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A numerical study for boundary layer current and sheet flow transport induced by a skewed asymmetric wave
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Xin CHEN1, *, Zichao ZHANG2, Fujun WANG1
Acta Oceanologica Sinica | 2018, 37(9) : 82 - 89
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Acta Oceanologica Sinica | 2018, 37(9): 82-89
• Marine Geology •
A numerical study for boundary layer current and sheet flow transport induced by a skewed asymmetric wave
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Xin CHEN1, *, Zichao ZHANG2, Fujun WANG1
Affiliations
  • 1 Beijing Engineering Research Center of Safety and Energy Saving Technology for Water Supply Network System, China Agricultural University, Beijing 100083, China
  • 2 School of Electric Power, North China University of Water Resources and Electric Power, Zhengzhou 450045, China
Published: 2018-09-25 doi: 10.1007/s13131-018-1267-4
Outline
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An analytical model with essential parameters given by a two-phase numerical model is utilized to study the net boundary layer current and sediment transport under skewed asymmetric oscillatory sheet flows. The analytical model is the first instantaneous type model that can consider phase-lag and asymmetric boundary layer development. The two-phase model supplies the essential phase-lead, instantaneous erosion depth and boundary layer development for the analytical model to enhance the understanding of velocity skewness and acceleration skewness in sediment flux and transport rate. The sediment transport difference between onshore and offshore stages caused by velocity skewness or acceleration skewness is shown to illustrate the determination of net sediment transport by the analytical model. In previous studies about sediment transport in skewed asymmetric sheet flows, the generation of net sediment transport is mainly concluded to the phase-lag effect. However, the phase-lag effect is shown important but not enough for the net sediment transport, while the skewed asymmetric boundary layer development generated net boundary layer current and mobile bed effect are key important in the transport process.

analytical model  /  boundary layer current  /  sediment transport  /  sheet flow  /  skewed asymmetric wave
Xin CHEN, Zichao ZHANG, Fujun WANG. A numerical study for boundary layer current and sheet flow transport induced by a skewed asymmetric wave[J]. Acta Oceanologica Sinica, 2018 , 37 (9) : 82 -89 . DOI: 10.1007/s13131-018-1267-4
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1 Introduction
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Oscillatory flows induced by short waves in nearshore zones are velocity-skewed with peaked narrow crest and flat wide trough in wave shoaling, and acceleration-skewed with steep frontal slope and gentle rear slope in a surf zone. In a sheet flow, a shear stress is large and a sediment concentration is high. The sediment transport is very essential under the velocity-skewed and acceleration-skewed (skewed asymmetric) oscillatory sheet flows because it is very important in the topography evolution. Knowledge of sediment transport in skewed asymmetric oscillatory sheet flows is required in coastal engineering which is related to velocity skewness and acceleration skewness.
The studies of the sediment transport under skewed asymmetric oscillatory sheet flows include purely velocity-skewed flows, purely acceleration-skewed flows and mixed flows with the velocity skewness and the acceleration skewness (Ruessink et al., 2009; Dong et al., 2013). Asymmetric development of a boundary layer thickness (Yuan and Madsen, 2015) and a large net sediment transport rate are observed with high velocity skewness (O’Donoghue and Wright, 2004b) or acceleration skewness (Watanabe and Sato, 2004). The generation of the net sediment transport is mainly attributed to the phase-lag both in velocity-skewed flows (Li et al., 2008) or acceleration-skewed flows (van der A et al., 2010). There are three types of phase-lag (Chen et al., 2018a): (1) a phase-shift which denotes the responded time of sediment movement in the sheet flow layer falling behind free stream velocity; (2) a phase-residual which denotes sediment entrained during the current half period, maintained in movement during deceleration stage, and transported after flow reversal; and (3) a phase-lead which denotes the leading time of bottom shear stress and boundary layer velocity to the free stream velocity. The above conclusions about the generation of net sediment transport (Li et al., 2008; van der A et al., 2010) and importance of each part of phase-lag are still not clear.
In the recent couple of decades, advanced empirical models considering phase-lag effects are established for sediment transport under skewed asymmetric oscillatory sheet flows. Most of the empirical models are half-periodic types following Dibajnia (1991) which only discontinuously consider the phase-residual above a critical value. To cover new measured data, more parameters related to the phase-residual are given (Dong et al., 2013), together with the acceleration and the asymmetric development of boundary layer thickness between onshore and offshore flow stages. Some empirical models are not easy to implement in engineering due to difficulties in calibrating abundant parameters. Some key factors are not considered, such as the impacts of the net current caused by the velocity skewness or acceleration skewness. Furthermore, half-periodic empirical models are lack in instantaneous sediment transport. To bridge the gap, advanced instantaneous models considering the phase-shift have been developed in the last decade (Gonzalez-Rodriguez and Madsen, 2007). However, the phase-lead and continuous appearance of the phase-residual are never seen in any instantaneous model, and the asymmetric boundary layer thickness is not either. None of the instantaneous models can be applied widely for sediment transport in velocity- and acceleration-skewed oscillatory sheet flows, and a proper exponent of a velocity power function for the instantaneous sediment transport rate should be selected accordingly (Chen et al., 2018b, c).
The sediment transport in the oscillatory sheet flow is a typical hyper-concentrated two-phase flow with a sediment volumetric concentration of about 0.08–0.60. To enhance the theory and relevant mechanism about the velocity skewness and the acceleration skewness, advanced two-phase numerical models are developed for skewed asymmetric oscillatory sheet flows in the recent two decades. A two-phase model is enabled to study instantaneous sediment transport and net sediment flux, and also erosion depth, phase-lag and boundary layer development. However, all variables in the two-phase model, such as phase-lag and boundary layer development, are hard to be separated for the actual effect of the velocity skewness or acceleration skewness in sediment transport. A basic problem is that the causes of the net sediment flux are still insufficiency known.
This study utilizes an instantaneous analytical model (Chen et al., 2018b, c) with the parameters settings from the two-phase model (Lee et al., 2016) for sediment transport in skewed asymmetric oscillatory sheet flows. Three types of the phase-lag and the boundary layer thickness are discussed. We will focus on the selection of exponents in the power function between the sediment transport rate and the velocity and skewed asymmetric characteristic. To isolate the effects of the velocity skewness and the acceleration skewness, we will examine (1) the purely velocity-skewed flows, and (2) the purely acceleration-skewed flows separately.
2 Methods and data
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The analytical model (Chen et al., 2018b, c) consists of velocity and concentration profiles and the sediment transport rate. The two-phase model (Lee et al., 2016) supplies essential parameters for the analytical model.
2.1 Model and data description
Following Abreu et al. (2010), the free stream velocity (U) of skewed asymmetric oscillatory flow is imaginary part of
$\begin{array}{l}V + U{\rm{i}} = F\sum\limits_{k = 0}^\infty {{r^{ - k}} \times } \exp \{ {\rm{i}}[(k + 1)\sigma (t - {t_0}) + k\chi ]\} ,\end{array}$
where F is the free steam velocity amplitude; i is the imaginary unit; r and χ are wave form parameters; t is the time; $\sigma = 2{\text{π}} /T$, is the angular frequency; T is the period; and t0=σ–1arcsin(r–1sinχ), forces the free stream velocity being 0 at the time 0. Figure 1 shows a typical process of the free stream velocity, where r>1 and –${\text{π}} $/2<χ<0; subscripts a and d denote acceleration and deceleration stages, respectively; subscripts c and t are the crest and trough stages, respectively; positive and negative symbols denote onshore and offshore directions, respectively.
The exponential approach of the velocity in the boundary layer is given by the following Nielsen and Guard (2010), that is,
${V_{\rm{b}}} + {U_{\rm{b}}}{\rm{i}} = (V + U{\rm{i}})\left\{ {1 - \exp \left[ { - (1 + \lambda {\rm{i}})\frac{{y + Z}}{\delta }} \right]} \right\}, $
$\begin{array}{l}{U_{\rm{b}}} = F\sum\limits_{k = 0}^\infty {{r^{ - k}}\left\{ {\sin [(k + 1)\sigma (t-t_o) + k\chi ] - } \right.} \\ \;\;\;\;\;\;\;\;\exp \left( { - \frac{{y + Z}}{\delta }} \right) \times \\\left. { \;\;\;\;\;\;\;\;\sin \left[ {(k + 1)\sigma (t-t_o) + k\chi - \lambda \frac{{y + Z}}{\delta }} \right]} \right\},\end{array}$
where λ is the phase-lead parameter on the immobile bed surface; subscript b denotes the boundary layer; y is the vertical coordinate originally located at the initial undisturbed bed; Z is the erosion depth; δ=δb/4.6; and δb is the turbulent boundary layer thickness.
Chen et al. (2018b, c) applied an ideally exponential approach (Chen et al., 2013) to the real sediment volumetric concentration (C), which is
$C = {C_{{\rm{max}}}}\exp \left[ { - (1 + \frac{y}{Z})} \right], $
where subscript max denotes the maximum and Cmax=0.6. Equation (4) considered the mass conservation, i.e., the integration of Eq. (4) above y=–Z equal ZCmax.
The integration of Eqs. (2) and (4) is
$\begin{array}{l}\int\limits_{ - Z}^\infty {(V + U{\rm{i}})\left\{ {1 - \exp \left[ { - (1 + \lambda {\rm{i}})\frac{{y + Z}}{\delta }} \right]} \right\} \times } \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{C_{{\rm{max}}}}\exp \left[ { - (1 + \frac{y}{Z})} \right]{\rm{d}}y \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; =\frac{{{C_{{\rm{max}}}}Z(V + U{\rm{i}})(1 + \lambda {\rm{i}})}}{{(1 + \lambda {\rm{i}}) + \delta /Z}}.\end{array}$
The instantaneous transport rate is taken from the imaginary part of Eq. (5), that is,
$q = {C_{{\rm{max}}}}Z\left[ {\frac{{U(1 + {\lambda ^2} + \delta /Z) + V\lambda \delta /Z}}{{{\lambda ^2} + {{(1 + \delta /Z)}^2}}}} \right].$
Equation (6) consists of the free stream velocity, the bottom velocity phase-lead, the boundary layer thickness and the erosion depth. The instantaneous q is usually approximated by q/qmax=sgn(U)|U/Umax|n, in which n is summarized in Chen et al. (2018b, c) which increases with the decrement of the phase-residual. Equation (6) is invalid for a progressive wave or wave-current flow, because Eq. (2) is only the analytical approach of the oscillatory flow (Chen et al., 2018c).
The model is available for analyzing the phase-lag and boundary layer development effect related to the velocity skewness and the acceleration skewness. λ, δb and Z are so far unknown and need calibration to determine. In the present study, an Eulerian two-phase model (Lee et al., 2016) is applied to determine λ, δb and Z. Here λ and δb are obtained by Eq. (3) using a least square method, with Ub and Z predicted by the two-phase model (Lee et al., 2016) that incorporates the rheological characteristic of sediment, and considers the enduring-contact, inertial, and fluid viscosity effects in a sediment pressure and stress for a wide range of particle Reynolds number. A k-ε turbulence model is adopted to compute the fluid Reynolds stresses, and a novel numerical scheme is proposed to avoid numerical instability caused by a high sediment concentration and allow the computation within and outside the sediment bed in Lee et al. (2016). Net q validation is shown in Fig. 2 for collected data in the oscillatory sheet flows which include purely velocity-skewed flows (χ=–${\text{π}} $/2) and purely acceleration-skewed flows (χ=0) (Dibajnia, 1991; Ribberink and Al-Salem, 1995; O’Donoghue and Wright, 2004b; Watanabe and Sato, 2004; Li et al., 2008; van der A et al., 2010; Dong et al., 2013). Good agreement between computation and experiment is obtained (Fig. 2).
2.2 Skewed asymmetric characteristic
In the velocity-skewed flow, the net sediment transport caused by a high velocity skewness is usually classified by the phase-lag. Sediment would be entrained very high by large Umax that they cannot completely settle down with small D during short T before flow reversal. Offshore net q is observed when the phase-residual is very obvious with large Umax, small D or short T. Onshore net q is observed when the phase-lag effect is small. On the basis of different phase-lag effects, Ribberink and Al-Salem (1995) explained the relation between net q and the velocity skewness, later Hassan and Ribberink (2010) added the influence of D and Umax.
In the acceleration-skewed flow, the onshore net q is attributed to the phase-lag related to the acceleration skewness (van der A et al., 2010) denoted by β=amax/(amax–amin), where a is the acceleration. This means from U=–Ut to U=Uc (Fig. 1), small Td, t corresponding to large acceleration makes much sediment entrained in Tt remain in movement after flow reversal to be carried away during Tc. The process is opposite from U=Uc to U=–Ut, so onshore net q is generated by different sediment amount between Tc and Tt. The onshore net q is enhanced when Umax and β increase or D decreases, which lead to increments in the phase-lag related to the acceleration skewness.
The three parts of the phase-lag and the boundary layer thickness can be seen in Eq. (6), where the phase-shift and the phase-residual are included in the erosion depth Z (O’Donoghue and Wright, 2004a). So the previous conclusions (Ribberink and Al-Salem, 1995; Hassan and Ribberink, 2010; van der A et al., 2010) about the phase-lag related to the velocity skewness and the acceleration skewness can be contained. Furthermore, the effect of the boundary layer thickness (δb) also can be shown by δ in Eq. (6), where a larger δb corresponds to a smaller transport rate for the same Z.
3 Results and discussion
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Generally, the following analysis is based on O’Donoghue and Wright (2004b) and van der A et al. (2010) cases in Table 1.
3.1 Purely velocity-skewed flows
To illustrate the phase-shift and the phase-residual, Fig. 3 shows Z with T=7.5 s, Ut=0.9 m/s and Umax=Uc=1.5 m/s, where “Exp.” and “Pres.” denote the experiment and the present computation respectively. The velocity skewness parameter is R=Uc/(Uc+Ut)=0.625. The phase-lag parameter Ψ being σΔ/w is the time ratio between the sediment falling down and the wave period introduced by Dohmen-Janssen (1999), where Δ is the sheet flow layer thickness and w is the sediment falling velocity. Ψ is 0.71 for D=0.13 mm case and is 0.10 for D=0.46 mm case. Accordingly, the phase-shift in Fig. 3a is about 0.08t/T, and larger than 0.02t/T in Fig. 3b, as the minimum Z moment falling behind t/T=0. The phase-residual for D=0.13 mm is also larger than that of D=0.46 mm: Z at Tt (t/T=0.71) is close to that at Tc (t/T=0.21) in Fig. 3a due to large phase-residual, while Z at Tt is obviously smaller than that at Tc in Fig. 3b. If the phase-residual reduces to 0, minimum Z at flow reversal (t/T=0.0 or 0.42) is 0, and Z/Zmax=Θ/Θmax. In D=0.13 mm case, Z/Zmax=Θ/Θmax cannot be used due to large phase-residual.
To explain the effect of boundary layer development, Fig. 4 shows the instantaneous δb. The boundary layer developments of Tc (t/T=0.0–0.42) and Tt (t/T=0.42–1.0) stages are asymmetric in the velocity-skewed flow. The shear stress and roughness near the flow crest (t/T=0.15–0.25) are much larger than that near the flow trough (t/T=0.6–0.8) due to Uc>>Ut, and δb (proportional to roughness) near the flow crest is larger than that near the flow trough.
The asymmetric development of δb caused by the velocity skewness leads to a net current in Eq. (3) at every y (Fig. 5). In Fig. 4a, δb in Tc is almost larger than that in Tt, so the Ub/U near flow crest is smaller than that near the flow trough in Eq. (3) based on y+Z, leading to offshore net Ub (Fig. 5). The offshore net Ub is observed (O’Donoghue and Wright, 2004a, b) and predicted by Eq. (3), which is also obtained based on y (Fig. 5) except onshore value in the sheet flow layer bottom for the mobile bed effect. Notice the velocity skewness parameter R>0.5, the lowest level mobilized by the strong Uc is immobile during Tt with a weak Ut, which is seen as the Z in Fig. 3. With the increment of D (0.13–0.46 mm), the phase-residual decreases and the Z difference between onshore flow crest and offshore flow trough increases, so the onshore net Ub based on y increases from Fig. 5a to Fig. 5b.
The net Ub causes net sediment flux ϕc=<Ub><C>, where the angle brackets denote the periodic average. The present computed ϕc, the wave related ϕw=<Ub′C′> and the total flux ϕ=ϕc+ϕw are shown in Fig. 5. The experimental (circle) net ϕ of D=0.13 mm (Fig. 5a) near the bed is clearly offshore, while the ϕ of D=0.46 mm (Fig. 5b) is generally onshore. The offshore ϕ of D=0.13 mm is concluded to large phase-lag in previous study, where obvious phase-lag makes a large amount of sediment entrained in Tc remain in suspension after flow reversal to be carried away at Tt contributing offshore ϕ. The onshore ϕ of D=0.46 mm is concluded to small phase-lag, where Z is proportional to U2 and ϕ is proportional to ZU, thus the net ϕ is proportional to the velocity skewness <U3>. The data for the D=0.46 mm are scattered due to C measurement uncertainty. However, the present prediction almost passes the centre of the data, and has the same shape as that fitted by O’Donoghue and Wright (2004b). Over all, the present prediction agrees reasonably with the experiment and Chen et al. (2018c).
For D=0.13 mm (Fig. 5a), offshore ϕc almost expands to the whole sheet flow layer accordingly, and closely coincides with net ϕ, which means the offshore net ϕ is mainly caused by net Ub, instead of previous only phase-lag. Large phase-residual only makes periodic Z and C almost constant, whereas the direction of net Ub caused by the asymmetric development of δb decides the net ϕ direction. While for D=0.46 mm, the phase-residual is much small, the Z near flow crest is much larger than that during Tt. In Fig. 5b, ϕc is onshore and relatively large near bottom of the sheet flow layer, and is offshore and relatively small almost above the initial bed corresponding to net Ub, because the C decreases along y. Generally, averaged net ϕ is dominated by ϕc in the bottom of sheet flow layer for D=0.46 mm due to the mobile bed effect. Offshore net Ub above the initial bed is not important for net ϕ in D=0.46 mm case, because it cannot expands to the bottom with high C.
Figure 6 shows the instantaneous q, with formulas including: (1) Ribberink (1998) without acceleration; (2) Nielsen (2006) and Gonzalez-Rodriguez and Madsen (2007) considering acceleration. The q magnitude decreases with the increment of D (Figs 6a and b) due to the decrement of Z. The phase-residual of D=0.13 mm is very large that Z and q near the flow trough (t/T=0.6–0.8) are close to flow crest (t/T=0.15–0.25), which is important to offshore net q. In addition, the asymmetric δb leads to a relatively large Ub in the offshore flow (Eq. (3)) and offshore net Ub (Fig. 5a). This process is clearly illustrated in Fig. 6c with a comparison of sgn(U)|U/Umax|n. q/qmax=U/Umax can be used with very large phase-residual for a symmetric δb (Chen et al., 2018b). In addition, the offshore Ub in D=0.13 mm is relatively enlarged because of relatively small δb (Eq. (3)), which results in q/qmax<U/Umax near the flow trough (t/T=0.6–0.8) (Fig. 6c). Thus, the integration of q/qmax is smaller than the integration of U/Umax, which is offshore and in agreement with flux in Fig. 5a (Chen et al., 2018c).
In Fig. 6b, all formulas can be adequately used. The suspended sediment amount and the phase-lag are much small in this case, thereby leading to small amplitude of q, and a much stronger onshore Z and q than those at offshore due to velocity skewness. With small phase-lag, the relation Z/D∝Θ can almost be used (Fig. 3b), and q/qmax=(U/Umax)3, which is close to the instantaneous formulas without phase-residual, can be used for approximation (Fig. 6d), corresponding to an onshore integration of q/qmax, i.e., (U/Umax)3 (Chen et al., 2018c). The velocity skewness makes a much stronger q at flow crest than that at flow trough and generates onshore net rate, in agreement with Fig. 5b that the offshore Ub caused by the asymmetric development of δb is not important due to the small phase-lag. The effect of the velocity skewness is contributed by the phase-lag and asymmetric δb. In turn, only present prediction agrees with experiment well, especially the tendency with D (Figs 6a and b) and obviously offshore q (Fig. 6a) caused by large phase-residual and asymmetric δb.
3.2 Purely acceleration-skewed flows
In the purely acceleration-skewed flows, the Z is first shown in Fig. 7 for the same T=6 s and Umax=Uc=Ut=1.3 m/s, where the phase-shift and the phase-residual are similar to Chen et al. (2018b). Ψ is 0.62 for D=0.15 mm case and is 0.08 for D=0.46 mm case. Correspondingly, the phase-shift for D=0.15 mm in Fig. 7a is about 0.07t/T, which is larger than 0.02t/T for D=0.46 mm in Fig. 7b. Owing to the phase-residual, the classical relation Z/Zmax=Θ/Θmax cannot be used in Fig. 7a. The minimum Z near flow reversal in Fig. 7a is also larger than that in Fig. 7b. The periodic variation of Z is small in Fig. 7a due to large phase-residual, whereas the variation in Fig. 7b is much obvious. In Fig. 7, Z near flow crest (t/T=0.18) is larger and more sediment is carried up than that near flow trough (t/T=0.82), because the shear stress at flow crest is larger than that at flow trough (Suntoyo et al., 2008).
Notice the boundary layer has short time (Ta, c) for U to reach Uc due to the acceleration skewness (Nielsen, 1992), but has much time (Ta, t) for U to reach –Ut. δb is proportional to the oscillatory flow orbital amplitude and related to acceleration time. Figure 8 shows the corresponding δb used in Eq. (3), where the developments of onshore acceleration and offshore acceleration stages are different due to the acceleration skewness, and δb near the flow crest is smaller than that near the flow trough. δb decreases near flow crest and increases near flow trough with the increment in β, which results in a large Ub near flow crest and a relatively small Ub near flow trough in Eq. (3), thereby leading to onshore net Ub based on y+Z and y (Fig. 9) in agreement with van der A et al. (2011) and Chen et al. (2018b). Onshore ϕc resulting from net Ub is also shown in Fig. 9, and can be illustrated by the mobile bed effect (Chen et al., 2018b) which is important for the onshore net sediment transport. The effect of acceleration is related to the development of the boundary layer (Chen et al., 2018b). The net Ub magnitude is larger than Chen et al. (2018b) because the turbulence asymmetry (Ruessink et al., 2011) is not considered in Eq. (3). However, Eq. (4) is based on a constant sediment mixing efficient assumption near the bed, which leads to a smaller suspended sediment concentration than Chen et al. (2018b), and the net sediment transport rate validation in Fig. 2 is still good.
The wave related ϕw and total flux ϕ are also shown in Fig. 9. Averaged ϕ profiles in Figes 9a and b have the same shape, which is onshore at every location. The averaged ϕ decreases with the increment of D at every location for the same acceleration parameter β. Also, the averaged ϕ decreases with the decrement of β for the same D in agreement with the development of the boundary layer related to the acceleration skewness, where the boundary layer difference between the onshore acceleration and the offshore acceleration stages are decreased by the decrement of β. Otherwise, the averaged ϕ is 0 without the acceleration skewness, and U reverts to sinusoidal flow. Onshore ϕc and ϕ expand to the whole sheet flow layer (Fig. 9) corresponding to the net Ub (Chen et al., 2018b). The conclusion is confirmed that onshore net Ub caused by the asymmetric development of δb plays a very important role in the total ϕ.
Following Figs 7–9, the present computed instantaneous q/qmax is shown in Fig. 10. The q/qmax in the onshore flow stage can be approximated by sgn(U)|U/Umax|n (Chen et al., 2018b). The q is not 0 at flow reversal due to the phase-lead. The same as Chen et al. (2018b), qc is always larger than qt, corresponding to the onshore net q. This is important for the estimation of onshore net q, which increases with decreasing D and increasing β (van der A et al., 2010). The less time developed δb at the flow crest is smaller than that at the flow trough (Ta, c<Ta, t) (Fig. 8). With the increment in β, δb decreases near the flow crest and increases near the flow trough, thereby enlarges the δb difference between the onshore acceleration and offshore acceleration stages. Thus, Ub near flow crest is larger than near the flow trough (Eq. (2)), and Z and q near the flow crest are larger with more sediment carried up than those near the flow trough (Fig. 7). The difference of δb, Ub and Z between the onshore acceleration and offshore acceleration stages are key factors for the onshore net q generated in the purely acceleration-skewed flow. The effect of acceleration skewness is confirmed mainly the results of the asymmetric development of the boundary layer between the onshore acceleration and offshore acceleration stages. Furthermore, n≥3 can be applied in Figs 10b and d. This is in agreement with (Chen et al., 2018b) when Z can be approximated by U2 in Fig. 7b, and coincides with bedload formulas without phase-lag, which implies that onshore net q still exists without phase-lag due to acceleration skewness.
4 Conclusions
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A qualitatively analytical model is utilized to study the sediment transport under skewed asymmetric oscillatory sheet flow conditions with the essential phase-lead, the boundary layer thickness and the erosion depth given by a two-phase model which contains the rheological characteristic of sediment, and the enduring-contact, inertial, and fluid viscosity effects in the sediment pressure and stress. The mass conservation, the skewed asymmetric free stream velocity, and the exponential approaches of the boundary layer velocity and the sediment concentration are contained in the analytical model.
The increment of exponents in the power function between the sediment transport rate and velocity is confirmed by the decrement of phase-lag effect. Net boundary layer current and flux are obtained, and the contributions of phase-lag and boundary layer development to sediment flux and sediment transport rate for skewed asymmetric oscillatory sheet flow are clearly shown. In conclusion for sediment transport in the purely velocity-skewed oscillatory sheet flow, the effect of the velocity skewness is the main cause of the phase-lag and asymmetry of the boundary layer between onshore and offshore stages. In conclusion for sediment transport in the purely acceleration-skewed oscillatory sheet flow, the effect of the acceleration skewness is the main cause of the asymmetry of the boundary layer between the onshore acceleration and offshore acceleration stages.
Funding
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  • The National Natural Science Foundation of China under contract Nos 51609244 and 51779258.
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Article Info
doi: 10.1007/s13131-018-1267-4
  • Receive Date：2017-11-12
  • Online Date：2026-04-14
  • Published：2018-09-25
Article Data
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History
  • Received：2017-11-12
  • Accepted：2018-02-26
Funding
The National Natural Science Foundation of China under contract Nos 51609244 and 51779258.
Affiliations
    1 Beijing Engineering Research Center of Safety and Energy Saving Technology for Water Supply Network System, China Agricultural University, Beijing 100083, China
    2 School of Electric Power, North China University of Water Resources and Electric Power, Zhengzhou 450045, China

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表12种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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