### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Model Formulation
- 3. Validation
- 4. Analysis of Streaming Generating Mechanisms
- 5. Discussion
- 6. Conclusion
- Appendix A:: Numerical Solution Method
- Appendix B:: Shape Expression
- Acknowledgments
- References
- Supporting Information

[1] The net current (streaming) in a turbulent bottom boundary layer under waves above a flat bed, identified as potentially relevant for sediment transport, is mainly determined by two competing mechanisms: an onshore streaming resulting from the horizontal non-uniformity of the velocity field under progressive free surface waves, and an offshore streaming related to the nonlinearity of the waveshape. The latter actually contains two contributions: oscillatory velocities under nonlinear waves are characterized in terms of velocity-skewness and acceleration-skewness (with pure velocity-skewness under Stokes waves and acceleration-skewness under steep sawtooth waves), and both separately induce offshore streaming. This paper describes a 1DV Reynolds-averaged boundary layer model with*k*-*ε*turbulence closure that includes all these streaming processes. The model is validated against measured period-averaged and time-dependent velocities, from 4 different well-documented laboratory experiments with these processes in isolation and in combination. Subsequently, the model is applied in a numerical study on the waveshape and free surface effects on streaming. The results show how the dimensionless parameters*kh* (relative water depth) and *A*/*k*_{N} (relative bed roughness) influence the (dimensionless) streaming velocity and shear stress and the balance between the mechanisms. For decreasing *kh*, the relative importance of waveshape streaming over progressive wave streaming increases, qualitatively consistent with earlier analytical modeling. Unlike earlier results, simulations for increased roughness (smaller *A*/*k*_{N}) show a shift of the streaming profile in onshore direction for all *kh.* Finally, the results are parameterized and the possible implications of the streaming processes on sediment transport are shortly discussed.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Model Formulation
- 3. Validation
- 4. Analysis of Streaming Generating Mechanisms
- 5. Discussion
- 6. Conclusion
- Appendix A:: Numerical Solution Method
- Appendix B:: Shape Expression
- Acknowledgments
- References
- Supporting Information

[2] The dynamics of water and sediment in the bottom boundary layer under waves in coastal seas are of key importance for the development of cross-shore and long-shore coastal profiles. Many recent studies on the complex interaction between wave motion and seabed emphasize the influence of the waveshape on bed shear stress, sediment transport and flow velocities, either focusing on velocity-skewness (present under waves with amplified crests), acceleration-skewness (present under waves with steep fronts) or both phenomena in joint occurrence (for references see*Ruessink et al.* [2009]). Experimental studies on waveshape effects have often been carried out in oscillating flow tunnels, with both fixed and mobile beds of various sand grain sizes, and special attention has been paid to the sheet-flow transport regime, where bed forms are washed away and the bed is turned into a moving sediment layer [*Ribberink et al.*, 2008]. An important observation from tunnel experiments in the sheet-flow regime is that under velocity-skewed flow over coarse grains the sediment transport is mainly onshore, but that net transport decreases with decreasing grain sizes and can even become negative for fine sand [*O*'*Donoghue and Wright*, 2004]. *Dohmen*-*Janssen and Hanes* [2002] and very recently *Schretlen et al.* [2011]carried out detailed full-scale wave flume experiments on sand transport by waves in the sheet-flow regime. These flume measurements show onshore instead of offshore transport of fine sand under 2nd order Stokes waves and larger transport rates for medium-sized sand compared to experiments with comparable velocity-skewness in oscillating flow tunnels. These different results for sediment transport emphasize the importance of a good understanding of the hydrodynamic differences between oscillating flow tunnels, with horizontally uniform oscillating pressure gradients, and wave flumes, with horizontally non-uniform pressure gradients and vertical motions due to the free surface.

[3] A remarkable free surface effect that potentially contributes to onshore (current related) sediment transport is the generation of a steady bottom boundary layer current in onshore direction [*Longuet*-*Higgins*, 1953]: the vicinity of the bed affects the phase of the horizontal and vertical orbital velocities. This introduces a wave-averaged downward transport of horizontal momentum that drives an onshore boundary layer current (here called ‘progressive wave streaming’). This process acts opposite to the net current that will be generated in a turbulent bottom boundary layer by a velocity-skewed or acceleration-skewed oscillation (‘waveshape streaming’). The latter mechanism, that can be present both in tunnels and flumes, is due to the different characteristics of the time-dependent turbulence during the on- and offshore phase of the wave, introducing a nonzero wave-averaged turbulent shear stress. This phenomenon was first predicted for velocity-skewed waves by*Trowbridge and Madsen* [1984b] and observed in tunnel experiments by *Ribberink and Al*-*Salem* [1995].

[4] It is the aim of this study to develop a carefully validated numerical model for the net currents in the turbulent wave boundary layer above a flat but hydraulically rough bed, and to develop more insights in the balance between the waveshape streaming and progressive wave streaming on the shoreface.

[5] The various streaming contributions have been modeled before by several authors: *Longuet*-*Higgins* [1958] predicted the onshore streaming under progressive waves analytically using a constant viscosity. *Johns* [1970]included height-dependency in the eddy viscosity and later*Johns* [1977] used a turbulent kinetic energy closure in a numerical study on the residual flow under linear waves. *Trowbridge and Madsen* [1984a] developed an analytical model with time dependent eddy viscosity. Their second order approach [*Trowbridge and Madsen*, 1984b] (TM84) jointly included 1) the advective terms of the momentum equation, 2) (forcing) free stream velocities determined with Stokes' 2nd order wave theory, and 3) an eddy viscosity being the product of a vertical length scale and the first three Fourier components of the shear velocity. This key development revealed the competition between onshore progressive wave streaming and offshore velocity-skewness streaming, with dominance of the latter for relatively long waves. Later work [*Trowbridge and Young*, 1989] and a recent coupling of the TM84 model with a bed load transport formula [*Gonzalez Rodriquez*, 2009, chapter 6] indeed showed a significant effect of progressive wave streaming on shear stress and net bed load transport. Due to the absence of detailed flume measurements and just tunnel data available for validation, progressive wave streaming was not included in most of the (one and two phase) numerical boundary layer models developed for research on shear stress and sediment transport under waves [e.g., *Davies and Li*, 1997; *Holmedal and Myrhaug*, 2006; *Conley et al.*, 2008; *Fuhrman et al.*, 2009a, 2009b; *Hassan and Ribberink*, 2010; *Hsu and Hanes*, 2004; *Li et al.*, 2008; *Ruessink et al.*, 2009]. Such models, both with one and two-equation (*k*-*ε* and *k*-*ω*) turbulence closures, are generally fairly well capable to reproduce the velocity-skewness streaming as measured in tunnels by*Ribberink and Al*-*Salem* [1995]. These Reynolds-averaged models have recently been supported by results of Direct Numerical Simulations [*Cavallaro et al.*, 2011], have been used in a 2D version to investigate slope effects in tunnels [*Fuhrman et al.*, 2009a] and have shown good reproduction of measured sediment transport rates in tunnels as well [e.g., *Ruessink et al.*, 2009; *Hassan and Ribberink*, 2010].

[6] To the author's knowledge, only a few studies [*Henderson et al.*, 2004; *Hsu et al.*, 2006; *Holmedal and Myrhaug*, 2009; *Yu et al.*, 2010] have presented numerical boundary layer models that include effects of the free surface and the waveshape on the boundary layer flow simultaneously. These studies demonstrate respectively the relevance of progressive wave streaming for onshore sandbar migration (first two references, validation on morphological field data), for streaming profile predictions (third reference, without data-model comparison) and for suspended sediment transport (fourth reference, validation on concentration profiles). Nevertheless, a detailed validation of the numerical models on net current measurements is still lacking until now.

[7] Considering the experimental observations and indications from the model studies, the research objectives in this study are: i) to validate the hydrodynamics of a numerical Reynolds-averaged boundary layer model, extended with free surface effects, using selected laboratory measurements of different types of wave boundary layer flow, ii) to apply this model to obtain insight in the balance between progressive wave streaming and waveshape streaming, and how this is affected by varying wave and bed conditions. Our model, basically an extension of the model used in*Ruessink et al.* [2009] and *Hassan and Ribberink* [2010], is described in section 2. The model validation on detailed velocity measurements above fixed beds is given in section 3. The balance between progressive wave streaming and velocity-skewness streaming is studied with a systematic numerical investigation of velocities and shear stresses insection 4. Section 5 gives a short outlook on the implications of modeling these streaming processes on sediment transport predictions. Section 6 summarizes the major conclusions of this study.

### 4. Analysis of Streaming Generating Mechanisms

- Top of page
- Abstract
- 1. Introduction
- 2. Model Formulation
- 3. Validation
- 4. Analysis of Streaming Generating Mechanisms
- 5. Discussion
- 6. Conclusion
- Appendix A:: Numerical Solution Method
- Appendix B:: Shape Expression
- Acknowledgments
- References
- Supporting Information

[26] We subsequently investigate how the observed direction and shape of the net current profiles can be attributed to the various streaming mechanisms and their potential competition. Next, we explore systematically how this competition will change for changing wave and bed conditions. Finally, we study the effects of the mean pressure gradient.

#### 4.1. Streaming Mechanisms in the Validation Cases

[27] We use our model to assess and distinguish the influence of the various mechanisms on the *U*_{0}profile. First, a ‘shape’-expression has been derived from the momentum balance (by period averaging and integration over*z*, see also Appendix B, overbar indicates period-averaging):

This shows the influence of the various momentum transferring mechanisms to the mean velocity gradient (note that the wave averaged viscosity is always positive) or more precise the current-related part of the mean shear stress. The terms on the right hand side show respectively the contributions from 1) mean momentum transport by vertical velocity (‘wave Reynolds stress’) driving the progressive wave streaming, 2) the wave-averaged pressure gradient, and 3) differences in turbulence between the on- and offshore phase of the wave driving the waveshape streaming (wave-related mean shear stress). Second, profiles of all these terms have been computed from the model results. A direct comparison of the four validation cases is possible after normalization. The vertical distance has been scaled by*δ**, an estimate for the thickness of the turbulent wave boundary layer [*Nielsen*, 1992; *Swart*, 1974]:

The stress contributions are scaled by the maximum bed shear stress *τ*_{b,m} exerted by a sinusoidal oscillatory flow with a velocity amplitude _{1∞} identical to the validation case [see *Fuhrman et al.*, 2009a]. Here, this *τ*_{b,m} was obtained from simulations, but can equally well be computed with *τ*_{b,m} = 1/2*ρf*_{w} _{1}^{2} and *f*_{w} according to (18). These results are shown in Figure 6.

[28] In Figures 6a–6d we observe the following (in order of increasing interest). No contribution of the wave Reynolds stress (Figure 6b) is present in the tunnel cases (case 2 and 3). A positive mean pressure gradient (Figure 6c) is present in the flume cases (case 1 and 4), a negative in the tunnel cases, consistent with the directions of the mass transport compensation currents. The waveshape related contribution (Figure 6d) is negative for all cases, running from a maximum negative value at or very near the bed toward zero around *δ*.*This contribution is not only present for the velocity-skewed oscillations/waves (case 2and 4), but also under the acceleration-skewed oscillation (case 3), albeit smaller. Also the practically linear wave in case1 shows a negative waveshape contribution (d). We ascribe this to the increased onshore and reduced offshore near bed velocities due to the positive progressive wave streaming, introducing a turbulence behavior like under velocity-skewed oscillation. The deviation of the wave Reynolds stress (b) from its free stream value has the same form in the two flume experiments: constant and positive close to the bed and subsequently twisting around zero with decreasing amplitude for increasing distance from the bed. Forcase4, the summation in panel (a) shows a clear competition between the contribution from the wave Reynolds stress and from the wave-related mean stress. At low levels, apparently the waveshape streaming wins and the velocity gradient is negative. At higher levels, the gradient becomes positive and subsequently negative again, under influence of the progressive wave streaming mechanism. This explains the velocity profile inFigure 4, where the negative velocity is the result of velocity-skewness streaming, but the bulb in positive direction follows from the progressive wave streaming. Note that the latter has its level of maximum influence on a higher level than the first.

#### 4.2. Influence of Changing Wave and Bed Conditions

[29] Under free surface waves in prototype situation, both streaming phenomena act simultaneously. However, their contribution can vary largely with varying wave conditions. When waves approach the shore, orbital velocities close to the bed will increase while the wave propagation velocity decreases. Therefore, progressive wave streaming may be expected to increase with decreasing depth. However, the waveshape will change simultaneously. Where waveshape streaming due to velocity-skewness is absent for linear waves offshore, it will also increase with decreasing depth. So it is not a priory clear which of the streaming mechanisms wins. Earlier analytical investigation of this balance by*Trowbridge and Madsen* [1984b]revealed a reversal of the streaming velocity at the edge of the bottom boundary layer from on- to offshore for relatively long waves.*Holmedal and Myrhaug*'s [2009]numerical simulations showed increasing importance of velocity-skewness streaming over progressive wave streaming for increasing wave periods, qualitatively consistent with*Trowbridge and Madsen* [1984b]. Here, we use the validated numerical model for a systematic quantitative investigation on the balance between the competing mechanisms for changing wave and bed conditions on the shoreface. These general insights in streaming are considered to be valuable for the development of adequate hydrodynamic input for practical sand transport formulae.

##### 4.2.1. The Non-dimensional Parameter Domain

[30] The hydrodynamics of the boundary layer above a flat horizontal bed under a free surface wave is completely described by the parameters *a*, *h*, *T*, *k*_{N}, *g* and *υ.*With six dimensional parameters and two fundamental dimensions, this situation can be described by combinations of four basically independent non-dimensional parameters, e.g.,*a*/*h*, *kh*, *A*/*k*_{N} and *Re*, respectively the relative wave amplitude, relative water depth, relative bed roughness and the (wave) Reynolds number. Note that other informative non-dimensional parameters can be derived from these 4 parameters, for instance the parameter _{1}/*c* that indicates the relative importance of the advective terms in the momentum equation (1), and the parameter *R*that describes the degree of velocity-skewness(16). In contrast with tunnel experiments, velocity-skewness*R* is not a free parameter under real free surface waves. It depends on the relative water depth *kh* and relative wave amplitude *a*/*h.* To describe the shape of the near bed velocity signal as function of these parameters, a wave theory or model is needed. Using 2nd order Stokes theory, see equation (8), *R* can be expressed as:

From the four non-dimensional parameters*a*/*h*, *kh*, *A*/*k*_{N} and *Re*, the first three are considered most relevant studying streaming and shear stress in a turbulent wave boundary layer potentially inducing sheet-flow: wave condition parameters*a*/*h* and *kh* give the forcing of the boundary layer model, parameter *A*/*k*_{N} directly influences the friction of the flow over the bed. Within the (rough) turbulent flow regime, the influence of *Re* on the boundary layer flow characteristics should diminish. Extensive tests on model behavior do confirm this and show that the area of *Re*-independent model results coincides quite well with the experimentally determined delineation of the rough turbulent flow regime (seeFigure 1). Restricting our exploration to this flow regime, we therefore couple the *Re* number to the relative roughness *A*/*k*_{N} with:

which is a line parallel to the turbulent delineation of *Jonsson* [1966] in Figure 1, inside the rough turbulent regime.

[31] We investigate the balance between velocity-skewness streaming and progressive wave streaming in the turbulent wave boundary layer for a domain spanned by the remaining parameters*a*/*h*, *kh* and *A*/*k*_{N}. Because we use second order Stokes theory to determine the oscillating free stream velocity (model input), cases outside the domain of applicability of this theory (wave breaking, too much nonlinearity) have been excluded from the further procedure. The used restrictions are: *a*/*h* < 0.4 and *R* < 0.625 (coincides more or less with Ursell number *U* = *HL*^{2}/*h*^{3} < 45, with *H* and *L* wave height and length respectively). See the delineation in Figure 7(top). Within these limits, cases have been defined (circles, same figure), and simulations have been carried out using the BL2-free-model for zero mean pressure gradient. Following*Trowbridge and Madsen* [1984b], the computed streaming velocity just outside the boundary layer is taken as a measure in the visualization of the results. Dependency on *a*/*h* is nearly completely removed from the visualization when the streaming is normalized as (*U*_{0}/ _{1})/( _{1}/*c*). This can be seen from Figure 7 (bottom), showing results for *A*/*k*_{N} = 100. Only at the outer edges of the domain, the surface formed by the numerical results is slightly bent in *a*/*h* direction (which is attributed to slight numerical inaccuracies in the extreme cases). Note that the *a*/*h* independency in the mentioned normalization reduces the normalized streaming to a function of *kh* and *A*/*k*_{N} only.

##### 4.2.2. Influence of Relative Water Depth *kh*

[32] Figure 8shows the non-dimensional streaming as function of*kh* for a single roughness. The results show a clear dependence on *kh*: streaming is positive at large *kh*, but decreases more and more for decreasing *kh.* Simulations with waveshape effect and progressive wave effect only, clarify these results: at relatively deep water (large *kh*) the non-dimensional streaming is completely determined by the free surface effect. For decreasing relative water depth (*kh*), the normalized progressive wave streaming stays nearly constant (also for strongly nonlinear waves, the contribution of higher harmonics to progressive wave streaming is small). However, the importance of waveshape effect relative to the free surface effect increases, resulting in a reversal from on- to offshore. This*kh* behavior is qualitatively consistent with the findings of *Trowbridge and Madsen* [1984b]. For *A*/*k*_{N} = 320, the numerical model gives the directional reversal close to *kh* = 0.8.

##### 4.2.3. Influence of Relative Bed Roughness *A*/*k*_{N}

[34] Figure 9shows model results for the non-dimensional streaming velocity for various values of*A*/*k*_{N}, together with the analytical results of *Trowbridge and Madsen* [1984b] (TM84). In the numerical results, the main influence of the roughness is that for all *kh* the streaming value shifts in negative direction for increasing *A*/*k*_{N}, with decreasing shifts for larger values of *A*/*k*_{N}. The results differ from TM84 in various ways. First, the simulated streaming velocities at *kh* = 3, approximating the streaming from progressive wave streaming only, are smaller and much less sensitive for *A*/*k*_{N}. Second, the numerically predicted *kh* value of streaming reversal is higher. Finally, at low values of *kh* the *A*/*k*_{N} influence is opposite in the two models. According to the analytical model results both streaming processes become stronger with increasing bed roughness (decreasing *A*/*k*_{N}). We conversely found almost no influence of the roughness on the offshore waveshape streaming. Like in validation case 4, this can be explained by the diffusive transport of t.k.e., which is included in the present model with turbulence memory and *k*-*ε* closure, and not in TM84, with an eddy viscosity being a function of the instantaneous shear velocity.

##### 4.2.4. Parameterizations

[35] Parameterizations of the numerical results may be helpful to include progressive wave streaming and waveshape streaming into practical sand transport formulae, that either use a free stream velocity moment (Bagnold-Bailard type) or bed shear stress (Meyer-Peter and Müller type) as hydrodynamic input [e.g.,*van Rijn*, 2007; *Nielsen*, 2006]. The results for the streaming at the top of a rough turbulent boundary layer can be parameterized as follows:

(with the first two terms parameterizing progressive wave streaming and the last term connected to waveshape streaming beneath Stokes waves).

[36] The current related mean bed shear stress and the contributions to it from the wave Reynolds stress and the waveshape effect (see equation (19)), have been studied just like the streaming velocities. When we normalize the contributions at the bed by *τ*_{b,m} _{1}/*c*, the results shows a *kh*-dependency similar toFigure 8, but now independent of *A*/*k*_{N}. Without a mean pressure gradient, the total mean bed shear stress is equal to the wave Reynolds stress . We found from the numerical simulations:

With , this gives:

which numerically confirms earlier analytical estimates for from energy dissipation *D*_{E} in a sinusoidal oscillation:

as applied before [*Nielsen*, 2006] to include progressive wave streaming in practical sand transport formulae.

#### 4.3. Effects of a Mean Pressure Gradient on Current and Stress

[37] In reality, the boundary layer may also be affected by a mean pressure gradient, related to return current, undertow or effects of wave transformation on a sloping beach. This mean pressure gradient is not included in the simulations (and parameterizations) of section 4.2. We explore the influence of a mean pressure gradient on the mean current and stress components with the numerical model. Based on case 4: Van Doorn, with a mean pressure gradient of 0.2 Pa/m, we define three additional cases: with respectively a strongly increased positive mean pressure gradient, a zero mean, and a strong negative mean pressure gradient. The results are shown in Figure 10. Figure 10e for *U*_{0} shows that the mean pressure gradients have large effects on both magnitude and shape of the *U*_{0} profile inside the wave boundary layer. Not only the extreme cases, but also the simulation with zero mean pressure gradient show significant differences with the validation case. Figures 10a–10d show the current related shear stress and the various contributions to it, see equation (19). The waveshape contribution (Figure 10d) decreases with increasing pressure gradient, which is according expectation: a negative mean current reduces the difference between on- and offshore turbulence beneath the velocity-skewed wave. The contribution of the pressure gradient (Figure 10c) is substantial: in the original validation case 4, with only a small return current, the contribution from the pressure gradient at the bed is already 1/3 of the wave Reynolds stress (Figure 10b) at the bed. We can also observe that the wave Reynolds stress (Figure 10b) at the bed is not affected by an adapted mean pressure gradient. So also with strong undertow or shoaling effects, the wave Reynolds stress contribution to the mean bed shear stress can be modeled with equation (25).

[38] Estimates of realistic mean pressure gradients, that not only depend on the local situation, might be obtained from wave properties, mass-fluxes and geometric information through undertow models. See also*Zhang et al.* [2011] who studied the wave boundary layer beneath shoaling and breaking waves, both generating mean pressure gradients, with a first order boundary layer model. The coupling to undertow models has not been tested here.

### 5. Discussion

- Top of page
- Abstract
- 1. Introduction
- 2. Model Formulation
- 3. Validation
- 4. Analysis of Streaming Generating Mechanisms
- 5. Discussion
- 6. Conclusion
- Appendix A:: Numerical Solution Method
- Appendix B:: Shape Expression
- Acknowledgments
- References
- Supporting Information

[39] The main motive for this hydrodynamic study on wave boundary layer streaming is its potential influence on total sediment transport and nearshore morphology. Progressive wave streaming might explain the differences found in sand transport between tunnel and flume experiments. This is especially relevant, because most morphodynamic models use shear stress and transport formulations primarily based on tunnel experiments, and also tend to under predict onshore transport in accreting conditions [*van Rijn et al.*, 2011]. To show the potential importance of progressive wave streaming for sediment transport, we apply the numerical model both with (BL2-version) and without free surface effects (BL1-version) for two conditions of the full scale flume experiments of*Dohmen*-*Janssen and Hanes* [2002] (MI and MH, both with grain size *d*_{50} = 0.24 mm). Following the example of *Gonzalez Rodriquez and Madsen* [2011], we use the simulated time-dependent bed shear stress*τ*_{b}(t) from both versions as input to a bed load sediment transport formula [*Nielsen*, 2006]:

where *q*_{s}(t) and *θ*(t) are the time dependent sediment transport and Shields parameter respectively. The latter is computed from the model results for *τ*_{b}(t) through:

To account for the higher roughness of the mobile bed, a bed roughness height *k*_{N} (model input) is used of the order of the maximum sheet flow layer thickness in these experiments (*k*_{N} = 20*D*_{50}). Figure 11 shows results for *τ*_{b}(t) and net transport rate 〈*q*_{s}〉 from BL1 and BL2. The predicted 〈*q*_{s}〉 increases with 40% in case MH and even 100% in case MI. So in the latter case, the contribution of progressive wave streaming to onshore transport is of the same order of magnitude as the contribution of velocity-skewness. In both cases, the measured 〈*q*_{s}〉 is approached the best with progressive wave streaming included. Note that the numerical framework of the present model, shown to have some important advantages over the analytical approach concerning the hydrodynamics (see 3.4), also allows to investigate the role of streaming for fine sands, with much more sand in suspension. The question whether streaming is the full explanation of the differences in transport found in tunnel and flume will be discussed both for medium and fine sized sands in a future article, including a systematic data-model comparison involving all available large scale flume data.

[40] Although the test cases 1 to 4 are represented by the model reasonably well, they still show sometimes small differences between the measured and computed mean and unsteady flow near the bed. The question could therefore be raised whether these inaccuracies may form a serious shortcoming of the model when applied to sediment transport predictions. What is the deviation in predicted sediment transport these errors might introduce and how does this compare to the effects of progressive wave streaming we pointed at before? To get an impression hereof, we study the influence of inaccuracies in mean and unsteady flow on the third-order velocity moment 〈*u*^{3}〉. We do this for case 4, for which near the bed (0–5 mm) the negative streaming was somewhat overpredicted and the phases of the harmonic components were underpredicted, the latter explained by the model's underestimation of the friction. We study 〈*u*^{3}〉 because in this region very close to the bed, it is reasonable to assume that *τ*_{b}(t) ∼ ∣*u*(*t*)∣*u*(*t*) and *q*_{s}(t) ∼ *τ*_{b}(*t*)*u*(*t*) ∼ *u*(*t*)^{3}(at least for medium-sized sand, neglecting phase-lags of suspended sediment) [see*Bailard*, 1981; *Ribberink and Al*-*Salem*, 1994]. Figure 12 shows 〈*u*^{3}〉 computed from the experimental data and as computed by the model (BL2). Next, 〈*u*^{3}〉 has also been computed from a simulation without progressive wave streaming (BL1), and also again from the BL2 model but now with the computed mean velocity near the bed (0–10 mm) replaced by an approximation of the measured mean current (−0.0025 m/s). In this way, possible differences between the first and the last computation can only be caused by inaccuracies in the simulated unsteady velocities. Figure 12 shows that the influence of unsteady flow inaccuracies on 〈*u*^{3}〉 is very small compared to steady flow inaccuracies, and the latter are much smaller than deviations introduced by neglecting progressive wave streaming (BL1). This underlines the primary importance of a good streaming prediction for sediment transport prediction in this case. At the same time, the present model performance in prediction of the near-bed unsteady flow seems to be sufficient.

[41] Concerning the validity of the model assumptions it should be noted that the results in section 4.2have been obtained using Stokes theory to determine the waveshape. Seaward of the surf zone, where waves are predominantly velocity skewed with limited nonlinearity and acceleration-skewness is nearly absent [*Ruessink et al.*, 2009], this approximation is valid and the presented results can be applied. Note that the model itself is very well able to deal with the effects of larger nonlinearity and acceleration-skewness on the boundary layer, as shown insection 3. So with a more advanced predictor of the waveshape, the model can also be applied in more shallow water and the surf zone. However, note that there also turbulence effects of (especially plunging) wave breakers may start to effect the boundary layer flow [*Fredsøe et al.*, 2003; *Scott et al.*, 2009].

[42] Finally, preliminary simulations with the present model including sediment and buoyancy-effects show a slight influence of suspended sediment on streaming, especially for fine sediment, most likely related to turbulence damping by density stratification. This asks for re-validation of the model on measured velocities above mobile beds when the contribution of progressive wave streaming to transport rates will be studied in more detail. Also here, reference is made to a future article which is focused on sediment transport prediction with the BL1 and BL2 models.

### 6. Conclusion

- Top of page
- Abstract
- 1. Introduction
- 2. Model Formulation
- 3. Validation
- 4. Analysis of Streaming Generating Mechanisms
- 5. Discussion
- 6. Conclusion
- Appendix A:: Numerical Solution Method
- Appendix B:: Shape Expression
- Acknowledgments
- References
- Supporting Information

[43] A numerical boundary layer model has been developed to investigate the net current and shear stress in the bottom boundary layer as determined by waveshape effects and free surface effects. The latter have been taken into account by inclusion of advection of momentum and turbulence properties into the 1DV-RANS model formulations and*k*-*ε* turbulence closure.

[44] The model has been validated with good agreement on a selection of experimental cases with different types of wave boundary layer flow. This fills a gap in literature on comparison of numerical models with measured mean wave boundary layer currents. The validation showed that both streaming processes, waveshape streaming and progressive wave streaming, need to be considered to reproduce the measurements. Besides, the turbulence memory in the model's (k-*ε*) turbulence closure and the presence of more harmonic velocity components contributes significantly to improved reproductions compared to earlier analytical modeling of streaming, e.g., the accurate reproduction of observed offshore current beneath acceleration-skewed waves where earlier analytical models failed.

[45] Subsequently, the model has been used to investigate the changing balance between offshore waveshape streaming and onshore progressive wave streaming for varying wave and bed conditions (section 4.2), by studying their contribution to the non-dimensional streaming velocity*U*_{0}*c*/ _{1}^{2} in the parameter space spanned by relative water depth *kh* and roughness parameter *A*/*k*_{N}. At relative deep water (large *kh*) the streaming is completely determined by the free surface effect. For decreasing relative water depth (*kh*), the normalized progressive wave streaming stays nearly constant, but the importance of waveshape effect relative to the free surface effect increases. The effect of bed roughness is less distinct. For increasing relative bed roughness (decreasing *A*/*k*_{N}), we found slightly stronger onshore progressive wave streaming. These model results have been parameterized in an expression for the streaming velocities at the top of the boundary layer as function of *kh* and *A*/*k*_{N}, see equation (23). The model results for the contribution of progressive wave streaming to the normalized mean bed shear stress do not show a roughness dependency and give a numerical confirmation of earlier analytical estimates hereof for sinusoidal waves, which are shown to apply also when a strong pressure gradient is present (section 4.3).

[46] Other insights obtained during this study are that the maximum offshore current resulting from velocity-skewness takes place on a lower level in the bottom boundary layer than the maximum onshore current from the progressive wave streaming. Therefore, layers with positive and negative shear (∂*U*_{0}/∂*z*) can generally be observed in the mean current profile when both mechanisms are active. Next, the effect from acceleration-skewness is basically the same as the effect from velocity-skewness: a difference in turbulence properties during on- and offshore movement results in an offshore mean current. However, the acceleration-skewness effect is smaller and the level of the maximum offshore current is closer to the bed.

[47] An exploration of the potential importance of the model results for sediment transport modeling is given in section 5, showing that increased bed shear stress due to progressive wave streaming leads to larger predicted sediment transport under waves, better matching the data. It is finally concluded that the validated numerical model provides a modeling framework for follow-up research on the question whether progressive wave streaming is the full explanation of the different sediment transport rates found in tunnel and flume experiments.