MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi 10.1155/2017/2614943 2614943 Research Article Numerical Study for Near-Bed Variables in Velocity-Skewed Oscillatory Sheet Flow Transport http://orcid.org/0000-0002-6244-7866 Chen Xin 1 Zhang Zichao 2 Li Yong 3 Shi Xiaoyan 1 Borthwick Alistair 1 Beijing Engineering Research Center of Safety and Energy Saving Technology for Water Supply Network System China Agricultural University Beijing 100083 China cau.edu.cn 2 School of Electric Power North China University of Water Resources and Electric Power Zhengzhou 450045 China ncwu.edu.cn 3 Tianjin Centre China Geological Survey Tianjin 300170 China 2017 2282017 2017 14 04 2017 02 07 2017 11 07 2017 2282017 2017 Copyright © 2017 Xin Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The near-bed sediment concentration and vertical sediment flux are important in sediment transport mechanics, but they are known much less than the horizontal sediment flux, especially for fine sediment in unsteady flows. A developed two-phase model is applied to study the near-bed sediment concentration, vertical sediment flux, and the relevant total sediment amount for the velocity-skewed oscillatory sheet flow transport. With the sediment concentration hindered fall velocity, the classical reference concentration formulas conducted by Engelund and Fredsoe (1976) and Zyserman and Fredsoe (1994) are utilized for the comparison with the two-phase model and illustration of the phase-lag and sediment size effects in near-bed sediment concentration and vertical sediment flux. The concentration and vertical flux predicted by the two-phase model agree well with experimental data and are better than empirical formulas. Furthermore, the sediment size effect for pick-up flux function over starved bed is shown to be quite different from that containing sufficient sediment in oscillatory flows.

National Natural Science Foundation of China 51609244 11472156 National Science-Technology Support Plan of China 2015BAD20B01
1. Introduction

The phenomenon of sediment entrainment and settlement is universal in river and coastal environment and extraordinarily important in the subaquatic geomorphology. The vertical sediment fluxes consist of pick-up flux due to turbulence, settling flux due to gravity, and their summation denoting the variation rate of total sediment amount. Associated with the sediment concentration, they represent the vertical sediment exchange strength and are important for the study of sediment settling down to deposition and picked up into suspension. Most sediment models need a type of near-bed condition associated with vertical sediment flux for bottom boundary conditions. So the predictions of near-bed concentration, vertical sediment flux, pick-up flux, and settling flux are very fundamental for such sediment models.

In fully developed steady flows, the pick-up flux and settling flux are in balance and the total vertical flux is zero. Numerous steady flow formulas have been carried out to predict the vertical sediment flux over loose sediment bed in developed equilibrium steady flows. Generally, some are straight driven by the pick-up function based on clear water flow over starved bed without considering the influence of suspended sediment [1, 2], so they are not appropriate in the flow already containing large amount of suspended sediment. The others are evaluated by the balance of pick-up and settling down according to the concentration and sediment fall velocity , and the relevant concentration denotes the vertical sediment flux strength.

The near-bed sediment concentration and vertical sediment fluxes in unsteady flows are much more complicated than that in steady flows due to the phase-lag effect, and formulas in unsteady flows are lacking. So the near-bed concentrations denoting vertical flux strength in steady flows  are applied and widely used in unsteady numerical models for sediment transport , and even their concentrations drop to zero near the flow reversal when the bottom shear stress is under a threshold. The knowledge of vertical sediment flux in unsteady flow is still necessary for numerical models’ boundary conditions in engineering.

In unsteady flows, the vertical sediment flux can be derived from the variation rate of sediment amount in movement , which can be obtained by single phase models or two-phase models. The single phase numerical models are dependent on the reference concentration or vertical flux as the bottom boundary conditions and cannot describe the vertical flux below reference height. The two-phase models  can obtain the details of reference concentration and vertical flux as a result of the turbulent upward effect and gravitational settlement. Yu et al.  study the vertical flux for medium and coarse sediment, but the model is not valid for fine sediment (diameter D<0.2 mm) when the phase-lag is obvious due to the sediment stress. In addition, the vertical sediment fluxes have not been extracted accurately enough due to the noise or error of modelling high concentration. Very little is known about the near-bed concentration and vertical flux for fine sediment and the phase-lag effects, which are important in the generation of net sediment transport, especially in velocity-skewed oscillatory flows with a sharp wave crest and gentle wave trough.

Present study utilizes a two-phase model to study the near-bed sediment concentration and vertical fluxes in velocity-skewed oscillatory sheet flows containing sufficient amount of sediment. The results mainly exhibit and explain the phase-lag characteristics in concentration and vertical flux at the reference height, near-immobile bed, and very low concentration area for difference sediment sizes.

2. Applied Models

Chen and Yu  two-phase model is utilized to obtain the sediment concentration and vertical fluxes. For comparison, several steady flow empirical formulas are also applied.

2.1. Two-Phase Flow Model

The two-phase model includes Reynolds averaged continuity and momentum equations:(1)1-αuf,j+αus,jxj=0,αt+αus,jxj=xjνstδsαxj,1-αuf,it+1-αuf,iuf,jxj=-1-αρfpxi+τfjxj+1-αgi-Fiρf,αus,it+αus,ius,jxj=-αρspxi+τsjxj+αgi+Fiρs,where α is the sediment volumetric concentration; u is the velocity; x is the Cartesian coordinate; t is the time; the subscripts f and s stand for the fluid and the sediment phases; the indices i,j=1,2 represent horizontal and vertical components and obey summation convention; νst is the sediment turbulent viscosity confirmed by fluid turbulent viscosity νft; δs=1 is the Schmidt number; p is the pressure; τ is the stress; ρ is the density; g is the body force; F is the interaction force. νst, νft, τ, and F can be seen in Chen and Yu . Equations (1) are closed by k-ε turbulent equations:(2)1-αkt+1-αkufjxj=xj1-ανfkkxj+1-αG-ε1-αεt+1-αεufjxj=xj1-ανfεεxj+1-αkC1Gε-C2ε2,where k is the turbulent kinetic energy (TKE); ε is the turbulent kinetic energy dissipation rate (DTKE); νfk=νft, νfε=νft/1.33, C1 = 1.44, and C2 = 1.92; G is the turbulence generation term .

Equations (1) and (2) are applied for sheet flows happening at maximum Shields parameter θ > 0.8–1.0 when large amounts of sediment are in movement. The computation domain covers the immobile bed and the water tunnel top. The initial bed is located at x2 = 0. The still water condition is assumed to be the initial condition. At the time t=0, all flow variables are zero, while α=0 at x2 > 0 and α=0.6 at x2 ≤ 0.

At the tunnel top, uf,1=U(t)=U0{0.8cos[2π(t/T-0.21)]+0.2cos[4π(t/T-0.21)]}, where U is the free stream (Figure 1); A=dU/dt; U0 and A0 are the amplitudes of U and A; T is the period; subscripts a and d denote acceleration and deceleration stages; scripts c and t denote the crest and trough durations. Positive (t/T = 0–0.42) and negative (t/T = 0.42–1) U denote onshore and offshore durations, respectively. The vertical gradient of all variables vanished, and the sediment flux is(3)νstδsαx2-us,2α=0.

Free stream velocity and acceleration.

Nonslip condition is assumed at the bottom that α=0.6 and the other variables are set to be zero.

In the lateral boundaries, the horizontal gradients of α, u, k, and ε are all zero. The flow is driven by a horizontal pressure gradient:(4)px1=-ρfdUdt.

2.2. Flux Definition and Empirical Formulas

The sediment amount in movement is defined by Nielsen et al. :(5)Qx2,t=x2αy,tdy.

The total vertical flux is(6)qtx2,t=Qx2,tt,and the downward settling flux is(7)qdx2,t=-αx2,twx2,t,where w is the fall velocity equal to -us,2. The pick-up flux is(8)qux2,t=qtx2,t-qdx2,t.The dimensionless equations (10)–(12) are Φ=qS-1gD3-0.5, where the specific gravity S=2.65. In the two-phase model, α and w are straight given by (1), and qu=-νst/δs(α/x2).

For qd comparison, the classical Engelund and Fredsoe  formula (EF76) and Zyserman and Fredsoe  formula (ZF94) for reference α defined at x2/D=2 and w=w01-αn are applied. w0 is the single sediment fall velocity in still water  and n is the same as Yu et al. . The EF76 formula is(9)α2D,t=0.61+λ3,(10)λ=κcrθ-θc0.013Sθ1-πμ6θ-θc4+πμ64-1/4,where D is the sediment diameter; λ is the linear concentration; κ=0.4 is the Karman constant; cr=2 is associated with the bed roughness; θc=0.05 is the critical Shields parameter; μ=0.6 is the dynamic friction coefficient. The ZF94 formula is (11)α2D,t=0.331θ-θc1.75cmcm+0.331θ-θc1.75,where cm=0.46. Figure 2 shows the reference α against θ for the two empirical formulas. The EF76 considers the dispersive stress from suspended sediment and drag of moving bed, so the reference α is mildly increased when θ>1 and is hindered mildly to be a constant. ZF94 is based on Garcia and Parker’s  prohibiting a particular trend toward a limitation for high sediment concentration. So the ZF94 also has a limitation of α and keeps a mild increment after θ>6 (Figure 2).

Reference α against θ in steady flow.

For qu comparison, van Rijn  pick-up function over a starved bed is applied:(12)Φu=quS-1gD3=3.3×10-4θθc-11.5S-1gD3νf020.1,where νf0 is the fluid molecular viscosity. qd in EF76 and ZF94 formulas and qu in van Rijn  drop to zero when θ<θc without phase-lag effect.

3. Results

Computations are based on the O’Donoghue and Wright  experiments in Table 1, where the phase-lag effect increases with the decrement of D. Results are shown for the classical reference height x2/D=2 first and second the near-immobile bed and low concentration areas.

Computation conditions.

Test case U 0 D (mm) T (s) w 0 (m/s)
Fine: FA5010 1.5 0.13 5.0 0.0126
Medium: MA5010 1.5 0.27 5.0 0.0387
Coarse: CA5010 1.5 0.46 5.0 0.0672
3.1. Reference Vertical Flux

The references α and Φd are shown in Figure 3. The absolute locations of x2/D=2 are slightly different and are very close to each other, so they can be treated as an approximation to the initial bed. The α distribution pivot, around which α is almost constant in a wave period, exists at the location a little below the initial bed . In Figures 3(a), 3(c), and 3(e), the location x2/D=2 is very close to the pivots, and both the maximum free stream flow and flow reversal would probably lead to α peaks. For the experiment (dot), the α difference between flow crest (t/T=0.21) and flow trough (t/T=0.71) is not obvious and even the crest flow strength is much larger than that at flow trough (Figure 3), because many sediments picked up at the flow crest are still in movement at the flow trough. Large D corresponds to small concentration magnitude due to the large fall velocity. α never decrease to zero in a period due to the phase-lag effect. The results of two-phase model exhibit clearly the decrement of concentration magnitude when D increases, and the computed α is also always above a proper nonzero value (Figures 3(a), 3(c), and 3(e)) due to the phase-lag. Some differences between two-phase model and experimental data exist from a qualitative perspective, probably due to the high concentration measurement uncertainties  without mass conservation.

Reference α and Φd.

α of FA5010

Φ d of FA5010

α of MA5010

Φ d of MA5010

α of CA5010

Φ d of CA5010

The EF76 and ZF94 are strictly in phase with U without any phase-shift. Mild increment at large shear stress (Figure 2) leads to a small concentration variation near onshore flow peak around t/T=0.21 (Figures 3(a) and 3(c)). There are also small concentration variations for EF76 and ZF94 near the offshore flow peak (t/T = 0.6~0.8) caused by the nearly constant U (Figure 1). When θ is less than a critical level, EF76 and ZF94 predicted α are unrealistic zero (Figure 2), because the formulas are derived without the phase-lags. Predicted concentrations of EF76 and ZF94 increase rapidly in the acceleration stage and decrease fast in the deceleration stage (Figures 3(a), 3(c), and 3(e)). Furthermore, their prediction between onshore and offshore are quite different because of the large different flow strength and bottom shear stress. Although both EF76 and ZF94 can predict the experiments near the flow trough (t/T = 0.6~0.8), they much overestimate the experiments near the flow crest (t/T=0.21). Generally, the EF76 and ZF94 are weaker than the two-phase model results at the flow reversal and flow crest. RMS errors in Table 2 for two-phase model are much less than EF76 and ZF94 formulas.

Root-mean-square (RMS) errors at x2/D=2.

Cases α Φ d
FA5010 MA5010 CA5010 FA5010 MA5010 CA5010
Two-phase model 0.051 0.039 0.042 0.037 0.038 0.053
EF76 0.138 0.094 0.086 0.091 0.100 0.118
ZF94 0.148 0.094 0.068 0.095 0.102 0.106

Corresponding to the concentration due to phase-lag, the experimental Φd never decreases to zero near the flow reversal in Figure 3, and the value near the flow crest is close to that near the flow trough. The two-phase model predicted computed Φd agree very well with the experiments for nonzero duration and the magnitude increment with D. Although the α magnitude decreases with the increment of D, the magnitude of Φd increases with the increment of D in Figure 3, because the flux is hindered by high concentration at small sediment size. Based on the same U0 = 1.5 m/s, the pick-up flux (Φu=-Φd) against D for ZF94 and ZF76 in steady flow is shown in Figure 4, where the dimensionless D is represented by w0/U0. The flux increases first and then decreases with D (w0/U0). The Φd magnitude would decrease with small α when D is large enough as the balanced pick-up flux (Φu=-Φd) in Figure 4, and finally it decreases to zero when reference α magnitude reduces to zero.

Near the flow crest, there are hollows in EF76 and ZF94 that predicted Φd which are far from the experiments (Figures 3(b), 3(d), and 3(f)) with overestimation of α (Figures 3(a), 3(c), and 3(e)), because the maximum Φd and optimal α exist in the two empirical formulas when we applied w=w01-αn, as seen in Figure 5. The maximum Φd approximately corresponds to optimal α=0.14, after which Φd decreases with the increment of α by the hindered fall velocity. Near the flow trough, both EF76 and ZF94 can predict Φd well due to the agreement of concentration. At flow reversal, both the EF76 and EF94 predicted Φd drop to zero because of unrealistic zero α (Figures 3(a), 3(c), and 3(e)) when θ is less than a critical value. Predicted Φd of EF76 and ZF94 changes rapidly near flow reversal with rapid change of α. Generally, the RMS errors for EF76 and ZF94 are much larger than two-phase model for Φd (Table 2).

Φ d of different D in steady flow.

Against the settling down induced by gravity, the pick-up effect induced by turbulence maintains the suspension of sediment, and the total vertical sediment flux (Φt) is the balance of pick-up flux (Φu) and settling flux (Φd). Figure 6 shows the two-phase model computed Φt and Φu at x2/D=2 and the immobile bed surface level denoted by Z (α=0.594). To illustrate their relationships availably and validate Φu indirectly, Φd in Figure 3 is included again. Magnitudes of Φu and Φd are around each other with opposite direction, and they are in periodic averaged balance with averaged Φt=0. As the difference of Φu and Φd, Φt is almost one magnitude smaller than Φu and Φd. Φu increases and reaches a local maximum value before flow crest (t/T=0.21) or flow trough (t/T=0.71) and decreases to a local minimum value before flow reversal both in the onshore and in the offshore durations. Correspondingly, Φt almost follows the same variation tendency as Φu. Similar process can be seen in Ribberink et al. , Nielsen et al. , Li et al. , and Bakhtyar et al. . Onshore magnitudes of Φu and Φt are larger than that at offshore, and the onshore variations are more rapid than offshore.

Vertical sediment flux and immobile bed surface.

Φ at x2/D=2, FA5010

Z , FA5010

Φ at x2/D=2, MA5010

Z , MA5010

Φ at x2/D=2, CA5010

Z , CA5010

Z almost varies together with Φt which decided the bed morphology in the present three cases in Figure 6. The bed morphology repeats periodically in oscillatory sheet flow conditions without horizontal variation. When Φt>0, Z decreases because sediment below the initial bed is picked up. Z decreases to minimum (t/T = 0.25–0.3) almost after Φt decreases to zero. While Φt<0, Z increase because sediment above the initial bed settles down. Z increases to local maximum (t/T = 0.45–0.5) almost after Φt increases to zero. Z never drops to zero due to the phase-lag effect. For comparison, Zmθ/θm is shown considering theoretical Z/Zm=θ/θm in steady flow, where Zm is the maximum Z. The difference between Z and Zmθ/θm denotes the phase-lag. There is a time phase-shift between Z and Zmθ/θm. For fine 0.13 mm, the phase-lag effect is the largest with the most obvious difference between Z and Zmθ/θm, while the phase-lag effect for 0.46 mm is the smallest because present Z almost coincides with Zmθ/θm.

Corresponding to Φd, the Φu magnitude also increases with the increment of D for the three sizes. But it does not increase with the increment of D monotonously, which is similar to Figure 4. In the two-phase model, Φu is obtained from the turbulence diffusion term -νst/δs(α/x2) modelled by the classical gradient transport assumption. There are two reasons for the smallest Φu amplitude with fine D = 0.13 mm in Figure 6: the concentration gradient and the turbulence intensity. With D = 0.13 mm, much sediment is picked up for suspension and α/x2 at x2/D=2 becomes smaller than D = 0.27–0.46 mm. When much sediment is already in suspension in the boundary layer, much turbulence energy is dissipated [15, 18] corresponding to small νst and restrains other sediments to be picked up.

For comparison without the effect of suspended hindered sediment, van Rijn  pick-up flux function is shown in Figure 7 under the same U0 = 1.5 m/s. van Rijn  pick-up function decreases with the increment of D, which is opposite to the flows containing sufficient sediment amount in Figures 4 and 5. The onshore magnitude is much larger than offshore magnitude without the effect of suspended hindered sediment. In the experiment , Φu did not drop to zero due to the shear (velocity overshoot) and turbulence that exist near flow reversal. But van Rijn  function based on steady flow drops to zero around flow reversal. Comparing the van Rijn pick-up function (Figure 7) with Φu given by two-phase model (Figure 6), their Φu magnitudes are the closest only for D = 0.46 mm when the suspension sediment amount is the smallest (Figure 8). Over all, the van Rijn  pick-up function is not valid for the oscillatory sheet flow when the maximal θ > 0.8–1.0 and sufficient sediment is in suspension.

Pick-up function of van Rijn.

Sediment amount above x2/D=2.

In developed oscillatory sheet flow, the averaged Φt is zero, and suspended sediment is quasi-steady saturated and hinders the pick-up process. The vertical fluxes display the sediment exchange in vertical direction, but they can not be used as the index for the sediment-laden capacity of oscillatory sheet flow. The sediment amount Q is in a quasi-steady status in the developed stage and can be used to estimate the sediment-laden capacity. Q above x2/D=2 is shown in Figure 8, where H = 0.5 m is the water depth. Q increases with the decrement of D. The variation rate of Q is Φt (see (6)), so Q reaches a local maximum value when positive Φt decreases to zero near the flow peak and reaches a local minimum value when negative Φt increases to zero near the flow reversal. Due to large phase-lag, the Q magnitude’s ratio between onshore (t/T = 0–0.42) and offshore (t/T = 0.42–1) for 0.13 mm case is 0.87, much larger than 0.41 for 0.46 mm case.

3.2. Other Vertical Fluxes

The near-bed flux at the locations above or below the reference height is essential for better understanding of the near-bed sediment exchange. Figure 9 shows the two-phase model computed vertical fluxes at x2 = 10 mm and x2 = −3.5 mm. −3.5 mm is in the main eroded area with high concentration and almost fully hindered w0. 10 mm is out of the sheet flow layer where particle interactions can be negligible and is still in the oscillatory boundary layer with small concentration and unhindered w0. Because the absolute locations of x2/D in the main eroded area or low concentration area for three sizes are far from each other, the vertical locations in Figure 9 are in absolute millimetres rather than as a function of the diameter. At x2=-3.5 mm, Φu, Φd, and Φt decrease with the increment of D. Two peak values exist in Figure 9(a) and onshore magnitude is larger than offshore, but only one peak value exists near the onshore flow crest in Figure 9(c), while in Figure 9(e) almost zero fluxes exist because −3.5 mm is nearly located at the immobile bed for D = 0.46 mm. At x2 = 10 mm, obvious phases for all the fluxes are shifted after x2/D=2. Both the maximum Φu and Φd are around 0.01, which are much larger than Φt. For fine and medium sediment (Figures 9(b) and 9(d)), their magnitudes of Φu, Φd, and Φt are similar to and larger than coarse sediment (Figure 9(f)). For coarse sediment, both Φu and Φd only exist near the onshore flow peak, and Φt almost disappear compared to the fine and medium sediment cases. Comparing the results in Figure 9 to that in Figure 6, all Φt, Φu, and Φd decrease from the initial bed toward the immobile bed and the top of tunnel. The decrement is also seen in Li et al.  and Bakhtyar et al. . All the fluxes finally decrease to zero when the location is down into the immobile bed or up to the top of tunnel.

Computed vertical flux at x2 = −3.5 mm and x2 = 10 mm.

FA5010, x2 = −3.5 mm

FA5010, x2 = 10 mm

MA5010, x2 = −3.5 mm

MA5010, x2 = 10 mm

CA5010, x2 = −3.5 mm

CA5010, x2 = 10 mm

Also, Q above the locations −3.5 mm and 10 mm are shown (Figure 10). At x2 = −3.5 mm and x2 = −10 mm, the variation amplitudes of each sediment sizes are much smaller than that at x2/D=2 (Figure 8) due to smaller Φt in Figure 9. Finally, the Q variation amplitudes decrease to zero in the immobile bed or tunnel top corresponding to Φt=0 (Figure 9).

Sediment amount above x2 = −3.5 mm and x2 = 10 mm.

Above x2 = −3.5 mm

Above x2 = 10 mm

4. Conclusions

The present work studies the near-bed conditions in velocity-skewed oscillatory sheet flows with a two-phase model and classical empirical formulas. The phase-lag and sediment size effects in concentration, vertical sediment flux, and sediment amount are focused on the area near the initial bed, the immobile bed, and the low concentration area.

Due to the phase-lag, the reference concentration and the settling flux never drop to zero in a period even at the flow reversal, and the relevant magnitudes between onshore and offshore are similar. The two-phase model computed reference concentration and settling flux by two-phase model agree with experiment very well and are better than empirical formulas. Under the same flows, reference concentration magnitudes decrease with the increment of sediment size due to the increment of fall velocity. While, for settling flux, as the product of concentration and fall velocity, the magnitude increases first and then decreases with the increment of sediment size.

The pick-up flux and settling flux are in periodic averaged balance in oscillatory sheet flow, and their magnitudes are around each other with opposite direction. The pick-up flux increases and reaches a local maximum value before flow peak and decreases to a local minimum value before flow reversal both in the onshore and in the offshore durations. Corresponding to the settling flux, the pick-up flux also increases first and then decreases with the increment of sediment size.

The sediment amount magnitude increases with the decrement of sediment size. Phase-lag effect increases the sediment amount magnitude’s ratio between onshore and offshore. As the sediment amount variation rate, the total vertical flux is much smaller than the pick-up flux and varies almost following the pick-up flux. The vertical sediment flux magnitude decreases from the initial bed toward immobile bed and top of tunnel.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grants no. 51609244 and no. 11472156) and National Science-Technology Support Plan of China (Grant no. 2015BAD20B01).

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