Journal of Geophysical Research: Oceans

Three-dimensional mixture simulations of flow over dynamic rippled beds


Corresponding author: A. M. Penko, Marine Geosciences Division, Naval Research Laboratory, Code 7434, 1005 Balch Blvd., Stennis Space Center, MS 39529, USA. (


[1] A three-dimensional mixture theory model for flow and sediment transport in the seafloor boundary layer, SedMix3D, simulated the flow over and the resulting sediment entrainment and evolution of rippled beds. SedMix3D treats the fluid-sediment mixture as a continuum of varying density and viscosity with the concentration of sediment and velocity of the mixture simulated by the Navier-Stokes equations coupled with a sediment flux equation for the mixture. Model validation was performed by comparing simulated time-dependent flow quantities and bulk flow statistics with measurements obtained in the laboratory under scaled forcing conditions. Two-dimensional planes extracted from a three-dimensional simulation were compared to observations made using planar Particle Image Velocimetry (PIV) in a laboratory flume. The simulated results of time-averaged velocities and time-dependent quantities of vorticity and swirling strength were in good agreement with the observations. The model was used to analyze the three-dimensionality of vortex formation and ejection produced by oscillatory flow over vortex ripples, a process that cannot be observed in the laboratory with planar PIV measurements. The three-dimensional simulated results showed that the swirling strength varied significantly in the cross-flow direction, indicating that the vortices formed and dissipated non-uniformly due to random fluctuations. Subsequently, an order of magnitude difference in offshore sediment flux was observed using two different methods to calculate sediment fluxes (spatially averaging and at a point). The results suggest that while a two-dimensional plane may be sufficient to examine the general hydrodynamics over ripples, three-dimensional analysis is necessary for a complete understanding of sediment transport.

1 Introduction

[2] Subaqueous bed forms are ubiquitous on the seafloor, along coastlines, in estuaries, and on riverbeds, affecting a wide range of phenomena including sediment transport, granular sorting, waves and circulation, benthic activity, and acoustic penetration into sediments. Since many of the processes driving sediment transport occur over small spatial and temporal scales, we believe small-scale bed form dynamics are directly linked to large-scale morphodynamics. However, the ephemeral and turbulent nature of bed forms inhibits our understanding of their impact on large-scale sedimentary and hydrodynamic processes. Consequently, bed forms are of interest to the coastal engineering, underwater acoustics, oceanography, riverine, and environmental communities.

[3] Of the many different types of bed forms (e.g., ripples, dunes, anti-dunes, megaripples), ripples can occur in areas seaward of the breaker zone and where waves have reformed after breaking on a sandbar [Nielsen, 1981; Hanes et al., 2001] and have been observed in situ when the Shields parameter is less than 1.0 [Fredsøe and Deigaard, 1992]. Ripples are classified as rolling grain ripples if the flow does not separate behind the ripple crests, and vortex ripples if vortices are generated as the flow reverses directions [Bagnold, 1946]. The process of vortex formation can entrain sediment and dissipate wave energy [Tunstall and Inman, 1975; Nakato et al., 1977; Ardhuin et al., 2002].

[4] Obtaining measurements at the spatial and temporal resolution necessary for the examination of small-scale turbulent fluctuations over vortex ripples is difficult. For example, an oscillating tray of sediment has been used to study three-dimensional bed forms generated by combined steady currents and oscillations induced by the tray motion [Lacy et al., 2007]. However, steady streaming effects are not produced by the device, and the tray motion induces a mean flow and an inertial force on the bed forms not found in the field [Scherer et al., 1999; Lacy et al., 2007]. Two-dimensional bed forms have been studied in detail in oscillatory flow tunnels that simulate horizontal flows [Foti and Blondeaux, 1995; O'Donoghue and Clubb, 2001; O'Donoghue et al., 2006; Ribberink et al., 2008], but three-dimensional bed form geometries are less common in one-dimensional flows. Recently, detailed information on bed form dynamics under surface gravity waves has been studied in both laboratory flumes [Landry et al., 2007; van der Werf et al., 2007, 2009; Nichols and Foster, 2009; Hurther and Thorne, 2011] and in the field [Traykovski et al., 1999; Hanes et al., 2001; Doucette, 2002a, 2002b]. However, limitations exist in laboratory flumes such as the inability to produce multi-directional waves and combined wave-current flows. In situ measurements of vector and scalar quantities of a three-dimensional domain are even more sparse, as the technology to take such measurements has only recently been developed and is still being refined [Doucette et al., 2002; Elsinga et al., 2006; Fouras et al., 2009].

[5] Previous numerical studies of the flow structure over static (or fixed) bed forms [Blondeaux and Vittori, 1991; Scandura et al., 2000; Barr et al., 2004; Chang and Scotti, 2003; Zedler and Street, 2006; Shimizu et al., 2001; Bhaganagar and Hsu, 2009] have broadened our knowledge of vorticity dynamics in oscillatory flow; however, the dynamic coupling of the fluid and sediment, affecting turbulence and resulting morphodynamics has been lacking. Because strong correlations exist between the sediment transport, flow field, and bed morphology, coupling these processes is necessary to fully understand bottom boundary layer dynamics. We hypothesize it is the three-dimensional small-scale turbulent fluctuations and the small-scale fluid-sediment interactions that govern much of the morphologic evolution of the seafloor. However, the disparate spatial and temporal scales of interest for vortex entrainment of sediment over a sand ripple compared with the transition of a natural ripple field in response to changing wave conditions, for example, limit our present ability to perform three-dimensional high-fidelity numerical simulations for the latter. Existing two-dimensional numerical models that solve the Reynolds-Averaged Navier-Stokes (RANS) equations have been used to simulate coupled bed form dynamics under unidirectional [Giri and Shimizu, 2006] and oscillatory flow [Marieu et al., 2008]. However, these formulations must parameterize the three-dimensional temporal and spatial fluctuations that drive boundary layer turbulence and ultimately, morphodynamics. While these types of models may be useful for engineering applications, a three-dimensional Navier-Stokes solver is necessary to resolve the turbulent structures and the coupled fluid-sediment interactions over an evolving bed.

[6] We use a three-dimensional mixture theory model, SedMix3D, that solves the unfiltered Navier-Stokes equations for a fluid-sediment mixture to simulate bottom boundary layer oscillatory flow over ripples [Penko et al., 2011]. Mixture theory treats the fluid and sediment phases as a single continuum with the inter- and intra-phase interactions (e.g., effective viscosity, hindered settling, diffusion) parameterized through closure relations. Results from a three-dimensional simulation performed with SedMix3D are compared with planar particle image velocimetry (PIV) data obtained in the laboratory under scaled forcing conditions. Model validation is performed using both time domain flow velocities and bulk statistics.

[7] Although the comparisons made here utilize velocities measured in a two-dimensional vertical plane, it is important to note that the results obtained from SedMix3D are three-dimensional. In the following section, the model formulation and the experimental conditions are described. Direct comparisons between the laboratory data and the simulation results are made in section 3. The three-dimensionality of the vorticity and resulting sediment suspension that is unobtainable from the data but simulated by the model is also discussed in section 3. The discussion focuses on addressing discrepancies between laboratory data and simulation results where they exist and the limitations of the PIV technique.

2 Methodology

2.1 SedMix3D

[8] The three-dimensional mixture model (SedMix3D) solves the unfiltered Navier-Stokes equations and a sediment flux equation for a fluid-sediment mixture resulting in the time-dependent sediment concentration and three-component velocity field in a three-dimensional domain [Penko et al., 2011]. SedMix3D treats the fluid-sediment mixture as a continuum with effective properties that parameterize the fluid-sediment and sediment-sediment interactions including a bulk-hindered settling velocity, a shear-induced diffusion, an effective viscosity, and a particle pressure. Grid spacing is on the order of a sediment grain diameter and time step O(10 − 5 s). The turbulent eddy length scale essential for sediment entrainment over vortex ripples is approximately an order of magnitude larger than the grid step, and therefore we believe any constraints relating grain length scales and eddy length scales should be satisfied with the current grid size.

[9] The numerical scheme is finite difference in the vertical with second-order central differences employed on a staggered grid and spectral in the horizontal. The boundary conditions for the velocities and concentration are periodic in the horizontal. The driving force on the mixture is spatially constant and time varying, and the v and w velocities are set to zero at the top of the domain. The model only simulates flow in the boundary layer (i.e., no free surface). A no-slip condition exists for the velocities at the bottom boundary of the domain; however, the sediment bed thickness was chosen such that the sediment never eroded to expose the bottom of the simulation domain and the mixture velocity is effectively zero deep within the bed near the boundary. The concentration is initialized at the approximate maximum packing concentration of sandy sediment (ϕ = 0.63) at the bottom boundary and zero at the top boundary. All simulations are fully three-dimensional. The flow is directed along the x axis, with the transverse flow direction along the y axis.

[10] The model equations are based on the mixture velocity (the velocity of the sediment and fluid as a continuum) as defined by mixture theory. SedMix3D is a one-phase continuum model, with a scalar quantity of sediment concentration that determines the mixture's bulk properties (density, viscosity, settling velocity) at every grid point. The mixture is treated as a continuum of varying density and viscosity with the sediment concentration and velocity of the mixture calculated by the Navier-Stokes equations for the mixture coupled with a sediment flux equation. The mixture continuity equation was derived by combining the fluid and sediment phase continuity equations:

display math(1)

where u is the mixture velocity and ρ is the mixture density,

display math(2)

where ϕ is the sediment volumetric concentration, and ρs and ρf are the sediment and fluid densities, respectively. The mixture momentum equation was derived from the sum of the fluid and sediment phase momentum equations:

display math(3)

where P is the mixture pressure, μ is the effective viscosity, F is the external driving force vector per unit volume, g is gravitational acceleration (981 cm s − 2 inline image), and Sb is the particle pressure parameterization equation (5). SedMix3D employs a modified Eilers equation [Eilers, 1941] to represent effective viscosity, μ, here scaled by the pure water viscosity, μf,

display math(4)

where [μ] is the intrinsic viscosity, a dimensionless parameter representing the sediment grain shape, and 0.0 ≤ ϕ ≤ 0.63, with the lower bound representing pure water and upper bound roughly corresponding to the maximum concentration of a packed sediment bed. The maximum value of the effective viscosity is fixed by specifying ϕm = 0.66. The intrinsic viscosity parameter, [μ], is 2.5 to represent spherical particles [Einstein, 1906]. The model will simulate mixtures from 0% volumetric concentration (pure fluid) to mixtures with the maximum concentration of sediment allowable in a given volume due to the spherical shape of the grains (here, we assume 63% volumetric concentration). When the concentration of sediment is zero, the model reduces to solving the Navier-Stokes equations for a pure fluid at a resolution of the selected grid size (O(1 mm)).

[11] A particle pressure is necessary for a mixture theory model to build and maintain bed forms. We assume the parameterization of particle pressure to be an exponential function of sediment concentration [Buyevich, 1999; Chen et al., 2003]. The particle pressure term in SedMix3D parameterizes the contact forces by applying a concentration dependent exponential damping function to the velocity to stabilize the bed:

display math(5)

where γ is 0.3. The bed stiffness factor, γ, determines the amount of particle pressure applied. The mixture velocity, u, is decreased an amount determined by the bed stiffness function equation (5). The function is based on the assumption that the particle pressure force becomes significant when ϕ >∼0.35, roughly corresponding to the onset of the enduring contact region [Fredsøe and Deigaard, 1992]. In concentrations less than∼0.35, the particle pressure is approximately zero.

[12] The concentration of sediment is modeled with a sediment flux equation [Nir and Acrivos, 1990] that balances the temporal gradients in sediment concentration with advection, gravity, and shear-induced diffusion,

display math(6)

where Wt is the concentration-dependent settling velocity [Richardson and Zaki, 1954],

display math(7)

Wt0 is the settling velocity of a single particle in a clear fluid and q is an empirical constant,

display math(8)

Rep is defined as the particle Reynolds number,

display math(9)

and d is the sediment grain size diameter. The shear-induced diffusion of sediment, D, is a function of grain size, volumetric concentration, and mixture stresses [Leighton and Acrivos, 1986],

display math(10)


display math(11)

and where α is an empirical constant found to be approximately 0.33 for large values of the Shields parameter ( 0.5 < θ < 30) by combining results from the dilute limit with measurements in dense concentration suspensions.

2.2 Experimental Facility and Conditions

[13] The experiments were performed in a wave flume at the Fluid Mechanics Laboratory at Delft University of Technology, the Netherlands (S. Rodriguez-Abudo et al., Spatial variability of the wave bottom boundary layer over movable rippled beds, manuscript in review). The experimental effort aimed to provide two-dimensional insight into the turbulent nature of the wave bottom boundary layer and coherent structures resulting from oscillating flows above mobile rippled beds. Of special interest was accurately resolving the flow field within ripple elements.

[14] The flume was 42 m in length, 0.8 m in width, and 1 m in height, with the bottom covered with a layer of artificial sediment. The sampling window (11 cm × 11 cm) was located approximately 29 m from the wave generator where the mean water level was about 0.31 m. Correct physical modeling of nearshore sediment transport in a laboratory setting requires scaling of the sediment particles [Henriquez et al., 2008]. In order to preserve the ratios of the Reynolds number, shear stress, and particle settling, Henriquez et al. [2008] followed the Froude scale for short waves and chose sediments with a specific gravity of 1.2 and mean grain diameter of 0.54 mm, based on the wave flume dimensions and the wavemaker specifications. The prescribed waves (H = 5 cm, T = 2 s) produced a maximum free-stream velocity (U ∞ ,max) of 13 cm s − 1 and a wave orbital excursion (A) of 4.1 cm. These sedimentary and hydrodynamic conditions produced a mobility number of approximately 16 and should result in equilibrium vortex ripples with a length of 5.4 cm and a height of 0.8 cm [Nielsen, 1981], very similar to the actual ripple dimensions recorded in the laboratory and in the simulation. The low density nature of the grains resulted in a dynamic and very responsive bed. Vortex formation and ejection were observed at every half-wave period. Sediment grains were actively picked up and entrained within these vortices [Rodriguez-Abudo and Foster, 2012].

[15] A Dantec Particle Image Velocimetry (PIV) system obtained two-dimensional vertical (x-z) plane optical images of the sampling window using a 120 mJ Nd-Yag pulsed laser synchronized with a 1 megapixel camera. The laser was located in a water-tight housing approximately 27 cm above the bed and the camera was positioned outside the flume, perpendicular to the oscillatory flow direction. The camera captured image pairs (10 ms time lag between pair members) of the sampling window for 60 s bursts at approximately 12 Hz. Suspended sediment, organic matter, and micro-bubbles acted as seeding agents in the water column. The artificial sediment had a particle Stokes number less than one, therefore, particles can be assumed to closely follow the streamlines of the flow. Particle displacements were calculated at the subpixel level using a three-point Gaussian estimator. Although no formal assessment of the velocity resolution at the subpixel level was performed, it has been shown that this technique achieves accuracies between 1/10 and 1/20 of a pixel [Raffel et al., 2007]. This method results in velocity resolutions between 0.55 and 1.1 mm s − 1. The velocity vectors were calculated by correlating the image pairs using 64 × 32 pixel interrogation windows with 50% overlap. The resulting spatial resolution of the vector field was 3.48 mm × 1.74 mm. Outliers were removed with a three-standard deviation filter and replaced with the local ensemble average. An Acoustic Doppler Velocimeter (ADV) time-synchronized with the PIV system measured the free-stream velocity approximately 15 cm above the bed.

3 Model Data Comparison Results

3.1 Simulation Initialization and Analysis

[16] The total simulation domain was 18.9 cm × 2.6 cm  × 14.2 cm with 256 × 32 × 192 grid points (Figure 1). The horizontal length of the domain was twice the length of the observations to reduce the effects of the periodic boundary conditions in the model. Only the center 10.6 cm of the simulation (and the full y and z dimensions) was included in the analysis (boxed area in Figure 1). The model is computationally expensive and the domain width was set to the minimum distance needed to sufficiently resolve turbulent structures in the simulation. An examination of the ensemble- and x-averaged autocovariance of the cross-flow (v) velocity showed that the velocities decorrelated within 2 cm in the y-direction (Figure 2). We calculated the decorrelation length scale by taking the autocovariance of the v velocities at z = 3.7 cm (∼2 mm above the ripple crest) shifted spatially in the y-direction, then ensemble-averaged in time and spatially averaged in the x-direction. The velocity was more correlated during flow acceleration; however, all phases were almost completely decorrelated after 2 cm. Consequently, we believe the domain width was sufficient for the size and two-dimensional shape of the ripples simulated here. We acknowledge that the domain width would need to be increased significantly to simulate three-dimensional ripples that might include bifurcations and terminations in the ripple field.

Figure 1.

Shown is the initial simulation bed profile in the simulation coordinate system. The total domain is 256 × 32 × 192 grid points corresponding to lengths of 18.9 cm  × 2.6 cm × 14.2 cm. Note that only the lower 6 cm is shown in the figure. The area inside the box is used in the comparisons.

Figure 2.

The figure shows the autocovariance of the v-velocity shifted in the y-direction. The autocovariance is ensemble- and x-averaged and plotted with phase and distance in y.

[17] The simulation bed profile was initialized with the observed instantaneous bed profiles wave-averaged over the first period from the PIV time series. A wave-average of the observed image intensities was necessary to reduce the noise in a single, instantaneous image and obtain a continuous bed profile. Slight (O(1 mm)) perturbations were added on the surface to create a rough bed and to break the symmetry of the simulation initialization. The fluid properties (ρf = 0.998 g cm − 3, μf = 1 × 10 − 2 g cm − 1 s − 1) and sediment parameters (d = 0.054 cm, ρs = 1.198 g cm − 3) were consistent with the observations. The free-stream velocity recorded by the ADV over 15 wave periods at z = 15 cm was used to calculate the external driving force per unit volume on the mixture (Figure 3),

display math(12)

where uADV is the velocity recorded by the ADV and the driving force acts only in the stream wise (x) direction.

Figure 3.

The velocity time series used to drive the model measured by the ADV (dotted lines) and the resulting spatially averaged free-stream velocity at the top of the simulation domain (solid lines) are plotted. Three periods (not shown) at the beginning of the free-stream velocity time series were discarded as spin-up periods for the model. Positive velocities are directed offshore. The black line indicates the beginning of the simulated and observed data used in the comparisons (t = 16 s).

[18] A useful way to examine the turbulent rotational flow over a rippled bed is to identify the closed rotational vortex structures by calculating the swirling strength [Zhou et al., 1999]. The swirling strength is calculated from the decomposition of the velocity gradient tensor in Cartesian coordinates, D = ∇ u,

display math(13)

where λr is the real eigenvalue with a corresponding eigenvector υr, and λcr ± λcii are the conjugate pair of the complex eigenvalues with complex eigenvectors υcr ± υcii. The local flow swirls on the plane formed by the vectors υcr ± υci and is stretched or compressed along the vector υr. Therefore, the amount of local swirling is quantified by the imaginary part of the complex eigenvalue pair (λci) and is referred to as the swirling strength of the vortex. Because the eigenvalue is complex only in regions with local circular or spiraling streamlines, this method excludes regions having vorticity but no spiraling motion, such as boundary-generated shear.

[19] An analysis of the variance of the swirling strength in the y-direction indicated that the simulated flow was not fully developed until t = 16 s (Figure 4). Therefore, only the last 14 s of the simulation and observations were used in the comparisons. The model is fully three-dimensional but the observations were limited to a single x-z plane. Consequently, a representative plane at each time step was selected by minimizing the root mean square deviation (RMSD) of the swirling strength, λci,rmsd, between the single observational plane and all x-z simulation planes,

display math(14)

where zf (x) > zbed(x) and λci was calculated using the method of Zhou et al. [1999]. This method produced one plane for each time step, resulting in multiple planes per simulation. The RMSD calculations only included the portion of the domain above the bed elevation, zbed(x). Figure 5 shows the y-averaged λci,rmsd and the standard deviation in the y-direction over the seven wave periods compared. The greatest deviations occurred at flow reversal with a maximum standard deviation 0.08 s − 1. Both ensemble- and time-averaged quantities of the minimized swirling strength RMSD x-z planes over the last seven wave periods were calculated and compared to the observations in the following section.

Figure 4.

Plotted is the spatially averaged variance of the simulated swirling strength in the y-direction throughout the time series. The black vertical line indicates the beginning of the simulated and observed data used in the comparisons (t = 16 s).

Figure 5.

Plotted is the y-averaged root mean square deviation of the swirling strength, λci,rmsd (solid line), between the single observational plane and all x-z planes in the simulation. The dotted lines represent the y-averaged swirling strength RMSD plus/minus the standard deviation in the y-direction.

3.2 Hydrodynamics

[20] Comparisons of the bulk and temporal flow statistics are presented in this section. The standard deviation of the simulated and observed horizontal velocity, σu, was calculated at each x and z grid point as

display math(15)

and is plotted in Figures 6a and 6c, respectively, where angled brackets denote a time average. A time average of the estimated bed profile for the simulation and observations is also plotted. The observed bed profile was determined by choosing a threshold of the time-averaged observed image intensities. The simulated bed profile was determined by time averaging the concentration and choosing a threshold concentration of 57% by volume as the concentration limit. Choosing a threshold concentration between 55% and 63% changed the location of the ripple profile by less than 3 mm. The actual location of the bed is arbitrary, not used in the model calculations, and plotted here only as a reference. Model estimations of σw were calculated similarly to equation (15) and were in good qualitative agreement with the observations in the troughs and over the crests of the ripples, but deviated slightly from the observations ( < 1 cm s − 1) on the ripple slopes and by 2 cm s − 1 very near the bed (within 0.3 cm) (Figures 6b and 6d).

Figure 6.

Contours of the standard deviation of the simulated (a) horizontal and (b) vertical velocity fields are plotted with the standard deviation of the observed (c) horizontal and (d) vertical velocity fields. The simulated and observed time-averaged bed profiles are included on the plots for reference.

[21] The time-averaged simulated horizontal velocities, inline image, also agreed well with the observations (Figures 7a and 7c), specifically over the left ripple. The simulated mean horizontal velocity over both ripples was symmetric and directed toward the ripple flanks; however, the observations exhibited less symmetry with mean horizontal velocities directed toward the flanks of the left ripple, but not the right ripple. The model was in better agreement with the observed time-averaged horizontal velocity in the left portion of the domain. The observations exhibited more symmetry in the plot of time-averaged vertical velocity (Figures 7b and 7d). The model agreed well with the observations throughout the entire domain, estimating average vertical velocities within 1 cm s − 1 of the observations in some locations.

Figure 7.

Contours of the time-averaged simulated (a) horizontal and (b) vertical velocity fields are plotted with the observed (c) horizontal and (d) vertical velocity fields. The simulated and observed time-averaged bed profiles are included on the plots for reference.

[22] The simulated and observed horizontal mean flow profiles at varying locations over the bed are plotted in Figure 8. The maximum difference between the simulated and observed mean flow was less than 2 cm s − 1 at all locations along the ripples. The simulated mean flow agreed well with the observations at every profile except at the horizontal boundaries, matching the maximum, minimum, and shape of the mean horizontal velocity profile. A 2 cm s − 1 difference between the simulated and observed horizontal mean velocity and different profile shape occurred within 1 cm of the bed at x = 1.5 and 9.8 cm, and could be attributed to an increased uncertainty near the edges of the PIV domain due to the inability to overlap the correlation windows at the domain edges.

Figure 8.

Profiles of the simulated (solid) and observed (dots) time-averaged horizontal velocity (inline image) are plotted at varying locations along the bed. The observed time-averaged bed profile is included on the plot for reference.

[23] A cross-spectral analysis between both the simulated and PIV observed horizontal velocities, and the velocity measured by the ADV (located 15 cm above the bed) was performed to compute the near bed phase leads (Figure 9). Positive contours indicate the simulated (Figure 9a) and PIV observed (Figure 9b) velocity leads the ADV velocity. The model and PIV observations illustrated similar phase differences 3–5 mm above the bed; however, the simulation showed a lead of 90° within 2 mm of the troughs and the PIV observations showed a phase lag of 10° approximately 1.5 cm above the ripple trough.

Figure 9.

Cross-spectral analysis shows the phase difference between the horizontal velocity recorded with the ADV and the (a) simulated and (b) PIV observed horizontal velocity, u. Positive contours indicate locations where the simulated or PIV observed velocities lead the ADV velocity. The simulated and observed time-averaged bed profiles are included on the plots for reference.

[24] Further analysis of phase is illustrated in the simulated and observed ensemble-averaged x-z plane vorticity fields, Ωy (Figure 10). The simulated vorticity plots (left column) showed the boundary layer shear on the face of the ripples at near maximum free-stream velocity (Figure 10b). At flow reversal, vortices were generated and ejected into the water column, and advected by the flow (Figures 10c and 10d). The vortices then dissipated after maximum flow and became less organized by the next flow reversal (Figure 10e). The observations (right column) also indicated vortex formation and ejection, and boundary layer shear. Note the observed vorticity magnitude only differed from the simulated vorticity by approximately 5%. The coherent vortex structures were less clearly defined in the observations than in the simulation. During acceleration/deceleration, the simulated vorticity agrees 20% better with the observations during offshore flow (flatter portion of the period, i.e., lower acceleration) than onshore flow (steeper portion, i.e., higher acceleration). The observed vortices generated during offshore flow were more coherent (typical of onshore migrating ripples) and therefore were in better agreement with the simulation. The model predicted the location, size, and rotational direction of the vortices for all phases fairly well. The shape of the vortex structure, including the vortex tails (the area of vorticity connecting the ejected vortex to the generation point, as in Figures 10c, 10d, and 10f) was also predicted well by the model.

Figure 10.

(a–f) Ensemble-averaged simulated (left column) and observed (right column) vorticity fields, Ωy (s − 1), at six phase locations of a wave are plotted in the contoured panels. The top panel is a plot of the ensemble-averaged free-stream velocity where the gray lines indicate the phase location. The flow is initially directed to the right (offshore). Positive (red) contours indicate a clockwise rotation. The simulated and observed time-averaged bed profiles are included on the plots for reference.

[25] In complex, three-dimensional and oscillatory flows, vortex structures are often difficult to distinguish from the vorticity due to boundary-generated shear. The swirling strength, λci [Zhou et al., 1999], is used to identify the coherent closed rotational vortex structures excluding the interference from boundary-generated shear equation (13). The method is effective in determining the location of vortex cores in boundary layer shear flow, but does not identify the rotational direction. The horizontal and vertical positions of the vortices and the timing relative to the wave phase predicted by the model were qualitatively similar to the observations (Figure 11). The strength of the vortices predicted by the model was only slightly greater than the observations. The observations showed more coherent vortices forming as the flow reversed from off to onshore flow (Figures 11c and 11d, right column) than from on to offshore flow (Figures 11e and 11f, right column), which may be characteristic of onshore ripple migration. However, the model predicted relatively symmetrical vortex size, shape, and generation frequency during both flow reversals. The x-z positions of the vortices were determined through manual inspection of the ensemble-averaged swirling strength (Figures 12b and 12c). Figure 12a shows the location in the wave phase with the symbol of corresponding size in Figures 12b and 12c. The simulated vortex positions were in good agreement with the observations during offshore flow. The vortex beginning and ending locations were within approximately 0.4 cm of each other (near the horizontal resolution of the PIV), and both the simulated and observed vortices had similar tracks during offshore flow. The shape of the simulated vortex path during onshore flow compared well with the observations; however, it was shifted about 1 cm horizontally and vertically from the observed vortex path. As the onshore flow decelerated, the simulated and observed vortices were ejected upward into the water column. The x and z locations of the simulated vortex centers differed from the observed vortex center locations by an average of 1.2 cm during onshore flow and 0.43 cm during offshore flow, which is on the order of the vortex structure diameter (∼1 cm).

Figure 11.

(a–f) Ensemble-averaged simulated (left column) and observed (right column) swirling strength fields, λci (s − 1), at six phase locations of a wave are plotted in the contoured panels. The top panel is a plot of the ensemble-averaged free-stream velocity where the gray lines indicate the phase location. The flow is initially directed to the right (offshore). Darker areas indicate coherent vortex structures. The simulated and observed time-averaged bed profiles are included on the plots for reference.

Figure 12.

(a) Positions in the wave phase of the simulated (crosses) and observed (circles) centers of the vortices generated during (b) offshore and (c) onshore flow. The vortices move in the direction of increasing symbol size. The size of symbol corresponds with the position in the wave phase of the equivalent size circle in the offshore/onshore portion of the wave phase in Figure 12a. Positive velocity is offshore. The arrow indicates the direction of the mean flow.

3.3 3D Vortex Structures and Sediment Entrainment

[26] Vortex structures have been shown to be highly three-dimensional [Blondeaux et al., 2004; Scandura et al., 2000; Zedler and Street, 2006] and dominate the hydrodynamics of oscillatory flow over ripples. Therefore, three-dimensional simulations are necessary to analyze the turbulent flow fields and sediment transport over bed forms. The model-observation comparisons demonstrated that the model captures the physics necessary to examine the three-dimensionality of the hydrodynamics over sand ripples that is difficult and costly to measure in the laboratory and virtually impossible to measure in the field. Accurate predictions of the vorticity field are imperative since the vortex structures play a significant role in sediment transport. Although the averaged quantities give some insight to the hydrodynamics, the turbulent structures are what drive the sediment transport over vortex ripples. The three-dimensional simulation illustrates the variance of the turbulent structures in the cross-flow direction and in time (Figure 4) that cannot be captured with point measurements or a two-dimensional observational plane obtained with PIV. Since vortex formation and shedding on the seafloor due to ocean waves are the dominant mechanisms for sediment pick-up and transport [van der Werf et al., 2007], successful simulations of the three-dimensional vortex structures may show a more realistic estimation of wave-induced sediment transport.

[27] Examining the simulated sediment concentration at different locations throughout the domain can quantify the effects of the three-dimensionality of vortex formation and ejection. We examined the simulated ensemble-averaged sediment concentration at two cross-flow locations and four vertical elevations (I–IV) above the ripple crest (Columns a and b) and trough (Columns c and d). The x- and y-locations and the z-elevations of the points are plotted in the plan and side views of the domain in the top row of Figure 13. The cross-flow points were chosen to be equidistant apart with one point near the center (a, c) and one near the (periodic) edge of the domain (b, d). The cross-flow variation of the simulation was analyzed by comparing the concentrations at the two y-locations along the crest and trough. Elevation points III and IV above the crest (a, b) were located in the very near-bed region with high sediment concentrations. Elevation IV (a, b) was within the packed bed region with a concentration of approximately 700 g L − 1 or 58% concentration by volume. Elevation III (a, b) was within the bed load region and exhibited a slight temporal variation of the concentration with the flow. All other points over the crest and trough were located in the suspended load region over the ripples, with sediment concentrations ranging from 0 to O(100) g L − 1.

Figure 13.

The figure plots the ensemble-averaged concentration at four elevations (I–IV) above two locations over the (a and b) ripple crest and two locations over the (c and d) ripple trough. The solid and dashed lines in the top left panel indicate the time-averaged x locations of the ripple crests and troughs in the domain, respectively.

[28] Above the crest (a, b), the first peak in concentration (a-b, I-II) was due to the entrainment of sediment in the vortex generated by the flow reversing from on to offshore. The suspended sediment at both the center (a) and edge (b) of the domain reached a maximum of about 10 g L − 1. The vortex was advected to the right (offshore) by the free-stream flow and was observed passing over the trough with a concentration peak around 120° in the center of the domain (c-d, I-IV). Over point c, sediment was observed at all four elevations and exhibited an expected phase lead near the bed; however, at the edge of the domain near the periodic boundary (d), the concentration peak is only observed closer to the bed (III-IV), illustrating a variation in suspended sediment concentration in the cross-flow direction. Next, the off to onshore flow reversal generated a vortex that entrained sediment as it was advected to the left (onshore) over the ripple crest and caused a concentration peak 220°–260° (a-b, II). We also observed the sediment entrainment from this vortex when it passed over the trough (c, d) between 260° and 320°. Cross-flow variability was again observed during onshore flow (200°–360°) over the ripple trough. Suspended sediment was observed at all four elevations near the periodic boundary (d); however, suspended sediment was only observed close to the bed (IV) in the center of the simulation domain (c). Sediment was entrained higher in the water column over point d during onshore flow, whereas it was entrained higher over point c during offshore flow. The entrainment event at 300° over point d exhibited the same phase lead near the bed also observed at 120° over point c. Additional cross-flow variations in the suspended sediment were observed over the trough at 250°. The concentration peaked at point d (IV), but not at point c (IV).

[29] The final peak in suspended sediment concentration over the crest (280°–360°) was entrained in the vortex generated over the neighboring offshore ripple by the off to onshore flow reversal and then was advected onshore (left) over points a and b. The timing of the suspended sediment concentration peaks are consistent with recent laboratory experiments on vortex formation and advection under asymmetrical waves [van der Werf et al., 2007; Hurther and Thorne, 2011].

[30] In general, the temporal variations of the ensemble-averaged concentration were similar at both cross-flow locations above the crest (a and b) due to accelerated flow and more uniform, coherent vortices. In the trough (c and d), the flow decelerated, causing random velocity fluctuations. The turbulent nature in the trough of the ripple caused the vortices to break up and dissipate at random, producing significant cross-flow variability in the suspended sediment. These results suggest that while choosing a single or several x-z planes is sufficient to analyze the hydrodynamics of the flow, it does not capture the whole picture regarding sediment transport. Also supporting this assertion are plots of the concentration and sediment flux profiles (Figure 14) time-, x-, and y-averaged (solid), time- and x-averaged at the x-z planes used in the model-data comparisons (dash-dot), and time-averaged at each of the four points (a, b, c, d) from Figure 13. The time-averaged elevations of the ripple crest and trough are located at z = 3.5 cm and z = 2.75 cm, respectively. The profiles illustrate the difference between analyzing a spatially averaged quantity versus a point quantity. The averaged concentration profiles (solid, dash-dot) had the same shape, with differences from about z = 4 cm to z = 4.5 cm. As observed in Figure 13, the concentration profiles were very similar over points a and b, and varied over points c and d. The differences in the concentrations at the crest and trough are expected; however, the most significant difference was observed in the sediment flux profiles.

Figure 14.

Plotted are the (a) vertical sediment concentration and (b) sediment flux profiles time-, x-, and y-averaged (solid), time- and x-averaged at the x − z plane locations chosen for the model-data comparison (dash-dot), and time-averaged at points a, b, c, and d from Figure 13. The time-averaged elevations of the ripple crest and trough are located at z = 3.5 cm and z = 2.75 cm, respectively.

[31] Typically, sediment fluxes are used to calculate net sediment transport, so their accuracy is imperative when calculating estimates of coastal sediment transport. Using two different methods to calculate the fluxes (time-, x-, and y-averaging (solid), and time- and x-averaging at the x-z planes used in the model-data comparisons (dash-dot)), an order of magnitude difference in the offshore sediment flux was observed (Figure 14b). This difference may be attributed to a considerable variation in the hydrodynamics in the cross-flow (y) direction further illustrated in Figure 15. The simulated three-dimensional swirling strength at six phases of the wave shows the three-dimensionality of the vortex generation and shedding (Figure 15). The blue isosurface of the swirling strength is where λci = 10 s − 1 and the swirling strength from λci = 10 s − 1 to λci = 15 s − 1 is contoured where the vortex intersects the domain edges. The figure shows the cross-flow variation in the vortex formation and ejection, specifically at off- to onshore flow deceleration and reversal (Figures 15c and 15d) and over the trough (Figures 15c–15e) in space. The swirling strength was also much stronger soon after maximum onshore flow (Figure 15e) than after maximum offshore flow (Figure 15b). The vortices from the previous off- to onshore flow reversal were observed passing over the troughs in Figure 15e, but not in Figure 15b. The slight asymmetry of the free-stream caused the differences in off- and onshore flow.

Figure 15.

The plot shows the isosurfaces of the three-dimensional swirling strength at six phase angles. The blue surface indicates a swirling strength of 10 s − 1 and the color contours represent the swirling strength inside a vortex that intersects the edge of the domain. The brown isosurface represents the instantaneous bed at 57% concentration by volume.

4 Discussion

[32] In general, the comparisons of the SedMix3D simulation results with the observations were in good agreement. Specifically, the simulated time-averaged and time-dependent flow quantities over the left ripple agreed well with the observations. The discussion includes an explanation of the observed velocity phase differences among the simulation, PIV, and ADV measurements and PIV sensitivity.

4.1 Boundary Layer Phase Lead

[33] Oscillating boundary layers are characterized by a velocity phase lead with respect to the free-stream [Stokes, 1851]. In the case of the wave bottom boundary layer, lower momentum fluid near the bed responds faster to pressure gradients than the free-stream, inducing a phase lead of roughly 10° in flat beds [Foster et al., 2000]. Near-bed phase leads over rippled beds are primarily due to vortex formation and have been measured to be 30°–90° ahead of the free-stream velocity [van der Werf et al., 2007; Hurther and Thorne, 2011]. In the simulation, we observed up to a 90° phase lead very near the bed in the troughs and a 30° phase lead over the crest, consistent with laboratory measurements. The simulated horizontal velocities were in phase with the ADV from z = 6 cm up to the top of the domain as expected. The model simulates the boundary layer only (i.e., no free surface) and was forced uniformly across the domain with the measured horizontal velocity from the ADV; therefore, the wavelength is effectively infinite and a phase difference between the simulated and measured ADV velocities in the free-stream will not be present.

[34] The PIV observations exhibited about a 9° phase difference with the ADV measurements at the free-stream (Figure 9); however, the phase difference over the width of the PIV window was consistent with linear theory. One can also observe a 2°–10° phase lead between the simulation and the PIV observations above z = 5 cm primarily due to the progressiveness of the waves. Additionally, the ADV sampling volume was not perfectly centered over the PIV window. It is not possible to determine the phase lead very near the bed with the PIV observations due to the large light reflections at the fluid-sediment interface. In the near bed region (up to about 0.3 cm above the bed), the high light intensity causes the tracer particles to become indistinguishable and results in a lower confidence in the velocity estimates in this region.

4.2 PIV Sensitivity

[35] The inherent ability of the model to calculate velocities with equal (and higher) resolution than the observations, even in the highly concentrated layer of moving sediment within 0.1–0.3 cm of the bed, may account for some of the discrepancies between the simulated and observed vorticity in the near-bed region (Figure 10). The model resolution is finer than the observed resolution and consequently may identify smaller-scale eddies that the observations do not capture with the coarser resolution. The PIV data exhibited more noise in the right side of the domain than the left side, possibly contributing to less coherent structures observed over the right ripple. In the laboratory experiment, the fluid was not seeded with tracer particles in conjunction with the PIV measurements. Depending on the sediment Stokes number, particles in the fluid will not always follow the streamlines of the flow. Therefore, capturing closed streamlines with PIV measurements is less likely with an unseeded fluid. An unseeded fluid will also cause gaps in the velocity measurements when suspended sediment is sparse, leading to random errors in the PIV velocity due to over-smoothing and variance. Edge effects from the image pair correlations may also account for the discrepancies in the velocity profile comparisons at the edges of the domain (Figure 8). Perspective errors associated with the viewing angle of the camera with respect to the angle of the laser sheet exist in PIV measurements. Near the edges and corners of the domain, this angle is greater, introducing a slight bias in the measurements. While it appears the diffusive effects are stronger in the observations, typically only very strong circulation events are recorded in situ [Nichols and Foster, 2007].

5 Conclusions

[36] SedMix3D is state-of-the-art in that it provides an unprecedented level of detail on sand ripple dynamics that exceeds field and laboratory technologies. The model can be used to analyze the complex, three-dimensionality of turbulent bottom boundary layer flow over dynamic rippled beds. While the comparisons of the time-averaged and time-dependent flow statistics of the two-dimensional plane from the three-dimensional simulation are in good agreement with the single observation plane, the two-dimensional analysis does not show the three-dimensionality of the vortex generation, dissipation, and suspended sediment concentration. A three-dimensional analysis of the simulated suspended sediment showed significant variations in concentration in the cross-flow direction. Using two different methods to calculate sediment fluxes (spatially averaging and at a point), an order of magnitude difference in offshore sediment flux was observed. This difference may be attributed to the cross-flow variation in the hydrodynamics caused by vortices dissipating non-uniformly due to the random turbulent fluctuations. The results suggest that while a two-dimensional plane may be sufficient to obtain a general idea of the hydrodynamics over vortex ripples, three-dimensional analysis is necessary for a complete understanding of sediment transport. SedMix3D allows for the examination of the three-dimensional turbulent vortices and the sediment transport induced by oscillatory flow over vortex ripples. Since few three-dimensional models and laboratory data sets exist yet, SedMix3D can be utilized to complete the three-dimensional information missing from existing field and modeling efforts.


[37] A. M. P. was supported by the Jerome and Isabella Karle Distinguished Scholar Fellowship Program at the Naval Research Laboratory. J. C. was supported under base funding to the Naval Research Laboratory from the Office of Naval Research. S. R. A. and D. L. F. were supported by the National Science Foundation (CTS-0348203). This work was supported in part by a grant of computer time from the DoD High Performance Computing Modernization Program at the NAVY, AFRL, and the ERDC DSRC.