Suspended load in flows on erodible bed

International Journal of Sediment Research 24 (2009) 315-324 Suspended load in flows on erodible bed Sujit K. BOSE1 and Subhasish DEY2* Abstract Steady state suspended-load of sediment transported in flow over erodible beds usually is treated by the advection-diffusion approach, though in recent years, it is being treated as a two-phase flow phenomenon incorporating kinetics of sediment particles. Among the advection-diffusion approaches, Rouse’s (1937) equation is the well-known, although a number of researchers in later periods have attempted to improve it by modifying the mixing length concept taking into account other aspects. In this paper, the advection-diffusion approach and associated logarithmic law of flow velocity are revisited. It is concluded from the logarithmic law that the Reynolds shear stress is a linear function of height above the bed, which reduces to bed shear stress in the case of a long horizontal channel. As a consequence, it is shown that the volumetric concentration of sediment is best approximated by the sum of two power laws of height above the bed. An equation is derived for the suspended-load transport rate in terms of elementary functions. Key Words: Sediment transport, Fluvial hydraulics, Suspended load, Open channel flow 1 Introduction In flow over a sediment bed, the sediment is partly transported as suspended load, where the sediment particles occasionally come in contact with the bed and are transported in suspension surrounded by the fluid. The theory of such flows is usually based on the Fick’s law of diffusion leading to an advectiondiffusion equation. Alternatively, the same equation has been shown by Foister and van de Ven (1980) to be equivalent to the long time stochastic analysis of the Langevin equation for the motion of particles under viscous drag in a shear flow and random disturbing forces. Subjecting the advection-diffusion equation to Reynolds averaging, the resulting equation expresses the conservation of mass flux in the vertical direction. The equation is a simple first-order differential equation (Eq. (1) in the next section). Rouse (1937) integrated the differential equation by assuming (1) the logarithmic law of flow velocity and (2) Reynolds shear stress diminishes linearly from the bed shear stress, τ0, to zero at the free surface. Rouse’s equation for the sediment concentration, C, was generally validated against the experimental data of Vanoni (1946) and was considered satisfactorily agreeable with the laboratory and field data (Graf, 1971). Rouse’s theory was critically examined by Einstein and Chien (1955) to derive an expression for flow velocity, u, in terms of Nikuradse’s equivalent sand roughness, ks, and sediment concentration, C, given by Rouse’s equation. Their equation of sediment transport rate contains definite integrals, which were expressed in terms of the Gamma function by Guo and Wood (1995). Hunt (1954) considered a mixture of sediment and fluid to derive a more complicated equation than that of Rouse. In recent times, improved theoretical approaches have been developed. Umeyama and Gerritsen (1992) and Umeyama (1992) introduced a new mixing length equation and concentration dependent Reynolds 1 Visiting Fellow, Department of Civil Engineering, Indian Institute of Technology, Kharagpur 721302, West Bengal, India 2 Chair Prof., Department of Civil Engineering, Indian Institute of Technology, Kharagpur 721302, West Bengal, India. Fax: +91 3222 282254, * Corresponding author, E-mail: sdey@iitkgp.ac.in Note: The original manuscript of this paper was received in July 2008. The revised version was received in April 2009. Discussion open until Sept. 2010. International Journal of Sediment Research, Vol.24, No. 3, 2009, pp. 315–324 - 315 - shear stress to incorporate the effect for upper part of the flow. Ni and Wang (1991) considered, in general, a high concentration and introduced a characteristic length for sediment motion in the vertical direction. The expression for C agreed with those obtained earlier by others for different cases. Kovacs (1998) generalized the mixing length concept for sediment laden flows and obtained a transcendental equation for C, which he iteratively solved by the form of C = byα (where b = a coefficient; and α = an exponent) valid for clear-water flow velocity. Such a form of solution was originally proposed by Montes and Ippen (1973) and described in Allen (1985), where this form was of particular interest. For the flow velocity, Kovacs (1998) gave a first-order differential equation that could be solved numerically. McLean (1991) pointed out that since the net transport of suspended sediment is calculated by integrating the product of the sediment concentration and velocity over the flow depth, the effects of density stratification by suspended sediment, as well as those due to bed forms and the mixture of sediment sizes are important. He treated these aspects from a theoretical viewpoint generalizing Rouse’s theory and McLean’s earlier work. The effect of bed roughness on sediment suspension was investigated by Mazumder et al. (2005) from both theoretical and experimental viewpoints. Wren et al. (2005) considered the effect of mobile beds on suspended sediment concentration adopting a two-phase flow approach. Cao et al. (1995) gave a new diffusion equation for sediment concentration demonstrating that previous formulations for low concentration were particular cases. Fu et al. (2005) incorporated a kinetic model for particles in the two-phase theory and obtained a diffusion equation which showed that concentration does not monotonically increase near bed. This concentration distribution was validated for medium and coarse sediments with the observations of Bouvard and Petkovic (1985), Wang and Ni (1990), and Wang and Qian (1989). Recently, Wang et al. (2008) gave a kinetic model based simulation for comprehensive analysis of suspended sediment transport under various conditions. In this paper, the concentration distribution of fine sediment in suspension in a steady-state flow over an erodible bed is considered for the case low concentration (less than 0.1) that is of common practical interest. The conservation principle of mass flux along the vertical is adequate for this purpose and the logarithmic law may be assumed for the streamwise flow velocity. The upper wake layer is similarly treated with a decrease in the von Karman constant. A logarithmic law implies a Reynolds shear stress that varies linearly with the height from the bed, possessing a gradient with respect to the vertical that equals in magnitude, the streamwise hydrostatic pressure gradient and the body force per unit volume on the fluid. The gradient vanishes altogether for a long horizontal channel with an invariant Reynolds shear stress nearly equal to the bed shear stress. Based on these equations, keeping in view the small variability of the inverse of the Schmidt number, ζ, it is shown that the concentration of suspended sediment can be expressed as a sum of two power law functions. The expression contains three parameters that can be iteratively adjusted for experimental data. 2 The sediment mass conservation equation In flow on an erodible bed carrying suspended load, the volume fraction, (i.e. concentration), C, of sediment is adequately described by the advection-diffusion equation. Taking the x-axis along the bed in the direction of flow and the y-axis vertically upwards, the steady-state equation can be integrated as follows (Graf, 1971): dC = constant = 0 (1) ws C + ε y dy where ws = terminal fall velocity of sediment particles; and εy = sediment diffusivity in the vertical direction. Since the left hand side vanishes at the free surface, y = h, where h is the flow depth, the constant of integration appearing in Eq. (1) equals zero. Equation (1) is in fact a sediment mass conservation equation, in as much as the mass flux εy∂C/∂y in the y-direction is balanced by the settling sediment flux –wsC. In Eq. (1), ws may be regarded as a constant as the fluid flow is assumed typically to be horizontal. On the other hand, εy is in general a function of y, depending on the eddy motion of the fluid flow. If the latter is represented by turbulent diffusivity or eddy viscosity, ε, defined by - 316 - International Journal of Sediment Research, Vol. 24, No. 3, 2009, pp. 315–324 τ = ρε du dy (2) where τ = Reynolds shear stress of the flow; ρ = mass density of the fluid; and u = time-averaged streamwise velocity at an elevation y, then it may be assumed that εy is proportional to ε (Graf, 1971): (3) ε y = βε where β = factor of proportionality or the inverse of Schmidt number. Intensive research has been carried out to quantify β. Carstens (1952) presented a mathematical expression for β by using the analog of the oscillatory motion of a spherical particle in a fluid, and obtained β ≤ 1 for all experimental cases. Other experimental investigations supported this conclusion with the observation β ≈ 1 for fine particles and β < 1 for coarse particles (Graf, 1971). On the other hand, Singamsetti (1966) found that the analyzed value of β in a sediment laden turbulent jet falls in the range of 1.2 – 1.5 and increases with an increase in boundary Reynolds number. Similarly, Jobson and Sayre (1979) reported experimental evidence that β in an open channel flow depends on turbulent characteristics and suggested possibilities of both β < 1 and β > 1. It is, therefore, concluded that the value of β is around unity, possibly varying slowly with y on a large scale such as the flow depth, as the variation with y is governed by the variable turbulence characteristics. The profile of velocity u depends on the turbulence generated in the fluid having sediment in suspension. A usual assumption is the logarithmic velocity profile based on the von Karman constant, κ = 0.41. Bose and Dey (2007) showed, in general, that since the turbulent Reynolds shear stress τ dominates the viscous shear stress υ∂ u /∂y in the momentum equation of flow (where υ = kinematic viscosity of the fluid), a profile like (4) u = uτ ( A + B ln y ) holds in terms of the slowly varying variable in y. In Eq. (4), uτ is the shear velocity, A is the integration constant, and B is a constant, such that for the clear-water condition, B = 1/κ. For sediment laden flows, it is suggested that the value of κ diminishes by virtue of diminishing turbulence. Hence, B and likewise the integration constant A must slowly vary with y, in as much as, the suspended sediment decreases concentration in the upward direction. Assuming that in general, the bed has a small inclination, θ, with the horizontal, the Reynolds shear stress, τ, in the fluid motion is related to u by the Reynolds averaged Navier-Stokes (RANS) steady-state equations 1 ∂ p 1 dτ d 2u (5a) + ⋅ +υ 2 0 = g sin θ − ⋅ ρ ∂x ρ dy dy 0 = g cos θ − 1 ∂p ∂ − (v' v ') ⋅ ∂y ρ ∂y (5b) where g = gravitational acceleration; p = time-averaged hydrostatic pressure; and v′ = fluctuation of vertical velocity. The over-bar of a quantity denotes the time-averaged value of the quantity. Integrating Eq. (5b), one gets p = p0 ( x ) + ρ(gycosθ – v′v′ ), where p0 is an integration function. Substituting p in Eq. (5a), one gets d p0 dτ d 2u ∂ (6) + ρυ 2 = − ρg sin θ + − ρ (v' v') dy dy ∂x dx In the shear flow under consideration, the vertical Reynolds stress, v′v′ , and its streamwise gradient are negligible. If streamwise gradient of v′v′ is neglected, the left hand and right hand sides of Eq. (6) become functions of y and x, respectively, and, thus, equal a constant, say –ρk, where k = –ρ-1d p0 /dx + gsinθ ≥ 0. In a long channel, the streamwise pressure gradient d p0 /dx is very small and the flow International Journal of Sediment Research, Vol.24, No. 3, 2009, pp. 315–324 - 317 - becomes mainly gravity driven. For decreasing values of θ, k diminishes linearly with θ. For an infinitely long horizontal channel, it reduces to zero. The solution of Eq. (6) in general is B du (7) τ = τ 0 − ρky − ρυ = τ 0 − ρυuτ ≈ τ 0 − ρky y dy where τ0 = bed shear stress. The contribution of the term containing viscosity υ is negligible in a fully developed turbulent flow. The vertical distribution of the Reynolds shear stress, τ, is, therefore, linear having a small gradient with respect to the downward vertical direction for a case with small pressure gradient and bed slope. In a long horizontal channel, τ is almost a constant over a considerable depth as was experimentally observed by Bose and Dey (2007). Apparently, this expression for τ is valid at distances away from the wall, in the logarithmic turbulent layer of the fluid flow. Writing τ0 = ρuτ2 and using Eqs. (7), (4) and (2), Eq. (3) yields the form βu ⎛ y ⎞ (8) ε y ≈ τ y ⎜1 − ⎟ B ⎝ d⎠ where d = uτ2/k, which is a large quantity for a small value of k, uv = shear velocity. Equation (8) is similar to the one used by Rouse (1937) for his sediment concentration formula, where it was assumed that d = h, that is the flow depth. This could be argued in the case d p0 /dx ≠ 0 and θ ≠ 0 for uniform flows, because the bed shear stress, τ0, is balanced by the body force of a fluid column over a unit area, that is τ0 = –hd p0 /dx + ρghsinθ = ρhk. This leads to the equation ρuτ2 = ρhk or d = k. The argument that d = h, however, does not hold in the case of negligible pressure gradient in a horizontal channel, in which case also τ0 > 0, because of the presence of the viscous sub-layer overlain by the buffer and the turbulent flow layers. Moreover, a consequence of the assumption that d = h is that εy vanishes near the free surface. This particular result, however, does not agree with the observation of natural rivers, where εy is found to be nonzero at the free surface (Chien and Wan, 1999). They argued that the diffusivity of fluid (or momentum exchange coefficient) is zero at free surface, but the diffusivity of sediment (or sediment exchange coefficient) is finite there. This is theoretically ascribed to the fact that the momentum exchange takes place due to Reynolds shear stress, τ = –ρ u ′v′ , or the turbulence correlation, u ′v′ /[( u ′u ′ )0.5( v′v′ )0.5], in which v′ is much smaller than u′, where u′ denotes the fluctuation of the streamwise velocity. Hence, even if u′ and v′ are uncorrelated, some sediment can still be transported by the turbulence fluctuations (u′, v′). For Eq. (8), however, if d > h, εy > 0 implying that the sediment diffusion could exist even near the free surface of the flow. Inserting Eq. (8) into Eq. (1), the conservation equation becomes ⎛ d ⎞ dy dC ⎟⎟ (9) = −ζ ⎜⎜ C ⎝d − y⎠ y where ζ = wsB/(βuτ). For a constant ζ, an integration of Eq. (9) yields the expression ζ C ⎛ a d − y⎞ ⎟ (10) = ⎜⎜ ⋅ Ca ⎝ d − a y ⎟⎠ where a = reference level; and Ca = reference concentration at y = a. The level, a, may be considered at the bottom of the turbulent layer, assuming a constant C = Ca within this thin layer. In this context, it is pertinent to point out that Rouse’s (1937) formula can be recovered, if d is replaced by the flow depth h. For applying Eq. (10) to a given sediment laden flow, one needs to estimate two parameters ζ and d from experimental data. Such an estimation is, however, difficult for the form of Eq. (10). An alternate approximate procedure is to set ζ[d/(d – y)] as a composite function ζ(y), which reduces to ζ in the case d → ∞ or k → 0. Such an approach has the advantage of accounting for similar dependency of the parameters ws, B, and β on y due to the variable concentration of suspended sediment in the vertical direction. Assuming ζ(y) to be a slowly varying function of y, one can separate a constant mean part, ζ0, from ζ(y), and decompose it as - 318 - International Journal of Sediment Research, Vol. 24, No. 3, 2009, pp. 315–324 ζ ( y ) = ζ 0 + η ( y ), ζ 0 > 0 (11) where η is the small perturbation part of ζ that slowly varies with y. Inserting Eq. (11) into Eq. (9) and integrating, one gets the following equation for the suspended sediment concentration: −ζ −ζ ⎡ y η ( y) ⎤ ⎛ y ⎞ ⎡ y η ( y) ⎤ C ⎛ y⎞ (12) dy ⎥ dy ⎥ ≈ ⎜ ⎟ ⎢1 − ∫ = ⎜ ⎟ exp ⎢− ∫ Ca ⎝ a ⎠ ⎦ ⎦ ⎝a⎠ ⎣ a y ⎣ a y Equation (12) contains an unknown integral that may be approximated by assuming a functional form δ0 ⎛ y⎞ (13) η ( y ) = −η 0δ 0 ⎜ ⎟ ⎝a⎠ where η0 > 0 and δ0 ≥ 0 are small constants. As stipulated, the function slowly varies with y due to smallness of δ0. Performing the integration in Eq. (12), one obtains the approximate form δ ζ −ζ −ζ C ⎛ y⎞ ⎡ ⎛ y⎞ ⎛ y⎞ ⎤ ⎛ y⎞ (14) = ⎜ ⎟ ⎢1 + η 0 ⎜ ⎟ ⎥ = ⎜ ⎟ + η 0 ⎜ ⎟ Ca ⎝ a ⎠ ⎢⎣ ⎝a⎠ ⎝ a ⎠ ⎥⎦ ⎝ a ⎠ where ζ1 = –ζ0 + δ0. The constants η0 and ζ1 are to be estimated from experimental observations. The leading term of Eq. (14) has the same form as that obtained by Montes and Ippen (1973). The approximation in Eq. (14) is expected to accurately hold for a long horizontal channel, because k → 0 and d → ∞ in such cases. Deviation from such ideal condition may result in a discrepancy between theoretical and experimental results for higher values of y that is near the free surface. This conclusion follows from the definition of ζ(y) with nonzero k or finite d, particularly when d is close to h. An equation more compact than Eq. (14) can alternatively be attempted by assuming the slowly varying function 0 0 0 0 0 1 ζ ⎛ y⎞ (15) ⎝a⎠ where |ξ| < 1 and λ are constants. Equation (9) then yields ⎧⎪⎛ y ⎞ζ ⎫⎪ C = exp[− λ ]⎨⎜ ⎟ − 1⎬ (16) Ca ⎪⎭ ⎪⎩⎝ a ⎠ However, it transpires that Eq. (16) is not a practical proposition, as the estimation of λ and ξ is rather difficult from experimental data. ζ ( y ) = λε ⎜ ⎟ 3 Determination of constants An iterative least-squares method of fitting of experimental data, C/Ca versus y/a, in Eq. (14) is applied. Dropping the perturbation term, the starting value of ζ0 is given by ln(C / Ca ) ζ0 = (17) ln(a / y ) For the experimental data, that are described in the last paragraph of this section, the estimation of ζ0 given in Eq. (17) is found to have a small systematic bias and error. This means that it varies slowly with y and a mean value may be taken as a starting approximation. Using this approximation, Eq. (14) can now be rewritten as ζ0 C ⎛ y⎞ ⎛ y⎞ ⎜ ⎟ − 1 = ln η 0 + δ 0 ln⎜ ⎟ Ca ⎝ a ⎠ ⎝a⎠ (18) From Eq. (18), the least-squares approximation of η 0 and δ0 can be obtained for given experimental data. In practice, by writing a computer code, incremental values to determine ζ0 were given starting from a slightly lower value of the average obtained earlier and picking the values of η0, ζ0 and ζ1 yielding the minimum least-squares residual for Eq. (18). In Figs. 1 and 2, the theoretical curves obtained from Eq. (14) are presented for the experimental data of Vanoni (1946) and Coleman (1986), respectively. There is some discrepancy between the experimental data and theoretical curves near the free surface in the case of Vanoni (1946) for ζ = 0.66, 0.52, and 0.34. International Journal of Sediment Research, Vol.24, No. 3, 2009, pp. 315–324 - 319 - This is due to change from convexity to concavity in the data distribution near the free surface, while the theoretical curves are strictly concave. The estimated values of constants (η0, ζ0, and ζ1) are listed in Table 1. In general, the values of the constants progressively diminish with decrease in ζ in the experimental data of Vanoni (1946) and Coleman (1986). The agreement between experimental data and curves obtained from Eq. (14) is satisfactory. Fig. 1 Comparison of C/Ca versus y/a obtained from Eq. (14) and Rouse (1937) using the experimental data of Vanoni (1946) - 320 - International Journal of Sediment Research, Vol. 24, No. 3, 2009, pp. 315–324 Fig. 2 Comparison of C/Ca versus y/a obtained from Eq. (14) and Rouse (1937) using the experimental data of Coleman (1986) Source Vanoni (1946) Coleman (1986) Table 1 Estimated values of constants for Eq. (14) fitted to the data Estimated η0 Estimated ζ0 ζ 1.46 0.721 –2.086 1.03 0.035 –1.589 0.89 0.299 –1.344 0.66 0.284 –0.988 0.52 0.214 –0.8 0.34 0.08 –0.607 0.712 2.118 –0.939 0.676 0.057 –0.848 Estimated ζ1 –2.443 –0.564 –1.432 –0.976 –0.809 –0.197 –1.178 –0.541 4 Suspended-load transport rate The quantity of interest in suspended-load theory is the suspended-load transport rate, qs (in volume per unit time and width), that is given by h qs = ∫ Cudy (19) a where C is of the form of Eq. (14). Regarding u , experiments show that it follows the logarithmic law of the form of Eq. (4) with B = 1/κ in the near-bed one-fifth of the flow depth (excluding the thin viscous sub-layer in the vicinity of the bed and a transition layer). For low concentration solutions, say C < 0.1, one may take B = 1/κ, when the von Karman constant, κ = 0.41. In terms of the depth-averaged velocity, U, the logarithmic law for the velocity distribution can be written as u U 1 1 ⎛ y ⎞ , a ≤ y ≤ h/5 (20) = + + ln ⎜ ⎟ uτ uτ κ κ ⎝ h ⎠ However, above the depth of h/5, there is an increase in velocity in a deep flow by a small amount. This increase in velocity is given by Coles’ law (Hinze, 1975), and the velocity profile is given by ` u = U + 1 − Π + 1 ln ⎛⎜ y ⎞⎟ + 2Π sin 2 ⎛⎜ π ⋅ y ⎞⎟ , h/5 < y ≤ h (21) uτ uτ κ κ κ ⎝ h ⎠ κ ⎝2 h⎠ where Π = Coles’ wake parameter whose value is approximately 0.09. In order to make further progress with Eq. (19), one can introduce the approximation International Journal of Sediment Research, Vol.24, No. 3, 2009, pp. 315–324 - 321 - ⎛π y⎞ ⎛ y⎞ sin 2 ⎜ ⋅ ⎟ ≈ 0.957 + 0.786 ln ⎜ ⎟ , h/5 < y ≤ h ⎝2 h⎠ ⎝h⎠ (22) Equation (21) then becomes u U 1 0.914Π 1 ⎛ y⎞ = + + + (1 + 1.572Π ) ln ⎜ ⎟ uτ uτ κ κ κ ⎝h⎠ (23) The form of Eq. (23) reiterates the fact that the velocity profile follows a logarithmic law. However, the new von Karman constant becomes approximately κ(1 – 1.572Π), which implies that the von Karman constant decreases from its traditional value of 0.41. This is in agreement with the observations of Einstein and Chien (1955) and Elata and Ippen (1961) regarding the diminution of κ due to increase in sediment concentration. Inserting Eqs. (14) and (23) in Eq. (19), the integral is evaluated as 1−ζ 1−ζ ⎤ ⎤ ⎫⎪ ⎛ U 1 1 h ⎞⎧⎪ 1 ⎡⎛ h ⎞ η 0 ⎡⎛ h ⎞ qs = ⎜⎜ + − ln ⎟⎟⎨ 1 − + ⎢⎜ ⎟ ⎢⎜ ⎟ − 1⎥ ⎬ ⎥ uτ Ca a ⎝ uτ κ κ a ⎠⎪⎩1 − ζ 0 ⎣⎢⎝ a ⎠ ⎦⎥ 1 − ζ 1 ⎣⎢⎝ a ⎠ ⎦⎥ ⎪⎭ 0 + 1−ζ 1 ⎡ 1 ⎛h⎞ ⎜ ⎟ ⎢ κ ⎣⎢1 − ζ 0 ⎝ a ⎠ 0 1−ζ 1 ⎛ h η ⎛h⎞ 1 ⎞ 1 ⎜⎜ ln − ⎟⎟ + + 0 ⎜ ⎟ 2 − − − ζ1 ⎝ a ⎠ a ζ ζ 1 ( 1 ) 1 0 ⎠ 0 ⎝ Π⎛ h ⎞⎡ 1 ⎛ h ⎞ ⎜ ⎟ ⎜ 0.914 − 1.572 ln ⎟ ⎢ κ ⎝ a ⎠ ⎣⎢1 − ζ 0 ⎝ a ⎠ 1−ζ 0 + + 1.572 1 1−ζ 0 1 Π ⎡ 1 ⎛h⎞ ⎛ h ⎜ ⎟ ⎜⎜ ln − ⎢ κ ⎣⎢1 − ζ 0 ⎝ a ⎠ ⎝ a 1 − ζ 0 ⎛ h η0 ⎤ 1 ⎞ ⎟⎟ + ⎜⎜ ln − 2 ⎥ ⎝ a 1 − ζ 1 ⎠ (1 − ζ 1 ) ⎦⎥ 1−ζ 1 η ⎛h⎞ 1 ⎞ ⎛ ⎜1 − 1−ζ ⎟ + 0 ⎜ ⎟ ⎝ 5 ⎠ 1− ζ1 ⎝ a ⎠ 0 1−ζ 0 ⎞ 1 ⎛ h ⎞ ⎟⎟ − ⎜ ⎟ ζ 0 ⎝ 5a ⎠ 1 − ⎠ 1 ⎞⎤ ⎛ ⎜1 − 1−ζ ⎟⎥ ⎝ 5 ⎠⎦⎥ ⎛ h 1 ⎜⎜ ln − a ζ0 5 1 − ⎝ 1 ⎞⎤ ⎟⎟⎥ ⎠⎦⎥ 1−ζ 1−ζ Π ⎡ 1 ⎛h⎞ ⎛ h 1 ⎞ 1 ⎛ h ⎞ ⎛ h 1 ⎞⎤ ⎟⎥ ⎟⎟ − (24) − ⎜ ⎟ ⎜⎜ ln ⎜ ⎟ ⎜⎜ ln − ⎢ κ ⎢⎣1 − ζ 1 ⎝ a ⎠ ⎝ a 1 − ζ 1 ⎠ 1 − ζ 1 ⎝ 5a ⎠ ⎝ 5a 1 − ζ 1 ⎟⎠⎥⎦ Equation (24) is of elementary form. On the other hand, Rouse’s (1937) theory leads to a result in terms of Gamma functions (Einstein and Chien, 1955; Guo and Wood, 1995). Alternatively, as at the free surface the logarithmic law of velocity with the von Karman constant κ = 0.41 does not hold, the following equation makes it possible to estimate the velocity near the free surface (Chien and Wan, 1999): + 1.572η 0 1 1 u max − u 2 ⎛h− y⎞ = arctanh ⎜ ⎟ uτ κ ⎝ h ⎠ 1.5 (25) where u max = maximum value of u , which occurs at y = h. Equation (19) for qs can be evaluated in this case as well, but in terms of complicated algebraic and arctanh functions. While comparing the suspended-load estimation given in Eq. (24) with the existing equations of Einstein (1950) and Brooks (1963) that are commonly used, Eq. (24) does not require any implicit integrals to be solved numerically or by the help of a chart, as must be done for solving the equations given by Einstein (1950) and Brooks (1963). 5 Conclusions The advection-diffusion approach and associated logarithmic law of flow velocity have been revisited. The logarithmic law implies that the Reynolds shear stress is a linear function of height above the bed, and the resulting volumetric concentration of sediment can be best approximated by a sum of power laws of the height above the bed. An equation is derived for the suspended-load transport rate and as shown to agree well with the available data. Acknowledgements The authors are grateful to the anonymous reviewers for their suggestions that helped to improve the content of the paper. 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International Journal of Sediment Research, Vol.24, No. 3, 2009, pp. 315–324 - 323 - Notations The following symbols are used in this paper: A = a constant; a = reference level; B = a constant; b = a coefficient; C = concentration; = reference concentration at level a; Ca d = uτ2/k; g = gravitational acceleration; h = flow depth; k = a constant; = Nikuradse’s equivalent sand roughness; ks p = time-averaged hydrostatic pressure; p0 = integration function; qs = suspended-load transport rate; u = time-averaged streamwise velocity; = fluctuation of streamwise velocity; u′ u max uτ v′ ws y α β δ0 ε εy η η0 κ λ υ Π θ ρ τ τ0 ζ ζ0 ζ1 - 324 - = = = = = = = = = = = = = = = = = = = = = = = maximum value of u ; shear velocity; fluctuation of vertical velocity; terminal fall velocity of sediment particles; vertical distance from bed; an exponent; factor of proportionality; a constant; turbulent diffusivity or eddy viscosity; sediment diffusivity; small perturbation part of ζ; a constant; von Karman constant; a constant; kinematic viscosity; Coles’ wake parameter; bed inclination; mass density of fluid; Reynolds shear stress; bed shear stress; Schmidt number; constant mean part of ζ; a constant. International Journal of Sediment Research, Vol. 24, No. 3, 2009, pp. 315–324