Geo-Marine Letters

, Volume 31, Issue 3, pp 189–201

Modeling wave–current bottom boundary layers beneath shoaling and breaking waves

Authors

  • Chi Zhang
    • State Key Laboratory of Hydrology-Water Resources and Hydraulic EngineeringHohai University
    • State Key Laboratory of Hydrology-Water Resources and Hydraulic EngineeringHohai University
  • Yigang Wang
    • Key Laboratory of Coastal Disaster and Defence of Ministry of EducationHohai University
  • Zeki Demirbilek
    • U.S. Army Engineer Research and Development Center, Coastal and Hydraulics Laboratory
Original

DOI: 10.1007/s00367-010-0224-9

Cite this article as:
Zhang, C., Zheng, J., Wang, Y. et al. Geo-Mar Lett (2011) 31: 189. doi:10.1007/s00367-010-0224-9

Abstract

The boundary layer characteristics beneath waves transforming on a natural beach are affected by both waves and wave-induced currents, and their predictability is more difficult and challenging than for those observed over a seabed of uniform depth. In this research, a first-order boundary layer model is developed to investigate the characteristics of bottom boundary layers in a wave–current coexisting environment beneath shoaling and breaking waves. The main difference between the present modeling approach and previous methods is in the mathematical formulation for the mean horizontal pressure gradient term in the governing equations for the cross-shore wave-induced currents. This term is obtained from the wave-averaged momentum equation, and its magnitude depends on the balance between the wave excess momentum flux gradient and the hydrostatic pressure gradient due to spatial variations in the wave field of propagating waves and mean water level fluctuations. A turbulence closure scheme is used with a modified low Reynolds number k-ε model. The model was validated with two published experimental datasets for normally incident shoaling and breaking waves over a sloping seabed. For shoaling waves, model results agree well with data for the instantaneous velocity profiles, oscillatory wave amplitudes, and mean velocity profiles. For breaking waves, a good agreement is obtained between model and data for the vertical distribution of mean shear stress. In particular, the model reproduced the local onshore mean flow near the bottom beneath shoaling waves, and the vertically decreasing pattern of mean shear stress beneath breaking waves. These successful demonstrations for wave–current bottom boundary layers are attributed to a novel formulation of the mean pressure gradient incorporated in the present model. The proposed new formulation plays an important role in modeling the boundary layer characteristics beneath shoaling and breaking waves, and ensuring that the present model is applicable to nearshore sediment transport and morphology evolution.

Introduction

Understanding the mechanics of the coastal bottom boundary layer (BBL) has attracted the attention of many scientists and engineers for several decades. BBL characteristics influence incipient sedimentation and particle suspension, the morphological evolution of shorelines, and the dissipation of surface waves, which affects engineering works in coastal regions for navigation and large-scale construction. Coexisting waves and currents in prototype environments control the characteristics of the BBL. The currents may be wave-induced on natural beaches, or from runoff in rivers or estuaries. When wave-generated near-bed flow velocities mobilize large quantities of sand, sheet flow conditions occur above the seabed. In such cases, sediment transport occurs within a thin fluid–sediment layer near the bed, depending on the magnitude and the direction of wave-induced currents and on the grain size, as well as the distribution of flow within the boundary layer. For this reason, the modeling of wave–current bottom boundary layers has been extensively studied since the 1950s using experimental and numerical models.

Many experiments have been conducted since the 1970s to investigate the characteristics of wave–current boundary layers above a horizontal bed for a wide range of flow conditions and bedforms (e.g., Bakker and Van Doorn 1978; Kemp and Simons 1982; Sleath 1991; Lodahl et al. 1998; Faraci et al. 2008). Waves and currents were commonly generated in oscillatory flow tunnels or wave flumes with recirculation systems, and velocity profiles, bed shear stresses, and flow turbulence intensities were measured. In contrast, and despite its particular importance for nearshore sediment transport and morphological changes, only a few measurements of the bottom boundary layers under waves propagating over a sloping bed have been conducted. In this case, currents are naturally generated by the transforming waves, and controlled BBL measurements are much more challenging. Kirkgoz (1989) used a laser Doppler velocimeter (LDV) to measure horizontal velocity profiles in the boundary layers of the transformation zone of plunging breakers. Deigaard et al. (1991) employed a hot film probe, and Cox et al. (1996) an LDV to measure the bed shear stress under breaking waves in the surf zone. These studies indicated that the mean shear stress in the surf zone decreased linearly from the wave trough level to the bottom, and the bed shear stress was directed offshore. Lin and Hwung (2002) measured BBL flow in the pre-breaking zone of shoaling waves, and found a local onshore mean current near the bottom. All the above mentioned BBL experiments beneath transforming waves were conducted over a planar sloping fixed bed, and effects of the movable bed and the bed form (e.g., rippled bed) for real beaches were not considered. Because the BBL experiments over movable/rippled sloping beds are rather scarce to date, more easily controlled planar sloping fixed-bed experiments appear as an ideal means of investigating the fundamental mechanisms of near-bed hydrodynamics for testing numerical models.

A variety of numerical models have been developed to describe the BBL system under combined wave and current conditions. The models vary widely in their complexity, and may be grouped as process-based models or empirical models. Using linearized boundary layer equations and a time-invariant linear eddy viscosity, Grant and Madsen (1979) proposed an analytical model for wave–current interactions. Eddy viscosity formulations have since been used by others (Trowbridge and Madsen 1984; Christoffersen and Jonsson 1985; Myrhaug and Slaattelid 1990; You 1994). In more refined modeling of the BBL, the focus has been on improvement of the turbulence closure, which led to the development of several one-equation models (Bakker and Van Doorn 1978; Davies et al. 1988; Soulsby et al. 1993; Madsen 1994) and two-equation models (Li and Davies 1996; Guizien et al. 2003; Holmedal et al. 2003, 2004; Henderson et al. 2004; Shi and Wang 2008). A detailed review can be found in Malarkey and Davies (1998). The current effect was included in some models by a mean horizontal pressure gradient term, either explicitly or implicitly in the governing equations.

The above models treated this mean pressure gradient term as independent of the wave parameters. Some models neglected this term, assuming that the current-related pressure gradient is very small compared to that induced by waves (Malarkey and Davies 1998; Dohmen-Janssen et al. 2001; Holmedal et al. 2003, 2004; Henderson et al. 2004), and a few formulated it as a function of a pre-known reference current velocity, such as the depth-averaged velocity (Fredsøe and Deigaard 1992; You 1994; Shi and Wang 2008), or specified it as a non-zero initial value (Davies and Li 1997; Holmedal and Myrhaug 2006). For example, Shi and Wang (2008) related the mean pressure gradient term to a shear current velocity empirically calculated from the depth-averaged mean velocity, and obtained a favorable comparison between modeling estimates and the experimental data of Bakker and Van Doorn (1978), for which the current was generated by a recirculation system over a horizontal bed and its velocity was pre-specified. These cases can represent the wave–runoff interactions in estuaries, where currents are generated independent of waves and devoid of any strong variation over the depth.

On natural beaches with non-uniform bathymetries, currents are generated by the transforming waves, and implementation of the abovementioned approaches is difficult because wave-induced currents are initially unknown and vary over the water column. These are different features from those of ambient currents, and require a different mathematical treatment of the mean horizontal pressure gradient term. Various methods may be used to calculate the threshold near-bed velocity for sediment incipience (e.g., the Reynolds stress technique, and the logarithmic profile method). The magnitude of net sediment transport is calculated by different methods assuming sediment transport consists of bed and suspended loads, as is commonly done. The bedload refers to all sediments maintaining occasional contact with the bed that move horizontally at a somewhat slower rate than the flow. The bed shear stress is controlled by the velocity in an effective near-bed layer, bed skin roughness, and bed form (Bartholdy et al. 2010). The suspended sediment is assumed to remain above the bed at all times, controlled mainly by the turbulence and transported horizontally at the fluid velocity. The sand activation depth is related to the wave height, wave obliquity, and bed slope (Bertin et al. 2008). The features of bedload and suspended sediment transport are outside the scope of this paper, but the characteristics of the BBL are paramount to their calculations, and are the main focus of this paper.

The interactions between waves and wave-induced cross-shore currents in the BBL beneath normally incident waves transforming over an arbitrary bathymetry are described in this study. The formulation of the current-based pressure gradient term is revisited in this paper, with an objective of developing a BBL model of general applicability to coastal problems. For waves transforming over a sloping bed, this objective is pursued by deriving the mean pressure gradient term from the wave-averaged momentum equation, without using a reference current velocity. We have implemented this formulation in a first-order boundary layer model, and investigated the model’s performance with two sets of experimental data extracted from Lin and Hwung (2002) and Cox and Kobayashi (1996). Since the present model is based on the first-order boundary layer equation, the boundary layer streaming and the horizontally non-uniform effects are not taken into account. The motivation for developing a boundary layer model is to couple it with a cross-shore sediment transport model, to predict beach profile evolution in coastal engineering applications. A number of studies have followed this approach, and have used a first- or second-order BBL model (Henderson et al. 2004; Hsu et al. 2006; Holmedal and Myrhaug 2009). The second-order BBL model would be more physically sound, since it can solve both the boundary layer streaming and the horizontally non-uniform flows. However, the underlying mechanisms of these complicated physical processes are still under investigation, and have not gained a universal understanding (Longuest-Higgins 1953; Deigaard and Fredsøe 1989; Rivero and Arcilla 1995; Blondeaux et al. 2002; Holmedal and Myrhaug 2009; Fuhrman et al. 2009) for their exact influences on nearshore sediment transport. Some second-order models that require much more computational effort are not showing better results than a first-order model (Hsu et al. 2006).

A related fundamental question is why is there a need for a different approach for wave-generated cross-shore flow in the treatment of BBL dynamics? This question arises from a perspective of the physics of bottom boundary layers, because there is a significant difference between modeling an ambient flow and a wave-induced cross-shore current. Thus, there is a need to provide a universal formulation for the current-related mean pressure gradient for transforming waves, similar to the formulations developed specifically for ambient flows. From the physical point of view, both ambient flows and wave-induced cross-shore currents can be regarded as time-invariant mean flows that affect the characteristics of BBLs. The same governing equations may be used for both flow types by adding a mean pressure gradient term to the original wave boundary layer equations. The major difference between the two current types is in their generation mechanisms and mathematical expressions. The ambient current is generated independent of waves. Examples are the current generated by a recirculation system in laboratories, and the runoff from a slope of the water surface upstream in rivers and estuaries. In many applications, an important feature of this type of current is that it is devoid of any strong vertical variation, and could be represented by a log profile. For ambient flows, the associated mean pressure gradient can thus be formulated as a function of the current itself (e.g., depth-averaged velocity).

For wave-induced cross-shore flows, a different mathematical treatment of the mean pressure gradient is required for the following reasons:
  1. 1.

    The mean pressure gradient may not be neglected in this case, otherwise some important physical processes, such as the undertow, cannot be represented.

     
  2. 2.

    The currents induced by wave transformation are closely tied to variation in the transforming wave parameters. Consequently, the formulation of the mean pressure gradient must be a function of wave parameters.

     
  3. 3.

    The current velocity is unknown a priori, and an initial input current cannot be specified to calculate the mean pressure gradient using traditional methods.

     
  4. 4.

    Wave-induced cross-shore currents generally exhibit a strong vertical variation both in velocity magnitude and direction through the water column. In the surf zone, the wave-induced cross-shore current is often directed offshore in the lower water column, and onshore in the upper water column. Outside the surf zone, the wave-induced cross-shore current is directed onshore near the bottom and at the surface, and offshore in the middle water column. Since the cross-shore current velocity is not regularly distributed over the water column, and could be directed onshore or offshore depending on the water depth, it is difficult to choose an appropriate reference current required for traditional approaches.

     
  5. 5.

    The depth-averaged mean cross-shore current is zero to maintain mass conservation in the cross-shore direction along a closed coast, where the undertow balances the onshore mass flux near the surface. In other words, there is a current at any given elevation in the water column, but there is no current from a depth-averaging perspective. Consequently, the depth-averaged current velocity cannot be used to estimate the mean pressure gradient, which is clearly not zero over the depth.

     

Methods

Governing equations

Some existing models have considered the interactions between ambient currents and progressive waves over a constant depth. Since there is no wave-induced current present in such models, they cannot represent a wave–current BBL over a sloping bed, where shoaling and breaking-induced currents dominate. We provide a theoretical formulation of the BBL characteristics under transforming (shoaling and breaking) waves over non-uniform depth, which is necessary for the modeling of coastal sediment transport and morphological changes.

We begin model formulation by assuming a horizontally uniform flow. The first-order momentum balance equation in the oscillatory bottom boundary layer reduces to
$$ \frac{{\partial u}}{{\partial t}} = - \frac{1}{\rho }\frac{{\partial p}}{{\partial x}} + \frac{1}{\rho }\frac{{\partial \tau }}{{\partial z}} $$
(1)
where x is the horizontal coordinate (positive onshore), z the vertical coordinate (positive upward with z = 0 at the bed), t the time, u the instantaneous horizontal velocity component, p the pressure, τ the horizontal shear stress, and ρ the water density. Following Malarkey and Davies (1998), these three variables can be separated into wave-induced and current-induced time-averaged components:
$$ \begin{array}{*{20}{c}} {u = \tilde{u} + \bar{u},} \hfill & {p = \tilde{p} + \bar{p},} \hfill & {\tau = \tilde{\tau } + \bar{\tau }} \hfill \\\end{array} $$
(2)
where ~ and – denote the wave and current component, respectively. We further assume that the horizontal pressure gradient is constant in the boundary layer, and equal to that just outside the boundary layer (free stream). With these assumptions, the following boundary layer approximation is obtained:
$$ - \frac{1}{\rho }\frac{{\partial \tilde{p}}}{{\partial x}} = - \frac{1}{\rho }\frac{{\partial {{\tilde{p}}_{\infty }}}}{{\partial x}} = \frac{{\partial {{\tilde{u}}_{\infty }}}}{{\partial t}} $$
(3)
$$ - \frac{1}{\rho }\frac{{\partial \bar{p}}}{{\partial x}} = - \frac{1}{\rho }\frac{{\partial {{\bar{p}}_{\infty }}}}{{\partial x}} = - \frac{1}{\rho }\frac{{\partial {{\bar{\tau }}_{\infty }}}}{{\partial z}} $$
(4)
where ∞ denotes the physical quantity at the upper limit of the boundary layer, and u is the near-bed free stream velocity. Equations 3 and 4 respectively represent the wave-induced and current-induced horizontal pressure gradients. Following Svendsen et al. (1987) and Stive and De Vriend (1994), the local wave-averaged horizontal momentum equation at the upper limit of the boundary layer under transforming waves can be written as
$$ \frac{{\partial {{\bar{\tau }}_{\infty }}}}{{\partial z}} = \frac{\partial }{{\partial x}}\rho \overline {({{\tilde{u}}_{\infty }}^2 - {{\tilde{w}}_{\infty }}^2)} + \frac{\partial }{{\partial x}}\rho g\bar{\eta } $$
(5)
where w is the vertical component of instantaneous velocity, g the gravitational acceleration, and \( \bar{\eta } \) the mean water surface elevation. The first term on the right side of Eq. 5 represents the cross-shore gradient of wave excess momentum flux due to spatial variation in the wave field, such as wave shoaling or breaking. The second term represents the gradient of the hydrostatic pressure resulting from associated wave setup/setdown. For consistency, the nonlinear term \( \rho \overline {\tilde{u}\tilde{w}} \) has been excluded here. As shown in Eq. 5, it is the local imbalance between these two terms that causes vertical variations in the mean shear stress and mean velocity. This concept has been widely used to solve the wave-averaged cross-shore mean current inside and outside the surf zone (e.g., Reniers et al. 2004; Ruessink et al. 2007; Zheng and Tang 2009; Zhang et al. 2009).
Zheng (2007) derived a set of vertically varying expressions for the wave excess momentum flux through the water column. Under normally incident linear waves and shallow water approximation, we obtain the following expression for the wave momentum flux below the wave trough:
$$ \rho \overline {({{\tilde{u}}_{\infty }}^2 - {{\tilde{w}}_{\infty }}^2)} = \frac{E}{h} = \frac{{\rho g{H^2}}}{{8h}} $$
(6)
where E, H, and h are respectively the wave energy density, local wave height, and water depth. Combining Eqs. 4, 5 and 6, the current-related mean horizontal pressure gradient can be expressed as
$$ - \frac{1}{\rho }\frac{{\partial \bar{p}}}{{\partial x}} = - \frac{{gH}}{{4h}}\frac{{\partial H}}{{\partial x}} - g\frac{{\partial \bar{\eta }}}{{\partial x}} $$
(7)

Real wave profiles in the nearshore are different from those of linear waves, and more similar to cnoidal wave profiles (Le Roux 2008; Demirbilek and Vincent 2008). According to Svendsen et al. (2003), the linear wave theory is adequate for the radiation stress gradient and the mean surface level variation across the surf zone. This observation was based on comparisons to a large set of experimental data obtained from the Large-scale Sediment Transport Facility (LSTF) at the US Army Engineer Research and Development Center. We use the linear wave theory in this study for its simplicity, but any nonlinear wave theory could also be used. The cross-shore distribution of wave height and mean surface elevation can be obtained from measurements or from numerical wave models (Nwogu and Demirbilek 2001; Le Roux et al. 2010).

For the case of a horizontal bottom, if the current is unaffected by the superimposed wave (e.g., it is driven by the slope of mean water level, as in natural situations), then the wave momentum flux gradient would vanish as there is no spatial variation of wave height \( \partial H/\partial x = 0 \). In this case, Eq. 7 reduces to
$$ - \frac{1}{\rho }\frac{{\partial \bar{p}}}{{\partial x}} = - g\frac{{\partial \bar{\eta }}}{{\partial x}} = \frac{1}{{\rho h}}\overline {{\tau_{\rm{b}}}} = \frac{{u_{{ * {\rm{c}}}}^2}}{h} $$
(8)
where \( \overline {{\tau_{\rm{b}}}} \) and \( {u_{{ * {\rm{c}}}}} \) are the mean bed shear stress and current shear velocity, respectively.
Substituting Eqs. 3 and 7 into Eq. 1, and introducing the Boussinesq approximation \( \tau = \rho ({v_{\rm{t}}} + v)\partial u/\partial z \), the governing equation for the wave–current boundary layer under transforming waves can be expressed as
$$ \frac{{\partial u}}{{\partial t}} = \frac{{\partial {{\tilde{u}}_{\infty }}}}{{\partial t}} - \left( {\frac{{gH}}{{4h}}\frac{{\partial H}}{{\partial x}} + g\frac{{\partial \bar{\eta }}}{{\partial x}}} \right) + \frac{\partial }{{\partial z}}[({v_{\rm{t}}} + v)\frac{{\partial u}}{{\partial z}}] $$
(9)
where vt and v are the turbulent eddy viscosity and the kinematic viscosity, respectively. Equation 9 includes spatial variations of the wave field and the mean surface elevation to account for the effects of cross-shore wave-induced currents on the bottom boundary layer. The formulation for the mean pressure gradient in Eq. 9 is a function of the wave parameters. The magnitude of this term depends on the imbalance between two competing physical processes: the wave excess momentum flux gradient due to wave transformation, and the hydrostatic pressure gradient due to the variation of the mean surface level. This concept has been used in undertow models, but not yet implemented in wave–current boundary layer models. The mean pressure gradient could be positive (onshore directed) or negative (offshore directed) depending on the specific wave-dynamic region, but it is not necessarily related to a reference current. The depth-averaged current, which is commonly a necessary input for previous models to calculate the mean pressure gradient, is not required as an input in our model. In the present formulation, non-zero values of the current-related horizontal pressure gradient can be obtained inside and outside the surf zone, even when the depth-averaged current velocity is zero.

Turbulence closure model

The turbulent eddy viscosity vt in Eq. 9 is obtained from a modified low Reynolds number k-ε model, referred to as the MKM model by Sana et al. (2007). The term ‘low Reynolds number’ implies that the model is applicable over the whole cross-stream dimension, including the low Reynolds number region known as the viscous sublayer (Schlichting 2000). For additional information, the reader is referred to Sana et al. (2007). The governing equations of the turbulence model are
$$ \frac{{\partial k}}{{\partial t}} = \frac{\partial }{{\partial z}}\left[ {\left( {\frac{{{v_{\rm{t}}}}}{{{\sigma_k}}} + v} \right)\frac{{\partial k}}{{\partial z}}} \right] + {v_{\rm{t}}}{\left( {\frac{{\partial u}}{{\partial z}}} \right)^2} - \varepsilon $$
(10)
$$ \frac{{\partial \varepsilon }}{{\partial t}} = \frac{\partial }{{\partial z}}\left[ {\left( {\frac{{{v_{\rm{t}}}}}{{{\sigma_{\varepsilon }}}} + v} \right)\frac{{\partial \varepsilon }}{{\partial z}}} \right] + {f_1}{C_{{\varepsilon 1}}}\frac{\varepsilon }{k}{v_{\rm{t}}}{\left( {\frac{{\partial u}}{{\partial z}}} \right)^2} - {f_2}{C_{{\varepsilon 2}}}\frac{{{\varepsilon^2}}}{k} $$
(11)
$$ {v_{\rm{t}}} = {f_{\mu }}{C_{\mu }}\frac{{{k^2}}}{\varepsilon } $$
(12)
where k and ε are the turbulent kinetic energy and the turbulent dissipation rate, respectively. The following values of model parameters were used: Cμ = 0.09, Cε1 = 1.4, Cε2 = 1.8, σk = 1.4, σε = 1.3, f1 = 1.0. The functions fμ and f2 are defined as
$$ {f_{\mu }} = \left( {1 + 3.45/\sqrt {{{R_{\rm{t}}}}} } \right) \times \left( {1 - \exp ( - {y^{ * }}/42.42)} \right) $$
(13)
$$ {f_2} = \left( {1 - 2 \times \exp ( - {R_{\rm{t}}}^2/36)/9} \right) \times {\left( {1 - \exp ( - {y^{ * }}/3.03)} \right)^2} $$
(14)
where Rt = k2/εv is the turbulence Reynolds number, and y* = ()1/4z/v the Kolmogorov’s length scale.

This form of k-ε model has been successfully used by Sana et al. (2007) to describe oscillatory boundary layers, and excellent results for turbulent flows due to bottom friction were reported. However, their model did not account for breaking-induced turbulence in the surf zone. In practice, empirical breaking formulas are often used (Zheng et al. 2008). The main difficulty lies with the parameterization of an external breaking turbulence source in the governing equations of the k-ε model. This shortcoming and its effects on the calculated results are discussed next.

Boundary conditions

The no-slip condition is applied at the bottom as
$$ \begin{array}{*{20}{c}} {u({z_0},t) = 0} \hfill & {({z_0} = {k_{\rm{s}}}/30)} \hfill \\\end{array} $$
(15)
where ks is the Nikuradse roughness. For turbulent flows, the bottom boundary conditions are given by Sana et al. (2007) as
$$ \begin{array}{*{20}{c}} {k({z_0},t) = 0,} \hfill & {\varepsilon ({z_0},t) = 2v{{\left( {\frac{{\partial \sqrt {k} }}{{\partial z}}} \right)}^2}} \hfill \\\end{array} $$
(16)
The velocity is equal to the prescribed free stream at the upper edge of the BBL designated as z = zmax, and we have
$$ u({z_{{\max }}},t) = {u_{\infty }} $$
(17)
The turbulent kinematic energy and its dissipation rate at the top of the boundary layer follow the zero-flux condition given by
$$ \begin{array}{*{20}{c}} {\frac{\partial }{{\partial z}}k({z_{{\max }}},t) = 0,} \hfill & {\frac{\partial }{{\partial z}}\varepsilon ({z_{{\max }}},t) = 0} \hfill \\\end{array} $$
(18)
where Eq. 18 poses a problem in the wave–current coexisting field because zmax is commonly situated within the current-dominating boundary layer. Consequently, the zero-flux boundary condition applied to the turbulent quantities is unrealistic. Holmedal et al. (2003) suggested that large vertical gradients of k and ε were confined to a very thin layer close to the bed, decaying rapidly with depth, and became very small at the outer boundary. Using the same boundary conditions as above, their calculated results provided good estimates of both velocity and turbulent quantities in the main part of the flow domain, and were not strongly affected by values of the eddy viscosity near the outer boundary. Good numerical results have been reported in similar treatment by Mellor (2002) and Guizien et al. (2003). Therefore, Eq. 18 is used in the present model, assuming that inaccuracies in the turbulence boundary conditions (Eqs. 15 to 18) will not significantly change the large-scale near-bed hydrodynamics.

Numerical implementation

A Crank-Nicholson-type implicit finite-difference scheme was employed to discretize the governing Eqs. 9 to 12. An iterative algorithm was applied to solve the nonlinear system of differential equations. Closer to the bed, where the vertical gradients are particularly high, a stretched vertical grid was used by allowing grid spacing to increase exponentially in the vertical direction. The specified grid spacing was Δz = (r–1)ri–1zmax/(rn–1), where r is the ratio between two consecutive grid points, i the grid node index, and n the total number of grid nodes (Suntoyo et al. 2008). The parameter r was chosen so that the distance from the bed to the first grid node is less than or equal to 0.1v/u*, a value that was selected on the basis of physical considerations for the thickness of the viscous sublayer for a smooth boundary (e.g., Foti and Scandura 2004). The computational convergence for u, k, and ε was achieved when the maximum relative difference was less than 0.00001 between any two consecutive iterations.

Results

Wave shoaling over a sloping bed

Lin and Hwung (2002) reported measurements of BBL flow for shoaling waves over a sloping bed. The experiments were conducted in a glass-walled wave flume at the Tainan Hydraulic Laboratory, National Cheng Kung University. The wave flume was 9.5 m long, 0.3 m wide, and 0.7 m deep. Monochromatic waves of height 0.053 m and period 1.41 s were generated by a flap-type wave maker, propagating over a smooth bed of 1:15 uniform slope. This has been referred to as case 1 in Lin and Hwung (2002). Time series of water surface elevation and the corresponding horizontal velocity in the bottom boundary layer were measured. Data was collected at ten cross-shore sections using wave gauges and an LDV system positioned on the sloping section of beach near the breaking point (see Fig. 1 for details).
https://static-content.springer.com/image/art%3A10.1007%2Fs00367-010-0224-9/MediaObjects/367_2010_224_Fig1_HTML.gif
Fig. 1

Sketch of wave flume and location of measurements (extracted from Lin and Hwung 2002)

The cross-shore distribution of boundary layer characteristics outside the surf zone was investigated by comparing the present model predictions to data at gauge locations P4, P8, and P10. In this case, the top boundary of the model was set to 0.005 m above the bottom where time series of the free stream velocity (u) were observed. The purpose of using measured free stream velocity at the top boundary is to remove potential effects of incorrect boundary conditions (if any other is employed), so that the role of the mean pressure gradient can be investigated without disturbance arising from other uncertainties in the boundary conditions. Calculations were made with 100 vertical grid cells and 300 time steps per wave period, and a Nikuradse roughness (ks) of 0.002 m that was estimated from the extrapolation of logarithmic velocity distribution measured above the bed. The values of the calculated mean pressure gradient \( ( - 1/\rho \,\partial \bar{P}/\partial x) \) at P4, P8, and P10 were 0.002, 0.005, and 0.01 m/s2, respectively. The value of the mean pressure gradient is always positive (onshore directed) under shoaling waves, and increases when approaching the breaking point.

Comparisons were made between observed and model-predicted instantaneous velocity profiles at eight instants of time (Fig. 2). Calculated model results with and without the mean horizontal pressure gradient term are respectively plotted by solid lines and dashed lines in order to highlight the critical role of this term in the accuracy of model estimates. The free stream velocity time series at the top of the boundary layer were also illustrated. The observed and modeled spatial and temporal variations in horizontal velocity in the bottom boundary layer reveal a clear offshore-directed tendency of the upper velocity profile at all gauge locations. This phenomenon is increasingly apparent for gauges P4 to P10, which is captured by the calculated results. The calculated upper velocity profiles without the mean pressure gradient are nearly uniform. In general, incorporation of the pressure gradient produces larger velocities at all gauge locations, and improves predictions within the boundary layer during onshore accelerating phases (phases A–C). The model underestimates the magnitudes and locations of overshooting at the crest half cycle. This discrepancy is more obvious closer to the surf zone (at P10). As shown below, this is a consequence of underestimations of the near-bed mean velocity and wave-induced boundary layer thickness.
https://static-content.springer.com/image/art%3A10.1007%2Fs00367-010-0224-9/MediaObjects/367_2010_224_Fig2_HTML.gif
Fig. 2

Comparison of instantaneous velocity profiles at gauges P4, P8, and P10 in the shoaling zone (case 1, from Lin and Hwung 2002), based on observations (dots) and simulations with (solid lines) and without (dashed lines) the mean horizontal pressure gradient

Comparison between observed and modeled data for wave velocity amplitudes at the three measurement locations (Fig. 3) reveals that the agreement is satisfactory, and the current-induced mean pressure gradient does not affect the calculated wave velocity amplitudes. The model accurately represents increasing wave velocity skewness caused by increasing nonlinear wave effects at the locations of gauges P4 to P10. The boundary layer thickness due to waves is under-predicted during onshore phases, and underestimation increases with decreasing distance to the breaking point (i.e., the largest underestimate occurs at the breaking point).
https://static-content.springer.com/image/art%3A10.1007%2Fs00367-010-0224-9/MediaObjects/367_2010_224_Fig3_HTML.gif
Fig. 3

Comparison of wave velocity amplitude profiles at gauges P4, P8, and P10 in the shoaling zone (cf. Lin and Hwung 2002), based on observations (dots) and simulations with (solid lines) and without (dashed lines) the mean horizontal pressure gradient

A remarkable effect of the mean pressure gradient is shown in Fig. 4 for the simulated mean velocity profiles and observations. Calculations based on the positive (onshore directed) mean pressure gradient successfully reproduced the values of local onshore mean velocity near the bottom. The predicted mean velocity profiles without the pressure gradient follow the offshore mean velocity at the grid boundary. However, the maximum near-bed onshore mean velocity is generally underestimated. The mean pressure gradient incorporated into the present model is essential since it dominates the direction of mean bed shear stresses (Table 1). The mean bed shear stress calculated with the mean pressure gradient is directed onshore, and increases toward the breaking point. The calculated mean bed shear stress without the mean pressure gradient is directed offshore, and decreases toward the breaking point.
https://static-content.springer.com/image/art%3A10.1007%2Fs00367-010-0224-9/MediaObjects/367_2010_224_Fig4_HTML.gif
Fig. 4

Comparison of mean velocity profiles at gauges P4, P8, and P10 in the shoaling zone (cf. Lin and Hwung 2002), based on observations (dots) and simulations with (solid lines) and without (dashed lines) the mean horizontal pressure gradient

Table 1

Calculated values of mean bed shear stress \( {\bar{\tau }_{\rm{b}}} \) (m2/s2) at three locations in the shoaling zone (cf. Lin and Hwung 2002)

 

P4

P8

P10

With the mean pressure gradient

0.0084

0.0141

0.0178

Without the mean pressure gradient

−0.0026

−0.004

−0.0062

Spilling waves in the surf zone

The experimental data of Cox and Kobayashi (1996) was used to investigate the model’s performance for representing bottom boundary layers under spilling breakers in the surf zone. The experiment was conducted in a wave tank in the Ocean Engineering Laboratory at the University of Delaware (Fig. 5). The wave flume was 33 m long, 0.6 m wide, and 1.5 m deep. Regular spilling waves with a period of 2.2 s were generated over a rough and 1:35 sloping bed. The rough bed consisted of a single layer of natural sand with a median grain diameter d50 = 1.0 mm glued to the plane slope. Free surface elevations and velocity profiles in the bottom boundary layers were measured along six vertical lines across the surf zone (referred to as L1 to L6 in Fig. 5, where the breaking point is at L2).
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Fig. 5

Experimental layout for spilling waves in the surf zone (extracted from Cox et al. 1996)

The data collected at locations L4 and L5 in the inner surf zone is compared to the results of the present model. The model was driven by the measured velocity time series obtained at 0.121 m above the bed for L4, and at 0.071 m above the bed for L5. Following Cox and Kobayashi (1996), the Nikuradse roughness was set at ks = 2d50. The temporal and spatial resolutions used for shoaling waves (case 1) were applied. The calculated mean horizontal pressure gradient \( ( - 1/\rho \,\partial \bar{P}/\partial x) \) is –0.03 m/s2 at L4 and –0.018 m/s2 at L5. These calculated values are close to the observed data (–0.026 m/s2 at L4 and –0.019 m/s2 at L5) of Cox and Kobayashi (1996).

Comparisons between observed and modeled vertical distributions of the mean velocity are shown in Fig. 6. The mean pressure gradient improves the simulated mean velocity profiles, especially in capturing the locations of the maximum velocity near the bed. The model without the mean pressure gradient fails to accurately reproduce the velocity magnitudes and the maximum velocity locations. However, the calculated results with the mean pressure gradient over-predict the vertical gradient of the near-bed velocity. The calculated vertical variation and magnitude of the mean shear stress compare much better to the observed data (Fig. 7). The measured mean shear stress decreases linearly from a positive (onshore directed) value at the up-wave boundary, and becomes negative (offshore directed) at the down-wave end, which are reproduced accurately by the model results only with the pressure term incorporated. The calculated vertical variation in the mean shear stress is insignificant if the mean pressure gradient is neglected. Although the predicted velocity is considerably different, a good agreement with the observed data is obtained for the mean shear stress. Figure 8 shows the comparison of the mean turbulent kinematic energy. The model results with the mean pressure gradient well predict the turbulence production close to the bottom, but underestimate the turbulence in the upper water column. The under-prediction of turbulence increases with increasing distance to the bottom. The calculated mean turbulent kinematic energy without the mean pressure gradient is much smaller than the measured data, and cannot be seen in this figure.
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Fig. 6

Comparison of mean velocity profiles at locations L4 and L5 in the inner surf zone (cf. Cox and Kobayashi 1996), based on observations (dots) and simulations with (solid lines) and without (dashed lines) the mean horizontal pressure gradient

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Fig. 7

Comparison of mean shear stress profiles at locations L4 and L5 in the inner surf zone (cf. Cox and Kobayashi 1996), based on observations (dots) and simulations with (solid lines) and without (dashed lines) the mean horizontal pressure gradient

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Fig. 8

Comparison of mean turbulent kinematic energy profiles at locations L4 and L5 in the inner surf zone (cf. Cox and Kobayashi 1996), based on observations (dots) and simulations with (solid lines) and without (dashed lines) the mean horizontal pressure gradient

Discussion

The difference between the proposed first-order boundary layer model and other existing models used to represent bottom boundary layers is in the formulation of the current-induced mean horizontal pressure gradient. We developed this term from the time-averaged momentum equation, and its importance depends on the relative dominance between the wave momentum flux gradient and hydrostatic pressure gradient.

For the case of shoaling waves, the positive (onshore directed) values of the mean horizontal pressure gradient indicate that the onshore hydrostatic pressure gradient induced by wave setdown is greater than the offshore momentum flux gradient caused by increasing wave heights due to shoaling. This trend continues toward the breaking point, and tends to drive an onshore mean current in the bottom boundary layer. The onshore mean pressure gradient leads to the offshore-directed tendency of the upper velocity profile in Fig. 2, which is consistent with the laboratory observation. The under-prediction of the boundary layer thickness (Fig. 3) by the present numerical model is due to the advection of breaking wave turbulence from the surf zone, since it increases with decreasing distance to the breaking point. The onshore mean horizontal pressure gradient term is necessary to reproduce both the onshore mean current near the bed (Fig. 4) and the onshore mean bed shear stress (Table 1) in the shoaling zone. Inclusion of the current-induced mean horizontal pressure gradient in nearshore modeling improves the estimates of wave-induced onshore sediment transport outside the surf zone, considering that many models for beach profile evolution struggle to simulate the beach recovery and the onshore sandbar migration (Ruessink et al. 2007). The underestimation of the maximum near-bed onshore mean velocity is due to the neglect of boundary layer streaming developing under strongly transforming waves close to the surf zone. This effect cannot be considered in the first-order boundary layer models that neglect the term \( \rho \overline {\tilde{u}\tilde{w}} \). However, the proposed model produced results in good agreement with two datasets, and it may be used in coastal projects. Future improvements should investigate the effects of the second-order terms and dynamic nonlinear wave characteristics that can affect the mean current and bed stress.

For the case of spilling waves, the negative (offshore directed) values of the mean horizontal pressure gradient indicate that the offshore hydrostatic pressure gradient due to wave setup is greater than the onshore wave momentum flux gradient due to the dissipation of wave energy. The offshore-directed mean pressure gradient is realistic since it has been recognized to drive the undertow in the surf zone (Dyhr-Nielsen and Sorensen 1970; Svendsen 1984). Furthermore, the good agreement obtained between the calculated and measured mean pressure gradient demonstrates that the pressure gradient is properly formulated. The term \( \rho \overline {\tilde{u}\tilde{w}} \) is shown to be insignificant in the inner surf zone, according to Cox and Kobayashi (1996). In principle, the present model is not applicable to the surf zone because the turbulence scheme is not suitable for modeling external turbulence generated by wave breaking. The consequence can be seen in Fig. 8, in which the model under-predicts the mean turbulent kinematic energy at higher elevations. This under-prediction increases with increasing distance above the bottom, due to the neglect of the breaking turbulence that penetrates downward from the surface. Nevertheless, the turbulence production close to the bed is well predicted, indicating that the near-bed turbulence in the BBL in a spilling surf zone results mainly from the bottom friction, and could be represented by a two-equation turbulence scheme in the present model. Based on this argument, the comparisons of the mean shear stress and the mean velocity can be interpreted as below.

The time-averaged forms of the governing Eq. 1 and the Boussinesq approximation read as
$$ \frac{{\partial \bar{\tau }}}{{\partial z}} = \frac{{\partial \bar{p}}}{{\partial x}} $$
(19)
$$ \bar{\tau } = \rho (\overline {{v_{\rm{t}}}} + v)\frac{{\partial \overline u }}{{\partial z}} $$
(20)
It is noticed that the vertical gradient of mean shear stress \( (\partial \bar{\tau }/\partial {\hbox{z)}} \) is equal to the mean horizontal pressure gradient \( (\partial \bar{p}/\partial x) \). By integrating Eqs. 19 and 20 from the bed (z0) to any given elevation (z), we obtain
$$ \bar{\tau }(z) = \overline {{\tau_{\rm{b}}}} + \frac{{\partial \bar{p}}}{{\partial x}}(z - {z_0}) $$
(21)
$$ \overline u (z) = \int_{{{z_{{0}}}}}^z {\frac{{\bar{\tau }(z)}}{{\rho (\overline {{v_{\rm{t}}}} (z) + v)}}} {\hbox{d}}z $$
(22)

It can be seen from Eq. 21 that an accurate estimate of mean shear stress \( (\bar{\tau }) \) in Fig. 7 would require accurate calculations of both the mean bed shear stress \( (\overline {{\tau_{\rm{b}}}} ) \) and the mean horizontal pressure gradient \( (\partial \bar{p}/\partial x) \). The correct estimate of the mean bed shear stress \( (\overline {{\tau_{\rm{b}}}} ) \) benefits from the accurate prediction of the near-bed turbulence shown in Fig. 8. The calculation of the mean horizontal pressure gradient \( (\partial \bar{p}/\partial x) \) using Eq. 7 provides an accurate prediction with a vertically decreasing pattern of the mean shear stress \( (\bar{\tau }) \) over the water column. Although the mean shear stress is estimated correctly, the under-predicted mean turbulent kinematic energy in the main water column (Fig. 8) results in an underestimation of the mean eddy viscosity \( (\overline {{v_{\rm{t}}}} ) \), and this over-predicts the mean velocity \( (\overline u ) \) in Fig. 6. These results indicate that if the mean horizontal pressure gradient term is correctly formulated, then the present model calculates the mean shear stress accurately at the bed, and also its vertical distribution beneath spilling waves in the surf zone. The mean horizontal pressure gradient is crucial since it leads to a reliable prediction of mean bed shear stress in the surf zone, which is important to simulate beach erosion and offshore sandbar migration. However, the mean velocity cannot be modeled accurately with the present turbulence scheme without taking into account the wave breaking-generated turbulence.

By including a mean horizontal pressure gradient term into the bottom boundary layer equation, we are able to provide a more realistic physical representation of the coexisting wave and wave-induced current in the present model. The model produces both the onshore-directed and the offshore-directed mean bed shear stresses beneath shoaling waves and breaking waves, respectively. The model has a potential of reproducing both the beach erosion and the beach recovery, when coupled with a sediment transport model.

Conclusions

In this paper, we present a new numerical model for bottom boundary layers beneath shoaling and breaking waves. On the basis of comparisons between the model results and measured data, the following conclusions are drawn:
  1. 1.

    The mean horizontal pressure gradient included in the present model worked well and provided reasonable estimates. Tests performed show that the mean pressure gradient is directed onshore under shoaling waves, and that it is directed offshore under breaking waves.

     
  2. 2.

    The mean pressure gradient plays an indispensable role in representing realistic bottom boundary layer characteristics beneath transforming waves. These include the onshore mean current near the bed under shoaling waves, and the vertically decreasing pattern of the mean shear stress under breaking waves. The proposed formulation shows considerable improvements in both numerical results and physical meanings of calculated engineering parameters.

     
  3. 3.

    For spilling breakers in the surf zone, the near-bed turbulence is calculated well by a k-ε turbulence scheme in the present model. The flow velocity cannot be accurately simulated without taking into account the breaking turbulence.

     
  4. 4.

    The present model is computationally efficient to calculate mean bed shear stresses inside and outside the surf zone, and has the potential to be applied to nearshore sediment transport and morphology evolution.

     
  5. 5.

    Future enhancements should consider including the effects of the second-order terms, the wave nonlinearity, and the breaking turbulence in the proposed model.

     

Acknowledgements

The work was supported by the National Natural Science Foundation of China (Grant No. 50979033), the Program for New Century Excellent Talents in University of China (NCET-07-0255), and the Special Research Funding of State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering (Grant No. 2009585812). The authors thank two anonymous reviewers for their valuable comments that have improved the quality of the paper.

Copyright information

© Springer-Verlag 2010