demonstrated to delimit the applicability region of mathematical river models, and the time scale of suspended sediment transport relative to that of the pertinent flow information is analyzed to address if the concept of sediment transport capacity is applicable. For shallow flows with high sediment concentrations, bed deformation may considerably affect the flow and a fully coupled model is normally required. In contrast, for deep flows at low sediment concentrations, a decoupled model is mostly justified. This pilot study of the time scales delivers a new theoretical basis, on which the interaction between flow, suspended sediment transport and bed deformation can be potentially better characterized. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Fluvial flow; Sediment transport; Riverbed deformation; Time scales; Sediment transport capacity; Alluvial river; Mathematical river modeling
1. Introduction The interactive processes of flow, sediment transport and bed deformation in alluvial rivers constitute a system of physical problems of significant interest in the fields of fluvial hydraulics and morphodynamics. Generally, the bottom boundary of fluvial flow, the bed, undergoes deformation due to non-equilibrium sediment transport, which in turn affects the flow. When the rate of bed deformation is significantly smaller than that of flow changes by orders of magnitude, the feedback impacts of bed deformation on the flow is justifiably negligible, and traditional decoupled mathematical river models are applicable. Otherwise, decoupled models can totally collapse, and fully cou*
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pled models are required in order to properly resolve the strong interactions between flow, sediment and the bed [2,3]. Irrespective of the observation, however, there has been no generally valid criterion measuring how fast the bed deforms compared to flow changes. Concurrently, sediment transport is often assumed to adapt to local hydraulic conditions instantly, and accordingly sediment transport rate is set to be equal to its capacity value that is commonly computed using local flow conditions. Yet, this assumption remains to be justified and its validity can be challenged, because a definitive description of the time required for sediment transport to adapt to local flow conditions is still missing. At the same time, a number of studies have clearly demonstrated phase differences between flow and sediment transport rate. This argument is correct for not only suspended sediment transport, but also bed load transport e.g., [25,20,22,23,32,13,14,10,28,26,29,1,9,15]. Therefore,
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Nomenclature A matrix defined in Eq. (2) c volumetric sediment concentration ce sediment transport capacity d sediment particle diameter D, E sediment deposition and entrainment fluxes F net flux of sediment exchange Fr Froude number g gravitational acceleration h flow depth m exponent n Manning roughness p bed sediment porosity q unit-width discharge qs unit-width sediment transport rate R matrix defined in Eq. (2) R1, R2, R3, R4 components of R in Eq. (2) Rp wd/m particle Reynolds number Sb bed slope Sf friction slope t time
sediment transport capacity formulations derived for steady flows are unable to reflect unsteady transport processes. Apparently, it is essential to achieve an improved understanding of the rates of changes of flow, sediment transport and bed deformation. Alternatively, the respective time scales of flow, sediment transport and bed deformation need to be properly defined. By time scale, we mean the time that governs the proprietary rate of change of a specific quantity describing the flow, sediment transport and bed deformation. Basically, it measures how fast a physical quantity of fluvial processes changes with time. The greater the time scale, the slower the physical quantity varies in time. Prior studies on the time scales of fluvial processes are based on the concept of sediment transport capacity. De Vries [6–8] studied the morphological time scale that characterizes the speed at which morphological processes take place. The morphological response exhibits either a wave character, which dominates over short distances, or a diffusion character prevailing over long distances. Accordingly, the morphological time scale was defined at two distinct levels respectively describing the wave and diffusion characteristics, as summarized by van Vuren [30]. Based on the St. Venant equations along with the concept of sediment transport capacity, Lyn [18] identified the multiple time scales of the system of flow, sediment transport and bed deformation. He showed that previous models, which reduce the number of conservation equations to be solved simultaneously from three to two under the ‘‘quasi-steady state’’ or ‘‘fixed bed’’ assumption, are unable to satisfy exactly either a general boundary condition or an arbitrary initial condition. Also, in situations with highly variable discharge
Tb Te Th Ts Tu T0 u U u* w x z a k1,2,3,4 m q q0 qw, qs
time scale of bed deformation time required for sediment transport to adapt to capacity time scale of flow depth time scale of sediment transport time scale of flow velocity time scale of quantity u3/ghw flow velocity matrix defined in Eq. (2) bed shear velocity settling velocity of a single particle in tranquil water streamwise coordinate bed elevation coefficient celerities kinematic viscosity of water density of water–sediment mixture density of saturated bed; and densities of water and sediment, respectively.
and sediment inputs, the aforementioned assumption is not justified. Nevertheless, it is recognized that the applicability of previous studies on the time scales of fluvial processes is limited as sediment transport can be far from capacity status, and sediment concentration in volume may not be far smaller than unity. The most telling case is that hyperconcentrated sediment-laden flood flows in the middle Yellow River and its tributaries (China) can still erode a huge amount of sediment from the bed, and sediment concentration during the flood events may well exceed the values that could be predicted by any existing sediment transport capacity formulations. Naturally the need for an improved understanding of the time scales of fluvial processes is evident, which is generally valid for non-capacity sediment transport phenomenon. This paper presents new formulations of the time scales of fluvial flow, suspended sediment transport and bed deformation under the framework of shallow water hydrodynamics and non-capacity sediment transport. It is substantiated under the theory of characteristics for the hyperbolic equations of conservation laws that govern the fluvial processes. The time scales of suspended sediment transport and bed deformation respectively relative to those of the pertinent flow information are analyzed. On this basis, the applicability region of mathematical river models is delimited, and the approximate applicability of the concept of sediment transport capacity addressed. The present study focuses on alluvial rivers carrying suspended sediment, whilst bed load-dominated rivers are reserved for future investigations, which involve distinct mechanisms.
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2. Governing equations
3. Definition of the time scales of fluvial processes
Consider one-dimensional (1D) flow in a channel with rectangular cross-sections of constant width. The shallow water hydrodynamics model for the flow, sediment transport and morphological evolution is built upon the mass and momentum conservation equations for the water–sediment mixture and the mass conservation equations respectively for sediment (carried in the flow) and bed material. Assuming the shape (correction) factors arising from the depth-averaging manipulation to be equal to unity, the complete governing equations [2] can be rewritten as
There can be different definitions of the time scales of fluvial processes as described by Eq. (1), which constitutes an inhomogeneous hyperbolic system. In general, fluvial flow, sediment transport and bed deformation are spaceand time-dependent. Even if the water flow is steady, sediment transport and bed deformation can change rather sharply in space, such as the sediment bores found in an experimental flume due to sediment overloading from the upstream [21]. Therefore it is attractive to achieve a spatially and temporally unified description of the time scales of fluvial processes. Given the above observation, the theory of characteristics is herewith opted for in defining the time scales. This option is physically sensible as disturbances in 1D fluvial flows are known to propagate along the characteristics [5,16]. The characteristic and compatibility equations in relation to Eq. (1) are pffiffiffiffiffi 8 dx=dt ¼ k1;2 ¼ u gh > > > rffiffiffi > > > g dh a23 dc g dz < du pffiffiffiffiffi pffiffiffiffiffi þ ð3a; bÞ dt h dt gh dt u gh dt > > ffiffi ffi r > > g a23 g > > : ¼ R2 pffiffiffiffiffi R4 R1 pffiffiffiffiffi R3 þ h u gh gh dx=dt ¼ k3 ¼ u ð4a; bÞ dc=dt ¼ R3 dx=dt ¼ k4 ¼ 0 ð5a; bÞ dz=dt ¼ R4
oU oU þA ¼R ot ox 2 3 2 u h 6u7 6g 6 7 6 U ¼ 6 7; A ¼ 6 4c5 40
ð1Þ h u
0 a23
3 0 g7 7 7; 05
0 u z 0 0 0 0 3 2 3 2 R1 F =ð1 pÞ 6 R 7 6 gS q Fu=qhð1 pÞ 7 7 6 27 6 f 0 R¼6 7¼6 7 4 R3 5 4 ð1 p cÞF =hð1 pÞ 5 R4
ð2a; b; cÞ
F =ð1 pÞ
where t is the time, x is the streamwise coordinate, h is the flow depth, u is the depth-averaged streamwise velocity, z is the bed elevation, c is the flux-averaged volumetric sediment concentration, g is the gravitational acceleration, Sf is the friction slope, p is the bed sediment porosity, F = E D is the net flux of sediment exchange between the flow and the bed, E, D are the sediment entrainment and deposition fluxes across the bottom boundary of flow, q = qw(1 c) + qsc is the density of the water–sediment mixture, q0 = qwp + qs(1 p) is the density of the saturated bed, a23 = (qs qw)gh/2q, and qw, qs are the densities of water and sediment, respectively. It is appreciated that the water–sediment mixture may behave as a non-Newtonian fluid at high sediment concentrations, and there has been a plethora of constitutive relationships describing the behavior e.g., [24,19,4,31]. Yet, in shallow water hydrodynamic models, it is implicitly incorporated in the term of bed shear stress (represented by friction slope in Eq. (2)), whilst the governing Eq. (1) remains correct. In applications, it is a common practice to calibrate the parameter(s) involved in determining the friction slope (Manning roughness, and when appropriate parameters describing the non-Newtonian behavior such as the yield stress). It is also noted that shallow water hydrodynamic models are applicable when the perturbation parameter defined as the ratio of the water depth to a characteristic length scale associated with the horizontal direction is far smaller than unity. This hierarchy of models resolves for the flow and sediment structure in the horizontal plane, rather than that along the water depth.
By virtue of Eq. (5) and the definition of bed slope Sb = oz/ox, the rate of change of bed elevation along the k1,2-characteristics can be determined by dz oz oz ¼ þ k1;2 ¼ R4 k1;2 S b ð6Þ dt k1;2 ot ox Similarly, the rate of change of sediment concentration along the k1,2-characteristics is expressed by dc oc oc ð7Þ ¼ þ k1;2 dt k1;2 ot ox and from Eq. (4b) dc oc oc ¼ þ k 3 ¼ R3 dt k3 ot ox Substituting Eq. (8) into Eq. (7), one has pffiffiffiffiffi oc dc oc ¼ R3 þ ðk1;2 k3 Þ ¼ R3 gh dt ox ox
ð8Þ
ð9Þ
k1;2
Substitution of Eqs. (6) and (9) into Eq. (3b) leads to 8 pffiffiffiffiffi > < dx=dt ¼ k1;2 ¼ u gh rffiffiffi rffiffiffi du g dh g oc > ¼ gS þ R R1 a23 ¼ R0 : 2 b dt h dt h ox
ð10a;bÞ
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It is rational to assume that the two terms in the LHS of Eq. (10b) are in general of the same order of magnitude as R0 in the RHS. Accordingly the time scale Tu of flow velocity can be defined along the k1,2-characteristics as u T u ¼ ð11Þ R
Again assuming that the two terms in the LHS of Eq. (10b) are of the same order of magnitude as R0 in the RHS, one can estimate the order of magnitude of u3/ghw along the k1,2-characteristics as follows: d u3 R0 ð3 FrÞFr2 ð17Þ dt ghw k1;2 w
Likewise, the time scale Th of flow depth can be defined and be related to Tu by
Therefore, the time scale T0 of quantity u3/ghw along the k1,2-characteristics can be defined as u3 =ghw u Tu ¼ ð18Þ T0 ¼ ¼ 2 R ð3 FrÞ absð3 FrÞ R0 ð3 FrÞFr =w 0
0
T h ¼ T u =Fr
ð12Þ pffiffiffiffiffi where Fr ¼ u= gh ¼ Froude number. Indeed, one can directly define the time scale of sediment transport based on Eq. (4). However the time scale so defined is associated with the k3-characteristics, and thus is not consistent with those of the flow as of Eqs. (11) and (12). Nevertheless, this inconsistency can be readily obviated. From Eq. (9), the time scale Ts of sediment transport in line with the k1,2-characteristics can be defined as c pffiffiffiffiffi ð13Þ T s ¼ R3 ghoc=ox
It has long been appreciated that sediment transport adapts to local flow conditions and tends to approach a capacity status (C = Ce) at which sediment transport is in equilibrium. The time scale Ts is a measure of the rate of this adaptation process, and yet the time Te required to achieve capacity status remains poorly understood from a quantitative sense. Here in complement to the time scale Ts, we aim to further substantiate the understanding of the adaptation process by defining ce c pffiffiffiffiffi ð14Þ T e ¼ R3 ghoc=ox Eq. (6) describes the lumped change of bed elevation along the k1,2-characteristics, which is determined by not only bed deformation, but also bed slope. Of particular interest in the present study is the rate of bed deformation due to non-equilibrium sediment transport as represented by oz/ot = R4, which is valid along any characteristics, including the k1,2-characteristics. Therefore, with the flow depth h as the appropriate length scale, the time scale Tb of bed deformation is defined as h ð15Þ jR4 j The respective time scales of the primitive variables (i.e., u, h, c, z) are defined along the k1,2-characteristics above. Yet, it may be necessary to define the time scales of deduced quantities composed of combinations of primitive variables. Here we take the non-dimensional quantity u3/ghw as an example, where w is the settling velocity of a single sediment particle in tranquil water (that can be calculated using for instance Zhang’s formula, see [33]). It is easy to derive that d u3 Fr2 du dh 3h u ð16Þ ¼ dt ghw dt dt wh Tb ¼
4. Auxiliary relationships Prior to proceeding to analyze the time scales of fluvial processes defined above, auxiliary relationships need to be introduced to close the governing equations. The conventional empirical relation is used to determine the friction slope involving the Manning roughness n n2 u2 S f ¼ 4=3 ð19Þ h Sediment exchange between the flow and the bed involves two distinct mechanisms, i.e., the upward bed sediment entrainment and the downward deposition due to gravitational action. The traditional and extensively used approach to specifying bed sediment entrainment flux is based on the premise that entrainment occurs always at the same rate as it does under equilibrium conditions. As in equilibrium conditions, the entrainment flux is equal to the deposition flux, and bed sediment entrainment flux can be computed using equilibrium near-bed sediment concentration ace and effective settling velocity w(1 ace)m of Richardson and Zaki [27]. Accordingly one has m ð20Þ E ¼ awce ð1 ace Þ where a is the coefficient, m ¼ 4:45Rp0:1 is the exponent denoting the effects of hindered settling due to high sediment concentrations, Rp wd/m is the particle Reynolds number, d is the sediment diameter, and m is the kinematic viscosity of water. The well tested and widely used semiempirical formula of Zhang and Xie [33] is employed to calculate suspended sediment transport capacity, which, following the interesting logarithmic-matching treatment of Guo [11], reads: ce ¼
1 ½u3 =ghw1:5 20qs 1 þ ½u3 =45ghw1:15
ð21Þ
The following relationship for sediment deposition flux, based on the formulation of hindered sediment settling velocity due to Richardson and Zaki [27], is introduced: D ¼ awcð1 acÞ
m
ð22Þ
The next flux F of sediment exchange determined from Eqs. (19) and (21) is essentially not new [17], except that the impact of hindered sediment settling has been incorporated with the Richardon and Zaki [27] formulation, which is
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known to be significant for high sediment concentrations. It is appreciated that existing formulations of sediment entrainment and deposition fluxes remain empirical, and the roughness parameter is hard to establish for natural rivers, especially as it is influenced by sediment. Thus Eqs. (19)–(22) represent only one set of the viable relationships closing the governing equations. 5. Evaluation of the time scales and their implications In evaluating the time scales of fluvial processes defined above, the following parameters are specified as: m = 1E6 m2/s, p = 0.4, qw = 1E+3 kg/m3, and qs = 2.65E+3 kg/m3. Other parameters are varied in different cases as shown in Table 1. 5.1. Basic features of the time scales Here we evaluate the basic features of the time scales defined above, corresponding to Case 1. Fig. 1(a) shows the contour of the time scale Tu of flow velocity in the Table 1 List of parameters for evaluation of the time scales Case no.
d (mm)
h (m)
n (s/m1/3)
a
Sb
hoc/ox
Fig. showing Tb/Th
1 2 3 4 5 6 7
0.1 0.1 0.1 0.04 0.1 0.1 0.1
10.0 3.0 3.0 3.0 3.0 3.0 3.0
0.03 0.03 0.025 0.025 0.025 0.025 0.025
1.0 1.0 1.0 1.0 1.2 1.0 1.0
2E4 2E4 2E4 2E4 2E4 5E4 2E4
1E5 1E5 1E5 1E5 1E5 1E5 3E5
Fig. Fig. Fig. Fig. Fig. Fig. Fig.
2 5 6 7 8 9 10
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u*–c plane, where u* is the bed shear velocity. The existence of double value of Tu is evident, as associated with respectively the k1- and k2-characteristics, which is ascribable to the influence of sediment exchange flux (F) and concentration gradient (oc/ox). At low sediment concentrations, say, c < 0.006, the time scale of flow is rather weakly dependent of sediment concentration, and the influence of bed shear velocity on Tu prevails. Interestingly, the maximum value of the time scale Tu appears around u* 0.14 m/s, which in essence corresponds to approximately steady and uniform flow (see Eq. (11)). As sediment concentration increases, its impact is considerable. The contour of the time scale Th of flow depth, shown in Fig. 1(b) exhibits considerable difference from that of Tu (Fig. 1a) as can be anticipated from Eq. (12). The rate of bed deformation can be measured with Tb, and it is advisable to evaluate it against the time scale of an appropriate flow variable (flow velocity or depth). Apparently, the most viable choice seems to be Th of the flow depth. In Fig. 2, the ratio of Tb to Th is illustrated in a contour format in the u*–c plane. Fig. 2 actually shows the rate at which the riverbed deforms, due to non-equilibrium sediment transport, relative to the change of the flow depth. The greater the value of the relative time scale Tb/ Th of bed deformation, the slower the bed deforms in relation to the flow, and the less bed deformation affects the flow. Thus, traditional decoupled mathematical river models tend to be applicable. In contrast, when Tb/Th is not sufficiently large, the feedback influences of bed deformation on the flow deserve full consideration and fully coupled mathematical river models must be used. Just for convenience, we tentatively use a few critical values (1000, 100, 10) of Tb/Th to distinguish the regimes with
Fig. 1. Contour of the time scale (a) Tu of flow velocity and (b) Th of flow depth, respectively along the k1 (solid line) and k2 (dashed line) characteristics, for Case 1.
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Fig. 2. Contour of Tb/Th for Case 1: the relative time scale of bed deformation respectively along the k1 (solid line) and k2 (dashed line) characteristics.
Table 2 Regimes of bed deformation Tb/Th
Tb/Th > 1000
1000 > Tb/Th > 100
100 > Tb/Th > 10
10 > Tb/Th
Regime of bed deformation Applicable mathematical river model
NBD: Negligible bed deformation Fixed-bed or decoupled mobile-bed model
WBD: Weak bed deformation
IBD: Intermediate bed deformation
Decoupled mobile-bed model
Fully coupled mobile-bed model
SBD: Strong bed deformation Fully coupled mobile-bed model
different relative time scale of bed deformation, and suggest the applicability of mathematical river models (see Table 2). Nevertheless, these critical values of Tb/Th need to be substantiated in further studies. In Fig. 2, it is demonstrated that the variation of Tb/Th exhibits a rather sophisticated pattern against sediment concentration and bed shear velocity. Firstly, consider the NBD regime with Tb/Th > 1000, which is in essence immediately around the curve of sediment transport capacity due to Eq. (21), see the thick solid line in Fig. 4. In the NBD regime, sediment transport is in a quasi-equilibrium state, bed deformation is negligibly weak compared to the change of flow depth, and fixed-bed or decoupled mobile-bed mathematical river models justifiably apply. Secondly, to the right-hand and lower part of the NBD regime (Fig. 2), u* > 0.0.092 m/s and c < ce, bed sediment entrainment dominates over deposition. The existence of a WBD, an IBD and a SBD regime is evident. Within the IBD and SBD regimes, the rate of bed deformation is not negligible compared against that of the change of flow depth, and naturally fully coupled mathematical river models are needed to properly resolve the strong interaction between flow, sediment transport and bed deformation. It is noted that the SBD regime is roughly around steady
and uniform flow state with u* 0.14 m/s, and its occurrence is ascribed to the infinite time scale Th of flow depth due to Eqs. (11) and (12). Physically, if ever the flow is steady and uniform, bed deformation (in this case, bed degradation as sediment entrainment overweighs over deposition) can be very active regardless of the low sediment concentration, which necessitates a fully coupled mathematical description of the fluvial processes. Presumably for the WBD, a decoupled mathematical model suffices. Thirdly, to the left-hand and upper part of the NBD regime (Fig. 2), c > ce, sediment deposition dominates over entrainment from the bed. In additional to a WBD regime with 100 < Tb/Th < 1000, there is quite an extensive area that is either an IBD or a SBD regime, with sediment concentration typically being higher than 0.001. In the IBD and SBD, the rate of bed deformation (in this case, bed aggradation as sediment deposition exceeds entrainment) cannot be ignored in comparison with that of the change of flow depth, and a fully coupled mathematical description of the fluvial processes is warranted. Fig. 3 is a plot of the ratio of the time scale Ts of sediment transport to the time scale T0 of the non-dimensional quantity u3/ghw, in a contour format in the u*–c plane. Of particular interest is that the value of Ts/T0 could be much
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Fig. 3. Contour of Ts/T0 for Case 1: the ratio of the time scale Ts of sediment transport to the time scale T0 of the non-dimensional quantity u3/ghw, respectively along (a) k1- and (b) k2-characteristics.
smaller than unity (e.g., Ts/T0 < 0.1), indicating that under favorable conditions (low sediment concentration and around the steady and uniform flow state at u* 0.14 m/s) sediment transport could adapt to local flow scenarios much faster than flow changes, and hence the concept of sediment transport capacity is approximately applicable as previously presumed. However, the applicability of the
concept of sediment transport capacity is dependent upon certain conditions, and so far remains to be justified on a physically rigorous basis. The present proposition of the respective time scales of fluvial processes provides such a device ascertaining when and where the concept of sediment transport capacity could be used. To substantiate this statement, Fig. 4 shows the contour of the ratio of the time
Fig. 4. Contour of Te/T0 for Case 1: the ratio of the time Te required for sediment transport to adapt to capacity status to the time scale T0 of the nondimensional quantity u3/ghw, respectively along the k1 (solid line) and k2 (dashed line) characteristics. The thick solid line denotes sediment transport capacity due to Eq. (21).
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Te required to adapt to capacity status of sediment transport to the time scale T0 of the non-dimensional quantity u3/ghw, along with the sediment transport capacity formulation as of Eq. (21). It is seen in Fig. 4 that in an area around the curve (thin dotted line) of sediment transport capacity, the value of Te/T0 is smaller than 0.1 (and could be smaller than 0.01, though not shown) along both the k1- and k2-characteristics. Alternatively, within this area, sediment transport can adapt to local flow conditions almost instantly and the concept of sediment transport capacity approximately applies. Accordingly, the area in which Te/T0 < 0.1, or Te/T0 < 1.0 can tentatively be referred to as the approximate applicability region of the concept of sediment transport capacity, depending on the desirable error tolerance. Huang et al. [12] propose an index of bed deformation rate to delimit the applicability of sediment transport capacity formulations. Yet the bed deformation rate was not normalized with any flow information, and thus is only of limited use. 5.2. Sensitivity of the time scales to parameter variation In relation to distinct cases with varied parameters d, h, Sb, hoc/ox, n and a, the main features of the time scales are qualitatively similar to Figs. 1–4. Here, we focus on the quantitative behavior of the relative time scale Tb/Th of bed deformation in response to parameter variation (Figs. 5–10). In Figs. 2 and 5 (Cases 1 and 2 in Table 1) it is seen that the flow depth may considerably affect the relative time scale of bed deformation. As flow depth decreases, the SBD region above the NBD appears to be enlarged in the u*–c plane. While the IBD region above the NBD seems
to experience only minor variation, the IBD region below the NBD is largely diminished due to the smaller flow depth. The major influence of a decreased roughness parameter (Figs. 5 and 6, corresponding to Cases 2 and 3 in Table 1) is that the IBD region below the NBD is extended, because the entrainment ability of the flow is enhanced as for specified flow depth and bed shear velocity (u*), the flow velocity increases due to roughness reduction. By Figs. 6 and 7 (Cases 3 and 4 in Table 1), it is indicated that the regimes of the relative time scale of bed deformation is rather sensitive to sediment particle size. As sediment diameter reduces from 0.1 mm to 0.04 mm, the SBD and IBD regions above the NBD are greatly diminished. This is not surprising, as above the NBD sediment deposition overweighs over entrainment, finer sediment deposition is much slower than coarser sediment, and especially finer sediment deposition can be substantially hindered at high concentrations. In contrast, the IBD region below the NBD is appreciably expanded in c but narrowed down in u*, when sediment diameter decreases. This results from the concurrent influences of the varied entrainment and deposition fluxes, as determined by Eqs. (20)–(22). Figs. 6 and 8 (Cases 3 and 5 in Table 1) show the impact on the relative time scale of bed deformation of the empirical parameter a in the closure of sediment exchange flux. A larger value of a seems to mainly lead to a wider IBD region below the NBD, whilst its impact on other regimes of the relative time scale of bed deformation is only marginal. The bed slope can also considerably affect the relative time scale of bed deformation, as demonstrated in Figs. 6
Fig. 5. Contour of Tb/Th for Case 2: the relative time scale of bed deformation respectively along the k1 (solid line) and k2 (dashed line) characteristics.
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Fig. 6. Contour of Tb/Th for Case 3: the relative time scale of bed deformation respectively along the k1 (solid line) and k2 (dashed line) characteristics.
Fig. 7. Contour of Tb/Th for Case 4: the relative time scale of bed deformation respectively along the k1 (solid line) and k2 (dashed line) characteristics.
and 9 (Cases 3 and 6 in Table 1). With all the remaining parameters kept unchanged, an increase in bed slope can result in contraction of the SBD and IBD regions above the NBD of the relative time scale of bed deformation, whilst the SBD and IBD regions below the NBD are appreciably extended. When purely the magnitude of the gradient of sediment concentration varies (Figs. 6 and 10, corresponding to Cases
3 and 7 in Table 1), the relative time scale of bed deformation sees more differentiation along the k1- and k2-characteristics. Thus, the regimes of bed deformation respectively along k1- and k2-characteristics are more differentiated. Alternatively, there are some scenarios for which bed deformation assumes a WBD or NBD regime along one of the characteristics (either k1 or k2), but a SBD or IBD regime along the other characteristics. The consequence of this
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Fig. 8. Contour of Tb/Th for Case 5: the relative time scale of bed deformation respectively along the k1 (solid line) and k2 (dashed line) characteristics.
Fig. 9. Contour of Tb/Th for Case 6: the relative time scale of bed deformation respectively along the k1 (solid line) and k2 (dashed line) characteristics.
feature is that the need for coupled mathematical modeling methodology is enhanced. Presumably, the forefronts of the sediment bores observed by Needham and Hey [21], and of the hyperconcentrated floods in the Yellow River and its tributaries are such scenarios. While a number of parameters may affect the regime of bed deformation, as demonstrated above in terms of the relative time scale of bed deformation, sediment concentration and flow depth assume paramount roles. At low sedi-
ment concentrations in deep flows, the SBD regime is very much less extensive than at high sediment concentrations in shallow flows. This observation has interesting implications for mathematical river modeling. For fluvial processes in the Yellow River and its tributaries, especially hyperconcentrated sediment-laden flood events, it is quite often that c > 0.01 and the flow is rather shallow (to the order of meters), and thus a fully coupled model would normally be needed, otherwise the modeling efforts could totally fail
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Fig. 10. Contour of Tb/Th for Case 7: the relative time scale of bed deformation respectively along the k1 (solid line) and k2 (dashed line) characteristics.
[3]. This also explains why a fully coupled model is required for other fluvial processes with active sediment transport and rapid bed deformation, such as dam-break flow over erodible bed [2]. However, the middle and lower reaches of the Yangtze River normally assume sediment concentrations lower than the order of magnitude of 10 kg/m3, i.e., c < 0.0038, and rather deep flows (to the order of tens of meters). It follows that the SBD regime, which necessitates the use of a fully coupled model is rather limited, and thus in most cases the application of existing decoupled models to the Yangtze River problems is justified. 6. Case analysis To further substantiate the significance of the present formulations of the time scales of fluvial processes, the response of an initially steady and uniform flow under equilibrium sediment transport to an unsteady sediment-laden flood event is analyzed based on numerical solutions. The numerical solutions are achieved by deploying the fully coupled mathematical modeling framework of Cao et al. [2,3], except that the sediment exchange flux is estimated according to Eqs. (20)–(22). The study case is designed to reflect typical processes in the Yangtze River. At the initial state, u = 1.2 m/s, h = 10 m, d = 0.05 mm, c = ce = 5.88E4, and bed slope is 4.18E5. At the upstream boundary, the unit discharge (q = uh) and sediment transport rate (qs = uhc) are specified as shown in Fig. 11, and at the downstream boundary water level is prescribed. The channel is 400 km long. Fig. 12 shows the variation with time of the time scale of bed deformation normalized with that of flow depth along the k1-characteristics (that along the k2-characteristics is
similar and thus not shown). The value of the relative time scale of bed deformation is consistently greater than 100, and the responsive process of the channel to the specified flood hydrographs is in the regime of WBD (Table 2). Not surprisingly, the differences between solutions (such as bed deformation, stage and unit discharge hydrographs, etc.) of the fully coupled modeling framework [2,3] and a decoupled model (e.g., which ignores R1 in the RHS side of Eq. (1)) are found to be negligible (not shown). This appears to justify the applicability of traditional decoupled models for Yangtze River problems. Succinctly, sediment transport capacity refers to the maximum possible amount of sediment (normally measured in terms of concentration or transport rate), which can be carried by the flow without causing essential bed deformation. Under capacity state, sediment transport is in equilibrium, i.e., the net flux of sediment exchange vanishes between flow and the bed. Only in steady and uniform flows would the concept of sediment transport capacity strictly apply. There has been a plethora of (semi-) empirical relationships for estimating sediment transport capacity (e.g., Eq. (21)). Natural stream flows are ubiquitously unsteady and non-uniform, thus the concept of sediment transport capacity may not be appropriate because the adaptation of sediment transport to local flow scenario may not be instantly fulfilled compared to flow changes in space and in time. Yet, one can always estimate ce with for instance Eq. (21) using local flow information. For general description, ce is hereafter defined as nominal sediment transport capacity. Under steady and uniform flows, the nominal sediment transport capacity is the physical sediment transport capacity. For a few decades, it has been attractive to estimate local sediment concentration by equating it to the nominal
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Fig. 11. Unit discharge and sediment transport rate specified at the upstream boundary.
Fig. 12. Tb/Th: the relative time scale of bed deformation along the k1-characteristics.
sediment transport capacity ce even in unsteady flood events, because only local flow information is involved in such estimation using for instance Eq. (21). In the reverse, some have attempted to analyze sediment transport capacity by equating it to measured local sediment concentration, and have suggested double or triple values of sediment transport capacity in unsteady flood events. However, this practice is theoretically not justified and may incur significant errors. For the present case, sediment con-
centration is found to deviate considerably from the nominal sediment transport capacity from Eq. (21). This is clear from Fig. 13, which illustrates the variation of sediment concentration, respectively at distinct cross-sections, against the non-dimensional parameter u3/ghw. To date, there seem to be few (if ever) general methods to delimit the difference between local sediment concentration and the nominal sediment transport capacity. The present formulations of the time scales deliver an easy-to-use
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Fig. 13. Variation of sediment concentration at a cross-section against the non-dimensional parameter u3/ghw progresses in a counterclockwise direction. The solid dots denote sediment transport capacity due to Eq. (21).
approach for this purpose. Fig. 14 shows the relative deviation of local sediment concentration from the nominal sediment transport capacity, defined as cd = (c ce)/ce, against the relative time Te/T0. While considerable scattering is seen as Te/T0 > 1, the value of cd invariably converges to zero as Te/T0 < 1. For the present case, jcdj < 10% at Te/T0 = 1. This finding is quite encouraging, and has interesting implications. Physically, as Te/T0 < 1, the adaptation of sediment
transport to local flow conditions is faster than flow changes, and with an error tolerance of 10%, local sediment concentration is approximately equal to the nominal sediment transport capacity, i.e., the concept of sediment transport capacity applies. It is noted that the gradient oc/ox of sediment concentration involved in the formulations of the time scales is not a prior known. This seems to cause difficulty in applications.
Fig. 14. Deviation of sediment concentration from the nominal sediment transport capacity against Te/T0 along (a) k1- and (b) k2-characteristics.
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However, it can be readily obviated by considering two successive cross-sections. The following procedure is suggested for applications, with given flow information to compute the time scales defined above. (a) Calculate the nominal sediment transport capacity ce at the two cross-sections xi and xi+1 by Eq. (21). (b) Assuming an error tolerance of 10%, estimate local sediment concentration at the two cross-sections by c ce(1 ± 0.1). (c) Assume that the two successive cross-sections have a common value of the gradient of sediment concentration, which can be calculated by oc/ox = (ci+1 ci)/ (xi+1 xi). (c) Using the sediment concentrations at the two crosssections and the gradient obtained above, calculate Te/T0 from the formulations above. (d) If Te/T0 < 1 is satisfied at both cross-sections, the above estimation of sediment concentration is acceptable. Otherwise, the concept of sediment transport capacity is unlikely to be applicable within the given error tolerance.
7. Conclusion The time scales of fluvial flow, suspended sediment transport and bed deformation are newly defined under the framework of shallow water hydrodynamics, non-capacity sediment transport and the theory of characteristics for the governing equations. The time scales of suspended sediment transport and bed deformation respectively relative to those of the flow are evaluated in detail, and the approximate applicability regions of decoupled mathematical river models, and of the concept of sediment transport capacity are addressed. Preliminary analysis of the time scales suggests that for shallow flows with high sediment concentrations (such as hyperconcentrated floods in the Yellow River and its tributaries), bed deformation may considerably affect the flow and a fully coupled model would normally be needed. In contrast, for deep flows at low sediment concentrations (e.g., the middle and lower reaches of the Yangtze River), a decoupled model is mostly justified. Also, the proposed relative time Te/T0 of the adaptation of suspended sediment transport to local flow conditions appears appropriate for evaluating the applicability of the concept of sediment transport capacity. This pilot study of the time scales of fluvial processes is conceptual in nature, and uncertainty is inevitable due to the roughness parameter approximating bed shear stress (including the non-Newtonian effect at high sediment concentrations) and also the formulation determining the net flux of sediment exchange. Nevertheless, the study facilitates a promising basis on which the rates of sediment transport adaptation to local flow conditions and of bed deformation, relative to the rates of flow changes, can be properly characterized, and the applicability of mathemat-
ical river models and the concept of sediment transport capacity can be ascertained. The present study is in principle extendable to the 2D shallow water hydrodynamic framework, provided that the second-order turbulent diffusion terms in the governing equations can be defined, in addition to the flow variables (velocity components, flow depth, sediment concentration and its gradient, bed slope). Extension of the methodology to alluvial rivers carrying bed load sediment is expected, whereas substantial study is required to quantify the sediment exchange flux between the flow and the bed with ameliorated accuracy. Acknowledgements The research reported is funded by the ‘‘973’’ National Key Basic Research and Development Program of China (under Grant No. 2003CB415200), and the Natural Science Foundation of China (under Grant No. 50459001). The constructive comments of the anonymous reviewers are appreciated. References [1] Alexandrov Y, Laronne JB, Reid I. Suspended sediment concentration and its variation with water discharge in a dryland ephemeral channel, northern Negev, Israel. J Arid Environ 2003;53:73–84. [2] Cao Z, Pender G, Wallis S, Carling P. Computational dam-break hydraulics over erodible sediment bed. J Hydraul Eng, ASCE 2004;130(7):689–703. [3] Cao Z, Pender G, Carling P. Shallow water hydrodynamic models for hyperconcentrated sediment-laden floods over erodible bed. Adv Water Resourc, Elsevier 2006;29(4):546–57. [4] Coussot P. Steady, laminar flow of concentrated mud suspensions in open channel. J Hydraul Res, IAHR 1994;32(4):535–60. [5] Cunge JA, Holly Jr FM, Verwey A. Practical aspects of computational river hydraulics. London: Pitman Pub.; 1980. [6] De Vries M. Considerations about non-steady bed-load transport in open channels. In: Proceedings of the 11th international congress, IAHR, Delft, vol. 3, 1965. pp. 381–3811. [7] De Vries M. River bed variations—aggradation and degradation. In: Proceedings of the internationt seminars of hydraulics of alluvial streams, IAHR, Delft, vol. 3, 1973. pp. 1–10. [8] De Vries M. A morphological time-scale for rivers. In: Proceedings of the 16th international congress, IAHR, Delft, vol. 2 (B3), 1975. pp. 17–23. [9] Graf WH, Qu Z. Flood hydrographs in open channels. Water Management, In: Proceedings of Institution of Civil Engineers UK, vol. 157(1), 2004. pp. 45–52. [10] Graf WH, Song T. Sediment transport in unsteady flow. In: Proceedings of the 26th congress, IAHR, London, 1995. pp. 480–5. [11] Guo J. Logarithmic matching and its application in computational hydraulics and sediment transport. J Hydraul Res, IAHR 2002;40(5):555–66. [12] Huang R, Wei Z, Zhao L, Li Y, Zheng C. Index of riverbed deformation and examination of sediment transport capacity formulations. Peoples’ Yellow River 2004;26(5):22–4 [in Chinese]. [13] Kuhnle RA. Bed load transport during rising and falling stages on two small streams. Earth Surf Process Landforms 1992;17:191–7. [14] Laronne JB, Reid I. Very high rates of bed load sediment transport in ephemeral desert rivers. Nature 1993;366:148–50. [15] Lee KT, Liu YL, Cheng KH. Experimental investigation of bed load transport processes under unsteady flow conditions. Hydrol Process 2004;18:2439–54.
Z. Cao et al. / Advances in Water Resources 30 (2007) 715–729 [16] LeVeque RJ. Finite volume methods for hyperbolic problems. England: Cambridge University Press; 2002. [17] Li Y. One-dimensional mathematical river models. In: Xie J, editor. River modelling. Beijing: China Water and Power Press; 1990. p. 86–120 [in Chinese, Chapter 3]. [18] Lyn DA. Unsteady sediment-transport modeling. J Hydr Eng, ASCE 1987;113(1):1–15. [19] Major JJ, Pierson TC. Debris flow rheology: experimental analysis of fine-grained slurries. Water Resourc Res 1992;28(3):841–57. [20] Marcus WA. Lag-time routing of suspended sediment concentrations during unsteady flow. Geol Soc Am Bull 1989;101:644–51. [21] Needham DJ, Hey RD. On nonlinear simple waves in alluvial river flows: a theory for sediment bores. Philos Trans Roy Soc Lond A – Math Phys Eng Sci 1991;334(1633):25–53. [22] Phillips BC, Sutherland AJ. Spatial lag effects in bed load sediment transport. J Hydraul Res, IAHR 1989;27(1):115–33. [23] Phillips BC, Sutherland AJ. Temporal lag effects in bed load sediment transport. J Hydraul Res, IAHR 1990;28(1):5–23. [24] Pierson TC, Scott KM. Downstream dilution of a lahar – transition from debris flow to hyperconcentrated streamflow. Water Resourc Res 1985;21(10):1511–24.
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[25] Reid I, Frostick LE, Layman JT. The incidence and nature of bed load transport during flood flows in coarse-grained alluvial channels. Earth Surf Process Landforms 1985;10:33–44. [26] Reid I, Laronne JB, Powell DM. Flash-flood and bed load dynamics of desert gravel-bed streams. Hydrol Process 1998;12:543–57. [27] Richardson J, Zaki W. Sedimentation and fluidization: I. Trans Inst Chem Eng 1954;32:35–53. [28] Song T, Graf WH. Experimental study of bed load transport in unsteady open-channel flow. Int J Sediment Res, IRTCES 1997;12(3):63–71. [29] Sutter RD, Verhoeven R, Krein A. Simulation of sediment transport during flood events: laboratory work and field experiments. Hydrol Sci J, IAHS 2001;46(4):599–610. [30] Van Vuren S. Stochastic modeling of river morphodynamics. PhD Thesis, Delft University of Technology, 2005. p. 275. [31] Wan Z, Wang Z. Hyperconcentrated flow. IAHR monograph. Rotterdam, The Netherlands: Balkema; 1994. [32] Williams GP. Sediment concentration versus water discharge during single hydrologic events in rivers. J Hydrol 1989;111:89–106. [33] Zhang R, Xie J. Sedimentation research in China – systematic selections. Beijing: China Water and Power Press; 1993.
Multiple time scales of alluvial rivers carrying suspended sediment and their implications for mathematical modeling Zhixian Cao *, Yitian Li, Zhiyuan Yue State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University, Wuhan, Hubei 430072, China Received 13 January 2006; received in revised form 27 May 2006; accepted 6 June 2006 Available online 17 August 2006
Abstract Flow, sediment transport and bed deformation in alluvial rivers normally exhibit multiple time scales. Enhanced knowledge of the time scales can facilitate better approaches to the understanding of the fluvial processes. Yet prior studies of the time scales are based upon the concept of sediment transport capacity at low concentrations, which however is not generally applicable. This paper presents new formulations of the time scales of fluvial flow, suspended sediment transport and bed deformation, under the framework of shallow water hydrodynamics, non-capacity sediment transport and the theory of characteristics for the hyperbolic governing equations. The time scale of bed deformation in relation to that of flow depth is demonstrated to delimit the applicability region of mathematical river models, and the time scale of suspended sediment transport relative to that of the pertinent flow information is analyzed to address if the concept of sediment transport capacity is applicable. For shallow flows with high sediment concentrations, bed deformation may considerably affect the flow and a fully coupled model is normally required. In contrast, for deep flows at low sediment concentrations, a decoupled model is mostly justified. This pilot study of the time scales delivers a new theoretical basis, on which the interaction between flow, suspended sediment transport and bed deformation can be potentially better characterized. Ó 2006 Elsevier Ltd. All rights reserved. Keywords: Fluvial flow; Sediment transport; Riverbed deformation; Time scales; Sediment transport capacity; Alluvial river; Mathematical river modeling
1. Introduction The interactive processes of flow, sediment transport and bed deformation in alluvial rivers constitute a system of physical problems of significant interest in the fields of fluvial hydraulics and morphodynamics. Generally, the bottom boundary of fluvial flow, the bed, undergoes deformation due to non-equilibrium sediment transport, which in turn affects the flow. When the rate of bed deformation is significantly smaller than that of flow changes by orders of magnitude, the feedback impacts of bed deformation on the flow is justifiably negligible, and traditional decoupled mathematical river models are applicable. Otherwise, decoupled models can totally collapse, and fully cou*
Corresponding author. Tel.: +86 27 68774409; fax: +86 27 68772310. E-mail address: [email protected] (Z. Cao).
0309-1708/$ - see front matter Ó 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.advwatres.2006.06.007
pled models are required in order to properly resolve the strong interactions between flow, sediment and the bed [2,3]. Irrespective of the observation, however, there has been no generally valid criterion measuring how fast the bed deforms compared to flow changes. Concurrently, sediment transport is often assumed to adapt to local hydraulic conditions instantly, and accordingly sediment transport rate is set to be equal to its capacity value that is commonly computed using local flow conditions. Yet, this assumption remains to be justified and its validity can be challenged, because a definitive description of the time required for sediment transport to adapt to local flow conditions is still missing. At the same time, a number of studies have clearly demonstrated phase differences between flow and sediment transport rate. This argument is correct for not only suspended sediment transport, but also bed load transport e.g., [25,20,22,23,32,13,14,10,28,26,29,1,9,15]. Therefore,
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Nomenclature A matrix defined in Eq. (2) c volumetric sediment concentration ce sediment transport capacity d sediment particle diameter D, E sediment deposition and entrainment fluxes F net flux of sediment exchange Fr Froude number g gravitational acceleration h flow depth m exponent n Manning roughness p bed sediment porosity q unit-width discharge qs unit-width sediment transport rate R matrix defined in Eq. (2) R1, R2, R3, R4 components of R in Eq. (2) Rp wd/m particle Reynolds number Sb bed slope Sf friction slope t time
sediment transport capacity formulations derived for steady flows are unable to reflect unsteady transport processes. Apparently, it is essential to achieve an improved understanding of the rates of changes of flow, sediment transport and bed deformation. Alternatively, the respective time scales of flow, sediment transport and bed deformation need to be properly defined. By time scale, we mean the time that governs the proprietary rate of change of a specific quantity describing the flow, sediment transport and bed deformation. Basically, it measures how fast a physical quantity of fluvial processes changes with time. The greater the time scale, the slower the physical quantity varies in time. Prior studies on the time scales of fluvial processes are based on the concept of sediment transport capacity. De Vries [6–8] studied the morphological time scale that characterizes the speed at which morphological processes take place. The morphological response exhibits either a wave character, which dominates over short distances, or a diffusion character prevailing over long distances. Accordingly, the morphological time scale was defined at two distinct levels respectively describing the wave and diffusion characteristics, as summarized by van Vuren [30]. Based on the St. Venant equations along with the concept of sediment transport capacity, Lyn [18] identified the multiple time scales of the system of flow, sediment transport and bed deformation. He showed that previous models, which reduce the number of conservation equations to be solved simultaneously from three to two under the ‘‘quasi-steady state’’ or ‘‘fixed bed’’ assumption, are unable to satisfy exactly either a general boundary condition or an arbitrary initial condition. Also, in situations with highly variable discharge
Tb Te Th Ts Tu T0 u U u* w x z a k1,2,3,4 m q q0 qw, qs
time scale of bed deformation time required for sediment transport to adapt to capacity time scale of flow depth time scale of sediment transport time scale of flow velocity time scale of quantity u3/ghw flow velocity matrix defined in Eq. (2) bed shear velocity settling velocity of a single particle in tranquil water streamwise coordinate bed elevation coefficient celerities kinematic viscosity of water density of water–sediment mixture density of saturated bed; and densities of water and sediment, respectively.
and sediment inputs, the aforementioned assumption is not justified. Nevertheless, it is recognized that the applicability of previous studies on the time scales of fluvial processes is limited as sediment transport can be far from capacity status, and sediment concentration in volume may not be far smaller than unity. The most telling case is that hyperconcentrated sediment-laden flood flows in the middle Yellow River and its tributaries (China) can still erode a huge amount of sediment from the bed, and sediment concentration during the flood events may well exceed the values that could be predicted by any existing sediment transport capacity formulations. Naturally the need for an improved understanding of the time scales of fluvial processes is evident, which is generally valid for non-capacity sediment transport phenomenon. This paper presents new formulations of the time scales of fluvial flow, suspended sediment transport and bed deformation under the framework of shallow water hydrodynamics and non-capacity sediment transport. It is substantiated under the theory of characteristics for the hyperbolic equations of conservation laws that govern the fluvial processes. The time scales of suspended sediment transport and bed deformation respectively relative to those of the pertinent flow information are analyzed. On this basis, the applicability region of mathematical river models is delimited, and the approximate applicability of the concept of sediment transport capacity addressed. The present study focuses on alluvial rivers carrying suspended sediment, whilst bed load-dominated rivers are reserved for future investigations, which involve distinct mechanisms.
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2. Governing equations
3. Definition of the time scales of fluvial processes
Consider one-dimensional (1D) flow in a channel with rectangular cross-sections of constant width. The shallow water hydrodynamics model for the flow, sediment transport and morphological evolution is built upon the mass and momentum conservation equations for the water–sediment mixture and the mass conservation equations respectively for sediment (carried in the flow) and bed material. Assuming the shape (correction) factors arising from the depth-averaging manipulation to be equal to unity, the complete governing equations [2] can be rewritten as
There can be different definitions of the time scales of fluvial processes as described by Eq. (1), which constitutes an inhomogeneous hyperbolic system. In general, fluvial flow, sediment transport and bed deformation are spaceand time-dependent. Even if the water flow is steady, sediment transport and bed deformation can change rather sharply in space, such as the sediment bores found in an experimental flume due to sediment overloading from the upstream [21]. Therefore it is attractive to achieve a spatially and temporally unified description of the time scales of fluvial processes. Given the above observation, the theory of characteristics is herewith opted for in defining the time scales. This option is physically sensible as disturbances in 1D fluvial flows are known to propagate along the characteristics [5,16]. The characteristic and compatibility equations in relation to Eq. (1) are pffiffiffiffiffi 8 dx=dt ¼ k1;2 ¼ u gh > > > rffiffiffi > > > g dh a23 dc g dz < du pffiffiffiffiffi pffiffiffiffiffi þ ð3a; bÞ dt h dt gh dt u gh dt > > ffiffi ffi r > > g a23 g > > : ¼ R2 pffiffiffiffiffi R4 R1 pffiffiffiffiffi R3 þ h u gh gh dx=dt ¼ k3 ¼ u ð4a; bÞ dc=dt ¼ R3 dx=dt ¼ k4 ¼ 0 ð5a; bÞ dz=dt ¼ R4
oU oU þA ¼R ot ox 2 3 2 u h 6u7 6g 6 7 6 U ¼ 6 7; A ¼ 6 4c5 40
ð1Þ h u
0 a23
3 0 g7 7 7; 05
0 u z 0 0 0 0 3 2 3 2 R1 F =ð1 pÞ 6 R 7 6 gS q Fu=qhð1 pÞ 7 7 6 27 6 f 0 R¼6 7¼6 7 4 R3 5 4 ð1 p cÞF =hð1 pÞ 5 R4
ð2a; b; cÞ
F =ð1 pÞ
where t is the time, x is the streamwise coordinate, h is the flow depth, u is the depth-averaged streamwise velocity, z is the bed elevation, c is the flux-averaged volumetric sediment concentration, g is the gravitational acceleration, Sf is the friction slope, p is the bed sediment porosity, F = E D is the net flux of sediment exchange between the flow and the bed, E, D are the sediment entrainment and deposition fluxes across the bottom boundary of flow, q = qw(1 c) + qsc is the density of the water–sediment mixture, q0 = qwp + qs(1 p) is the density of the saturated bed, a23 = (qs qw)gh/2q, and qw, qs are the densities of water and sediment, respectively. It is appreciated that the water–sediment mixture may behave as a non-Newtonian fluid at high sediment concentrations, and there has been a plethora of constitutive relationships describing the behavior e.g., [24,19,4,31]. Yet, in shallow water hydrodynamic models, it is implicitly incorporated in the term of bed shear stress (represented by friction slope in Eq. (2)), whilst the governing Eq. (1) remains correct. In applications, it is a common practice to calibrate the parameter(s) involved in determining the friction slope (Manning roughness, and when appropriate parameters describing the non-Newtonian behavior such as the yield stress). It is also noted that shallow water hydrodynamic models are applicable when the perturbation parameter defined as the ratio of the water depth to a characteristic length scale associated with the horizontal direction is far smaller than unity. This hierarchy of models resolves for the flow and sediment structure in the horizontal plane, rather than that along the water depth.
By virtue of Eq. (5) and the definition of bed slope Sb = oz/ox, the rate of change of bed elevation along the k1,2-characteristics can be determined by dz oz oz ¼ þ k1;2 ¼ R4 k1;2 S b ð6Þ dt k1;2 ot ox Similarly, the rate of change of sediment concentration along the k1,2-characteristics is expressed by dc oc oc ð7Þ ¼ þ k1;2 dt k1;2 ot ox and from Eq. (4b) dc oc oc ¼ þ k 3 ¼ R3 dt k3 ot ox Substituting Eq. (8) into Eq. (7), one has pffiffiffiffiffi oc dc oc ¼ R3 þ ðk1;2 k3 Þ ¼ R3 gh dt ox ox
ð8Þ
ð9Þ
k1;2
Substitution of Eqs. (6) and (9) into Eq. (3b) leads to 8 pffiffiffiffiffi > < dx=dt ¼ k1;2 ¼ u gh rffiffiffi rffiffiffi du g dh g oc > ¼ gS þ R R1 a23 ¼ R0 : 2 b dt h dt h ox
ð10a;bÞ
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It is rational to assume that the two terms in the LHS of Eq. (10b) are in general of the same order of magnitude as R0 in the RHS. Accordingly the time scale Tu of flow velocity can be defined along the k1,2-characteristics as u T u ¼ ð11Þ R
Again assuming that the two terms in the LHS of Eq. (10b) are of the same order of magnitude as R0 in the RHS, one can estimate the order of magnitude of u3/ghw along the k1,2-characteristics as follows: d u3 R0 ð3 FrÞFr2 ð17Þ dt ghw k1;2 w
Likewise, the time scale Th of flow depth can be defined and be related to Tu by
Therefore, the time scale T0 of quantity u3/ghw along the k1,2-characteristics can be defined as u3 =ghw u Tu ¼ ð18Þ T0 ¼ ¼ 2 R ð3 FrÞ absð3 FrÞ R0 ð3 FrÞFr =w 0
0
T h ¼ T u =Fr
ð12Þ pffiffiffiffiffi where Fr ¼ u= gh ¼ Froude number. Indeed, one can directly define the time scale of sediment transport based on Eq. (4). However the time scale so defined is associated with the k3-characteristics, and thus is not consistent with those of the flow as of Eqs. (11) and (12). Nevertheless, this inconsistency can be readily obviated. From Eq. (9), the time scale Ts of sediment transport in line with the k1,2-characteristics can be defined as c pffiffiffiffiffi ð13Þ T s ¼ R3 ghoc=ox
It has long been appreciated that sediment transport adapts to local flow conditions and tends to approach a capacity status (C = Ce) at which sediment transport is in equilibrium. The time scale Ts is a measure of the rate of this adaptation process, and yet the time Te required to achieve capacity status remains poorly understood from a quantitative sense. Here in complement to the time scale Ts, we aim to further substantiate the understanding of the adaptation process by defining ce c pffiffiffiffiffi ð14Þ T e ¼ R3 ghoc=ox Eq. (6) describes the lumped change of bed elevation along the k1,2-characteristics, which is determined by not only bed deformation, but also bed slope. Of particular interest in the present study is the rate of bed deformation due to non-equilibrium sediment transport as represented by oz/ot = R4, which is valid along any characteristics, including the k1,2-characteristics. Therefore, with the flow depth h as the appropriate length scale, the time scale Tb of bed deformation is defined as h ð15Þ jR4 j The respective time scales of the primitive variables (i.e., u, h, c, z) are defined along the k1,2-characteristics above. Yet, it may be necessary to define the time scales of deduced quantities composed of combinations of primitive variables. Here we take the non-dimensional quantity u3/ghw as an example, where w is the settling velocity of a single sediment particle in tranquil water (that can be calculated using for instance Zhang’s formula, see [33]). It is easy to derive that d u3 Fr2 du dh 3h u ð16Þ ¼ dt ghw dt dt wh Tb ¼
4. Auxiliary relationships Prior to proceeding to analyze the time scales of fluvial processes defined above, auxiliary relationships need to be introduced to close the governing equations. The conventional empirical relation is used to determine the friction slope involving the Manning roughness n n2 u2 S f ¼ 4=3 ð19Þ h Sediment exchange between the flow and the bed involves two distinct mechanisms, i.e., the upward bed sediment entrainment and the downward deposition due to gravitational action. The traditional and extensively used approach to specifying bed sediment entrainment flux is based on the premise that entrainment occurs always at the same rate as it does under equilibrium conditions. As in equilibrium conditions, the entrainment flux is equal to the deposition flux, and bed sediment entrainment flux can be computed using equilibrium near-bed sediment concentration ace and effective settling velocity w(1 ace)m of Richardson and Zaki [27]. Accordingly one has m ð20Þ E ¼ awce ð1 ace Þ where a is the coefficient, m ¼ 4:45Rp0:1 is the exponent denoting the effects of hindered settling due to high sediment concentrations, Rp wd/m is the particle Reynolds number, d is the sediment diameter, and m is the kinematic viscosity of water. The well tested and widely used semiempirical formula of Zhang and Xie [33] is employed to calculate suspended sediment transport capacity, which, following the interesting logarithmic-matching treatment of Guo [11], reads: ce ¼
1 ½u3 =ghw1:5 20qs 1 þ ½u3 =45ghw1:15
ð21Þ
The following relationship for sediment deposition flux, based on the formulation of hindered sediment settling velocity due to Richardson and Zaki [27], is introduced: D ¼ awcð1 acÞ
m
ð22Þ
The next flux F of sediment exchange determined from Eqs. (19) and (21) is essentially not new [17], except that the impact of hindered sediment settling has been incorporated with the Richardon and Zaki [27] formulation, which is
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known to be significant for high sediment concentrations. It is appreciated that existing formulations of sediment entrainment and deposition fluxes remain empirical, and the roughness parameter is hard to establish for natural rivers, especially as it is influenced by sediment. Thus Eqs. (19)–(22) represent only one set of the viable relationships closing the governing equations. 5. Evaluation of the time scales and their implications In evaluating the time scales of fluvial processes defined above, the following parameters are specified as: m = 1E6 m2/s, p = 0.4, qw = 1E+3 kg/m3, and qs = 2.65E+3 kg/m3. Other parameters are varied in different cases as shown in Table 1. 5.1. Basic features of the time scales Here we evaluate the basic features of the time scales defined above, corresponding to Case 1. Fig. 1(a) shows the contour of the time scale Tu of flow velocity in the Table 1 List of parameters for evaluation of the time scales Case no.
d (mm)
h (m)
n (s/m1/3)
a
Sb
hoc/ox
Fig. showing Tb/Th
1 2 3 4 5 6 7
0.1 0.1 0.1 0.04 0.1 0.1 0.1
10.0 3.0 3.0 3.0 3.0 3.0 3.0
0.03 0.03 0.025 0.025 0.025 0.025 0.025
1.0 1.0 1.0 1.0 1.2 1.0 1.0
2E4 2E4 2E4 2E4 2E4 5E4 2E4
1E5 1E5 1E5 1E5 1E5 1E5 3E5
Fig. Fig. Fig. Fig. Fig. Fig. Fig.
2 5 6 7 8 9 10
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u*–c plane, where u* is the bed shear velocity. The existence of double value of Tu is evident, as associated with respectively the k1- and k2-characteristics, which is ascribable to the influence of sediment exchange flux (F) and concentration gradient (oc/ox). At low sediment concentrations, say, c < 0.006, the time scale of flow is rather weakly dependent of sediment concentration, and the influence of bed shear velocity on Tu prevails. Interestingly, the maximum value of the time scale Tu appears around u* 0.14 m/s, which in essence corresponds to approximately steady and uniform flow (see Eq. (11)). As sediment concentration increases, its impact is considerable. The contour of the time scale Th of flow depth, shown in Fig. 1(b) exhibits considerable difference from that of Tu (Fig. 1a) as can be anticipated from Eq. (12). The rate of bed deformation can be measured with Tb, and it is advisable to evaluate it against the time scale of an appropriate flow variable (flow velocity or depth). Apparently, the most viable choice seems to be Th of the flow depth. In Fig. 2, the ratio of Tb to Th is illustrated in a contour format in the u*–c plane. Fig. 2 actually shows the rate at which the riverbed deforms, due to non-equilibrium sediment transport, relative to the change of the flow depth. The greater the value of the relative time scale Tb/ Th of bed deformation, the slower the bed deforms in relation to the flow, and the less bed deformation affects the flow. Thus, traditional decoupled mathematical river models tend to be applicable. In contrast, when Tb/Th is not sufficiently large, the feedback influences of bed deformation on the flow deserve full consideration and fully coupled mathematical river models must be used. Just for convenience, we tentatively use a few critical values (1000, 100, 10) of Tb/Th to distinguish the regimes with
Fig. 1. Contour of the time scale (a) Tu of flow velocity and (b) Th of flow depth, respectively along the k1 (solid line) and k2 (dashed line) characteristics, for Case 1.
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Fig. 2. Contour of Tb/Th for Case 1: the relative time scale of bed deformation respectively along the k1 (solid line) and k2 (dashed line) characteristics.
Table 2 Regimes of bed deformation Tb/Th
Tb/Th > 1000
1000 > Tb/Th > 100
100 > Tb/Th > 10
10 > Tb/Th
Regime of bed deformation Applicable mathematical river model
NBD: Negligible bed deformation Fixed-bed or decoupled mobile-bed model
WBD: Weak bed deformation
IBD: Intermediate bed deformation
Decoupled mobile-bed model
Fully coupled mobile-bed model
SBD: Strong bed deformation Fully coupled mobile-bed model
different relative time scale of bed deformation, and suggest the applicability of mathematical river models (see Table 2). Nevertheless, these critical values of Tb/Th need to be substantiated in further studies. In Fig. 2, it is demonstrated that the variation of Tb/Th exhibits a rather sophisticated pattern against sediment concentration and bed shear velocity. Firstly, consider the NBD regime with Tb/Th > 1000, which is in essence immediately around the curve of sediment transport capacity due to Eq. (21), see the thick solid line in Fig. 4. In the NBD regime, sediment transport is in a quasi-equilibrium state, bed deformation is negligibly weak compared to the change of flow depth, and fixed-bed or decoupled mobile-bed mathematical river models justifiably apply. Secondly, to the right-hand and lower part of the NBD regime (Fig. 2), u* > 0.0.092 m/s and c < ce, bed sediment entrainment dominates over deposition. The existence of a WBD, an IBD and a SBD regime is evident. Within the IBD and SBD regimes, the rate of bed deformation is not negligible compared against that of the change of flow depth, and naturally fully coupled mathematical river models are needed to properly resolve the strong interaction between flow, sediment transport and bed deformation. It is noted that the SBD regime is roughly around steady
and uniform flow state with u* 0.14 m/s, and its occurrence is ascribed to the infinite time scale Th of flow depth due to Eqs. (11) and (12). Physically, if ever the flow is steady and uniform, bed deformation (in this case, bed degradation as sediment entrainment overweighs over deposition) can be very active regardless of the low sediment concentration, which necessitates a fully coupled mathematical description of the fluvial processes. Presumably for the WBD, a decoupled mathematical model suffices. Thirdly, to the left-hand and upper part of the NBD regime (Fig. 2), c > ce, sediment deposition dominates over entrainment from the bed. In additional to a WBD regime with 100 < Tb/Th < 1000, there is quite an extensive area that is either an IBD or a SBD regime, with sediment concentration typically being higher than 0.001. In the IBD and SBD, the rate of bed deformation (in this case, bed aggradation as sediment deposition exceeds entrainment) cannot be ignored in comparison with that of the change of flow depth, and a fully coupled mathematical description of the fluvial processes is warranted. Fig. 3 is a plot of the ratio of the time scale Ts of sediment transport to the time scale T0 of the non-dimensional quantity u3/ghw, in a contour format in the u*–c plane. Of particular interest is that the value of Ts/T0 could be much
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Fig. 3. Contour of Ts/T0 for Case 1: the ratio of the time scale Ts of sediment transport to the time scale T0 of the non-dimensional quantity u3/ghw, respectively along (a) k1- and (b) k2-characteristics.
smaller than unity (e.g., Ts/T0 < 0.1), indicating that under favorable conditions (low sediment concentration and around the steady and uniform flow state at u* 0.14 m/s) sediment transport could adapt to local flow scenarios much faster than flow changes, and hence the concept of sediment transport capacity is approximately applicable as previously presumed. However, the applicability of the
concept of sediment transport capacity is dependent upon certain conditions, and so far remains to be justified on a physically rigorous basis. The present proposition of the respective time scales of fluvial processes provides such a device ascertaining when and where the concept of sediment transport capacity could be used. To substantiate this statement, Fig. 4 shows the contour of the ratio of the time
Fig. 4. Contour of Te/T0 for Case 1: the ratio of the time Te required for sediment transport to adapt to capacity status to the time scale T0 of the nondimensional quantity u3/ghw, respectively along the k1 (solid line) and k2 (dashed line) characteristics. The thick solid line denotes sediment transport capacity due to Eq. (21).
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Te required to adapt to capacity status of sediment transport to the time scale T0 of the non-dimensional quantity u3/ghw, along with the sediment transport capacity formulation as of Eq. (21). It is seen in Fig. 4 that in an area around the curve (thin dotted line) of sediment transport capacity, the value of Te/T0 is smaller than 0.1 (and could be smaller than 0.01, though not shown) along both the k1- and k2-characteristics. Alternatively, within this area, sediment transport can adapt to local flow conditions almost instantly and the concept of sediment transport capacity approximately applies. Accordingly, the area in which Te/T0 < 0.1, or Te/T0 < 1.0 can tentatively be referred to as the approximate applicability region of the concept of sediment transport capacity, depending on the desirable error tolerance. Huang et al. [12] propose an index of bed deformation rate to delimit the applicability of sediment transport capacity formulations. Yet the bed deformation rate was not normalized with any flow information, and thus is only of limited use. 5.2. Sensitivity of the time scales to parameter variation In relation to distinct cases with varied parameters d, h, Sb, hoc/ox, n and a, the main features of the time scales are qualitatively similar to Figs. 1–4. Here, we focus on the quantitative behavior of the relative time scale Tb/Th of bed deformation in response to parameter variation (Figs. 5–10). In Figs. 2 and 5 (Cases 1 and 2 in Table 1) it is seen that the flow depth may considerably affect the relative time scale of bed deformation. As flow depth decreases, the SBD region above the NBD appears to be enlarged in the u*–c plane. While the IBD region above the NBD seems
to experience only minor variation, the IBD region below the NBD is largely diminished due to the smaller flow depth. The major influence of a decreased roughness parameter (Figs. 5 and 6, corresponding to Cases 2 and 3 in Table 1) is that the IBD region below the NBD is extended, because the entrainment ability of the flow is enhanced as for specified flow depth and bed shear velocity (u*), the flow velocity increases due to roughness reduction. By Figs. 6 and 7 (Cases 3 and 4 in Table 1), it is indicated that the regimes of the relative time scale of bed deformation is rather sensitive to sediment particle size. As sediment diameter reduces from 0.1 mm to 0.04 mm, the SBD and IBD regions above the NBD are greatly diminished. This is not surprising, as above the NBD sediment deposition overweighs over entrainment, finer sediment deposition is much slower than coarser sediment, and especially finer sediment deposition can be substantially hindered at high concentrations. In contrast, the IBD region below the NBD is appreciably expanded in c but narrowed down in u*, when sediment diameter decreases. This results from the concurrent influences of the varied entrainment and deposition fluxes, as determined by Eqs. (20)–(22). Figs. 6 and 8 (Cases 3 and 5 in Table 1) show the impact on the relative time scale of bed deformation of the empirical parameter a in the closure of sediment exchange flux. A larger value of a seems to mainly lead to a wider IBD region below the NBD, whilst its impact on other regimes of the relative time scale of bed deformation is only marginal. The bed slope can also considerably affect the relative time scale of bed deformation, as demonstrated in Figs. 6
Fig. 5. Contour of Tb/Th for Case 2: the relative time scale of bed deformation respectively along the k1 (solid line) and k2 (dashed line) characteristics.
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Fig. 6. Contour of Tb/Th for Case 3: the relative time scale of bed deformation respectively along the k1 (solid line) and k2 (dashed line) characteristics.
Fig. 7. Contour of Tb/Th for Case 4: the relative time scale of bed deformation respectively along the k1 (solid line) and k2 (dashed line) characteristics.
and 9 (Cases 3 and 6 in Table 1). With all the remaining parameters kept unchanged, an increase in bed slope can result in contraction of the SBD and IBD regions above the NBD of the relative time scale of bed deformation, whilst the SBD and IBD regions below the NBD are appreciably extended. When purely the magnitude of the gradient of sediment concentration varies (Figs. 6 and 10, corresponding to Cases
3 and 7 in Table 1), the relative time scale of bed deformation sees more differentiation along the k1- and k2-characteristics. Thus, the regimes of bed deformation respectively along k1- and k2-characteristics are more differentiated. Alternatively, there are some scenarios for which bed deformation assumes a WBD or NBD regime along one of the characteristics (either k1 or k2), but a SBD or IBD regime along the other characteristics. The consequence of this
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Fig. 8. Contour of Tb/Th for Case 5: the relative time scale of bed deformation respectively along the k1 (solid line) and k2 (dashed line) characteristics.
Fig. 9. Contour of Tb/Th for Case 6: the relative time scale of bed deformation respectively along the k1 (solid line) and k2 (dashed line) characteristics.
feature is that the need for coupled mathematical modeling methodology is enhanced. Presumably, the forefronts of the sediment bores observed by Needham and Hey [21], and of the hyperconcentrated floods in the Yellow River and its tributaries are such scenarios. While a number of parameters may affect the regime of bed deformation, as demonstrated above in terms of the relative time scale of bed deformation, sediment concentration and flow depth assume paramount roles. At low sedi-
ment concentrations in deep flows, the SBD regime is very much less extensive than at high sediment concentrations in shallow flows. This observation has interesting implications for mathematical river modeling. For fluvial processes in the Yellow River and its tributaries, especially hyperconcentrated sediment-laden flood events, it is quite often that c > 0.01 and the flow is rather shallow (to the order of meters), and thus a fully coupled model would normally be needed, otherwise the modeling efforts could totally fail
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Fig. 10. Contour of Tb/Th for Case 7: the relative time scale of bed deformation respectively along the k1 (solid line) and k2 (dashed line) characteristics.
[3]. This also explains why a fully coupled model is required for other fluvial processes with active sediment transport and rapid bed deformation, such as dam-break flow over erodible bed [2]. However, the middle and lower reaches of the Yangtze River normally assume sediment concentrations lower than the order of magnitude of 10 kg/m3, i.e., c < 0.0038, and rather deep flows (to the order of tens of meters). It follows that the SBD regime, which necessitates the use of a fully coupled model is rather limited, and thus in most cases the application of existing decoupled models to the Yangtze River problems is justified. 6. Case analysis To further substantiate the significance of the present formulations of the time scales of fluvial processes, the response of an initially steady and uniform flow under equilibrium sediment transport to an unsteady sediment-laden flood event is analyzed based on numerical solutions. The numerical solutions are achieved by deploying the fully coupled mathematical modeling framework of Cao et al. [2,3], except that the sediment exchange flux is estimated according to Eqs. (20)–(22). The study case is designed to reflect typical processes in the Yangtze River. At the initial state, u = 1.2 m/s, h = 10 m, d = 0.05 mm, c = ce = 5.88E4, and bed slope is 4.18E5. At the upstream boundary, the unit discharge (q = uh) and sediment transport rate (qs = uhc) are specified as shown in Fig. 11, and at the downstream boundary water level is prescribed. The channel is 400 km long. Fig. 12 shows the variation with time of the time scale of bed deformation normalized with that of flow depth along the k1-characteristics (that along the k2-characteristics is
similar and thus not shown). The value of the relative time scale of bed deformation is consistently greater than 100, and the responsive process of the channel to the specified flood hydrographs is in the regime of WBD (Table 2). Not surprisingly, the differences between solutions (such as bed deformation, stage and unit discharge hydrographs, etc.) of the fully coupled modeling framework [2,3] and a decoupled model (e.g., which ignores R1 in the RHS side of Eq. (1)) are found to be negligible (not shown). This appears to justify the applicability of traditional decoupled models for Yangtze River problems. Succinctly, sediment transport capacity refers to the maximum possible amount of sediment (normally measured in terms of concentration or transport rate), which can be carried by the flow without causing essential bed deformation. Under capacity state, sediment transport is in equilibrium, i.e., the net flux of sediment exchange vanishes between flow and the bed. Only in steady and uniform flows would the concept of sediment transport capacity strictly apply. There has been a plethora of (semi-) empirical relationships for estimating sediment transport capacity (e.g., Eq. (21)). Natural stream flows are ubiquitously unsteady and non-uniform, thus the concept of sediment transport capacity may not be appropriate because the adaptation of sediment transport to local flow scenario may not be instantly fulfilled compared to flow changes in space and in time. Yet, one can always estimate ce with for instance Eq. (21) using local flow information. For general description, ce is hereafter defined as nominal sediment transport capacity. Under steady and uniform flows, the nominal sediment transport capacity is the physical sediment transport capacity. For a few decades, it has been attractive to estimate local sediment concentration by equating it to the nominal
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Fig. 11. Unit discharge and sediment transport rate specified at the upstream boundary.
Fig. 12. Tb/Th: the relative time scale of bed deformation along the k1-characteristics.
sediment transport capacity ce even in unsteady flood events, because only local flow information is involved in such estimation using for instance Eq. (21). In the reverse, some have attempted to analyze sediment transport capacity by equating it to measured local sediment concentration, and have suggested double or triple values of sediment transport capacity in unsteady flood events. However, this practice is theoretically not justified and may incur significant errors. For the present case, sediment con-
centration is found to deviate considerably from the nominal sediment transport capacity from Eq. (21). This is clear from Fig. 13, which illustrates the variation of sediment concentration, respectively at distinct cross-sections, against the non-dimensional parameter u3/ghw. To date, there seem to be few (if ever) general methods to delimit the difference between local sediment concentration and the nominal sediment transport capacity. The present formulations of the time scales deliver an easy-to-use
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Fig. 13. Variation of sediment concentration at a cross-section against the non-dimensional parameter u3/ghw progresses in a counterclockwise direction. The solid dots denote sediment transport capacity due to Eq. (21).
approach for this purpose. Fig. 14 shows the relative deviation of local sediment concentration from the nominal sediment transport capacity, defined as cd = (c ce)/ce, against the relative time Te/T0. While considerable scattering is seen as Te/T0 > 1, the value of cd invariably converges to zero as Te/T0 < 1. For the present case, jcdj < 10% at Te/T0 = 1. This finding is quite encouraging, and has interesting implications. Physically, as Te/T0 < 1, the adaptation of sediment
transport to local flow conditions is faster than flow changes, and with an error tolerance of 10%, local sediment concentration is approximately equal to the nominal sediment transport capacity, i.e., the concept of sediment transport capacity applies. It is noted that the gradient oc/ox of sediment concentration involved in the formulations of the time scales is not a prior known. This seems to cause difficulty in applications.
Fig. 14. Deviation of sediment concentration from the nominal sediment transport capacity against Te/T0 along (a) k1- and (b) k2-characteristics.
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However, it can be readily obviated by considering two successive cross-sections. The following procedure is suggested for applications, with given flow information to compute the time scales defined above. (a) Calculate the nominal sediment transport capacity ce at the two cross-sections xi and xi+1 by Eq. (21). (b) Assuming an error tolerance of 10%, estimate local sediment concentration at the two cross-sections by c ce(1 ± 0.1). (c) Assume that the two successive cross-sections have a common value of the gradient of sediment concentration, which can be calculated by oc/ox = (ci+1 ci)/ (xi+1 xi). (c) Using the sediment concentrations at the two crosssections and the gradient obtained above, calculate Te/T0 from the formulations above. (d) If Te/T0 < 1 is satisfied at both cross-sections, the above estimation of sediment concentration is acceptable. Otherwise, the concept of sediment transport capacity is unlikely to be applicable within the given error tolerance.
7. Conclusion The time scales of fluvial flow, suspended sediment transport and bed deformation are newly defined under the framework of shallow water hydrodynamics, non-capacity sediment transport and the theory of characteristics for the governing equations. The time scales of suspended sediment transport and bed deformation respectively relative to those of the flow are evaluated in detail, and the approximate applicability regions of decoupled mathematical river models, and of the concept of sediment transport capacity are addressed. Preliminary analysis of the time scales suggests that for shallow flows with high sediment concentrations (such as hyperconcentrated floods in the Yellow River and its tributaries), bed deformation may considerably affect the flow and a fully coupled model would normally be needed. In contrast, for deep flows at low sediment concentrations (e.g., the middle and lower reaches of the Yangtze River), a decoupled model is mostly justified. Also, the proposed relative time Te/T0 of the adaptation of suspended sediment transport to local flow conditions appears appropriate for evaluating the applicability of the concept of sediment transport capacity. This pilot study of the time scales of fluvial processes is conceptual in nature, and uncertainty is inevitable due to the roughness parameter approximating bed shear stress (including the non-Newtonian effect at high sediment concentrations) and also the formulation determining the net flux of sediment exchange. Nevertheless, the study facilitates a promising basis on which the rates of sediment transport adaptation to local flow conditions and of bed deformation, relative to the rates of flow changes, can be properly characterized, and the applicability of mathemat-
ical river models and the concept of sediment transport capacity can be ascertained. The present study is in principle extendable to the 2D shallow water hydrodynamic framework, provided that the second-order turbulent diffusion terms in the governing equations can be defined, in addition to the flow variables (velocity components, flow depth, sediment concentration and its gradient, bed slope). Extension of the methodology to alluvial rivers carrying bed load sediment is expected, whereas substantial study is required to quantify the sediment exchange flux between the flow and the bed with ameliorated accuracy. Acknowledgements The research reported is funded by the ‘‘973’’ National Key Basic Research and Development Program of China (under Grant No. 2003CB415200), and the Natural Science Foundation of China (under Grant No. 50459001). The constructive comments of the anonymous reviewers are appreciated. References [1] Alexandrov Y, Laronne JB, Reid I. Suspended sediment concentration and its variation with water discharge in a dryland ephemeral channel, northern Negev, Israel. J Arid Environ 2003;53:73–84. [2] Cao Z, Pender G, Wallis S, Carling P. Computational dam-break hydraulics over erodible sediment bed. J Hydraul Eng, ASCE 2004;130(7):689–703. [3] Cao Z, Pender G, Carling P. Shallow water hydrodynamic models for hyperconcentrated sediment-laden floods over erodible bed. Adv Water Resourc, Elsevier 2006;29(4):546–57. [4] Coussot P. Steady, laminar flow of concentrated mud suspensions in open channel. J Hydraul Res, IAHR 1994;32(4):535–60. [5] Cunge JA, Holly Jr FM, Verwey A. Practical aspects of computational river hydraulics. London: Pitman Pub.; 1980. [6] De Vries M. Considerations about non-steady bed-load transport in open channels. In: Proceedings of the 11th international congress, IAHR, Delft, vol. 3, 1965. pp. 381–3811. [7] De Vries M. River bed variations—aggradation and degradation. In: Proceedings of the internationt seminars of hydraulics of alluvial streams, IAHR, Delft, vol. 3, 1973. pp. 1–10. [8] De Vries M. A morphological time-scale for rivers. In: Proceedings of the 16th international congress, IAHR, Delft, vol. 2 (B3), 1975. pp. 17–23. [9] Graf WH, Qu Z. Flood hydrographs in open channels. Water Management, In: Proceedings of Institution of Civil Engineers UK, vol. 157(1), 2004. pp. 45–52. [10] Graf WH, Song T. Sediment transport in unsteady flow. In: Proceedings of the 26th congress, IAHR, London, 1995. pp. 480–5. [11] Guo J. Logarithmic matching and its application in computational hydraulics and sediment transport. J Hydraul Res, IAHR 2002;40(5):555–66. [12] Huang R, Wei Z, Zhao L, Li Y, Zheng C. Index of riverbed deformation and examination of sediment transport capacity formulations. Peoples’ Yellow River 2004;26(5):22–4 [in Chinese]. [13] Kuhnle RA. Bed load transport during rising and falling stages on two small streams. Earth Surf Process Landforms 1992;17:191–7. [14] Laronne JB, Reid I. Very high rates of bed load sediment transport in ephemeral desert rivers. Nature 1993;366:148–50. [15] Lee KT, Liu YL, Cheng KH. Experimental investigation of bed load transport processes under unsteady flow conditions. Hydrol Process 2004;18:2439–54.
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