The seabed boundary layer beneath waves opposing and following a current

Continental Shelf Research 65 (2013) 27–44

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The seabed boundary layer beneath waves opposing and following a current Lars Erik Holmedal n, Jona Johari, Dag Myrhaug Department of Marine Technology, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway

art ic l e i nf o

a b s t r a c t

Article history: Received 13 September 2012 Received in revised form 3 June 2013 Accepted 6 June 2013 Available online 15 June 2013

The effect of streaming on the sea bed boundary layer flow beneath combined waves and current has been investigated for waves following and opposing a current, taking both linear waves and second order Stokes waves into account. This flow results from an interaction between the classical wave–current seabed boundary layer mechanism and two different streaming mechanisms. The classical wave–current seabed boundary layer mechanism leads to a reduced mean velocity relative to the current alone; the two streaming mechanisms are streaming caused by turbulence asymmetry in successive wave halfcycles (beneath asymmetric forcing) and streaming caused by the presence of a vertical wave velocity within the seabed boundary layer as earlier explained by Longuet-Higgins. The effect of wave asymmetry, wave length to water depth ratio, magnitude of current and bottom roughness have been investigated by numerical simulations for realistic physical situations. Mean Eulerian quantities as well as the mass transport (wave-averaged Lagrangian velocity) are presented. The resulting sediment dynamics near the ocean bottom has been investigated; results for both suspended load and bedload are presented. For wave dominated situations, the velocity profiles beneath opposing waves and current are substantially different from those beneath following waves and current, both for linear and second order Stokes waves. As the flow becomes current dominated, the velocity profiles beneath opposing and following waves and current approach the velocity profile beneath horizontally uniform sinusoidal forcing. The net transport of suspended sediments and bedload is in the direction of the current both for waves following and opposing the current. For wave dominated situations the net sediment transport is strongly affected by the streaming mechanisms, both for linear and second order Stokes waves and current. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Wave–current interaction Oscillatory boundary layer Streaming Waves Sediment transport Sheet flow

1. Introduction In coastal waters of intermediate and shallow water depths the surface waves induce water particle trajectories from the free surface to the bottom. Tidal and wind induced currents interact with the wave oscillation, yielding a turbulent flow over a significant part of the water column. Because of the bottom friction a turbulent bottom boundary layer exists in the vicinity of the sea bed. The flow within this boundary layer, which is rough turbulent, is responsible for the inception of motion or transport of sea bed material, either as bedload or as suspended load. This material includes sediments, chemical compounds, as well as biological material such as fish larvae and plankton. For intermediate and shallow water depths the near-bottom water particle trajectories are ellipses where the horizontal axis is much larger than the vertical axis. Hence the classical way of modeling this flow is to neglect the vertical movement and to consider the flow to be horizontally uniform. The pioneer wave– current interaction model by Grant and Madsen (1979) used this

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assumption applying a time-invariant eddy viscosity to account for the turbulence. Since then a range of similar models for the wave–current interaction within the sea bed boundary layer have been published; see e.g. Holmedal et al. (2003) for a review. However, ocean surface waves are progressive, and thus a small vertical wave velocity exists in the flow. The existence of this vertical wave-induced velocity gives rise to a weak mass transport within the oscillatory bottom boundary layer. This happens because of the friction within the sea bed boundary layer leading to the vorticity and turbulence created within the oscillating boundary layer being transported upwards from the bottom with time (Batchelor, 1967). As a result, the vertical and horizontal velocity components are not 901 out of phase within this layer (as they are in potential flow), and the vertical velocity component combines with the horizontal velocity component through the convective terms in the governing boundary layer equations, giving rise to a non-zero wave-averaged drift within the oscillatory boundary layer. This effect is commonly referred to as steady streaming and was first explained by Longuet-Higgins (1953) for oscillating bottom boundary layers beneath gravity waves. Steady boundary layer streaming (Eulerian drift) is also caused by turbulence asymmetry in successive wave half-cycles beneath asymmetric waves. This is described in detail by Scandura (2007)

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L.E. Holmedal et al. / Continental Shelf Research 65 (2013) 27–44

and Davies and Li (1997) for flows over a smooth bottom, and flows over a rough bottom, respectively. This kind of streaming was first measured in an oscillating water tunnel in the rough turbulent flow regime by Ribberink and Al-Salem (1995) who showed reversing velocities near the bed beneath asymmetric boundary layer forcing. Earlier Trowbridge and Madsen (1984) showed that when the boundary layer is subjected to asymmetric forcing, the steady time-averaged velocity can be against the direction of wave propagation for very long waves. Deigaard et al. (1999) found similar results for long bound waves. Negative steady streaming velocities are also found in oscillating bottom boundary layers subjected to asymmetric horizontally uniform forcing by Davies and Li (1997), rough bottom; Holmedal and Myrhaug (2006), rough bottom; Scandura (2007), smooth bottom. The interactions between the Longuet-Higgins streaming and the asymmetry streaming for the seabed boundary layer beneath waves (without a current) were investigated in detail by Holmedal and Myrhaug (2009). Here the seabed boundary layer beneath both sinusoidal waves and second order Stokes waves, as well as horizontally uniform bottom boundary layers with asymmetric forcing were investigated for realistic physical situations. They found that the streaming velocities beneath sinusoidal waves (Longuet-Higgins streaming) are always in the direction of wave propagation, while the streaming velocities in horizontally uniform boundary layers with asymmetric forcing are against the wave propagation direction. Thus the effect of asymmetry in second order Stokes waves is to reduce the streaming velocity in the direction of wave propagation or to induce a streaming velocity against the direction of wave propagation for long waves (relative to the water depth). Furthermore, they showed that the Longuet-Higgins streaming decreases as the wave length increases for a given water depth, and the effect of wave asymmetry can dominate, leading to a steady streaming against the wave propagation. It was found that the boundary layer streaming leads to a wave-averaged transport of suspended sediments and bedload in the direction of wave propagation. Yu et al. (2010) investigated the sediment transport beneath asymmetrical wave groups using a two-phase model originally developed by Hsu et al. (2004). Both Longuet-Higgins streaming and asymmetric streaming were included in this study. They found that for very skewed waves, the non-linearity of the wave form accounted for most of the sediment transport, while for waves with less skewness the Longuet-Higgins streaming may become the dominant mechanism for sediment transport. Moreover, they found the sediment transport to be in the direction of wave propagation for all their cases, which is consistent with the result of Holmedal and Myrhaug (2009). Recently Fuhrman et al. (2013, Fig. 18) reported that Longuet-Higgins streaming (and related convective term effects) is capable of promoting onshore sediment transport, even for highly skewed waves on fine sands. This is in contrast to oscillating tunnel conditions (i.e. no convective terms) which yield offshore transport under such conditions. This result is supported by measurements (Schretlen et al., 2011) beneath progressive waves, and it is consistent with the findings of Blondeaux et al. (2012) who compared predictions of the sediment transport rates for oscillating tunnels with those beneath progressive waves. Other streaming mechanisms exist, such as streaming due to spatially variable roughness (Fuhrman et al., 2011) and streaming due to bed slopes (Fuhrman et al., 2009a, 2009b; Zhang et al., 2010; Scandura and Foti, 2011). However, these mechanisms will not be considered here. Fuhrman et al. (2009b) also investigated the sediment transport beneath asymmetric horizontally uniform boundary layer forcing over a flat bed. They found that the increase of sediment transport with increasing wave asymmetry was consistent with the findings of Holmedal and Myrhaug (2006). Ruessink et al. (2009, 2011) have presented further results on the measured and calculated sediment transport beneath

asymmetric horizontally uniform forcing over a flat bottom. Gonzalez-Rodriguez and Madsen (2011) presented an analytical model for the flow beneath asymmetric horizontally uniform forcing (including combined waves and current) over a flat bottom. Existing measurements of wave-averaged velocity profiles and bedload by, among others, Ribberink and Al-Salem (1995) and Deigaard and Fredsøe (1989), were well predicted by this model. This paper will focus on the effect of streaming on the combined wave–current seabed boundary layer. It is well known that for a large range of physical situations in the ocean, the wave–current seabed boundary layer is dominated by the wave-induced motions. For these physical cases, the streaming mechanisms strongly affect the wave–current seabed boundary layer flow. The classical wave– current seabed boundary layer theory, which assumes the near-bed flow to be horizontally uniform beneath symmetric forcing, does not distinguish between waves following or opposing the current. However, there is an interaction between the Longuet-Higgins streaming mechanism, the asymmetry streaming mechanism (for second order Stokes waves) and the classical wave–current interaction mechanism leading to the reduction of the mean (waveaveraged) near-bed current in the presence of waves. These interactions address important differences between the wave–current boundary layer flows in closed tunnels and real wave–current boundary layers beneath combined progressive waves and current. It appears that for a range of physical situations in the ocean, the mean flow beneath opposing waves and current is substantially different from the mean flow beneath following waves and current. Overall, the present work yields new insight into wave–current seabed boundary layers, and represents an extension of the work by Holmedal and Myrhaug (2009).

2. Model formulation For seabed boundary layers a turbulence model which accounts for the bottom roughness is required. Here a standard high Reynolds number k
ϵ model is chosen using a logarithmic wall law near the bottom. This standard model was developed by Launder and Spalding (1974) for plane jets, mixing layers and unidirectional steady flow close to the wall; more recently it has also been applied successfully to predict turbulent oscillating boundary layer flows over rough bottoms. The present model has been successfully applied to predict both the seabed wave boundary layer as well as the seabed wave–current boundary layer, including sediment concentration and sediment fluxes for sheet flow conditions. The measurements by Jensen et al. (1989, Test number 13) were well predicted (Justesen, 1991; Holmedal et al., 2003), including the instantaneous boundary layer velocity, shear stress and turbulent kinetic energy profiles, as well as the bottom friction velocity. Holmedal and Myrhaug (2006) also applied the standard high Reynolds number k
ϵ model to predict periodic boundary layer flows due to asymmetric forcing for horizontally uniform flows; the experimental data by Ribberink and Al-Salem (1995) were overall well predicted. Furthermore, the wave–current measurements by Dohmen-Janssen et al. (2001) (velocities, sediment concentrations and sediment fluxes) were well predicted by Holmedal et al. (2004). It is thus expected that the present model will yield satisfactory predictions when applied on the wave– current boundary layer near the ocean bottom. A further review of the capacity of these kind of models to predict the seabed boundary layer flow can be found in Cavarallo et al. (2010). 2.1. Governing equations Wave-induced mass transport in bottom boundary layers over an infinitely long flat bottom is considered. The horizontal

L.E. Holmedal et al. / Continental Shelf Research 65 (2013) 27–44

coordinate at the bottom is given as x, whilst the vertical coordinate z gives the distance from the bottom. The bottom is fixed at z ¼ z0 ¼ kN =30, where kN is the equivalent Nikuradse roughness. The limits of the horizontal coordinate x is such that x ¼0 at the start of the wave length, and x ¼ λ at the end of the wave length. For intermediate and shallow water depths, the water particle trajectories are ellipses where the horizontal axis is much larger than the vertical axis. Hence the boundary layer approximation applies, and the simplified Reynolds-averaged equations for conservation of the mean momentum and mass become
 ∂u ∂ðu2 Þ ∂ðuwÞ 1 ∂p ∂ ∂u þ þ ¼− þ νT ; ð1Þ ∂t ∂x ∂z ρ ∂x ∂z ∂z ∂u ∂w þ ¼ 0; ∂x ∂z

ð2Þ

where u is the horizontal velocity component, w is the vertical velocity component, p is the pressure, ρ is the density of the water, t denotes time and νT is the kinematic eddy viscosity. The turbulence closure is provided by a k-ϵ model. Subjected to the boundary layer approximation, these transport equations are given by (see e.g. Rodi, 1993)

2 ∂k ∂ðukÞ ∂ðwkÞ ∂ νT ∂k ∂u þ þ ¼ þ νT −ϵ; ð3Þ ∂t ∂x ∂z ∂z sk ∂z ∂z

2 ∂ϵ ∂ðuϵÞ ∂ðwϵÞ ∂ νT ∂ϵ ϵ ∂u ϵ2 þ þ ¼ þ cϵ1 νT −cϵ2 ; ∂t ∂x ∂z ∂z sϵ ∂z k ∂z k

ð4Þ

where k is the turbulent kinetic energy and ϵ is the turbulent dissipation rate. Here Eq. (2) has been applied to write Eqs. (1), (3) and (4) in conservative form. The kinematic eddy viscosity is given by 2

νT ¼ c 1

k : ϵ

ð5Þ

The standard values of the model constants (given in Rodi, 1993) have been adopted, i.e. (c1, cϵ1 , cϵ2 , sk , sϵ )¼(0.09, 1.44, 1.92, 1.00, 1.30). The instantaneous dimensionless bedload transport Φ is a function of the instantaneous dimensionless sea bed shear stress (Shields parameter) θ and is given by a formula by Nielsen (1992) Φ ¼ 12θ

1=2

θ ðθ−θc Þ jθj

ð6Þ

where Φ¼ θ¼

qb 3

ðgðs−1Þd50 Þ1=2 τb ρgðs−1Þd50

ð7Þ ð8Þ

Here qb is the instantaneous dimensional bedload transport, τb is the dimensional instantaneous sea bed shear stress, g is the gravity acceleration, s¼2.65 is the density ratio between the bottom sediments and the water, and d50 is the median grain size diameter. The critical Shields parameter θc ¼ 0:05 must be exceeded for bedload transport to take place. By using the boundary layer approximation, the equation for the sediment concentration c is written as
 ∂c ∂c ∂c ∂ ∂c þ u þ ðw−ws Þ ¼ ϵs ð9Þ ∂t ∂x ∂z ∂z ∂z ϵs ¼ ν T þ ν

ð10Þ

Here ϵs is the sediment diffusivity, ws is the settling velocity of the median sand grains in still water, and ν is the laminar kinematic viscosity of water. Here the laminar viscosity has been included in

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the sediment diffusivity in order to stabilize the numerical scheme; this model is described in more detail in Holmedal et al. (2004). 2.2. Simplification of equations In order to simplify the mathematical solution of Eqs. (1)–(4) and (9) the relation ∂=∂x ¼ −ð1=cp Þ∂=∂t where cp is the wave celerity, is applied. This is an approximation which is only valid for weakly decreasing waves (i.e. the wave height decay over a wave length due to the energy dissipation is small); this will be discussed further in conjunction with Eq. (14), and in Section 3. This approximation leads to the two-dimensional boundary layer equations (i.e. Eqs. (1), (3), (4) and (9)) being reduced to spatially one-dimensional equations. Physically this transformation implies a mapping from two spatial dimensions to one spatial dimension. The length of the physical twodimensional space is one wave length, and the height is zmax −z0 ; in one dimension the height is zmax −z0 . The results obtained in one dimension can be mapped back to the physical two-dimensional space. As a consequence of this simplification the vertical velocity component is found from the continuity equation and is evaluated as Z z Z ∂u 1 z ∂u dz ¼ dz ð11Þ w¼− cp z ¼ z0 ∂t z ¼ z0 ∂x and inserted into Eqs. (1), (3), (4) and (9). Here w¼ 0 at z ¼ z0 has been utilized (see Eq. (15)). The relation ∂=∂x ¼ −ð1=cp Þ∂=∂t is based on the physical assumption of permanent wave form; this assumption has been discussed in detail by Henderson et al. (2004) and appears to be physically sound when the momentum fluxes are weak, corresponding to the condition that the horizontal linear wave velocity amplitude is much smaller than the wave celerity (see Henderson et al., 2004 for details). If a boundary layer flow quantity ϕ beneath second order Stokes wave forcing can be expressed as ϕðx; z; tÞ ¼ a1 ðzÞ cos ðkp x−ωt−ψ 1 ðzÞÞ þ a2 ðzÞ cos 2ðkp x−ωt−ψ 2 ðzÞÞ

ð12Þ

then the relation ∂ϕ 1 ∂ϕ ¼− ∂x cp ∂t

ð13Þ

holds; here cp ¼ ω=kp , where kp ¼ 2π=λ is the wave number and ω is the wave frequency. Furthermore, a1 and a2 are coefficients and ψ 1 and ψ 2 are phase angles; these quantities are allowed to vary with z. Note that this assumption is possible since the dispersion relation ω2 ¼ gkp tanhðkp hÞ is the same for second order Stokes waves as for linear waves (here h is the water depth). Hence the assumption in Eq. (12) is consistent with second order Stokes boundary layer forcing. For linear waves Eq. (13) is valid with a2 ðzÞ ¼ 0 in Eq. (12). It should be noted that Eqs. (12) and (13) imply periodic lateral boundary conditions in the physical two-dimensional space. These periodic boundary conditions are justified as a reasonable approximation despite the fact that there will be a small decay of the wave height H due to the dissipation in the boundary layer. This decay can be estimated as dH 1 H2 ¼− f dx 3 h2 e

ð14Þ

2 where fe is the energy loss factor defined by τ^ b ¼ 0:5ρf e U^ 0 , where ^ U 0 is the near-bottom horizontal wave velocity amplitude outside the boundary layer, and τ^ b is the bottom shear stress amplitude; see Fredsøe and Deigaard (1992, Chapter 2.5) for a detailed derivation; f e ¼ 0:00836 for A=kN ¼ 1000 is taken from Table 2.1

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in that Chapter. Here A ¼ U^ 0 =ω is the near-bottom wave excursion amplitude. In Section 3 it will be shown that the wave decay is very small for the physical conditions to be discussed. 2.3. Boundary conditions 2.3.1. At the bottom The sea bed is assumed to be hydraulically rough. At the theoretical bed level z0 ¼ kN =30 the no-slip condition is applied, i.e. u ¼ 0; w ¼ 0

ð15Þ

For the turbulence field near the bed, the boundary conditions are given in a standard manner (see e.g. Rodi, 1993). By assuming local equilibrium between production and dissipation, the boundary conditions become  ∂u .pffiffiffiffiffi   k ¼ νT   c1 ð16Þ ∂z 3=2

ϵ ¼ ðc1 Þ3=4

k κz0

ð17Þ

where the flow is assumed to be parallel to the wall in the close vicinity of the bottom, i.e. on the computational node nearest the bottom. Here κ ¼ 0:4. The reference sediment concentration ca near the sea bed is obtained from the instantaneous Shields parameter θ by the Zyserman and Fredsøe (1994) formula ca ¼

0:331ðθ−θc Þ

1:75

1 þ 0:720ðθ−θc Þ1:75

at

z ¼ za ¼ 2d50

ð18Þ

2.3.2. At the edge of the boundary layer For waves alone, the upper edge of the flow domain, z ¼ zmax , is chosen such that the boundary layer effects have disappeared. The Neumann condition is applied on the velocity, i.e. ∂u ¼0 ∂z

ð19Þ

For waves plus current, the Dirichlet condition is applied on the velocity, i.e. u ¼ U0 þ Uc

∂k ¼0 ∂z

ð21Þ

∂ϵ ¼0 ∂z

ð22Þ

The applicability of the zero flux conditions of the turbulent quantities to waves plus current is discussed in Holmedal et al. (2003); it appears that these conditions yield a realistic representation of the sea bed boundary layer near the sea bed, which is of primary interest in this investigation. A zero flux condition is also imposed on the sediment concentration ∂c þ ws c ¼ 0 ∂z

ð23Þ

Following Fredsøe et al. (1985), this vertical flux condition can be simplified; because of the limited vertical extent of the wave boundary layer, Eq. (23) will degenerate to c-0

when z-∞

2.4. Forcing function Since the boundary layer approximation applies, the horizontal pressure gradient is considered to be constant through the boundary layer. Hence the horizontal pressure gradient ∂p=∂x is taken from the near-bottom free stream (potential flow) velocity field ðU 0 ; W 0 Þ, where U0 is the horizontal wave velocity component and W0 is the vertical wave velocity component of the free stream velocity. Hence −

1 ∂p ∂U 0 ∂U 0 1 ∂pc ¼ þ U0 − ρ ∂x ∂t ∂x ρ ∂x

ð24Þ

ð25Þ

where ∂pc =∂x represents the constant horizontal pressure gradient due to the current. Generally, for wave dominated sea situations, this pressure gradient is at least two orders of magnitude smaller than the pressure gradient associated with the waves (Davies et al., 1988). Thus it is consistent to neglect this term for wave dominated sea situations. Moreover, the term W 0 ∂U 0 =∂z has been neglected since both W0 and ∂U 0 =∂z are small quantities and thus this term is much smaller than the other terms in Eq. (25). In this work the free stream velocity field ðU 0 ; W 0 Þ is given from second order Stokes theory by (see e.g. Dean and Dalrymple, 1991) U 0 ðx; z; tÞ ¼ 7a 7

ð20Þ

where U0 is the harmonic component of the velocity and Uc is the mean current velocity at the outer boundary zmax . Zero flux conditions are imposed for the turbulent quantities, giving

νT

It should be noted that this simplified boundary condition yields zero sediment concentration for z 4 zmax , which is not necessarily realistic in the ocean where e.g. small ripples or large scale vortices might transport sand higher up (than z ¼ zmax ) in the water column. However, such mechanisms are excluded in the present study. Furthermore, the wave–current sheet flow measurements by Dohmen-Janssen et al. (2001) (including velocities, sediment concentrations and sediment fluxes in a combined wave–current boundary layer, conducted in an oscillating water tunnel), were well predicted by Holmedal et al. (2004) using Eq. (24). Hence Eq. (24) is implemented at z ¼ zmax in the present work. For sheet flow conditions, which will be studied in this work, this boundary condition is considered to be acceptable.

gkp coshðkp zÞ cos ð∓kp x þ ωtÞ ω coshðkp hÞ

3 a2 ωkp coshð2kp zÞ cos 2ð∓kp x þ ωtÞ 4 4 sinh ðkp hÞ

W 0 ðx; z; tÞ ¼ 7 a 7

ð26Þ

gkp sinhðkp zÞ sin ð∓kp x þ ωtÞ ω coshðkp hÞ

3 a2 ωkp sinhð2kp zÞ sin 2ð∓kp x þ ωtÞ 4 4 sinh ðkp hÞ

ð27Þ

Here the upper sign represents waves propagating in the positive x-direction, i.e. following the current from the left to the right, while the lower sign represents waves propagating in the negative x-direction, i.e. opposing the current. Moreover, a is the free surface linear wave amplitude. These free stream velocities are evaluated at z ¼ zmax to yield the near-bottom velocities which are forcing the sea bed boundary layer. 2.5. Numerical method and initial conditions A finite difference method was used to solve the parabolic Eqs. (1), (3), (4) and (9), using second order central differences in space. Geometric stretching of the mesh was applied to obtain a fine resolution close to the bed. This grid setting is based on previous experience when comparing experimental data with simulations. For a given horizontal position beneath the wave, the equations were integrated along vertical lines. Here the vertical mesh was staggered such that the turbulent quantities k and ϵ are stored at the boundaries of the velocity u cells. The sediment concentration

L.E. Holmedal et al. / Continental Shelf Research 65 (2013) 27–44

c cells are a subset of the velocity u cells, since the near-bed boundary condition for c is imposed at a given distance above the rough bottom. For each vertical line the spatial discretization of the equations for the horizontal velocity component, turbulent kinetic energy and dissipation rate give a set of stiff differentialintegral equations that were integrated simultaneously in time by the integrator VODE (Brown et al., 1989). Then the sediment equation was solved applying the eddy viscosity obtained from the turbulent flow, using the integrator VODE, thus taking advantage of the fact that the velocity is independent of the sediment concentration in the present setting. The integral on the right hand side of Eq. (11) was evaluated using a trapezoidal method which is consistent with the spatial discretization of the governing equations. There are no unique initial conditions for this set of parabolic equations, but for any reasonable initial values the solution reaches a steady state solution after a time interval of transients. Thus small positive values of the mean turbulence and flow quantities were initially seeded, and the equations were integrated in time until the flow was fully developed. It should be noted that the time interval of transients is long for sea bed boundary layers with asymmetric forcing, as shown earlier for horizontally uniform flow by Holmedal and Myrhaug (2006). In order to establish a fully developed flow (in the sense that wave-averaged quantities remain the same after successive wave periods), a spin-up time of 800 wave periods was applied. Previous experience shows that with this grid structure 100 vertical grid cells are sufficient for resolving the boundary layer. However, 200 vertical grid cells were used in most of the simulations. Some simulations with 100 vertical grid cells were carried out with a maximum deviation of 2% for the wave-averaged Eulerian velocities, showing that the grid convergence is sufficient for capturing the steady streaming velocity with a reasonable degree of accuracy.

3. Results and discussion This paper presents the flow and sediment transport within the ocean wave–current bottom boundary layer for realistic wave conditions, bottom roughnesses and water depth. Ocean surface waves with an amplitude of a¼ 1.22 m and a period of 6 s propagate over a flat rough bottom, in combination with a current specified as U c ¼ 0:1 m=s at zmax ¼ 0:25 m above the bottom. The water depth is 8 m, the resulting wave length is 45 m, and the bottom roughnesses are z0 ¼ 1  10−5 , 3  10−5 , 6  10−5 , 1  10−4 and 2:3  10−4 m, corresponding to A=kN ¼ 3000, 1000, 500, 300 and 130, respectively. If the empirical formula kN ¼ 2:5d50 is applied, these roughnesses would correspond to fine sand, medium sand, coarse sand, very coarse sand and gravel, respectively (Soulsby, 1997, Fig. 4). However, mobile bed effects such as formation of ripples are neglected in the present study; thus z0 represents a given roughness over a flat bottom. Flow over ripples represents a different flow regime as shown by Inman and Bowen (1962) who provided measurements showing that vortex asymmetry over the ripples caused sediment to be transported against the mean flow. However, the investigation of this flow regime is beyond the scope of this work. Predictions of the sediment transport will be given in Section 3.3 for sheet flow conditions. The given wave condition represents intermediate water depth (kp h ¼ 1:11) with wave steepness akp ¼ 0:17. The near-bottom potential flow is approximated by second order Stokes theory. In the forthcoming the predictions are conducted with the bottom roughness z0 ¼ 6  10−5 m unless otherwise specified. Moreover, Uc is specified at zmax ¼ 0:25 m above the bottom; a discussion of the implications of these boundary conditions is given in Section 3.1.2.

31

Application of Eq. (14) shows that the wave decay over the wave length for waves without a current is about 2% for A=kN ¼ 130 and less than 1% for A=kN ¼ 300. Since the current friction factor is much smaller than the wave friction factor for seabed boundary layers (see e.g. Soulsby, 1997), this order of magnitude estimate is valid for the present wave–current seabed boundary layer as well. Thus periodic lateral boundary conditions are reasonable to apply for this case; this is also the case for A=kN 4 300 for which the wave decay is smaller due to smaller friction (less rough bottom). Throughout the paper the dispersion relation for waves alone has been applied, neglecting the effect of the current on the wave number. For wave dominated flow, which is the main focus of this work, this effect is weak and thus the application of the dispersion relation for waves alone is a reasonable approximation. The validity of this approximation was checked by comparing the mean velocity profiles obtained using the waves alone dispersion relation with those obtained by using the wave–current dispersion relation ðω−kp UÞ2 ¼ gkp tanhðkp hÞ, where U is the depth-averaged current (see e.g. Jonsson, 1990 for details regarding this dispersion relation). Here U was set equal to U c in the wave–current dispersion relation (as an approximation). The comparisons were made for sinusoidal propagating waves and current with a wave period of 6 s and for A=kN ¼ 500. For following waves and current, the maximum deviations of the mean velocities (relative to those obtained from the waves alone dispersion relation) were 1.5%, 1.7% and 16% for U c ¼ 0:1 m=s, 0.25 m/s and 0.5 m/s, respectively. Similar deviations were found for opposing waves and current, except where the backflow (relative to the current) vanishes, i.e. where the mean velocity changes from small negative to small positive values. In this region the magnitude of the mean velocity is very small and the mean vertical velocity gradient is large. An additional check was carried out for A=kN ¼ 130 with a wave period of 6 s and U c ¼ 0:1 m=s; here the maximum deviation found in the mean velocities were 1.6 percent for following waves and current. Overall these predictions confirm that using the waves alone dispersion relation is a reasonable approximation except for U c ¼ 0:5 m=s. Strictly, the current U used in the wave– current dispersion relation should be the depth-averaged current over the water column. Since this current is not known apriori (only the water column within the wave–current boundary layer is considered here), the application of the waves alone dispersion relation simplifies the predictions substantially. Moreover, further simulations reveal that a small change in U c leads to deviations of the mean velocity of the same order of magnitude as those introduced by using the waves alone dispersion relation. Hereafter the following terminology will be used: ‘waves’ denote propagating waves where the vertical velocity component is included; ‘horizontally uniform symmetric forcing’ and ‘horizontally uniform asymmetric forcing’ will be used for cases where the vertical velocity component is neglected (i.e. horizontally uniform flow). Here ‘symmetric’ refers to sinusoidal forcing in time; ‘asymmetric’ refers to second order Stokes forcing in time. 3.1. Mean (wave-averaged) Eulerian velocities and related quantities 3.1.1. Waves alone Holmedal and Myrhaug (2009) investigated the streaming velocities, mass transport and sediment transport in the seabed boundary layer beneath waves alone. They considered waves propagating to the right only. For the sake of completeness a few corresponding results will be given here for both waves propagating to the right and to the left. These results form the background for understanding the effect of streaming on the wave–current seabed boundary layer. The boundary layer forcings are given in Eqs. (26) and (27); the limiting cases of horizontally uniform

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forcing are also given from these equations. It will be shown below that all the boundary layer quantities beneath waves propagating towards the left are symmetric with the corresponding quantities propagating towards the right with respect to the vertical axis, as expected. As pointed out by Holmedal and Myrhaug (2009), steady streaming in near-bed ocean flows is caused by both wave asymmetry (i.e. by asymmetry of turbulent fluctuations in successive wave half-cycles), as first explained by Trowbridge and Madsen (1984) (and later in further detail by Scandura (2007)) and by the presence of a small vertical wave velocity, as explained by Longuet-Higgins (1953). In order to isolate the effect of asymmetric forcing, the boundary layer streaming velocity is predicted by neglecting the vertical wave velocity and retaining the asymmetry (Stokes second order) of the boundary layer forcing. This has been applied both to waves propagating towards the right and the left. The resulting steady streaming velocities (i.e. mean Eulerian velocities) are shown in Fig. 1 for T p ¼ 6s. It should be noted that these streaming velocities are in the opposite direction of the wave propagation direction, as also found by Holmedal and Myrhaug (2009). A similar result was also obtained by Scandura (2007) from a DNS of a horizontally uniform (in terms of mean turbulent quantities) oscillatory boundary layer flow subjected to asymmetric boundary layer forcing. This flow was in the transitional to turbulent flow regime for a flat smooth bottom subjected to infinitesimal disturbances. Scandura (2007) showed that this streaming velocity was caused by the asymmetry of the boundary layer forcing due to different characteristics of the turbulence in successive half-cycles of the periodic (but asymmetric) boundary layer forcing. In the present work, which deals with fully rough turbulent flow, the effect of the Reynolds stresses is accounted for by a k-ϵ eddy viscosity model. In this setting the resulting time-averaged velocity profile depends on the phase difference (during the wave cycle) between the eddy viscosity and the vertical velocity gradient due to the Boussinesq approximation. Further details are given in Trowbridge (1983), Davies and Li (1997), Holmedal and Myrhaug (2006) for horizontally uniform flow with asymmetric forcing, and by Holmedal and Myrhaug (2009) for propagating second order Stokes waves. Experiments conducted by Ribberink and Al-Salem (1995) in an oscillating water tunnel (horizontally uniform flow) also revealed a negative wave-averaged horizontal velocity near the bottom (see e.g Holmedal and Myrhaug,

Fig. 1. Mean horizontal boundary layer velocity (i.e. steady streaming velocity) for asymmetric forcing and horizontally uniform velocity (the vertical velocity is zero) for waves towards the right (full line) and waves towards the left (dashed line). Note that the streaming velocity is in the opposite direction of the waves when the vertical velocity is neglected.

Fig. 2. Mean Eulerian horizontal boundary layer velocity (i.e. steady streaming velocity) for asymmetric and symmetric forcing when the vertical mean turbulent velocity is included. Full lines: beneath Stokes waves. Dashed lines: beneath linear waves. The positive velocity profiles are beneath waves propagating towards the right; the negative velocity profiles are beneath waves propagating towards the left. Note that the streaming velocity is in the same direction as the waves when the vertical velocity is taken into account.

2006 for a more detailed discussion). It should be noted that a zero wave-averaged velocity was obtained for sinusoidal forcing of the horizontally uniform boundary layer by both Scandura (2007) and Holmedal and Myrhaug (2009). Fig. 2 shows the streaming velocities for waves propagating towards the right and the left for T p ¼ 6 s. It appears that when the vertical velocity, which is always present beneath real ocean waves, is taken into account, the streaming velocity is in the direction of wave propagation for both propagating linear and second order Stokes waves. This velocity is caused by the term ∂ðuwÞ=∂z which acts as a depth-varying force pushing the flow in the direction of wave propagation. Thus the Longuet-Higgins streaming dominates the present case. It is observed that the magnitude of the streaming velocity beneath sinusoidal waves is larger than beneath second order Stokes waves. This means that the effect of asymmetry here is to reduce the streaming. This is physically sound, since the asymmetric forcing for the horizontally uniform boundary layer in Fig. 1 leads to a streaming velocity against the wave propagation direction. The mean product −uw beneath second order Stokes waves is shown in Fig. 3 for both waves propagating towards the right and the left for T p ¼ 6 s. In the close vicinity of the bottom −uw is zero as expected, since w approaches zero there. The profile of −uw through the boundary layer agrees qualitatively with the corresponding profile obtained by Longuet-Higgins (see e.g. Fig. 1.4.2 in Nielsen, 1992), but the magnitude is larger for turbulent flow than for laminar flow. Holmedal and Myrhaug (2009, Fig. 6) investigated the behavior of −uw for different roughnesses and found that the magnitude of −uw increases as the roughness increases, leading to a larger streaming velocity beneath waves alone.

3.1.2. Waves following and opposing a current Here we describe how the Longuet-Higgins streaming mechanism and the asymmetric streaming mechanism described in the previous paragraph affect the wave–current seabed boundary layer. Fig. 4 shows the mean Eulerian horizontal velocity profiles beneath waves following a current for T p ¼ 6 s and U c ¼ 0:1 m=s, where the current is directed towards the right. The velocity

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Fig. 3. −uw  103 versus z−z0 for linear sinusoidal waves. Full line: beneath waves propagating towards the right. Dashed line: beneath waves propagating towards the left.

33

second order Stokes waves, respectively. The mean velocity beneath linear waves is reduced compared to the current alone velocity, but it is substantially larger than the velocity beneath symmetric horizontally uniform boundary forcing. Here there is an interaction between the Longuet-Higgins streaming (shown in Fig. 2 for waves alone) and the classical wave–current mechanism. The Longuet-Higgins streaming mechanism forces the flow (via ∂ðuwÞ=∂z) in the direction of wave propagation (i.e. in the direction of the current), while the classical wave–current mechanism leads to a reduction of the mean velocity in the presence of waves relative to the velocity obtained beneath the current alone. Finally, the mean velocity beneath second order Stokes waves is smaller than the mean velocity beneath linear waves, but larger than the mean velocity beneath symmetric horizontally uniform forcing. Here there are three different mechanisms interacting with each other: the Longuet-Higgins streaming mechanism forces the flow in the direction of wave propagation (i.e. in the direction of the current); the asymmetry streaming mechanism forces the flow against the wave propagation direction (i.e. against the current), and the classical wave–current mechanism leads to a reduction of the velocity relative to the velocity beneath current alone. Fig. 5 shows the mean Eulerian horizontal velocity profiles beneath waves opposing a current for T p ¼ 6 s and U c ¼ 0:1 m=s; the current is directed towards the right and the waves propagate towards the left. The velocity profile beneath the current alone is included. Since the classical theory does not distinguish between opposing and following waves and current, the mean velocity profile beneath symmetric horizontally uniform forcing is identical to that for waves following the current (Fig. 4). For the other situations, however, the interaction between the streaming mechanisms and the classical wave–current mechanism leads to mean velocity profiles which are substantially different from those beneath following waves and current. The velocity beneath asymmetric horizontally uniform forcing is larger than beneath symmetric horizontally uniform forcing due to the asymmetry streaming mechanism, which acts against the wave propagation direction, forcing the flow in the direction of the current. The interaction between the asymmetry streaming mechanism and the classical wave–current mechanism ensures the reduction of the velocity compared to the velocity beneath the current alone. The mean velocity beneath linear waves is dominated by the Longuet-Higgins streaming mechanism which forces the flow in the direction of wave propagation, which here is against the

Fig. 4. Mean Eulerian horizontal velocity profiles beneath waves following a current. The velocity profile for current alone is given for comparison.

profile beneath the current alone is included for reference. The reduction of the mean velocity in the presence of waves due to the wave–current interaction is clearly shown for all the cases considered. This mechanism is commonly explained physically by that the waves act as an extra artificial bottom roughness (apparent roughness) leading to an extra reduction of the current (Grant and Madsen, 1979). In the forthcoming this will be referred to as the classical wave–current mechanism. For the classical case of symmetric horizontally uniform boundary layer forcing, this is the only such mechanism acting on the flow. For the case of asymmetric horizontally uniform boundary layer forcing, there exists an asymmetry streaming mechanism in addition which forces the flow towards the opposite direction of the wave propagation (as shown in Fig. 1), i.e. in the opposite direction of the current. This asymmetry streaming mechanism interacts with the classical wave–current mechanism, leading to the mean velocity being further reduced relative to the current alone. It should be noted that the symmetric and asymmetric horizontally uniform forcings here represent the limiting cases of infinitely long linear and

Fig. 5. Mean Eulerian horizontal velocity profiles beneath waves opposing a current. The velocity profile for current alone is given for comparison.

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current. It appears that the interaction between the LonguetHiggins streaming mechanism and the classical wave–current mechanism leads to a mean velocity profile which is strongly reduced compared to the velocity profile beneath the current alone. Moreover, the velocity is directed against the current in the close vicinity of the bottom. For the mean velocity beneath second order Stokes waves, the Longuet-Higgins streaming mechanism forces the flow against the current direction (in the direction of wave propagation) while the asymmetric streaming mechanism forces the flow in the direction of the current (against the wave propagation direction); the classical wave–current mechanism ensures the reduction of the mean velocity relative to the velocity beneath current alone. The Longuet-Higgins streaming mechanism provides the strongest forcing of the mean flow also here, but the presence of the asymmetry streaming mechanism leads to a modification of the mean velocity compared to the mean velocity beneath linear waves and current; the mean velocity is directed against the current in the close vicinity of the bottom, but the magnitude is smaller than beneath linear progressive waves. For pure wave driven seabed boundary layers the existence of an irrotational free stream velocity outside the boundary layer enables a consistent coupling using Neumann conditions sufficiently far away from the seabed. In contrast, wave–current seabed boundary layer models require a coupling within a sheared layer at a given distance zmax from the seabed, since the mean velocity profile (or current) may exist over the entire water column from the seabed to the ocean surface. Either a Dirichlet condition on the velocity is specified, or ∂u=∂z must be specified at zmax . The latter approach is complicated and has not proved very successful (see Justesen, 1988 for more details). Holmedal et al. (2003, 2004) found a good agreement between predictions and measurements by, among others, Dohmen-Janssen (2001) of the wave–current boundary layer (horizontally uniform flow), using the Dirichlet velocity condition at zmax given in Eq. (20). Here the mean velocity profiles and sediment concentration profiles were reasonably well predicted. For horizontally uniform flow it is easy to relate the specified mean velocity U c at zmax to the ocean current by e.g. fitting a logarithmic profile. When the effect of streaming is taken into account the relation between U c and the ocean current is less obvious. Due to the nonlinear interaction between the classical wave–current mechanism and the two streaming mechanisms it is not possible to separate the ocean current from the streaming induced current near zmax . Since the streaming mechanisms here force the flow in the direction of the wave propagation, the ocean current required to satisfy the Dirichlet velocity condition at zmax is not the same for opposing and following waves. The near-bed mean backflow (relative to the current) for opposing waves and current induces a larger friction than for following waves and current, and thus a larger ocean current is required for opposing waves than for following waves and current to satisfy Eq. (20). This means that the effect of the streaming mechanisms for opposing waves and current might be underpredicted because the flow is more current dominated for opposing waves and current than for following waves and current. A possible solution would be to predict the velocity through the entire water column from the seabed to the ocean surface, but this requires a non-hydrostatic approach and preferably a more sophisticated turbulence model which is beyond the scope of the present work. Another possibility would be to substitute U c with U c þ U s and U c −U s in Eq. (20) for following and opposing waves and current, respectively. Here U s is the magnitude of the streaming induced mean velocity at zmax for waves alone. This procedure might lead to the current conditions for following and opposing waves being more equal, but it will not be exactly equal because the streaming induced part of the current is not equal to the streaming induced mean velocity for waves alone, due to the interaction between the streaming mechanisms

Table 1 Physical wave parameters for Figs. 6–8. The rows in this table correspond to those in Figs. 6–8. Wave period T p ðsÞ

kp h

R for asymmetric forcing

6 8 10 12

1.11 0.74 0.60 0.49

0.53 0.55 0.56 0.58

and the classical wave–current mechanism. Furthermore, U s varies with the wave conditions, making the presentation of the results more demanding. This work aims to show the difference between the horizontally uniform wave–current boundary layer flow and the corresponding flow where the effect of streaming is accounted for. Thus the same method as previously used for horizontally uniform flows (see e.g. Holmedal et al., 2003, 2004) is applied here to investigate the effect of streaming. Despite the weaknesses of the present approach, the effect of streaming on the combined wave–current seabed boundary layer is clearly shown, which is the main focus of this work. Overall, Figs. 4 and 5 show that the mean velocities beneath linear and second order Stokes waves opposing the current, are substantially different from the mean velocities beneath linear and second order Stokes waves following the current. This difference is caused by the streaming mechanisms interacting with the classical wave–current mechanism, and is not captured by the classical wave–current seabed boundary layer models which only consider horizontally uniform symmetric boundary layer forcing. The results obtained in Figs. 4 and 5 are important for transport of e. g. pollutants and fish larvae near the sea bottom in shallow and intermediate near-coastal waters. The present results have been obtained for a physical situation where the effect of streaming is substantial. In the forthcoming the effect of increasing the wave period and the specified near-bed current will be investigated. 3.1.3. Effect of wave asymmetry and of kp h as well as the magnitude of the specified near-bed current The effects of streaming on the seabed boundary layer current have been investigated by increasing the wave period T p and by increasing the specified current Uc at 25 cm above the bottom, keeping all the other physical parameters constant. Here wave periods are 6, 8, 10 and 12 s; U c ¼ 0.10, 0.25 and 0.50 m/s. A similar investigation was conducted by Holmedal and Myrhaug (2009) to reveal the effect of wave asymmetry on the streaming velocities in the seabed boundary layer beneath waves alone. Here the purpose is to investigate the effect of the streaming mechanisms on the wave–current seabed boundary layer over a realistic physical parameter range. The increase of wave periods leads (via the dispersion relation) to increased wave lengths such that kp h decreases from 1.1 for T p ¼ 6 s to 0.49 for T p ¼ 12 s. It should be noticed that since A=kN is constant, increasing the wave period changes the wave velocity amplitudes, and thereby the asymmetry of the waves. The asymmetry factor R of the waves is defined as R¼

U wc U wc þ U wt

ð28Þ

where Uwc is the crest velocity outside the boundary layer and Uwt is the magnitude of the trough velocity. It should be noted that here it is not possible to isolate the effect of wave asymmetry from the effect of changing kp h. As the wave period (and thereby the wave length) increases, kp h becomes smaller (implying more shallow water conditions), and the wave asymmetry R increases (for asymmetric forcing). The asymmetry factor R varies from 0.53 for T p ¼ 6 s to 0.58 for T p ¼ 12 s; the details are given in Table 1.

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The resulting mean Eulerian velocity profiles are shown in Figs. 6–8 for U c ¼ 0:10, 0.25 and 0.50 m/s, respectively. Each of these figures are organized into four rows and three columns: from the upper to the lower row T p ¼ 6 s, 8 s, 10 s and 12 s, respectively. In all the three columns the velocity profile beneath the current alone is given for comparison. Furthermore, the left column shows the mean velocity beneath sinusoidal horizontally uniform forcing (i.e. the vertical velocity is neglected). This remains the same regardless of whether the waves are following or opposing the current. The mid column shows the mean velocity beneath linear waves opposing and following the current; the right column shows the mean velocity beneath second order Stokes waves opposing and following the current. This visualizes the differences between the mean velocity profiles beneath opposing and following waves and current. The left column of Fig. 6 shows that as T p increases, the resulting near-bed mean velocity increases, i.e. the wave–current seabed boundary layer becomes more current dominated. The mid column shows the mean velocities beneath propagating linear waves following and opposing the current. These mean velocities result from the interaction between the Longuet-Higgins streaming and the classical wave–current mechanism as explained above for Figs. 4 and 5. It is observed that the difference between the mean velocities beneath following and opposing waves decreases as T p increases (and thereby kp h decreases). Moreover, the difference between these mean velocities and the mean velocity beneath sinusoidal horizontally uniform forcing decreases as T p increases. These results are consistent with the findings of Holmedal and Myrhaug (2009) who found that the seabed streaming velocities beneath waves alone decrease as T p increases because the ratio between the vertical and horizontal wave velocity becomes smaller as kp h decreases. As the effect of streaming becomes smaller, the mean velocities beneath following and opposing waves become more similar, and the deviation from the mean velocities beneath sinusoidal horizontally uniform forcing becomes smaller. The right column of Fig. 6 shows the mean velocities beneath second order Stokes waves following and opposing the current; these are caused by the interaction between the Longuet-Higgins streaming, the asymmetry streaming mechanism and the classical wave–current mechanism as explained above for Figs. 4 and 5. These mean velocities follow the same trend as those beneath linear waves following and opposing the current except that the asymmetry streaming mechanism provides an additional forcing of the flow against the direction of wave propagation. As pointed out by Holmedal and Myrhaug (2009), the asymmetry streaming becomes stronger relative to the Longuet-Higgins streaming when the wave period increases, since the ratio between the vertical and horizontal wave velocity becomes smaller as kp h decreases. The streaming velocities beneath second order Stokes waves are affected by two simultaneous effects as the wave period increases: the increase of the asymmetry factor R increases the asymmetry streaming, and the decrease of kp h reduces the magnitude of the Longuet-Higgins streaming. Hence both of these effects reduce the importance of the Longuet-Higgins streaming (i.e. the effect of ∂ðuwÞ=∂z). For T p ¼ 10 s and T p ¼ 12 s the asymmetry streaming dominates the Longuet-Higgins streaming, leading to the mean velocity beneath opposing waves and current being slightly larger than the mean velocity beneath following waves and current. Figs. 7 and 8 show results corresponding to those in Fig. 6, for U c ¼ 0:25 m=s and U c ¼ 0:5 m=s, respectively. The results show the same trend as in Fig. 6; the wave–current seabed boundary layer becomes more current-dominated and the effect of the asymmetry streaming becomes more important relative to the effect of the Longuet-Higgins streaming as the wave period increases. In Fig. 7 the difference between the mean velocities beneath following and

35

opposing waves and current becomes small for T p ¼ 10 s and T p ¼ 12 s. However, these mean velocities differ from the mean velocity beneath horizontally uniform symmetric forcing also for these wave periods. In Fig. 8 the mean velocities beneath following waves and current are not very different from the corresponding mean velocities beneath opposing waves and current, for all the wave periods. For T p ¼ 12 s there is only a small difference between these mean velocities and the mean velocity beneath horizontally uniform forcing. It appears that for these larger wave periods and stronger currents, the classical wave–current interaction is dominating the streaming mechanisms, leading to almost similar velocity profiles beneath waves opposing and following the current. As the wave–current seabed boundary layer becomes even more current dominated, the mean velocity profiles beneath following and opposing waves and current will approach the mean velocity profile beneath horizontally uniform symmetric forcing. It should be noted that for the strong current in Fig. 8, the neglection of the effect of the current on the dispersion relation leads to a less accurate prediction of the mean velocity as discussed previously in Section 3. However, since the classical wave–current interaction is dominating the streaming mechanisms for this case, the inaccurate prediction of the wave number is not significant. The apparent roughness is commonly interpreted as an additional roughness experienced by the current in the presence of waves. It is a measure of the effect of the waves on the mean current profile; the more wave-dominated the flow is, the stronger is the effect of the waves on the current. The apparent roughness can be estimated by e.g. linear extrapolation of the semilogarithmic mean velocity profile back to zero velocity. Measurements by Klopman (1994) indicate that the apparent roughness is larger for opposing waves than for following waves and current. For the more current dominated cases shown in Figs. 7 and 8, our simulations support this observation. However, for many of the wave dominated situations presented in Fig. 6, the mean velocity in the close vicinity of the bed opposes the current causing the concept of apparent roughness to collapse (see also Fig. 5). Hence the concept of apparent roughness cannot be generally used for the most wave dominated situations within the wave–current boundary layer. Fig. 9 shows the mean Eulerian horizontal velocity profiles for A=kN ¼ 130, 300, 500, 1000 and 3000, for following linear waves and second order Stokes waves and current (Fig. 9a and b, respectively); for T p ¼ 6 s and U c ¼ 0:1 m=s. Here A=kN ¼ 130 represents the largest bottom roughness and A=kN ¼ 3000 represents the smallest bottom roughness. Holmedal and Myrhaug (2009) found that the maximum steady streaming velocity beneath waves alone increases as the bottom roughness increases. Fig. 9a and b show the opposite behavior; here the magnitude of the mean velocity profile decreases as the bottom roughness increases. This is caused by the classical wave–current mechanism which leads to a reduction of the mean velocity near the bed compared to the velocity profile beneath current alone; the larger roughness, the larger reduction. This mechanism appears to dominate the streaming mechanisms here. For following second order Stokes waves and current (Fig. 9b) the presence of the asymmetry streaming mechanism leads to slightly smaller mean velocities than beneath following linear waves and current. Fig. 10 shows the mean Eulerian horizontal velocity profiles for A=kN ¼ 130, 300, 500, 1000 and 3000, for opposing linear waves and current and opposing second order Stokes waves and current (Fig. 10a and b, respectively); here T p ¼ 6 s and U c ¼ 0:1 m=s. It appears that for opposing waves and current the LonguetHiggins streaming mechanism, forcing the flow in the direction of the wave propagation, is dominating. This forcing increases with the roughness (see e.g. Holmedal and Myrhaug, 2009; Fig. 6)

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Fig. 6. Mean Eulerian horizontal boundary layer velocity for current alone (C), waves following (W+C) and opposing (W-C) a current with Uc ¼0.1 m/s. The left column shows the mean velocity beneath horizontally uniform forcing as Tp and the asymmetry increases from top towards bottom of the figure. The mid column shows the mean velocity beneath sinusoidal waves as Tp increases from top towards bottom. The right column shows the mean velocity beneath second order Stokes waves as Tp and the asymmetry increases from top towards bottom. The asymmetry factor R is given in each subfigure; corresponding values of kph are given in Table 1.

explaining the behavior observed in Fig. 10 a and b. In the close vicinity of the bed the velocity becomes negative (i.e. opposing the current Uc) for the most rough bottoms. For opposing second order Stokes waves and current (Fig. 10b) the presence of the

asymmetry streaming mechanism, which tend to force the flow in the opposite direction of wave propagation (i.e. in the current direction), leads to slight modifications of the mean velocity profiles.

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Fig. 7. Mean Eulerian horizontal boundary layer velocity for current alone (C), waves following (W+C) and opposing (W-C) a current with Uc ¼ 0.25 m/s. The left column shows the mean velocity beneath horizontally uniform forcing as Tp and the asymmetry increases from top towards bottom of the figure. The mid column shows the mean velocity beneath sinusoidal waves as Tp increases from top towards bottom. The right column shows the mean velocity beneath second order Stokes waves as Tp and the asymmetry increases from top towards bottom. The asymmetry factor R is given in each subfigure; corresponding values of kph are given in Table 1.

3.2. Lagrangian velocity and particle trajectories The effect of the water particle trajectories, represented by the Stokes drift, must be taken into account to find the actual mass transport beneath real waves. The particle trajectories in the sea bed boundary layer are propagating ellipses (see Holmedal and Myrhaug, 2009, Fig. 12). When evaluating the mass transport

(i.e. the wave-averaged horizontal Lagrangian velocity) over a wave period, it is necessary to follow the fluid particle along its trajectory through the wave period. For the special case of horizontally uniform flow the mass transport is found by averaging the Eulerian flow field because the particle trajectories do not change in space. For realistic near-bottom flows in the ocean, however, the spatial changes of the particle trajectories must be

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Fig. 8. Mean Eulerian horizontal boundary layer velocity for current alone (C), waves following (W+C) and opposing (W-C) a current with Uc ¼ 0.5 m/s. The left column shows the mean velocity beneath horizontally uniform forcing as Tp and the asymmetry increases from top towards bottom of the figure. The mid column shows the mean velocity beneath sinusoidal waves as Tp increases from top towards bottom. The right column shows the mean velocity beneath second order Stokes waves as Tp and the asymmetry increases from top towards bottom. The asymmetry factor R is given in each subfigure; corresponding values of kph are given in Table 1.


Z taken into account (Batchelor, 1967). This is done by a Taylor expansion of the Eulerian flow field (u,w). The result, to second order accuracy, is
Z uL ¼ u þ

 uL dt

∂u þ ∂x


Z

 wL dt

∂u ∂z

ð29Þ

wL ¼ w þ

 uL dt

∂w þ ∂x


Z

 wL dt

∂w ∂z

ð30Þ

R R where uL dt and wL dt are the horizontal and vertical displacements, respectively. Here the wave-average of uL (i.e. uL ) is commonly referred to as the mass transport, representing the

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39

Fig. 9. Mean Eulerian horizontal boundary layer velocities for five different roughness conditions beneath linear waves and second order Stokes waves following a current. (a) Following sinusoidal waves and (b) Following Stokes waves.

mean velocity of a fluid particle averaged over a wave period. It is consistent with second order accuracy to replace uL and wL with u and w in the integrands of Eqs. (30) and (31) (see Mei, 1989, Chapter 9.2 for a detailed outline). Thus the velocity components along the particle trajectories can be evaluated from
Z uL ¼ u þ

 u dt


Z wL ¼ w þ

∂u þ ∂x

 u dt


Z

∂w þ ∂x

 w dt


Z

∂u ∂z

 w dt

∂w ∂z

ð31Þ

ð32Þ

Fig. 11 shows the mass transport velocity (i.e. the waveaveraged Lagrangian velocity) profiles for A=kN ¼ 130, 300, 500, 1000 and 3000, beneath linear waves following a current (Fig. 11a) and second order Stokes waves following a current (Fig. 11b). Here T p ¼ 6 s and U c ¼ 0:1 m=s. The mass transport velocity profiles follow the trend of the corresponding Eulerian velocity profiles shown in Fig. 9. Holmedal and Myrhaug (2009, Fig. 5) found that for waves alone (without a current) the mass transport velocity at the edge of the wave seabed boundary layer is about 0.08 m/s for second order Stokes waves and about 0.095 m/s for linear waves, i.e. slightly smaller than the specified current velocity Uc at 25 cm above the bottom. They also found that this mass transport velocity (at the edge of the wave seabed boundary layer) was larger than the Stokes drift velocity for waves alone (evaluated at the bottom). With the presence of the current, the mass transport velocity at 25 cm above the bottom is about 0.14 m/s. Hence this velocity is larger than both the mass transport velocity beneath waves alone and the specified current U c ¼ 0:1 m=s. It is observed that the mass transport velocity beneath second order Stokes waves and current is not very different from that beneath linear R waves and current. This is probably due to the ð w dtÞ∂u=∂z term which becomes large because of the current. The corresponding mass transport velocity profiles for waves opposing a current are shown in Fig. 12. The mass transport velocity is smaller near the bed and larger farther away from the bed compared with the corresponding mass transport velocity profiles for following waves and current in Fig. 11. The reduction of the velocity near the bed is caused by the Eulerian part of the mass transport velocity (u), while the enhancement farther away from the bed is caused by the R vertical velocity gradient, via, the term ð wdtÞ∂u=∂z. It appears

that these profiles are almost similar for linear waves and current and for second order Stokes waves and current. It should be noted that the mass transport is in the direction of the current for both opposing and following waves for this physical choice of parameters. The mean Eulerian velocity in the seabed boundary layer is important for finding the flux of pollutants or biological material near the bottom of the ocean. For e.g. pollutants or plankton which are neutrally buoyant and following the flow, the mean Eulerian velocity profiles give the flux through the seabed boundary layer. The mean mass transport velocity gives the velocity of the fluid particles. This gives information of the spreading of pollutants or biological material such as fish larvae or phytoplankton for shallow waters. This information is of particular relevance in near-coastal waters, to monitor the spreading of pollutants and biological material. 3.3. Effect on sediment transport If the resulting bottom shear stress is strong enough to move the sea bed material, or to bring it into suspension, then the nonzero mean velocity will cause a net transport of this material over time. This transport may take place either as net transport of suspended sediments or bedload. Fig. 13 shows the Eulerian wave-averaged suspended sediment flux profiles for following and opposing waves and current for T p ¼ 6 s and U c ¼ 0:1 m=s for the three median sand grain diameters d50 ¼ 0:13 mm, 0.21 mm and 0.32 mm. Here both linear and second order Stokes waves (following and opposing the current) are considered. The corresponding settling velocities are ws ¼ 0:0119 m=s, 0.026 m/s and 0.0429 m/s, respectively; these are taken from Dohmen-Janssen et al. (2001). By inspection the maximum Shields number is larger than 0.8 for all the simulations presented in this Section, and thus the sediment transport takes place as sheet flow (Soulsby, 1997). It should be noted that a range of different measurements of the sediment concentration and sediment flux under sheet flow conditions (horizontally uniform flow) were successfully predicted by Holmedal et al. (2004) using the present approach. Fig. 13 shows that the mean sediment flux increases as the median sediment diameter decreases, as expected. It appears that even though the Eulerian mean velocity is substantially different for waves following and opposing the current (as shown in Figs. 9 and 10), the sediment flux is less

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Fig. 10. Mean Eulerian horizontal boundary layer velocities for five different roughness conditions beneath linear waves and second order Stokes waves following a current. (a) Opposing sinusoidal waves and (b) opposing Stokes waves.

Fig. 11. Mean mass transport velocities (i.e. wave-averaged Lagrangian velocities) for five different roughness conditions beneath linear waves and second order Stokes waves following a current. (a) Following sinusoidal waves and (b) following Stokes waves.

different. However, Fig. 13 shows a larger sediment flux for opposing waves and current than for following waves and current. This difference is largest for d50 ¼ 0:13 mm and smaller for d50 ¼ 0:21 mm and d50 ¼ 0:32 mm. In order to understand this it is important to appreciate that the sediment transport results from the dimensionless bottom shear stress (Shields parameter), which is again a result of the boundary layer velocity. For the present wave dominated case with T p ¼ 6 s and U c ¼ 0:1 m=s, the LonguetHiggins streaming is strong. It appears that the phase difference between the horizontal and vertical velocity components is different for opposing waves and current than for following waves and current. As an example, Fig. 14a shows ∂ðuwÞ=∂z versus time beneath linear waves and current for d50 ¼ 0:21 mm. It appears that the magnitude of ∂ðuwÞ=∂z (and hence the Longuet-Higgins streaming) is larger for opposing than for following waves and current. This induces a slightly larger horizontal near-bed velocity for opposing than for following waves and current, as shown in Fig. 14b. Here the horizontal velocity is given at the nearest grid point above bed. Subsequently, this leads to a slightly larger

friction velocity uf for opposing waves and current than for following pffiffiffiffiffiffiffiffiffi waves and current, as shown in Fig. 14c. Here uf ¼ τb =ρ where τb is the bottom shear stress. It is observed that the crest of uf beneath opposing waves and current is larger than the corresponding crest beneath following waves and current, thus explaining the larger sediment fluxes for opposing waves and current. This also leads to a larger bedload transport beneath opposing waves and current than for following waves and current for this particular case. These results can also be viewed in light of the discussion in Section 3.1.2 of the Dirichlet velocity condition on top of the boundary layer (Eq. (20)): since the flow beneath opposing waves and current is more current dominated than the flow beneath following waves and current, it seems physically sound that the net sediment transport beneath opposing waves and current is slightly larger as well. The results presented in Fig. 13 are quantified in Tables 2 and 3 Rz which show the mean suspended sediment transport 2dmax uc dz, 50 the mean bedload transport qb and the total sediment transport (the mean suspended sediment transport plus the mean bedload

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41

Fig. 12. Mean mass transport velocities (i.e. wave-averaged Lagrangian velocities) for five different roughness conditions beneath linear waves and second order Stokes waves opposing a current. (a) Opposing sinusoidal waves and (b) opposing Stokes waves.

transport) for d50 ¼ 0:13, 0.21 and 0.32 mm for following and opposing linear waves and current as well as second order Stokes waves and current. It appears that both the mean suspended sediment transport and the mean bedload transport (and thereby the net sediment transport) are larger for opposing waves and current than for following waves and current. This is the case for both linear waves and second order Stokes waves, and for all the three sediment median grain diameters. Overall, both the suspended sediment transport and the bedload transport are in the direction of the current, even when the mean Eulerian velocity in the near vicinity of the bottom is directed against the current (opposing waves). Hence the net total sediment transport is in the direction of the current both for opposing and following waves.

3.3.1. Comparison with measurements To the authors knowledge existing measurements of near-bed wave–current boundary layer streaming are only available from field measurements over rippled beds or from wave flumes. Unfortunately, the characteristic size of the roughness elements glued to the bottom of the flume in these measurements is of the same order of magnitude as the near-bottom wave excursion amplitude. Hence form drag becomes important, and this effect is not accounted for in the present model, where the roughness is assumed to affect the turbulence intensity only (taken into account by the average roughness concept). Moreover, in wave flumes the near-bed wave-induced velocity amplitudes are small; thus the Longuet-Higgins streaming mechanism is weak compared to the wave dominated situations in the ocean. Nevertheless, comparisons with wave flume measurements of the wave–current seabed boundary layer flow, using the present model, were presented in Johari et al. (2011); the mean bottom boundary layer velocity profiles measured by Kemp and Simons (1982, 1983), Klopman (1994) and Umeyama (2005) were fairly well predicted except in the close vicinity of the bottom. However, since the physical mechanisms in the close vicinity of the bottom are governed by form drag rather than skin friction, and nearbed wave-induced velocity amplitudes are so small that the LonguetHiggins streaming mechanism is weak, the effect of streaming is hardly detectable in these measurements. In order to measure these effects relevant for flows in the ocean, measurements in very large wave flumes are required, where realistic ocean conditions such as wave height, wave period and water depth can be reproduced.

4. Summary and conclusions The effect of streaming on the sea bed boundary layer beneath combined waves and current has been investigated for waves following and opposing a current. Both linear waves and second order Stokes waves are considered. This flow results from an interaction between the classical wave–current seabed boundary layer interaction as described by Grant and Madsen (1979), and two streaming mechanisms. These two streaming mechanisms are streaming caused by turbulence asymmetry in successive wave half-cycles (beneath asymmetric forcing), and streaming caused by the presence of a vertical wave velocity within the seabed boundary layer as earlier explained by Longuet-Higgins (1953). The effects of wave asymmetry, wave length to water depth ratio, magnitude of current and bottom roughness have been investigated for realistic physical situations. Mean Eulerian quantities as well as the mass transport (wave-averaged Lagrangian velocity) are presented. The resulting sediment dynamics near the ocean bottom has been investigated; results for both suspended load and bedload are presented. It appears that the seabed boundary layer flow is substantially different beneath opposing waves and current than beneath following waves and current. The main results are as follows:

 The Longuet-Higgins streaming mechanism forces the flow in





the direction of wave propagation; the asymmetry streaming mechanism forces the flow against the direction of wave propagation, and the classical wave–current mechanism leads to a reduction of the mean (wave-averaged) velocity relative to that beneath the current alone. For waves following the current, the mean velocity is larger than the mean velocity beneath horizontally uniform symmetric forcing, but smaller than the mean velocity beneath the current alone. This applies for both linear and second order Stokes waves. Due to the asymmetry streaming mechanism, the mean velocity beneath second order Stokes waves is slightly smaller than beneath linear waves and current. For waves opposing the current, the mean velocity is smaller than the mean velocity beneath horizontally uniform symmetric forcing, and also smaller than beneath current alone. This applies for both linear and second order Stokes waves. In the close vicinity of the bottom, the mean velocity can be against the current. This behavior is caused by the Longuet-

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Fig. 13. Sediment flux for three different sand grain sizes for following and opposing linear waves and Stokes waves. (a) Sediment flux for d50 ¼0.13 mm and (b) sediment flux for d50 ¼ 0.21 mm and (c) sediment flux for d50 ¼0.32 mm.



Higgins streaming mechanism which forces the flow against the current. Due to the asymmetry streaming mechanism, the magnitude of the mean velocity beneath second order Stokes waves is slightly smaller than beneath linear waves and current. The flow in the wave–current boundary layer was made more current dominated by increasing the wave period and by increasing the specified current, leading to a weaker effect of the Longuet-Higgins streaming. Here the classical wave– current interaction dominates the streaming mechanisms, leading to the velocity profiles beneath waves opposing and following the current being more similar. For the most current dominated situations, these mean velocity profiles





approach the mean velocity profile beneath horizontally uniform symmetric forcing. In the close vicinity of the bottom the mass transport velocity beneath waves opposing the current is smaller than the corresponding mass velocity beneath following waves and current. This is consistent with the Eulerian mean velocities. Farther away from the bottom the mass transport velocity beneath opposing waves is larger than for following waves. This is caused by the larger horizontal velocity gradient in this region, observed from the mean Eulerian velocity profiles. An increase of the bottom roughness leads to a decrease of the mean wave–current boundary layer velocity both for the mean Eulerian velocities and the mass transport; here the classical

L.E. Holmedal et al. / Continental Shelf Research 65 (2013) 27–44

43

Fig. 14. Quantities for following and opposing linear waves: (a) ð1=10Þ  ð∂ðuwÞ=∂tÞ evaluated at the first gridpoint above the bottom, (b) the velocity component u at the first gridpoint above the bottom and (c) The friction velocity uf. Here d50 ¼0.21 mm and t¼ 6 s.

Table 2 Mean bedload transport, suspended sediment transport and total sediment transport ðq total Þ for linear waves and Stokes second order waves following a current. Here U c ¼ 0:1 m=s and T p ¼ 6 s. Waves

Linear waves Linear waves Linear waves Stokes second order waves Stokes second order waves Stokes second order waves



d50 (mm)

R

0.13 0.21 0.32 0.13

R zmax

Table 3 Mean bedload transport, suspended sediment transport and total sediment transport ðq total Þ for linear waves and Stokes second order waves opposing a current. Here U c ¼ 0:1 m=s and T p ¼ 6 s.

qb (mm2/s)

q total (mm2/s)

Waves

(mm2/s)

0.50 0.50 0.50 0.53

10.2 11.3 12.9 13.0

73.9 23.6 14.1 63.9

84.1 34.9 27.0 76.9

0.21

0.53

14.5

24.8

39.3

Linear waves Linear waves Linear waves Stokes second order waves Stokes second order waves Stokes second order waves

0.32

0.53

16.6

15.6

32.2

2d50

uc dz

wave–current interaction mechanism dominates. This mechanism leads to a reduction of the mean velocity beneath combined waves and current relative to that beneath current alone; the rougher the bottom, the larger the reduction. The net sediment transport is in the direction of the current both for following and opposing waves; for both linear and second order Stokes waves. For linear waves and current the net sediment transport is larger for opposing waves and current than for following waves and current. This might be

d50 (mm)

0.13 0.21 0.32 0.13 0.21 0.32

R

0.50 0.50 0.50 0.53 0.53 0.53

qb (mm2/s)

17.9 19.7 22.5 14.1 15.5 17.6

R zmax

(mm2/s)

q total (mm2/s)

122 31.7 20.7 116 28.8 18.1

139.9 51.4 43.2 130.1 44.3 35.7

2d50

uc dz

related to the Dirichlet velocity condition on top of the boundary layer applied in the present work, which yields a more current dominated situation for opposing waves and current than for following waves and current.

Acknowledgments This work was carried out as a part of the projects ‘Wave– current Interactions and Transport Mechanisms in the Ocean’ and

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