Effective Nikuradse Roughness on the Mobile Plan Bed

Nikuradse roughness (ks) is very important in the sediment transport prediction because it is related to the evaluations of the velocity distribution, shear stress and erosion depth. Dimensionless Nikuradse roughness (ks/D, where D is the sediment diameter) is usually given 1–2.5 on the immobile plan bed or at low shear stress. But it behaves differently on the mobile plan bed at high shear stress with much sediment picked up to movement when the Shields parameter (Θ) is larger than 0.8–1.0. The effective Nikuradse roughness on the mobile plan bed was derived indirectly from the erosion depth correlated to the mobile plan bed thickness considering the mass conservation in the present study. The proposed erosion depth confirmed the relation to the Shields parameters with an extra factor consisting of suspended sediment and its damping to turbulence. The decrement of the erosion depth caused by the increment of the sediment diameter at large shear stress was obtained, which was usually absent in classical empirical formulas based on the bedload theory. Good agreement with experiments was achieved by the present prediction of the Nikuradse roughness, erosion depth and sediment transport rate. Discussion was mainly focused on the prediction improvement caused by considering the impact of suspended sediment and its damping to turbulence.


Introduction
Sediment transport in sheet flow (Θ≥0.8-1.0) has been concerned very much due to many engineering problems caused by high stress, hyper concentrated concentration and large sediment transport rate. The sediment is mainly moved in a very thin mobile layer close to the initial bed in sheet flow transport, where the relevant mobile plan bed thickness and flow velocity are crucial to predicting the exchange of energy, nutrients, sediment and other species (Liu et al., 2017). Many studies are conducted under steady sheet flow (Sumer et al., 1996, Wilson, 2005, and unsteady oscillatory sheet flow (O'Donoghue and Wright, 2004a;Dong et al., 2013). The sediment transport rate is usually obtained by the product of the mobile plan bed thickness and averaged velocity, both of which are related to the Nikuradse roughness. So the prediction of the sediment transport rate largely depends on the accuracy of the Nikuradse roughness.
The mobile plan bed thickness is usually expressed by the erosion depth or sheet flow layer thickness. Erosion depth is defined as the distance between initial undisturbed bed level and the top of non-moving sediment bed level, below which is the immobile layer. Sheet flow layer thickness is commonly defined as the distance between the top of the non-moving sediment bed level and the level with sediment volumetric concentration (the ratio of the sediment volume to the total volume, Chen et al., 2018b) equaling 0.08 where the averaged particle-spacing is almost one particle diameter and particle interactions are negligible. Erosion depth and sheet flow layer thickness are inherently the same parameters due to the mass conservation (Chen et al., 2013). Since the mobile plan bed thickness is very critical to understand the sediment transport in estuary and coastal areas, a lot of works have been carried out, and are divided into models predicting the erosion depth (Flores andSleath, 1998, O'Donoghue andWright, 2004a), sheet flow layer thickness (Sumer et al., 1996;Lanckriet and Puleo, 2015), both (Dohmen-Janssen et al., 2001) and even containing the mass conservation (Chen et al., 2013). Linear relationship between dimensionless mobile plan bed thickness and Shields para-meter is acceptable widely, but it usually lacks of suspension sediment effect. Unrealistic increment of mobile plan bed thickness is predicted by increasing diameter at high shear stress (Dohmen-Janssen et al., 2001). While the sheet flow layer thickness for fine sediment is found about twice large as that of coarse sediment in O' Donoghue and Wright (2004b) under the same oscillatory flows, Dong and Sato (2010) and Dong et al. (2013) interpolate mobile plan bed thickness expressions by diameters. These studies imply that suspended sediment effect should be considered to reflect the real physics process at high shear stress.
Associated with the mobile plan bed thickness, Nikuradse roughness is very important in the prediction of the mobile plan bed thickness and velocity distribution. The integration of total two-phase momentum equation (Chen et al., 2011) can be used to obtain the bottom shear stress and the effective roughness as Nielsen and Guard (2010) when the velocity profile is known. However it would not be a convenient method because measurement difficulties exist in the rather thin sheet flow layer of minimeters to centimeters, where contained bed level varies with time during the wave period (O'Donoghue and Wright, 2004a), and high sediment concentration gradient requires accurate position of measurement probes. Instead much theory work is done based on the bedload sediment transport theory. With the compatibility of velocity at the top of sheet flow layer by rough-boundary equations, Wilson (1989) and later Wilson (2005) proposed a linear relationship between the Nikuradse roughness, sheet flow layer thickness and Shields parameter. Because the friction factor used to evaluate the Shields parameters in sheet flow also includes Nikuradse roughness, the predictions need iterations. Linear relation is also used by Ribberink (1998) with periodic averaged Shields parameter to predict roughness in oscillatory sheet flow, and suspension sediment is expected to have a minor influence on roughness. Predicted roughness is related little with the sediment diameter even if erosion depth and suspended sediment change at high shear stresses. As obvious sediment size effect is shown in experimental studies (Dohmen-Janssen et al., 2001;O'Donoghue and Wright, 2004b), Nikuradse roughness is interpolated (Dong et al., 2013) according to the sediment size to predict the sediment transport rate. To improve the accuracy, further work is necessary when the suspended sediment effect is obvious.
The present study tried to develop a method to predict effective Nikuradse roughness for sheet flow transport at high shear stress when the suspended sediment effect should not be neglected. Erosion depth is derived in Section 2, with the consideration of the suspended sediment and the damping of turbulence. The effective Nikuradse roughness was estimated by the relation to the sheet flow thickness, which in turn was the erosion depth by the mass conservation. Predictions of Nikuradse roughness, erosion depth and sediment transport rate are shown in Sections 3 and 4 when the suspended sediment and its damping to turbulence were considered.

Method derivation
Definition of the domain is shown in Fig. 1, where δ is the sheet flow layer thickness; Δ is the erosion depth; α is the sediment volumetric concentration; α m denotes the maximum value; y is the vertical direction; y=0 is set at the initial bed surface; h is the water depth; θ is the incline angle.

Estimation of erosion depth
First we consider the steady flow in an open channel. When the shear stress is very small that sediment is mainly transported as the bedload, the vertical inter-granular stress τ y at the immobile layer surface is balanced by the submerged specific gravity of sediment: where ρ is the water density; S is the specific gravity of sediment; g is the gravitational acceleration. In Bagnold (1954), the horizontal inter-granular stress at the immobile bed surface is τ x =τ y tanϕ, where ϕ is the sediment internal friction angle. Thus, τ x is balanced by the shear stress τ x = ρfU 2 /2 = ρu * 2 , where f is the wave friction factor; U is the free stream velocity; u * is the friction velocity. D does not appear in Eq.
(2) because the suspended sediment is neglected. In engineering, the commonly used α m =0.6 and ϕ=30° can be applied. With the definition of the Shields parameter (Θ), the dimensionless erosion depth in Eq.
(2) becomes The actual coefficient in Eq. (4) is 2.9. Here the approximate 3 is applied by considering the suggestion in Flores and Sleath (1998) based on the bedload theory, when the pressure gradient and inertia terms are negligible in unsteady oscillatory flows. CHEN Xin et al. China Ocean Eng., 2018, Vol. 32, No. 6, P. 730-736 731 2.2 Effect of the turbulence suspension The bedload-dominated Eq. (4) is inaccurate for high shear stress condition because of the sediment suspension. In Eq. (4) the increment of D leads to constant or even increased Δ due to f (Dohmen-Janssen et al., 2001). Notice the major suspended mechanism is the submerged gravity being balanced by the drag force, or particle falling down being balanced by the turbulent suspension. In present study we consider the suspended sediment effect with the suspension index w/u * (Rouse, 1937); where w is the sediment fall velocity. This index represents the suspended sediment amount in many concentration distribution formulas. Sumer et al. (1996) suggested that sediment is mainly transported as the bedload when the index w/u * >1. To account for the effect of suspended sediment, the effective erosion depth in Eq. (4), found by balancing the weight per unit area of mobile sediment with the fluid-sediment and inter-granular stresses, is assumed to be Δw/u * when w/u * <1. By replacing it to the left of Eq. (4), the effective erosion depth becomes (5) is in agreement with the energy balance between the suspension cost and shear stress generation ρ(S-1)gwΔ ητ x u * , as suggested in Chien and Wan (1999). η is the energy efficiency for suspended sediment. Furthermore, we consider the effect of large amount of suspended sediment to the turbulence when the shear stress is large, i.e. the damp of turbulence by suspended sediment. Under the same flows, a small dimensionless fall velocity index w[(S-1)gD] -0.5 corresponds to a large amount of suspended sediment and large damping of turbulence (Sumer et al., 1996). In van Rijn (1993), w[(S-1)gD] -0.5 <1 is valid for small sediment, and w[(S-1)gD] -0.5 ≈1 is valid for large sediment. Being applied to the right side of Eq. (5), the modification to the effective erosion depth caused by the turbulence damping of the suspended sediment is introduced with Eq. (6) considering suspended sediment and its damping to turbulence is valid at Θ>1 and Δ/D>Θ/(α m cosθtanϕ) when suspended sediment is obvious. Considering sheet flow appears at Θ>0.8-1, so it can be treated as a modification to sheet flow. Finally the total expression is Eq. (7) has been fitted by Chen et al. (2013) in sheet flow for large shear stress. Comparing with the results from the bedload theory, Eq. (7) is still the classical function of Shields parameter. Due to the rightmost extra factor denoting the contribution of suspended sediment and its damping to turbulence, the realistic decrement of the erosion depth caused by the increased sediment diameter at large shear stress is obtained and in agreement with Dohmen-Janssen et al. (2001) and Dong and Sato (2010).

Relation with the erosion depth
Effective roughness over a mobile plan bed is different from a fixed bed when large amount of sediment are picked up. At high shear stress Θ>1, Effective Nikuradse roughness is linear with sheet flow layer thickness δ as that suggested by Wilson (1989). Considering mass conservation, the sediment concentration profile can be approximated by an exponential or power law (Chen et al., 2013) in unsteady oscillatory sheet flow. The top of sheet flow layer is located at y=Δ by set α=0.08 and α m =0.6. Thus δ=2Δ can be obtained (Chen et al., 2018b(Chen et al., , 2018c. Effective Nikuradse roughness is also linear with erosion depth at high shear stress. Wilson (1989) proposed ks/D≈0.5δ/D, and later (Wilson, 2005) proposed ks/D≈0.33δ/D, on the basis of the bedload transport theory where turbulence suspension is expected to have a minor influence. The parameter difference before δ/D may be generated by the variations of the actual α m and ϕ. But a reprehensive parameter is suggested for a convenient application in engineering. A simply averaged relation should be given to investigate the suspended sediment and its damping to turbulence in ks/D and Δ/D, i.e., ks/D≈0.415δ/ D=0.83Δ/D. Thus, ks/D=2.5Θ 1.5 /cosθ is obtained according to Eq. (7). The simple average value cannot generate an error that is higher than one order of magnitude because the actual α m is generally in the range of 0.5-0.65 and ϕ is generally in the range of 12°-40°.
At low shear stress Θ≤1, the suspended sediment effect is neglected in the erosion depth and ks/D can be given constant of 2.5 as that in O' Donoghue and Wright (2004b) and Liu and Sato (2005), with the continuity at Θ=1. Thus  .
Erosion depth and sheet flow layer thickness are not constants in an unsteady oscillatory flow induced by wave. The periodic averaged Θ is much realistic (Ribberink, 1998) and thus, where the operate '−' denotes the periodic average. Eq. (9) is still a function of the Shields parameter, with an extra factor denoting the suspended sediment and its damping to turbulence, which is in agreement with Wilson (2005) conclusion of a rapid increment of roughness at high shear stress. In the present study, such increment is caused by the rapid increment of Δ denoted by . Θ f is necessary for in unsteady oscillatory flows in Eq. (4), which is given by Ribberink (1998)

Θ Θ
where A is the orbital excursion. Because ks contains and contains f, the iteration is necessary in Eq. (10). For convenience, an explicit approximation to Eq. (10) is given in Fig. 2.

Effective Nikuradse roughness
Because the effective Nikuradse roughness only behaves differently at high shear stress, the following results only focus on the plan beds, especially sheet flows when ripples are swept away. The studies for steady and unsteady oscillatory sheet flows were separated due to different method for Θ.

Steady flows
Collected plan bed data were under a wide range of sediment and flow conditions (see Table 1). Other formulas used for comparison included: (1) linear relation of Wilson (1989), Ribberink (1998) and Wilson (2005); (2) high order power law function obtained by Yalin (1992) with mathematical fitting in steady flow. Validation was first shown in Fig. 3 for 192 cases driven by the pressure gradient in a closed-conduit on a plan bed (Wilson, 1989(Wilson, , 2005Wilson and Nnadi, 1990). Averaged slope of ks/D against Θ seemed increased with the increase of Θ. The increment of slope was caused by the rapid increment of the erosion depth and suspended sediment amount as represented by Θ 0.5 in Eq. (9). In the range of 1≤Θ≤4, Eq. (9) was somewhat coinciding with Yalin (1992) fitting, and the coefficient of 5 (Wilson, 1989) was somewhat larger for most data. While in the range of 4≤Θ≤8, the coefficient of 5 was somewhat smaller for most data, and the coefficient of 3.3 (Wilson, 2005) was worse than 5. Data scatters would be caused by the measurement uncertainty. However, only Eq. (9) passed the center of them with extra factor Θ 0.5 denoting the suspended sediment and its damping to turbulence.
The large shear stress and incline angle could be seen in estuaries, so the validation in incline flumes is shown in Fig. 4.

Fig. 2. Friction factor approximated by new roughness.
CHEN Xin et al. China Ocean Eng., 2018, Vol. 32, No. 6, P. 730-736 log-log coordinate covering three orders of magnitudes. Factor 1/cosθ was in agreement with the observed increasing slope with the increased θ . Good agreement between the prediction and experiment for Eq. (9) could be seen again. For a quantitative comparison, Table 2 shows the average logarithm errors defined as Er={∑[log 2 (PERD/EXP)] 2 /n} 1/2 , and P 2 was the percentage of the numbers for |log 2 (PERD/EXP)|<1 as data between the twice-derivation error dashed-dotted lines in Fig. 4, where 'PERD' and 'EXP' respectively denoted the prediction and experiment. Whether on the plan bed or in the incline flume, the present averaged logarithm error was the smallest, and the percentage of data between twice-derivation error lines was the largest.
3.2 Unsteady oscillatory sheet flows There were quite a little direct data available about the effective Nikuradse roughness in unsteady oscillatory sheet flows. Associated with the effective Nikuradse roughness (ks=0.83Δ), the erosion depth data in unsteady oscillatory sheet flows predicted by Eq. (7) were investigated instead. Collected in Table 3 were 87 oscillatory sheet flow cases, of which some were under combined wave and current (Dohmen-Janssen et al., 2001). Data covered U m =1.68-2.20 m/s, wave period T=2.0-7.5 s, D=0.13-0.41 mm and S = 2.64-2.65.
The effective Nikuradse roughness was validated by the relation to the erosion depth including the suspended sediment and turbulence damping by the suspended sediment denoted by max(Θ 0.5 , 1). Since the comparison between different formulas had been made (Figs. 3-4), Fig. 5 compared the predictions of the maximum erosion depth by Eq.
(4) and Eq. (7). An underestimation of the experiments at high shear stress for Θ>3 (values which are larger than 9) was seen when max(Θ 0.5 , 1) was neglected (Fig. 5a). While the prediction was in good agreement with the previous experiments in Fig. 5b with factor max(Θ 0.5 , 1). The corresponding errors were Er=0.72, P 2 =82.76% in Fig. 5a; and Er=0.53, P 2 =96.55% in Fig. 5b. Prediction was improved much by accounting for the contribution of suspended sediment and its damping to turbulence at high shear stress.

Sediment transport rate
According to Wilson (1984), the velocity at the top of the sheet flow layer was found about 8.2u * . By assuming the average sediment velocity about its half value, the dimensionless sediment transport rate in the sheet flow layer is nearly Considering the critical Shields parameter Θ cr into Eq. (12), the transport rate is Sediment transport rate decreases with the increment of the sediment diameter at large shear stress in Eq. (13). If the contribution of max(Θ 0.5 , 1) is neglected, Eq. (13) reverts to the classical bedload form: In steady flows (Table 4), collected 230 plan bed data for validation were under S=1. 14-2.68, D=0.19-10.5 mm and Θ=0.09-7.71. Predicted Φ against Θ is shown in Fig. 6, where solid and dashed lines were computed by Eq. (13) and Eq. (14). Prediction uncertainty could be caused by the applied erosion depth and averaged velocity, so Wilson (1984) velocity and the present erosion depth and effective Nikuradse roughness were confirmed accurate enough by the good agreement between experiments and Eq. (14). The dashed Eq. (13) without max(Θ 0.5 , 1) seemed smaller than data at large shear stress Θ>1, which was probably the contribution of suspended sediment and its damping to turbulence covered by Θ 0.5 . Larger sediment transport rate would be caused by the contribution due to larger erosion depth. If the Eq. (13) was still applied for sediment transport rates at large shear stress, best parameters should be larger than the original 7.2, as that seen in Ribberink (1998) andNielsen (2006).
In unsteady oscillatory sheet flows, sediment transport rates of the second-order Stokes flow from O'Donoghue and Wright (2004b) were applied for validation. Phase-lag effect is not obvious for the medium (D=0.27 mm) or coarse (D=0.46 mm) sediment. Fig. 7 shows the instantaneous Φ against t/T, where t is the time. Θ m =2.2 in Fig. 7a is larger than 1.5 in Fig. 7b. Difference between solid and dashed lines is larger in Fig. 7a than that in Fig. 7b due to Θ 0.5 . Both predictions in Fig. 7b agree with experiments very well because Θ m =1.5 is not large enough and suspended sediment effect is not obvious. But for larger Θ m =2.2 in Fig. 7a when more sediment is in suspension, solid line is better due to the consideration of suspended sediment and its damping to turbulence. However, the present Eq. (14) is still not enough for instantaneous Φ in large phase-lag case as D=0.13 mm in O' Donoghue and Wright (2004b), because the phase-residual or phase-shift in erosion depth have not been presented in Eq. (7). Phase-residual can generate an extra offshore Φ (Chen et al., 2018a(Chen et al., , 2018c, and phase-shift makes Φ fall behind U.

Conclusions
Erosion depth for the sheet flow transport was obtained with the consideration of the suspended sediment and the damping of turbulence. The effective Nikuradse roughness on the mobile plan bed at high shear stress was obtained by a linear relation to the proposed erosion depth due to the mass conservation. The dimensionless erosion depth and effective Nikuradse roughness at high shear stress were the classical functions of the Shields parameters, with its power 1/2 denoting the suspended sediment and its damping to turbulence. Hence, the erosion depth and effective Nikuradse roughness at large shear stress decrease realistically by the Table 4 Plan bed data for the sediment transport rate validation in steady flows Author S D (mm) Θ Number Guy et al. (1966) 2.65 0.19-0.93 0.091-1.12 48 Nnadi and  1.14-2.67 0.67-3.94 0.806-7.71 105 Smart (1984) 2.67-2.68 2.0-10.5 0.098-3.35 77 Fig. 6. Prediction of the sediment transport rate in steady flows. Fig. 7. Prediction of the sediment transport rate in unsteady oscillatory sheet flows.
CHEN Xin et al. China Ocean Eng., 2018, Vol. 32, No. 6, P. 730-736 increment of the sediment diameter. The sediment transport rate in the sheet flow layer, which could revert to the same basic part as the classical bedload type formula, can also be driven by the erosion depth associated with the effective Nikuradse roughness. Results were classified into steady and unsteady oscillatory sheet flows due to different methods for the Shields parameter. Iteration is necessary for the effective Nikurasde roughness and periodic averaged Shields parameter in unsteady oscillatory sheet flows. Predictions were in good agreement with widely collected plan bed data under both steady and unsteady oscillatory sheet flows for the present effective Nikuradse roughness, erosion depth and sediment transport rate. The realistic variations of the effective Nikuradse roughness, erosion depth and sediment transport rate caused by the sediment diameter at large shear stress can be obtained. The present erosion depth and effective Nikuradse roughness considering the contribution of the suspended sediment and its damping to turbulence were confirmed accurate enough.