Explicit approximation for velocity and sediment flux above mobile sediment bed beneath current and asymmetric wave

Coastal Engineering 157 (2020) 103635

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Explicit approximation for velocity and sediment flux above mobile sediment bed beneath current and asymmetric wave Xin Chen *, Xinyu Hu Beijing Engineering Research Center of Safety and Energy Saving Technology for Water Supply Network System, China Agricultural University, Beijing, 100083, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Asymmetric wave Explicit approximation Mobile sediment bed Velocity distribution Wave and current

An explicit approximation is proposed for velocity distribution and sediment flux above mobile sediment bed beneath current and asymmetric wave. The predicted velocity is decomposed into an oscillatory part and a current part such that the interaction between current and asymmetric wave can be performed directly. The oscillatory part consists of a free stream velocity and a defect function which includes phase lead, mobile sediment bed level and wave boundary layer thickness. The oscillatory part is influenced by the current due to the changes of both mobile sediment bed level and wave boundary layer thickness. The current part is influenced by a wave eddy viscosity in the wave boundary layer and by an apparent wave roughness of the combined wavecurrent flow outside the wave boundary layer. Eight free variables are required for the approximation, and iterative bottom shear stress and roughness height are applied for the wave boundary layer thickness and mobile sediment bed level. The mobile sediment bed level takes account of the effects of phase lag (phase shift and residual), acceleration and flow asymmetry. Therefore, the mobile bed effect, instantaneous velocity, net velocity and sediment flux in the wave boundary layer can be reasonably presented by the explicit approximation beneath current and asymmetric wave.

1. Introduction Oscillatory sheet-flow occurs when velocity magnitude beneath wave and current is strong enough. Plenty of sediments are in movement and bed level is mobile in the oscillatory sheet-flow. Sediment transport is concentrated in a thin layer of several centimeters near the mobile sediment bed (O’Donoghue and Wright, 2004a, 2004b). The mass and energy exchanges are very intense and the sediment concentration is extremely high. Even a small extra current can result in a large net sediment transport rate (Dohmen-Janssen and Hanes, 2005). The sheet-flow contributes a lot to coastal beach formation and evolution due to large bottom shear stress and sediment transport rate. Many studies have been conducted to obtain the velocity distribution of wave-current induced sheet-flow. One type is theoretical and empirical study based on experiment (McLean et al., 2001; Ruessink et al., 2011), the other type is numerical study (Yu et al., 2010; Fuhrman et al., 2013; Holmedal et al., 2013; Chen et al., 2018b) such as two-phase model. The free stream velocity of sheet-flow becomes asymmetric when wave propagates into surf and swash zones. Mobile sediment bed level, wave boundary layer and turbulence development are asymmetric

among onshore, offshore, acceleration and deceleration flow stages. The asymmetry results in mobile bed effect (Davies and Li, 1997; Ruessink et al., 2011), net wave boundary layer velocity and net sediment transport rate (Chen et al., 2018b). The net wave boundary layer ve­ locity plays a very important role in the net sediment transport rate. Fuhrman et al. (2013) and Kranenburg et al. (2013) demonstrate that positive net velocity caused by progressive wave can reverse the offshore net sediment transport back to onshore even in case of velocity-skewed wave and fine sediment. When a current is imposed to the asymmetric wave, the combined wave-current velocity becomes much complex. Wave eddy viscosity decreases the current velocity in the wave bound­ ary layer. Current changes the asymmetries in the bottom shear stress, roughness height, sediment bed level, flow turbulence, wave boundary layer thickness and velocity distribution. Then additional net wave boundary layer velocity and net sediment transport rate are generated (Chen et al., 2018b). Velocity distribution of sheet-flow induced by wave-current is usu­ ally predicted by two-phase models (Yu et al., 2010; Chen et al., 2018b) which can well perform the sediment concentration, turbulence and their asymmetric distributions (Liu et al., 2006; Li et al., 2008; Chen

* Corresponding author. E-mail address: [email protected] (X. Chen). https://doi.org/10.1016/j.coastaleng.2020.103635 Received 1 July 2019; Received in revised form 8 December 2019; Accepted 11 January 2020 Available online 16 January 2020 0378-3839/© 2020 Elsevier B.V. All rights reserved.

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et al., 2011) and illustrate the generation of net wave boundary layer velocity and net sediment transport rate (Chen et al., 2018b). De­ velopments of sediment stress and turbulence closure are essential and meaningful for the model accuracy, but they become more and more complex (Yu et al., 2010; Si et al., 2019; Shi et al., 2019). An explicit approximation for velocity distribution beneath wave and current is desired for the efficiency of engineering design and model verification. However, it is a challenge to represent complicated sheet-flow dynamics beneath current and asymmetric wave. At least, there are quite different bottom shear stress, roughness height, wave boundary layer thickness, effects of phase lag (phase shift and residual) and acceleration over a mobile sediment bed than those over a fixed bed. The early explicit approximations of wave boundary layer velocity distribution essentially start from the classical model like Jonsson (1966). They are mainly about fixed bed based on assumptions of eddy viscosity distributions. The simplest explicit approximation can be ob­ tained by a constant eddy viscosity and phase lead of π/4 (Nielsen, 1992). In a turbulent flow, the phase lead is much smaller than π/4 (O’Donoghue and Wright, 2004a; Ruessink et al., 2011). An approxi­ mation of sinusoidal turbulent flow is suggested by Nielsen (1992) for a rough and fixed bed, and is further applied to the phase lead of much smaller than π/4 (Nielsen and Guard, 2010). The velocity phase lead increases toward the bed and is always considered as a constant at the fixed bed. The experiences over the fixed bed are expected to be used for the mobile sediment bed. But the phase lead neither always increases toward the bed (Malarkey et al., 2009; Ruessink et al., 2011) nor maintains a constant (O’Donoghue and Wright, 2004a; van der A et al., 2009) over the mobile sediment bed. The approximation beneath cur­ rent and asymmetric wave above the mobile sediment bed is much more complex than a sinusoidal flow above the fixed bed and has not been conducted. When a current is imposed to an asymmetric wave, the current changes the asymmetry in the total eddy viscosity distribution which is required for a current velocity distribution. The viscosity can be given by a simple one layer model in Fredsøe and Deigaard (1992) and Nielsen (1992) or complex three or four layers model (Yuan and Mad­ sen, 2015). This viscosity increases rapidly with distance from the bot­ tom, and maintains a constant in most of the wave boundary layer according to Yuan and Madsen (2015). Mobile sediment bed level can be defined as the location where the sediment concentration reaches 0.99 of the maximum concentration or velocity reduces to 0.01 of the maximum free stream velocity. The mobile sediment bed level is very crucial to the approximation of ve­ locity distribution because it defines the velocity bottom. The explicit approximation for the mobile sediment bed level has been conducted in the recent quarter-century (Dick and Sleath, 1992; Zala-Flores and Sleath, 1998) mainly based on bedload transport theory. The early ap­ proximations can perform neither bed level near flow reversal of comparatively large phase lag (phase shift and residual) case nor bed relation between onshore and offshore stages of asymmetric wave case (O’Donoghue and Wright, 2004a; Chen et al., 2018a). Besides, many other factors have to be considered in the accurate approximation of mobile sediment bed level in current and asymmetric wave induced sheet-flow. Bottom shear stress is larger corresponding to a large ac­ celeration than that of a small acceleration (Dong et al., 2013). The roughness height in a sheet-flow is much larger than that of a fixed bed (Wilson et al., 1995). Such an advanced explicit approximation of mo­ bile sediment bed level has been developed by Chen et al. (2018a) for the combined current and asymmetric wave. This approximation also considers the flow asymmetry and effects of phase lag and acceleration. This study aims to obtain an explicit approximation for the velocity distribution of sheet-flow induced by combined current and asymmetric wave. This approximation for complicated dynamics of sheet-flow beneath current and asymmetric wave is a challenge. The velocity is separated into an oscillatory part and a current part to isolate the interaction between current and asymmetric wave. On the basis of Chen et al.’s (2018a) mobile sediment bed level, the velocity distribution is

influenced by phase lag, acceleration modification and flow asymmetry. Classical mobile bed effect and net boundary layer velocity are expected to be realistically modeled. 2. Explicit approximation The sketch of the approximation is shown in Fig. 1, where the sheetflow velocity (UB) is divided into an oscillatory part (UBw) and a current part (UBc). Eight variables are required, i.e. amplitude of free stream velocity (U0), wave period (T), wave form variables (r and φ), sediment specific gravity (s), sediment diameter (D), current velocity (u) and its location (y0). 2.1. Free stream velocity The free stream velocity (U) of asymmetric wave can be given by the imaginary part of N X

WðtÞ ¼

Wk expfi½kωðt

t0 Þ þ φk �g ¼ V þ iU;

(1)

k¼1

where W is a complex velocity; t is the time; subscript k is the order of oscillatory frequency; N is the total number of components; i is the imaginary unit; ω (¼2π/T) is the angular frequency; T is the wave period; t0 forces U(0) ¼ 0; Wk and φk are parameters. Abreu et al.’s (2010) approximation [N ¼ ∞, Wk ¼ U0(1-1/n2)n1 k, t0 ¼ -ω 1arcsin (n 1sinφ), φk¼(k-1)φ] represents most wave shapes and covers existing theories of asymmetric waves, i.e. � � ∞ 1 X 1 WðtÞ ¼ U0 1 expfi½kωðt t0 Þ þ ðk 1Þφ�g n2 k¼1 nk 1 (2) U0 ðn2 1Þexp½iωðt t0 Þ� ¼ 2 ; n n expfi½ωðt t0 Þ þ φ�g pffiffiffiffiffiffiffiffiffiffiffiffiffi where n¼(1þ 1 r2 )/r. In a pure acceleration-skewed wave, φ ¼ 0 and the third moment of dU(t)/dt is positive. A free stream velocity for the combined current and acceleration-skewed wave is shown in Fig. 2, where subscripts c and t respectively denote flow crest and flow trough; subscripts a and d respectively denote acceleration and deceleration; the positive and negative symbols respectively denote onshore and offshore directions; times t1 to t4 correspond to U(t1) ¼ Uc, U(t3) ¼ -Ut and U(t2) ¼ U(t4) ¼ u. Considering the flow asymmetry in velocity and acceleration, Fig. 2 is divided into 4 quarters of pseudo-sinusoidal wave by t1–t4 with representative T ¼ 4[Tac, Tdc, Tat, Tdt] and U0 ¼ [Uc-u, Uc-u, Utþu, Utþu]. Equation (1) is also valid for the sawtooth wave [N ¼ ∞, Wk ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi U0k 1rkarctan 1(r/ 1 r2 ), t0 ¼ 0, φk ¼ 0] which corresponds to

Fig. 1. The approximation of sheet-flow velocity beneath current and asym­ metric wave. 2

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Coastal Engineering 157 (2020) 103635

where g is the gravitational acceleration; f [¼(U0fw þ ufc)/(U0þu)] is the friction factor for combined wave and current; fc is current friction factor; fw is the wave friction factor linearly interpolated between adjacent flow peak and reversal (Chen et al., 2018a). fw at the flow peak and reversal is given by Wilson et al. (1995) and modified by acceler­ ation (Dong et al., 2013). Below the mobile bed level, sheet-flow velocity is zero, i.e. UB(y < -Z, t) ¼ UBw(y < -Z,t) ¼ UBc(y < -Z,t) ¼ 0. Above y ¼ -Z, the momentum equation is � � ∂UB ∂U ∂ ∂U ν B ; ¼ þ P0 þ (6) ∂t ∂t ∂y ∂y where subscript B denotes the boundary layer; P0 is the constant force driving the current; y is the vertical coordinate originally located at the initial undisturbed bed; ν is the total eddy viscosity. Equation (6) in­ cludes an oscillatory force (∂U/∂t) and a constant force (P0). The oscil­ latory part (UBw) is separated to balance ∂U/∂t, i.e. � � ∂UBw ∂U ∂ ∂U ν Bw : ¼ þ (7) ∂t ∂t ∂y ∂y

Fig. 2. The free stream velocity of a combined current and acceleration-skewed wave case (r ¼ 0.5 and φ ¼ 0).

�

�X ∞ r pffiffiffiffiffiffiffiffiffiffiffi k 1 rk expðikωtÞ 2 1 r k¼1 � � r ¼ U0 arctan 1 pffiffiffiffiffiffiffiffiffi2ffiffi ln½1 r cosðωtÞ ir sinðωtÞ� 1 r � � �� �� � r 1 � r sinðωtÞ ln 1þr2 2r cosðωtÞ iarctan ¼ U0 arctan 1 pffiffiffiffiffiffiffiffiffi2ffiffi : 2 1 r cosðωtÞ 1 r (3) U0 arctan

1

Above the wave boundary layer, ∂UB/∂t ¼ ∂UBw/∂t ¼ ∂U/∂t and ∂UBc/ ∂t ¼ 0. In the wave boundary layer, the velocity oscillation (∂UB/∂t) is assumed mainly resulting from ∂UBw/∂t, and thus ∂UBc/∂t is neglected. Therefore, P0 and ∂UBc/∂y are left by comparing Equations (6) and (7), i. e.

�

The imaginary parts of Equation (2) and Equation (3) for pure acceleration-skewed wave are shown in Fig. 3, where (1-1/n2) and pffiffiffiffiffiffiffiffiffiffiffi arctan 1(r/ 1 r2 ) respectively force their amplitudes to be U0. Their difference increases with the increment in r and is obvious when r rea­ ches 0.7.

0 ¼ P0 þ

Chen et al. (2018c) follows Nielsen and Guard (2010) to obtain an approximation for pure oscillatory flow over the mobile sediment bed as a product of the free stream velocity and phase lead defect function, i.e. W{1-exp[-(1þαi)Y}. Since Nielsen’s (1992) study shows that the profile of oscillatory part (WB) is hardly changed by the imposed current, Chen et al.’s (2018c) approximation is tried to apply for Equation (7). Dif­ ference exists in harmonic components of the phase lead, as shown in Malarkey et al. (2009) and van der A et al. (2011). Therefore, Chen et al.’s (2018c) approximation is written as

Accurate mobile sediment bed level is of key importance. The mobile sediment bed level is described by the erosion depth (Z) of Chen et al. (2018a) with the consideration of suspended sediment, phase residual, phase shift and flow asymmetry, i.e. � � Zðωt þ Ψ Þ Um ¼ ðβ1 Θm þ β2 ΘÞ 1 þ 0:011 ; (4) D w

8 N X > > Wk expfi½kωðt t0 Þ þ φk �gf1 exp½ ð1 þ αk iÞY�g ¼ VBw þ iUBw > WB ðy; tÞ ¼ > > k¼1 > > > N < X ; VBw ¼ Wk fcos½kωðt t0 Þ þ φk � expð YÞcos½kωðt t0 Þ þ φk αk Y�g > > k¼1 > > > N > > > U ¼ X W fsin½kωðt t Þ þ φ � expð YÞsin½kωðt t Þ þ φ : α Y�g k

0

k

0

k

(8)

2.3. Oscillatory part

2.2. Mobile sediment bed level

Bw

�

∂ ∂U ν Bc : ∂y ∂y

(9)

k

k¼1

where α is the phase lead parameter on the mobile sediment bed level y ¼ -Z; αk is the kth component of phase lead; Y ¼ 4.61[(y þ Z)/δB]p; 4.61 forces exp(-Y) ¼ 0.01 at y þ Z ¼ δB; p is the exponent of defect function; δB is the wave boundary layer thickness given by Fredsøe and Deigaard (1992), i.e. � �0:82 δB ðtÞ A ¼ 0:09 ; (10) kN kN

where Ψ (¼2ωZm/w) represents the time ratio between sediment falling down and the wave period, and gives the phase shift of Z falling behind U; subscript m denotes the maximum value; β1 denotes the phase re­ sidual; β1 ¼ 3.2exp(-0.2/Ψ); β2 (¼3.2-β1) denotes the periodic variation of Z; w is the sediment falling velocity; Zm [ ¼ 3.2DΘm(1 þ 0.011Um/w)] is the maximum Z; Θm is the maximum Θ; Um is the maximum U. The Shields parameter Θ is ΘðtÞ ¼

fU 2 ; 2ðs 1ÞgD

(5)

where A is the wave orbital amplitude; kN is the roughness height over a mobile sediment bed. The representative A is calculated for each stage 3

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approximation is much complex. Since the tidal current is usually obvious in the coastal area, a simple one layer model of Nielsen (1992) for relatively strong currents is suggested for the present engineering application, i.e. ν ¼ κufcδB. Thus, the current part (UBc) in the wave boundary layer is UBc 1 y þ Z ¼ ; y þ Z � δB ; ufc κ δB

(14)

where ufc is the current friction velocity, κ ¼ 0.4 is the Karman constant. Outside the boundary layer, the viscosity is only caused by the current, and the current part is estimated by � � UBc 1 yþZ ; y þ Z > δB : ¼ log 30 (15) kw ufc κ kw is Grant and Madsen’s (1979) apparent ‘wave roughness’ in the combined wave-current flow. Considering the continuity of Equations (14–15) at y þ Z ¼ δB, kw ¼ 11.04δB. The steady current u is imposed to y ¼ y0 outside the wave boundary layer, thus

Fig. 3. Sketch of free stream velocity in pure acceleration-skewed wave (φ ¼ 0).

based on the peak velocity and flow time, i.e. A(t1,2) ¼ 2(Uc-u)Tac,dc/π; A (t3,4) ¼ 2(Utþu)Tat,dt/π. A(t) is linearly interpolated between adjacent times t1 to t4. kN/D ¼ 5Θ (Wilson et al., 1995) with the lower limit 2.5 (Nielsen, 1992). kN is much larger than 2.5D in a sheet-flow. Iterations are necessary for f and kN. Nielsen’s (1992) result shows that p varies from unity for very rough flows to about 0.33 for smooth turbulent flows. For an engineering application, the averaged p ¼ 0.8 in rough flows is simply used. The vertical gradient of Equation (9) is

N X 8 ∂VBw pY > expð YÞ Wk fcos½kωðt t0 Þ þ φk αk Y� αk sin½kωðt ¼ > > yþZ > ∂y k¼1 > > > > > > > N qffiffiffiffiffiffiffiffiffiffiffiffiffi X > pY > > > 1 þ α2k Wk cos½kωðt t0 Þ þ φk αk Y þ θk � ¼ expð YÞ > > < yþZ k¼1

> N > X > ∂UBw pY > > expð YÞ Wk fsin½kωðt t0 Þ þ φk αk Y� þ αk cos½kωðt ¼ > > > yþZ ∂y > k¼1 > > > > > > N qffiffiffiffiffiffiffiffiffiffiffiffiffi > X > pY : 1 þ α2k Wk sin½kωðt t0 Þ þ φk αk Y þ θk � ¼ expð YÞ yþZ k¼1

ufc ¼

expð YÞWk sin½kωðt

Uk ¼ Wk sin½kωðt

t0 Þ þ φk �:

t0 Þ þ φk

αk Y�;

(16)

If the wave is weak enough that kw < kN, the logarithmic Equation (15) is applied to the whole sheet-flow layer, and kN is forced to kw. Finally, the sheet-flow velocity consists of Equations (9), (14) and (15), i. e. UB¼UBw þ UBc, as shown in Fig. 1. The oscillatory part consists of a free stream velocity and a defect function which is influenced by cur­

t0 Þ þ φk

αk Y�g

; t0 Þ þ φk

(11)

αk Y�g

rent. The oscillatory part is changed by the current due to the changes of both erosion depth and wave boundary layer thickness. The current part is reduced by the wave eddy viscosity within δB. UBc calculated from Equation (14) may be smaller than the velocity given by a three or four layers model of Yuan and Madsen (2015) next to the mobile sediment bed level. However, the underestimation of current velocity cannot evidently change the net velocity and sediment trans­ port simulation because the current velocity is minimal in the area. The present approximation shows Nielsen’s (1992) viscosity can be applied for |u/U0|>0.1–0.2. A weak current case (|u/U0|<0.1–0.2) can be pre­ dicted by a linear eddy viscosity (Fredsøe and Deigaard, 1992) corre­ sponding to different Equation (14) (16).

where θk ¼ arctanαk. The averaged phase lead is about 0.1π in a tur­ bulent sediment-laden flow, and the variation is mainly within �0.02π (Chen et al., 2018b). The influence of current to the phase lead is still not clear. Therefore, the averaged 0.1π is selected by default for a simple engineering application, and αk ¼ tan(0.1kπ). The kth component of UBw is UBk ¼ Uk

κu �: � log 2:72 y0δþZ B

(12) (13)

When a current exists, Equation (9) is directly influenced by Y con­ sisting of the Z and δB. The change of ν is represented by the changes of both δB and kN.

3. Results and discussion The approximation is compared with cases in Table 1, which in­ cludes sinusoidal (φ ¼ 0, r ¼ 0), pure acceleration-skewed (φ ¼ 0, 0 < r < 0.8) and mixed-skewness waves (-π/2<φ < 0, 0 < r < 0.8). U of case 7 is given by Equation (3) and the other cases are given by Equation (2). The result under velocity-skewed wave (φ ¼ -π/2, 0 < r < 0.8) is not included in this paper because that was done in Chen et al. (2018c).

2.4. Current part The current part is obtained from Equation (8) with eddy viscosity assumptions. In the wave boundary layer, eddy viscosity caused by the wave reduces the current part. The total eddy viscosity can be given by a complex three or four layers model (Yuan and Madsen, 2015), and the 4

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Table 1 Cases for the verification of velocity approximation. Authors

Cases

U0 (m/s)

T (s)

r

φ

s

D (mm)

u (m/s)

Li et al. (2008) McLean et al. (2001) McLean et al. (2001) Ruessink et al. (2011) Ruessink et al. (2011) Ruessink et al. (2011) van der A et al. (2009)

1 2 3 4 5 6 7

1.26 1.39 1.52 1.20 1.20 1.20 1.07

6.0 7.2 7.2 7.0 7.0 7.0 7.0

0.0 0.0 0.0 0.3 0.5 0.3 0.5

0 0 0 0 -π/4 0 0

2.65 2.65 2.65 2.65 2.65 2.65 2.65

0.13 0.12 0.24 0.20 0.20 0.20 0.15

0.0 0.3 0.33 0.0 0.0 0.4 0.0

y0 (m) 0.25 0.25 0.30

Fig. 4. Sinusoidal velocity verification for U0 ¼ 1.26 m/s, T ¼ 6.0s, r ¼ 0.0, φ ¼ 0, s ¼ 2.65, and D ¼ 0.13 mm.

Fig. 5. Predicted velocity profiles for T ¼ 7.2s, r ¼ 0.0, φ ¼ 0, s ¼ 2.65, u ¼ 0.22U0 and y0 ¼ 0.25 m. (1) case 2, U0 ¼ 1.39 m/s, D ¼ 0.12 mm; (2) case 3, U0 ¼ 1.52 m/ s, D ¼ 0.24 mm.

3.1. Instantaneous velocity

exists near the flow crest or trough if y is very close to the maximum mobile sediment bed level. At the flow reversal (t/T ¼ 0 and 0.5), UBw6¼0 due to the existence of phase lead, and the classical overshoot phe­ nomenon is clear because UBw is larger at y ¼ 2.5 mm than at the upper (y ¼ 12.5 mm) and lower (y ¼ -1.45 mm) locations. The default α ¼ tan (0.1π) is acceptable for this case because there is not obvious phase shift between the approximation and experiment. Fig. 5 shows the predicted velocity profiles beneath current and si­ nusoidal wave cases 2–3, where u/U0 ¼ 0.22. Eight moments are located at U(t)/U0 reaching -0.78, -0.28, 0.22, 0.72 and 1.22. The velocity profiles start from different sediment bed levels below the initial bed (y ¼ 0 m). The sediment bed levels do not return to zero due to the phase residual denoted by β1. Case 2 involves finer sediments than case 3 and the flow conditions are not very different. The phase residual for case 2

UBw is first verified by Li et al. (2008) sinusoidal case 1 in Fig. 4 at locations below the initial bed (y ¼ -1.45 mm), above the initial bed (y ¼ 2.5 mm) and near the top of wave boundary layer (y ¼ 12.5 mm). At every location, the explicit approximation (solid line, E.A.) is in agree­ ment with the experimental data (symbols) for the amplitude and phase of velocity. This agreement denotes the reasonability of mobile sediment bed level from Equation (4). The amplitude of UBw/U0 increases with the increment of y, from about 0.3 at y ¼ -1.45 mm to about 0.9 at y ¼ 12.5 mm. Periodic velocity variations at the three points follow the free stream velocity and have good symmetry. The velocity variation is not a sinusoidal shape below the initial bed (y ¼ -1.45 mm) due to the influ­ ence of mobile sediment bed level. Sediment movement (velocity) only 5

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Fig. 6. Velocity verification for T ¼ 7.2s, r ¼ 0.0, φ ¼ 0, s ¼ 2.65, u ¼ 0.22U0 and y0 ¼ 0.25 m. (1–7) case 2, U0 ¼ 1.39 m/s, D ¼ 0.12 mm; (8–10) case 3, U0 ¼ 1.52 m/s, D ¼ 0.24 mm.

(β1 ¼ 2.64) is much larger than case 3 (β1 ¼ 1.44). Therefore, the sediment bed level variation, which is denoted by β2 in the explicit approximation, is smaller for case 2 than case 3, and the difference of sediment bed level between flow crest and trough is smaller for case 2 than case 3. The classical overshoot phenomena is clear when U(t)/U0 reaches 0.22. The wave boundary layer thickness is less than 0.04 m for every case, and larger for case 3 than case 2 due to a large U0. Above the wave boundary layer (y > 0.04 m), the velocity profiles are logarithmic in Fig. 5, and two profiles coincide with each other at U(t)/U0 ¼ [-0.28, 0.22, 0.72]. The two profiles are separated in the wave boundary layer by the phase lead. Fig. 6 shows the velocity verification within the wave boundary layer (y < 0.03–0.04 m) for cases 2–3. At every location, the explicit approximation (solid line, E.A.) for case 2 is also consistent with the experimental data (symbols, Exp.) as case 1 in Fig. 4. The consistency is not so clear for case 3 near t/T ¼ 0.4–0.45 probably because the mobile sediment bed level (y ¼ -Z) is underestimated there. UBc is largely restricted by the wave eddy viscosity in the wave boundary layer

according to Equation (14). Therefore, the velocity amplitude ratio be­ tween onshore and offshore stages is much smaller than Uc/Ut (¼1.56) for every location in Fig. 6 and is close to 1 below the initial bed (y ¼ 0 m). The sediment bed level for case 3 at the flow crest is obviously lower than that at the flow trough due to small phase residual, thus the velocity amplitude ratio between onshore and offshore stages is relatively larger for case 3 than case 2 below the initial bed. In addition, even the onshore and offshore stages of U are respectively about 0.57T and 0.43T, the onshore stages of UB are all smaller than 0.57T and offshore stages of UB are all larger than 0.43T in the wave boundary layer for cases 2–3. The explicit approximation for case 3 is also compared with the prediction by Yu et al.’s (2010) two-phase model (dash-dotted line, Yu.) in Fig. 6(8–10). The two-phase model overestimates the experiment near the flow crest and its velocity amplitude ratio between onshore and offshore stages is closer to Uc/Ut (¼1.56) than the ratio of present approximation. The two-phase model behaviour is worse than the explicit approximation probably because of its difficulties in modelling complex sediment bed level and concentration. Underestimation of 6

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Fig. 7. U, kN and δB for T ¼ 7.0s, U0 ¼ 1.2 m/s, s ¼ 2.65 and D ¼ 0.2 mm (1, 3, 5) r ¼ 0.3, φ ¼ 0; (2, 4, 6) r ¼ 0.5, φ ¼ -π/4.

sediment concentration in the sheet-flow layer for case 3 (Yu et al., 2010) results in an underestimation of sediment stress, and thus an overestimation of sediment velocity. Yu et al.’s (2010) model is not very valid for fine sediment (D < 0.2 mm) due to the limitation of sediment stress term. Therefore, the two-phase model would be still worse than the explicit approximation for case 2. A development of sediment stress is complex and challenging (Si et al., 2019; Shi et al., 2019), and thus Yu et al.’s (2010) model is not adopted in Fig. 6(1–7). Since the present approximation can reasonably predict the experiment, the default α, p, δB and Z are acceptable for the explicit approximation in engineering practices. The acceleration asymmetry is considered in the approximation. It results in asymmetric roughness height and wave boundary layer thickness and can generate net velocity and sediment flux. Fig. 7 shows the free stream velocity, roughness height and wave boundary layer thickness for cases 4–6. The wave boundary layer for case 4 has a short time (Tac) to develop before the flow crest. Very short Tac results in relatively small δB near the flow crest in Fig. 7(5) according to Equation (10) and relatively large bottom shear stress (Θ) (Nielsen, 1992; Suntoyo et al., 2008; Yuan and Madsen, 2015). The acceleration modification of Dong et al. (2013) for fw in Equation (5) recognizes the relatively large Θ denoted by kN ¼ 5ΘD near the flow crest in Fig. 7(3). By contrast, the wave boundary layer has sufficient time (Tat) to develop before the flow trough, which leads to a larger δB near the flow trough than that near the flow crest in Fig. 7(5) in agreement with van der A et al. (2011). The bottom straight parts of curves in Fig. 7(3-4) represent 2.5D and in Fig. 7(5–6) represent 0.09A0.82(2.5D)0.18. In Fig. 7(1–2), only the acceleration asymmetry exists in case 4, and the velocity asymmetry exists beside the acceleration asymmetry in case 5. The velocity magnitude at the flow crest is much larger than that at the flow trough (Uc/Ut ¼ 1.5) for case 5 in Fig. 7(2). Magnitudes of kN and δB are posi­ tively correlated to the velocity magnitude according to their defini­ tions. When the change of A is not significant from case 4 to case 5, the magnitudes of kN and δB are larger at the flow crest and smaller at the

flow trough for case 5 than for case 4. The velocity magnitude and turbulence are enhanced at the flow crest and weakened at the flow trough when u increases from -0.4 to 0.4 m/s. Correspondingly, kN and δB are increased at the flow crest and decreased at the flow trough in Fig. 7(3–6) by u ¼ -0.4 0.4 m/s. Based on kN and δB, Fig. 8 shows the predicted velocity data (E.A) of cases 4–6 which reasonably agree with the experiments (Exp). The first column is |UBw/U0| at the flow crest and trough. |UBw/U0| of case 4 [Fig. 8(1)] is asymmetric due to the acceleration asymmetry in free stream velocity. The predicted |UBw/U0| at the flow crest is larger than that at the flow trough for case 4 and agrees with experiments of Sun­ toyo et al. (2008) and van der A et al. (2011) because δB at the flow crest is smaller than that at the flow trough. For case 4, the mobile sediment bed level at the flow crest, which corresponds to a large shear stress, is lower and more sediment is in movement than at the flow trough in Fig. 8(1). The corresponding sediment flux predicted by the predicted | UBw/U0| at the flow crest would be larger than that at the flow trough and agrees with the two-phase model results of Yu et al. (2010). When an offshore current is imposed in case 6 [Fig. 8(3)], the velocity magnitude and δB are decreased at the flow crest and increased at the flow trough, as shown in Fig. 7. Therefore, the difference of |UBw/U0| between the flow crest and flow trough is enlarged in case 6. The velocity difference of |UBw/U0| also exists in Fig. 8(2) for case 5 when δB are almost the same at the flow crest and trough, because the mobile sediment bed level is lower at the flow crest than at the flow trough. The second to fourth columns of Fig. 8 are the phase leads of UBw, UB1 and UB2, i.e. Arg(UBw)-Arg(U), Arg(UB1)-Arg(U1) and Arg(UB2)-Arg(U2). The experimental (Exp.) phase lead of UBw, which is determined from a cross-spectral analysis, reaches a maximum (<0.1π) near y ¼ 6 mm and decreases along -y direction at the bottom. The decrement of phase lead along -y direction is similar to Malarkey et al.’s (2009) result influenced by mobile sediment bed. The decrement phenomenon is only observed in the mobile sediment bed case, where the mobile sediment bed level falls behind U a time phase of Ψ according to Equation (4). Beside the 7

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Fig. 8. Velocity verification for U0 ¼ 1.2 m/s, T ¼ 7.0s, s ¼ 2.65, and D ¼ 0.2 mm (1) case 4, r ¼ 0.3, φ ¼ 0, u ¼ 0 m/s; (2) case 5, r ¼ 0.5, φ ¼ -π/4, u ¼ 0 m/s; (3) case 6, r ¼ 0.3, φ ¼ 0, u ¼ -0.4 m/s, y0 ¼ 0.3 m.

mobile sediment bed level (y ¼ -Z), the time phase given by cross-spectral analysis or maximum UBw is almost the same as Z. The approximation (E.A) is determined from the phase of flow reversal (UBw ¼ 0) as O’Donoghue and Wright’s (2004b) experiments. This approxi­ mation includes one part determined by αk above the mobile sediment bed level, and another part coinciding with the mobile sediment bed level. Therefore, two solid lines intersect in the second to fourth columns of Fig. 8. The predicted phase leads of the 1st and 2nd fundamental harmonic velocities at the mobile sediment bed level are default 0.1π. They increase quickly from zero to the maximum along the -y direction within the wave boundary layer. The predicted phase leads of the two fundamental harmonic velocities are consistent with the fixed bed ex­ periments of van der A et al. (2011), where the phase leads of different harmonics at the mobile sediment bed level are close to each other. The predicted phase lead of UBw at the mobile sediment bed level is slightly larger than 0.1π due to high order components, αk ¼ tan(0.1kπ)>tan (0.1π). If αk ¼ tan(0.1π), then it converges to 0.1π. Temporal-spatial velocity contours are helpful for the detailed illustration of continuous parameters, such as the phase lead and

onshore/offshore stage which are important in net velocity and sedi­ ment transport. Fig. 9 shows the contours of UBw/U0 and its first two harmonic components (UB1/U0 and UB2/U0) for case 4. The magnitudes decrease very fast with components’ harmonic order. The magnitude of UB2/U0 is much smaller than that of UB1/U0 which is close to UBw/U0. High order components can be neglected and UBw � UB1þUB2. The bottom solid and dash-dotted lines indicate the predicted and measured mobile sediment bed level (y ¼ -Z), respectively. The bed level is always below the initial bed (y ¼ 0 m) due to the phase residual. The approx­ imation can reasonably predict the bed level. The wave boundary layer thickness is about 0.02–0.04 m for case 4, as shown in Fig. 7. The hor­ izontal and vertical densities of velocity contours in Fig. 9(1) denote the magnitudes of acceleration and velocity gradient, respectively. The ve­ locity contours are almost parallel to the y coordinate at the upper layer (y > 0.02–0.04 m) where the pressure gradient is balanced by flow ac­ celeration. They are more crowded near the flow reversals (t/T ¼ 0 and 0.5) than the flow crest and trough (t/T ¼ 0.21 and 0.79). Except near the flow reversals (t/T ¼ 0.0 and 0.5), the velocity contours gradually become parallel to the mobile sediment bed level at the bottom layer (y 8

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Fig. 9. Predicted velocity contours for case 4, U0 ¼ 1.2 m/s, T ¼ 7.0s, r ¼ 0.3, φ ¼ 0, s ¼ 2.65, and D ¼ 0.2 mm (1) UBw/U0; (2) UB1/U0; (3) UB2/U0.

< 0.02–0.04 m) where the pressure gradient is mainly balanced by bottom shear stress (velocity gradient). The characteristics of contours for the first two harmonic components in Fig. 9(2–3) are similar as those of UBw in Fig. 9(1). The wave boundary layer asymmetry, phase lead and phase shift contribute to asymmetry in mobile sediment bed level between the Tc and Tt stages. The velocity symmetry between the onshore and offshore

stages is then lost near the mobile sediment bed level, i.e. UBw/U0, UB1/ U0 and UB2/U0 at the time 0.5 þ t/T are unequal to those at 0.5-t/T. The values and vertical gradient (contour density) of UBw/U0 at the bottom layer are larger near the flow crest (t/T ¼ 0.21) than near the flow trough (t/T ¼ 0.79). The reason is that the wave boundary layer thick­ ness of case 4 [Fig. 7(5)] near the flow crest is considerably smaller than that near the flow trough. The phase lead corresponds to left offset of the 9

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Fig. 10. Predicted velocity contours for case 6, U0 ¼ 1.2 m/s, T ¼ 7.0s, r ¼ 0.3, φ ¼ 0, s ¼ 2.65, D ¼ 0.2 mm, u ¼ -0.4 m/s and y0 ¼ 0.3 m.

contours at the bottom layer. The classical overshoot phenomenon is clearly denoted by the consistently left offset of the contours beside the mobile sediment bed level. The maximum left offset before the flow reversals (t/T ¼ 0 and 0.5) is about 0.05 t/T corresponding to the default phase lead 0.1π. Almost near the flow reversals (t/T ¼ 0/0.5 � 0.05), the local maximum velocity along the y direction is at the bottom layer. At other times, the local maximum is at the upper layer. After the results of case 4 without a current (UBc�0), Fig. 10 illus­ trates the contours of UB/U0 for case 6 beneath current and pure acceleration-skewed wave. The predicted bottom contour of UB/U0 ¼ 0 (solid line) adequately coincides with the mobile sediment bed level (dash-dotted line). The phase residual for case 6 (β1 ¼ 1.22) is larger than that for case 4 (β1 ¼ 0.89) and results in a small variation of nonzero Z within a period. δB is reduced near the flow crest and increased near the flow trough for case 6, as shown in Fig. 7(5). The velocity contours are left-inclined during Tdc-Tat and right-inclined during Tat-Tdt to the y coordinate at the upper layer (y>δB) where the pressure gradient is balanced by oscillatory acceleration and current shear stress. They are more crowded near the moment of U ¼ u (t/T ¼ 0 and 0.5) than the flow crest and trough (t/T ¼ 0.21 and 0.79). Except near the moment of U ¼ u (t/T ¼ 0.0 and 0.5), the velocity contours gradually become parallel to the mobile sediment bed level at the bottom layer (y<δB) where the pressure gradient is mainly balanced by bottom shear stress. The bottom velocity contours are more crowded near the flow crest than near the flow trough because δB is small near the flow crest [Fig. 7(5)]. The ve­ locity overshoot is also denoted by the contours’ left offset beside the mobile sediment bed level, as shown in Fig. 9. For case 6 [Fig. 7(1)], the stages of the onshore and offshore U are respectively about 0.39T and

0.61T. But the offshore current is largely restricted at the bottom, as the restriction by wave eddy viscosity illustrated in Fig. 6. In comparison with Fig. 9(1), the offshore current only leads to the maximum left offset in Fig. 10 a little smaller than 0.05 t/T before t/T ¼ 0 and a little larger than 0.05 t/T before t/T ¼ 0.5. Beside the mobile sediment bed level, the stage of onshore UB is evidently larger than 0.39T and the stage of offshore UB is evidently smaller than 0.61T. 3.2. Net velocity Net velocity contributes a lot to net sediment transport (O’Donoghue and Wright, 2004b; Malarkey et al., 2009). Net UB (Total) of the explicit approximation for cases 2–3 is shown in Fig. 11 as the solid line. The net UB consists of a net oscillatory (UBw, dash-dotted line) and a net current (UBc, dashed line) part. The net current part is not logarithmic due to a restriction of the wave eddy viscosity. The imposed current generates significant flow asymmetry (Uc/Ut ¼ 1.56) to cases 2–3. Therefore, a classical mobile bed effect (Ruessink et al., 2011) exists near the bottom of mobile sediment bed level and corresponds to an onshore net UB (Chen et al., 2018b) because many sediments picked up by a strong flow crest cannot move near the flow trough. The mobile bed effect is less obvious for case 2 than for case 3 because the large phase residual weakens the difference of mobile sediment bed level between onshore and offshore flow stages. Above the area influenced by the mobile bed effect, the net oscillatory part is offshore, as in the results for velocity-skewed flow reported by Davies and Li (1997) and O’Donoghue and Wright (2004b). An onshore current enhances the turbulence at the onshore stage and decreases the turbulence at the offshore stage. Such

Fig. 11. Net UB for s ¼ 2.65, T ¼ 7.2s, φ ¼ 0, r ¼ 0, y0 ¼ 0.25 m and u ¼ 0.22U0. (1) case 2, U0 ¼ 1.39 m/s, D ¼ 0.12 mm; (2) case 3, U0 ¼ 1.52 m/s, D ¼ 0.24 mm. 10

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Fig. 12. Net current part for s ¼ 2.65, D ¼ 0.2 mm, T ¼ 7.0s, U0 ¼ 1.2 m/s, r ¼ 0.3, φ ¼ 0 and y0 ¼ 0.3 m. (1) u ¼ -0.4 m/s; (2) u ¼ -0.2 m/s; (3) u ¼ 0.2 m/s; (4) u ¼ 0.4 m/s.

Fig. 13. Net velocity for U0 ¼ 1.2 m/s, T ¼ 7.0s, r ¼ 0.3, φ ¼ 0, s ¼ 2.65, D ¼ 0.2 mm and y0 ¼ 0.3 m. (1) u ¼ -0.4 m/s, on y; (2) u ¼ -0.4 m/s, on y þ Z; (3) u ¼ 0 m/s, on y; (4) u ¼ 0 m/s, on y þ Z; (5) u ¼ 0.4 m/s, on y; (6) u ¼ 0.4 m/s, on y þ Z.

turbulence asymmetry causes the offshore net oscillatory part (Davies and Li, 1997; Ruessink et al., 2011) which is mainly restricted in the wave boundary layer. In the present approximation, the roughness height and wave boundary layer thickness are larger near the flow crest than those near the flow trough due to the onshore current. Relatively large wave boundary layer thickness corresponds to relatively small boundary layer velocity near the flow crest according to exp(-Y) in Equation (9), and finally causes the offshore net oscillatory part (Chen

et al., 2018b). In comparison with the relatively stable oscillatory part, the current part is largely reduced by the wave eddy viscosity. The net current part of Equations (14–15) for case 6 is illustrated in Fig. 12(1) as the solid line denoted by ‘C þ W’. For comparison, an equivalent current velocity (C only) without the influence of wave eddy viscosity is provided by log­ arithm Equation (15) with kw replaced by kN and ufc felt by UBc(y0,t) ¼ u. In addition, the results for u ¼ �0.2 and 0.4 m/s are shown in Fig. 12 11

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(2–4). The net current part (C þ W) of a combined wave and current is smaller than the equivalent current velocity (C only) without the in­ fluence of wave eddy viscosity. The current part (C þ W) and equivalent current velocity (C only) of the same u cases meet at the lowest mobile sediment bed level, and they meet again at y0 ¼ 0.3 m when UBc(y0,t) ¼ u. The lowest mobile sediment bed level for u > 0 is lower than that for u < 0 because in pure acceleration-skewed waves the wave friction factor for u > 0 is increased by Dong et al.’s (2013) acceleration modification at the flow crest and for u < 0 is decreased at the flow trough. The difference between the ‘C þ W’ and ‘C only’ at the same u decreases when the current magnitude decreases from �0.4 m/s to �0.2 m/s. They coincide with each other when u ¼ 0. The net UB (Tot) generated by acceleration-skewed wave is illus­ trated in Fig. 13, where the oscillatory part’s first two harmonic com­ ponents (UB1, 1st and UB2, 2nd) are shown as the dash-dotted and dashed lines. Case 4 corresponds to Fig. 13(3–4). The net UB is onshore and penetrates the full wave boundary layer for case 4 in Fig. 13(3). The first and second harmonic components play a major and minor role in the net UB, respectively. Ruessink et al. (2011) points out that friction on the sidewalls induces small three dimensional secondary flows and may result in the spurious onshore UB near the top (circles). Near the mobile sediment bed level, the predicted onshore net UB agrees with Ruessink et al.’s (2011) experiment where the offshore turbulence is stronger than the onshore turbulence, but is opposite to the clear water flow experi­ ments (van der A et al., 2011; Yuan and Madsen, 2015) over fixed beds. As mentioned in Fig. 7 8 for case 4, the mobile sediment bed level corresponding to a large shear stress is lower at the flow crest than at the flow trough. The mobile bed effect contributes a lot to the onshore net UB. Such onshore net UB agrees with the onshore net sediment transport rate (Watanabe and Sato, 2004; van der A et al., 2010) which is decided by net UB when phase lag is large enough (Chen et al., 2018b). The net UB without the mobile bed effect is shown in Fig. 13(4) based on y þ Z, where UB�0 at y þ Z ¼ 0. In comparison with Fig. 13(3), the onshore magnitude of net UB decreases in Fig. 13(4), but is still opposite to the experiments over fixed beds due to the turbulence asymmetry. The net UB in Fig. 13 is different from the current velocity in Fig. 12 due to the strong net oscillatory part. The net oscillatory part is changed by the current and also restricted in the wave boundary layer when a

strong current is imposed. When an offshore current of -0.4 m/s is imposed to the wave [Fig. 13(1)], the mobile bed effect generates an offshore net UB at the bottom, and turbulence asymmetry enhances the onshore net oscillatory part at the upper wave boundary layer (y ¼ 0.01–0.04 m), i.e. the velocity magnitude (U), roughness height (kN) and wave boundary layer thickness (δB) decrease at the onshore stage and increase at the offshore stage, as shown in Fig. 7. The onshore net oscillatory part based on y þ Z is enhanced without the mobile bed effect in Fig. 13(2). When an onshore current of 0.4 m/s is imposed to the wave [Fig. 13(5)], the mobile bed effect generates an onshore net UB at the bottom, and turbulence asymmetry reverses the onshore net oscillatory part offshore at the upper wave boundary layer (y ¼ 0.01–0.04 m), i.e. the velocity magnitude (U), roughness height (kN) and wave boundary layer thickness (δB) increase at the onshore stage and decrease at the offshore stage. The offshore net oscillatory part based on y þ Z pene­ trates the full wave boundary layer without the mobile bed effect in Fig. 13(6). The direction change of the net oscillatory part against u (¼-0.4–0.4 m/s) agrees with the experiment of Yuan and Madsen (2015) and the numerical result of Chen et al. (2018b). The UB/U0 axis of Fig. 13(1) is different from the others to cover the experimental data. For case 6, the explicit approximation apparently underestimates net UB to more than half of Ruessink et al.’s (2011) experiment at the same elevations. One reason is that the vertical structure of net UB in Ruessink et al.’s (2011) tunnel is quite compli­ cated, with net UB going to zero at the bed and at the top. Vertically averaged net UB is u (-0.33U0 ¼ -0.4 m/s) and net UB is over u at y0 ¼ 0.3 m in this experiment. The experiment reaches u at about y ¼ 0.05 m [Fig. 13(1)] which is much smaller than y0 ¼ 0.3 m. The explicit approximation reaches u at y0 ¼ 0.3 m but not y ¼ 0.05 m for case 6 in Fig. 13(1), and is smaller than the logarithmic profile of Fig. 12(1) below y0 according to Nielsen (1992) and Fredsøe and Deigaard (1992). The experiment is larger than the logarithmic profile of Fig. 12(1) above y ¼ 0.025 m ≪ y0. In addition, compared to a complex viscosity distribution of Yuan and Madsen (2015), the approximation with a constant viscosity distribution would underestimate current near the bottom, as mentioned in the last paragraph of Section 2.4. The net UB results from the competition of exp(-Y) and αkY in Equation (9) in the present approximation. Y consists of Z and δB. The

Fig. 14. Predicted UB for case 7, U0 ¼ 1.07 m/s, T ¼ 7.0s, r ¼ 0.5, φ ¼ 0, s ¼ 2.65, and D ¼ 0.15 mm (1) UB verification, αk ¼ tan(0.1kπ); (2) net UB, αk ¼ tan(0.1kπ); (3) UB verification, αk ¼ tan(0.2kπ-|t/T-0.5| � 0.2kπ); (4) net UB, αk ¼ tan(0.2kπ-|t/T-0.5| � 0.2kπ). 12

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Fig. 15. Net velocity for U0 ¼ 1.2 m/s, T ¼ 7.0s, r ¼ 0.5, φ ¼ -π/4, s ¼ 2.65, D ¼ 0.2 mm and y0 ¼ 0.3 m. (1) u ¼ -0.4 m/s, on y; (2) u ¼ -0.4 m/s, on y þ Z; (3) u ¼ 0 m/s, on y; (4) u ¼ 0 m/s, on y þ Z; (5) u ¼ 0.4 m/s, on y; (6) u ¼ 0.4 m/s, on y þ Z.

variation of Z results in the above mentioned mobile bed effect. The variation of δB refers to the above mentioned asymmetry of turbulence intensity. Besides, the effect of αk is illustrated in the following Fig. 14 for case 7. Phase lead at the mobile sediment bed level is set the default 0.1π. However, it is not a constant in time and space for experiments (symbols) in Fig. 14(1/3), also for velocity-skewed cases in O’Donoghue and Wright (2004a). At the initial bed (y ¼ 0 mm), it increases first from about 0.1π (0.05 t/T) at offshore-onshore reversal (t/T ¼ 0.0) to 0.2π (0.1 t/T) at onshore-offshore reversal (t/T ¼ 0.5) and then decreases back to about 0.1π at t/T ¼ 1.0. At the other locations, it also increases first and then decreases back. The predicted net UB is onshore in Fig. 14 (2) with the constant αk ¼ tan(0.1kπ), like Ruessink et al.’s (2011) case 4 in Fig. 13(3). When the phase lead is simply and linearly given according to the variation at the initial bed, i.e. αk ¼ tan(0.2kπ-|t/T-0.5| � 0.2kπ), the phase agreements between solid lines and symbols near t/T ¼ 0.5 are improved in Fig. 14(3). The stage of onshore flow is increased and the stage of offshore flow is reduced for each solid line in Fig. 14(3). Correspondingly, the net UB is reversed offshore at the upper wave boundary layer (y ¼ 0.01–0.04 m) in Fig. 14(4) which agrees with the experiments of van der A et al. (2011) and Yuan and Madsen (2015). A phase lead of 0.2π near t/T ¼ 0.5 is still not large enough at y ¼ 9 and 19 mm. Increment of the phase lead near t/T ¼ 0.5 increases the stage of offshore flow and reduces the stage of onshore flow. Therefore, the actual offshore velocity is larger than the approximation in Fig. 14(4). The velocity amplitude is underestimated by the approximation at the initial bed (y ¼ 0 mm). As mentioned by van der A et al. (2009), the PIV measured velocities deep in the sheet-flow layer are relatively high at

the bottom region. Since sediment flux approximation for mixed-skewness wave shapes would attract a lot of interest, the relevant net UB generated by case 5 is illustrated in Fig. 15(3–4). The legend and axis follow Fig. 13. The mobile sediment bed level for case 5 is lower at the flow crest than flow trough [Fig. 9(2)], and thus the mobile bed effect leads to a bottom onshore net UB in Fig. 15 (3), as case 4 in Fig. 13(3). The net UB is reversed offshore at about y ¼ 0.005–0.035 m, and agrees with the 2nd Stokes wave experiments of O’Donoghue and Wright (2004b) where the onshore turbulence is stronger than the offshore turbulence. However, the approximation underestimates the offshore net UB of Ruessink et al. (2011) probably due to the applied constant phase lead, as the illus­ tration in Fig. 14. The net UB without the mobile bed effect is shown in Fig. 15(4) where the onshore magnitude decreases and offshore magnitude increases. The results of u ¼ �0.4 m/s are also given in Fig. 15, and the changes of net velocity (UB, UB1, UB2) against current are similar to those shown in Fig. 13. In comparison with Fig. 13(1–2), the offshore magnitude of net UB at the bottom decreases in Fig. 15(1), and onshore magnitudes of net oscillatory part and UB also decrease in Fig. 15(1–2) because the turbulence asymmetry (denoted by Ut/Uc) decreases from 2.0 to 1.35. In comparison with Fig. 13(5–6), the onshore magnitude of net UB at the bottom decreases in Fig. 15(5) because the phase lag (β1) increases from 1.52 to 2.05, and the offshore magnitudes of net oscillatory part and UB increase in Fig. 15(5–6) because the tur­ bulence asymmetry (denoted by Uc/Ut) increases from 2.0 to 3.14.

13

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Fig. 16. Net sediment flux for U0 ¼ 1.2 m/s, T ¼ 7.0s, s ¼ 2.65, D ¼ 0.2 mm and y0 ¼ 0.3 m. (1) r ¼ 0.3, φ ¼ 0, u ¼ 0 m/s; (2) r ¼ 0.5, φ ¼ -π/4, u ¼ 0 m/s; (3) r ¼ 0.3, φ ¼ 0, u ¼ -0.4 m/s.

3.3. Sediment flux

Wright, 2004a), around which the sediment concentration is more or less constant and qw passes zero flux. Both the predicted and experi­ mental qw pass zero flux almost at y ¼ 0 m in Fig. 16.

Following the above net UB, the corresponding sediment flux is shown in Fig. 16, where q is the net sediment flux; qc is the net UB part, a product of net UB and period-averaged sediment concentration; and qw is the wave part (¼q-qc). The sediment concentration is the same expo­ nential distribution as Chen et al. (2018c) where more details about q, qc and qw can be seen. The explicit approximation (E.A.) predicts the experimental (Exp.) sediment flux of acceleration-skewed case 4 and mixed-skewness case 5 reasonably. Net sediment transport rates for cases 4 and 5 (37.0 and 89.8 mm2/s) are adequately predicted (38.9 and 105.7 mm2/s). Here, q in sheet-flow is mainly transported near the initial bed (O’Donoghue and Wright, 2004b). The underestimations of onshore net UB for case 4 (y ¼ 0.01–0.04 m) and offshore net UB for case 5 (y ¼ 0.005–0.035 m) hardly influence the approximation of sediment flux because the sediment concentration is relatively small at the upper wave boundary layer. According to Chen et al. (2018c), q is mainly decided by qc below the initial bed and by qw above the initial bed for the two cases. The explicit approximation underestimates the offshore sediment flux (q and qc) of current and acceleration-skewed case 6 due to the apparent underestimation of net UB in Fig. 13(1) above the initial bed. The net sediment transport rate for case 6 (-236.6 mm2/s) is correspondingly underestimated (-119.3 mm2/s). However, the approximation of qw for case 6 is still generally well. Sediment con­ centration pivot exists generally close to and below y ¼ 0 m (O’Do­ noghue and Wright, 2004a). Sediment concentration variations are opposite to each other above and below the pivot (O’Donoghue and

4. Conclusions An explicit model is proposed to predict the velocity distribution and sediment flux for sheet-flow induced by current and asymmetric wave. The mobile sediment bed level used for the model performs the effect of phase lag (phase shift and residual) and acceleration. The oscillatory and current parts are separated to isolate the wave-current interaction. The oscillatory part is a product of the free stream velocity and defect function. The defect function consists of phase lead, mobile sediment bed level and wave boundary layer thickness. When a current is imposed to the wave, the mobile sediment bed level and wave boundary layer thickness are changed and thus the oscillatory part as well. At the same time, the current part is reduced by the wave eddy viscosity in the wave boundary layer and changed by an apparent wave roughness of the total wave-current flow outside the wave boundary layer. Eight free variables are required for the approximation, in which four are about free stream velocity, two are about sediment characteristic and two are about cur­ rent. Flow asymmetry is presented by a division of the free stream ve­ locity into 4 quarters of pseudo-sinusoidal flow. Iterations are necessary for the bed shear stress and roughness height used for the prediction of mobile bed level and wave boundary layer thickness. The verification of the model, which is of significance in engineering practices, uses available laboratory data covering different wave shapes. 14

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Coastal Engineering 157 (2020) 103635

The model is first applied for sinusoidal waves. The predicted ve­ locity is reasonably consistent with the experiment, which indicates the rationality of mobile sediment bed level, phase lead and wave boundary layer thickness. When an onshore current is imposed to the sinusoidal wave, the current part is not logarithmic due to a restriction by wave eddy viscosity, and the classical mobile bed effect leads to a bottom onshore net velocity. The onshore current enhances the onshore turbu­ lence and decreases the offshore turbulence. Therefore, turbulence asymmetry appears and results in an offshore net oscillatory part ve­ locity above the area of mobile bed effect within the wave boundary layer. In the model, the onshore current generates larger roughness height and wave boundary layer thickness near the flow crest than those near the flow trough. Relatively large wave boundary layer thickness near the flow crest corresponds to relatively small boundary layer ve­ locity and finally results in the offshore net oscillatory part velocity. In the pure acceleration-skewed wave case, the velocity contours are almost parallel to the vertical coordinate at the upper boundary layer. Except near the flow reversals, the velocity contours gradually become parallel to the mobile sediment bed level at the bottom boundary layer. The predicted net velocity is onshore and agrees with experiment where the offshore turbulence is stronger than the onshore turbulence. The net velocity results from the mobile sediment bed level, boundary layer thickness and phase lead in the approximation. The mobile sediment bed level is lower at the flow crest with a large shear stress than at the flow trough, and then results in the mobile bed effect which contributes a lot to the onshore net velocity. The wave boundary layer thickness refers to the turbulence asymmetry and generates a larger boundary layer ve­ locity at the flow crest than at the flow trough. The default constant phase lead cannot change the onshore net velocity. When the phase lead increases first and then decreases back as an actual variation, the stage of offshore flow is increased and the stage of onshore flow is reduced. Correspondingly, the net velocity is reversed offshore at the upper wave boundary layer and agrees with the experiments over fixed beds. In the combined wave-current case, the net velocity in the wave boundary layer is different from the net current part due to the net oscillatory part. An onshore/offshore current increases/decreases the velocity magnitude, roughness height and wave boundary layer thick­ ness at the onshore stage and decreases/increases them at the offshore stage. When a strong offshore current is imposed to the asymmetric wave, the mobile bed effect generates an offshore net velocity at the bottom, and turbulence asymmetry enhances the onshore net oscillatory part at the upper wave boundary layer. In this case, the onshore net oscillatory part is further enhanced without the mobile bed effect. When a strong onshore current is imposed to the asymmetric wave, the mobile bed effect generates an onshore net velocity at the bottom, and turbu­ lence asymmetry reverses the onshore net oscillatory part offshore at the upper wave boundary layer. In this case, the offshore net oscillatory part further penetrates the full wave boundary layer without the mobile bed effect. The direction change of net oscillatory part against current agrees with the existing experimental and numerical results. In addition, net sediment flux and sediment transport rate of sheet-flow induced by asymmetric wave even with a current can be generally predicted by the model with an exponential distribution of sediment concentration. Net sediment flux is mainly decided by the net velocity part below the initial bed and by the wave part above the initial bed. The predicted net sediment flux is hardly influenced by the underestimation of net velocity at the upper wave boundary layer due to relatively small sediment concentration there.

We have revised it critically for important intellectual content and approved the final version to be published. We agree to be accountable for all aspects of the work in ensuring that questions related to the accuracy or integrity of any part of the work are appropriately investigated and resolved. All persons who have made substantial contributions to the work, including those who provided editing and writing assistance but who are not authors, are named in the Acknowledgments section of the manuscript. We deeply appreciate your consideration of our manuscript, and we look forward to receiving comments from the reviewers. If you have any queries, please don’t hesitate to contact me at the E-mail: chenx@cau. edu.cn. Declaration of competing interest We have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the work submitted. We declare that we have no financial and personal relationships with other people or or­ ganizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled “Explicit approximation for velocity and sediment flux above mobile sediment bed beneath current and asymmetric wave” (ID: CENG_2019_268). We deeply appreciate your consideration of our manuscript, and we look forward to receiving comments from the reviewers. If you have any queries, please don’t hesitate to contact me at the E-mail: chenx@cau. edu.cn. Acknowledgement The project is supported by National Natural Science Foundation of China (Grant Nos. 41961144014, 51609244 and 51836010) and Chi­ nese Universities Scientific Fund (Grant No. 2019TC133). Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.coastaleng.2020.103635. References Abreu, T., Silva, P.A., Sancho, F., Temperville, A., 2010. Analytical approximate wave form for asymmetric waves. Coast Eng. 57 (7), 656–667. Chen, X., Li, Y., Niu, X., Chen, D., Yu, X., 2011. A two-phase approach to wave-induced sediment transport under sheet flow conditions. Coast Eng. 58 (11), 1072–1088. Chen, X., Li, Y., Wang, F., 2018a. Mobile bed thickness in skewed asymmetric oscillatory sheet flows. Acta Mech. Sin. 34 (2), 257–265. Chen, X., Li, Y., Chen, G., Wang, F., Tang, X., 2018b. Generation of net sediment transport by velocity skewness in oscillatory sheet flow. Adv. Water Resour. 111, 395–405. Chen, X., Wang, F., Tang, X., Qiu, L., 2018c. Sediment flux based model of instantaneous sediment transport due to pure velocity-skewed oscillatory sheet flow with boundary layer stream. Coast Eng. 138, 210–219. Davies, A.G., Li, Z., 1997. Modelling sediment transport beneath regular symmetrical and asymmetrical waves above a plane bed. Cont. Shelf Res. 17, 555–582. Dick, J., Sleath, J., 1992. Sediment transport in oscillatory sheet flow. J. Geophys. Res. 97, 5745–5758. Dohmen-Janssen, C.M., Hanes, D.M., 2005. Sheetflow and suspended sediment due to wave groups in a large wave flume. Cont. Shelf Res. 25 (3), 333–347. Dong, L., Sato, S., Liu, H., 2013. A sheet flow sediment transport model for skewedasymmetric waves combined with strong opposite currents. Coast Eng. 71, 87–101. Fredsøe, J., Deigaard, R., 1992. Mechanics of Coastal Sediment Transport. World Scientific Publication. Fuhrman, D.R., Schløer, S., Sterner, J., 2013. RANS-based simulation of turbulent wave boundary layer and sheet-flow sediment transport processes. Coast Eng. 73, 151–166. Grant, W.D., Madsen, O.S., 1979. Combined wave and current interaction with a rough bottom. J. Geophys. Res. 84 (C4), 1797–1808.

Author statement Manuscript title: Explicit approximation for velocity and sediment flux above mobile sediment bed beneath current and asymmetric wave (ID: CENG_2019_268). Both authors have made substantial contributions to the work, which includes the acquisition analysis and interpretation of data. 15

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