1. Introduction
Ripples as an amazing type of coastal bedform have been attracting interest of the scientific community for more than a century (Pedocchi and Garcia 2009a). They may be categorized into orbital ripples, anorbital ripples, and suborbital ripples, according to their horizontal scale as compared to the near-bed orbital excursion of water particles caused by wave motion (Greenwood and Davis 1984; Pedocchi and Garcia 2009a). It may also be essential to distinguish between the rolling grain ripples and the vortex ripples following Bagnold (1946). The vortex ripples, with their shapes and dimensions jointly determined by the wave-induced near-bed orbital motion of water and the properties of the bed material (Bagnold 1946; Rousseaux 2006), play a major role in the various near-bed dynamic processes in the coastal waters and have been investigated by many researchers around the world.
The vortex ripples are always accompanied by a periodic formation–ejection process of vortices that then have a trapping–lifting effect on bed material (Zedler and Street 2006). They dominate the wave boundary layer structure as well as the entrainment mechanism of bed material and thus contribute substantially to the wave energy dissipation and the sediment transport rate from an overall point of view. Darwin (1883) and Ayrton (1910) carried out the earliest experiments to demonstrate the close relation between the ripples and vortices formed in between them. Bagnold (1946) tried to predict the ripple geometry from the sediment properties and the hydrodynamic conditions with his experiments involving an oscillating tray covered with sediments in a quiescent water tank. Following the boom of coastal engineering research in the 1950s, a large number of experiments have been done in either oscillatory flow tunnels (Carstens et al. 1969; Mogridge and Kamphuis 1972; Lofquist 1978; Sato et al. 1984; Sato and Horikawa 1986; Ribberink and Al-Salem 1994; O’Donoghue and Clubb 2001; Dumas et al. 2005; O’Donoghue et al. 2006; van der Werf et al. 2007; Pedocchi and Garcia 2009b) or wave flumes (Kennedy and Falcon 1965; Dingier and Inman 1976; Faraci and Foti 2002; Thorne et al. 2002; Williams et al. 2004) to look into the various aspects of the phenomenon, and many empirical formulas have been proposed (Camenen 2009; Pedocchi and Garcia 2009a) to describe the characteristics of ripples and the effects of ripples on either fluid flow or sediment transport.
Numerical study on the flow conditions and the sediment motion around a vortex ripple can be traced to Longuet-Higgins (1981) who adopted the discrete vortex method (DVM) to show the periodic formation–ejection process of vortices around a ripple. Longuet-Higgins’s (1981) approach was followed by the studies of Sleath (1982), Blondeaux and Vittori (1990, 1991), Hansen et al. (1994), and Malarkey and Davies (2002). The discrete vortex method is advantageous for a presentational description of the vortex formation–ejection process around a ripple, but struggles to represent the flow when it is highly turbulent. Sato et al. (1986) suggested a finite-difference model on a body-fitted curvilinear grid for wave-induced flow over a vortex ripple based on the so called ψ–Ω (streamfunction and vorticity) formulation of fluid flow and k–ε turbulence modeling. The motion of suspended particles in their study was simulated by means of the Monte Carlo method. Sato et al.’s (1986) approach was followed by the work of Aydin and Shuto (1988), Blondeaux and Vittori (1991), and van der Werf et al. (2008). As computational fluid dynamics (CFD) became popular in the last decades, most of the recent investigations (Tsujimoto et al. 1991; Kim et al. 2000; Sleath 2000; Andersen and Faraci 2003; Li and O’Connor 2007) chose to solve the fluid flow around ripples in the original variables (velocity and pressure), though different authors utilized different methods to deal with the turbulence. The standard CFD approach adopts the Reynolds-averaged Navier–Stokes equations (RANS) as the governing equations of the fluid phase. However, a relatively recent study by Chang and Scotti (2003) showed that large-eddy simulation (LES) might be necessary at least when accurate results on the turbulent behavior of the flow are required. LES in a boundary-fitted curvilinear coordinate system was proposed by Chou and Fringer (2010). It is common in almost all CFD approaches that an additional advection–diffusion equation on the sediment concentration must be solved with the bottom boundary condition given by an empirical formula for the pickup rate or the reference concentration. Such an empirical formula can hardly represent the real physics in general and is usually a source of inaccuracy.
A generally valid numerical simulation of the flow and sediment motion around ripples relies on an effective two-phase flow model that can deal with the hyperconcentrated sediment-laden flow near an erodible bed. Although two-phase flow models have been extensively developed in the recent years (Elghobashi and Abou-Arab 1983; Hsu et al. 2004; Longo 2005; Bakhtyar et al. 2009; Jha and Bombardelli 2009; Chen et al. 2011a,b), an application of such advanced models to the sediment-laden oscillatory flow over ripples has not yet been attempted. The present study is aimed at showing the effectiveness of the model proposed by the authors (Chen et al. 2011b) when applied to the flow and the sediment motion over an erodible ripple bed under wave action. In the following sections, we first give a brief description of the numerical model. Then, we present the numerical results and compare them with available experimental data.
2. Numerical model
a. Basic equations and numerical methods


























For numerical solutions, (1)–(4) along with (6) and (7) are discretized over a staggered grid by means of the finite-volume method, while the convection and the diffusion terms are treated with the third-order QUICK scheme and the second-order central difference scheme, respectively. Time stepping follows the standard strategy of computational fluid dynamics; that is, the continuity equation for the sediment phase is solved first to obtain the sediment concentration. Then, the momentum equations are used to predict the velocities. Thus, the pressure can be obtained by solving a pressure correction equation derived from the overall mass conservation equation. As a consequence, the velocity components can be corrected to ensure the conservation of momentum more accurately. As long as the volumetric concentration and the velocity of each phase are determined, the discretized equations of (6) and (7) readily yield the values of the turbulence kinetic energy and the turbulence kinetic energy dissipation rate.
The two-phase flow model described above has been shown to be valid for the wave-induced sheet flows (Chen et al. 2011a) and is applicable over the whole depth from the undisturbed sandy bed below the maximum depth of scour to the height where sediment particles can hardly reach. The model is advantageous also because it does not require an identification of the interface between moving and unmoving layers. There is also no need to separate the bed load and the suspended load. Neither a reference concentration nor a pickup function should be prescribed over the ripple bed as is usually necessary in a conventional advection–diffusion model.
b. Definition of problem
We focus on the dynamic process of intrawave water and sediment motion over regular sand ripples with known equilibrium profiles. One may expect a description on the deformation process of the ripple as a consequence of the interactions between the flow and ripple as in Chou and Fringer (2010). That is certainly important if we are interested in the relation between the ripple geometry and flow conditions, in the evolution of the ripple geometry, or in the translation of ripples in the direction of wave motion. However, if we limit our interest to the effects of the ripple on the intrawave flow and sediment motion, the geometry of the ripple rather than its gradual deformation should play the dominant role (Andersen 1999). For this reason and also for consistency with the experimental cases considered in this study, we adopt the physical model shown in Fig. 1. Taking two ripples in the horizontal direction is to assure the spatial repeatability of the numerical results.
Definition of problem for oscillatory, flow-induced sediment motion over vortex ripples.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0031.1









3. Results and discussions
a. Computational conditions
The cases of interest in the present study are carefully selected from a large number of laboratory investigations in the literature on the effects of the vortex ripples under oscillatory flow conditions, including Fredsoe et al.’s (1999) and Sleath’s (2000) experiments that focused on the periodic flow of the fluid phase and Nakato et al.’s (1977), Villaret and Perrier’s (1992), Steetzel’s (1984), Williams et al.’s (1998), and van der Werf et al.’s (2007, 2008) experiments that studied the periodic variation of the sediment concentration. Table 1 summarizes the representative parameters of the cases referred to in this study, including the ripples shapes, where T is the wave period, Um is the maximum free-stream velocity, and D50 is the median sediment diameter.
Representative parameters of cases studied.
In all the computations, the domain is fixed to two ripple lengths in the horizontal direction and to match the experimental water depth in the vertical direction. We adopt the rectangular grid for convenience. To obtain the details of the sediment motion near the bottom with reasonable efforts of computation, a varying grid size is considered; that is, we let the vertical grid size be equivalent to the sediment diameter nearby the bottom and increase linearly in the upward direction. The horizontal grid size is also set to be fine near the ripple crest and increases toward the ripple trough. Eventually, the vertical grid size varies from a minimum of 0.1 mm near the bottom to a maximum of 5.0 mm at the top, and the horizontal grid size varies from a minimum of 0.7 mm over the ripple crest to a maximum of 5.0 mm over the ripple trough. To ensure a steady-state solution, the computation for each case is continued for 10 to 20 wave periods until the relative difference of the concentration and velocity of each phase between two adjacent cycles reached a prescribed level of 10−4. The last cycle of the computation is then taken as the final result. It is found that the relative difference between the corresponding points over the two ripples included in the computational domain is also less than a level of 10−4 as long as the prescribed accuracy between the two adjacent cycles is reached. We then take the average of the computational results over the two ripples to represent the fluid and the sediment motion in the present study. The computational time required for a typical case is about 15 to 20 h by a desktop computer with 3.5-GHz CPU.
b. Flow of fluid phase
We first consider Fredsoe et al.’s (1999) case, which concerns clear water flow over a solid bed of fixed ripples. The free-stream velocity corresponds to a second-order Stokes wave-induced bottom flow and is given by
Comparison of the horizontal velocity at different cross sections over vortex ripples computed by the two-phase flow model with Fredsoe et al.’s (1999) experimental measurements. Variation of the velocity profile within a period is divided into two columns for clarity.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0031.1
To study the oscillatory flow over sandy ripples, we applied our numerical model to the experimental case of Sleath (2000). In this case, the bed is deformable while the free streamflow is sinusoidal. Figure 3 is a comparison of the computed vertical variation of the horizontal velocity over the crest (x = λ/2), the trough (x = 0), and a midway point
Comparison of the horizontal velocity at different cross sections over vortex ripples computed by the two-phase flow model with Sleath’s (2000) experimental measurements.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0031.1
c. Periodic vortex shedding due to ripples
Figure 4 presents the computed flow field within a wave period of Nakato et al.’s (1977) experimental case. A vortex formation–ejection process, as reported to a different extent in the previous studies (Ranasoma and Sleath 1992; Hansen et al. 1994; Malarkey and Davies 2002; Zedler and Street 2006), is clearly shown. Shortly after t/T = 0 when the free-stream velocity becomes positive, a clockwise vortex starts to form near the ripple crest next to the right-hand (lee) side below the previous anticlockwise vortex (Fig. 4b). As the free streamflow accelerates, the clockwise vortex grows in both size and strength (Figs. 4b–d). When the free streamflow decelerates, a rise of the vortex can be observed, while a strong reverse flow beneath the vortex develops (Figs. 4e–f). The vortex arises slowly at the beginning, but it is quickly ejected to its highest position after t/T = 5/12. As the free streamflow changes its direction at t/T = 1/2, the clockwise vortex is finally swept away while a new anticlockwise vortex starts to form at the left-hand (lee) side below the clockwise vortex (Fig. 4g), and the process repeats itself. The life cycle of vortex and its surrounding flow can be illustrated in Fig. 5.
Computed velocity of Nakato et al.’s (1977) experimental case.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0031.1
Sketch of vortex formation–ejection process.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0031.1
For a quantitative description of the vortex formation–ejection process, we define the center of the vortex at a moment as the point where the magnitude of velocity
Trajectory of vortex center and temporal variation of vortex scale and strength.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0031.1
The vortex dynamics may also be discussed by means of the swirling strength defined by Chong et al. (1990) and Zhou et al. (1999). In a two-dimensional flow, the swirling strength can be expressed by
Computed nondimensional swirling strength
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0031.1
For a direct comparison of the numerical results on the periodic development and decay of vortex with experiments, case Mr5b63 of van der Werf et al. (2007, 2008) is also studied. The free-stream velocity in this case is
Comparison of (left) computed nondimensional vorticity
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0031.1
d. Process of sediment entrainment
Figure 9 shows the intrawave variation of the sediment volumetric concentration obtained with the present numerical model for Nakato et al.’s (1977) experimental case. The contours in the figure correspond to constant values of the volumetric concentration. Note that the layer with concentration varying from 0.05 to its maximum value of 0.6 is too thin to be distinguished. The numerical results indicate that the process of sediment motion over a ripple bed is very different from the typical situation over a flat bottom, which is simply characterized by an intensive entrainment of sediment under the peak velocity of the oscillatory flow and an evident settling of sediment when the free-stream velocity changes its direction. Because the ripples play a dominate role, a clear process of sediment trapping and lifting can be observed in Fig. 9. Shortly after t/T = 0, when a vortex starts to form at the right-hand (lee) side of the ripple crest, a relatively high concentration region also appears at this position (Fig. 9b). Then, the high concentration region gradually expands as the scale and strength of the vortex increases since sediments are heavily entrained from the bed and are trapped in the vortex (Figs. 9c–f). Finally, the sediment cloud is ejected to a higher position over the crest and is swept toward the left-hand side of the ripple crest after the flow reverses (Fig. 9g). At the same time when the flow reverses, a relatively high concentration region starts to appear at the left-hand (lee) side of the ripple crest (Fig. 9i), and the process repeats itself. The sediment trapping–lifting process can also be sketched in Fig. 10. It is closely related to the vortex formation–ejection process depicted in Figs. 4, 5, 6, 7, and 8 and is thus a unique phenomenon over sandy ripples.
Computed sediment volumetric concentration of Nakato et al.’s (1977) experimental case.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0031.1
Sketch of sediment trapping–lifting process.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0031.1
Figure 11 compares the computed temporal variation of the sediment volumetric concentration with the measurement of Nakato et al. (1977) at six points as indicated in Fig. 12 by PC1, PC2, PC3, PT1, PT2, and PT3. Among these points, PC1, PC2, and PC3 are located over the ripple crest, while PT1, PT2, and PT3 are located over the ripple trough. Reasonable agreement between the computational results and experimental data is obviously obtained. As it is expected, the magnitude of volumetric concentration reduces with height in the vertical direction from PC1 to PC3 and from PT1 to PT3. Both the computation and measurement in Fig. 11 show that the period of variation of the sediment concentration is half of the period of the free-stream oscillatory flow. This, however, is only true right over the ripple crest or trough because of the symmetry of the ripple geometry.
Comparison of computed sediment volumetric concentration with Nakato et al.’s (1977) experimental measurement.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0031.1
Concentration measurement points in Nakato et al.’s (1977) experimental case.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0031.1
It is of interest to note that the temporal variation of the sediment volumetric concentration has four peaks within a period of the oscillatory flow as shown in Fig. 11. Referring to Figs. 4, 7, and 9, we understand that two of these peaks correspond to the ejection of the vortex, and the other two correspond to sweeping of the ejected vortex by the reversed flow.
For a further verification, the numerical results on the intrawave variation of the sediment volumetric concentration of van der Werf et al.’s (2007, 2008) case Mr5b63 are compared with the measurements in Fig. 13. The agreement between the present computation and van der Werf et al.’s (2007, 2008) experiment is shown to be generally good. Some phase difference may be observed between the computed and measured results at t/T = 2/3 for y/Δ > 1.5. This difference, however, may not be a problem of the numerical model because it was also reported in van der Werf et al.’s (2008) study. It is necessary to emphasize that the intrawave variation of the sediment cloud in Fig. 13 has a very close correlation with the vorticity contours for the same problem as shown in Fig. 8.
Comparison of the (left) computed sediment volumetric concentration with (right) van der Werf et al.’s (2007, 2008) experimental measurement.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0031.1
e. Net sediment transport
To show the feasibility of our numerical model for the prediction of the regional topography change because of sediment motion, we pay some attention to the vertical distribution of the sediment volumetric concentration averaged over a period of the free streamflow and over a wavelength of the ripple as well. Figure 14 shows the comparison of the computed results with the experimental data of Williams et al. (1998), Villaret and Perrier (1992), and Steetzel (1984). In all cases, the overall agreement between the present computation and experiment is reasonably good. It is worthwhile to point out that the diameter of sediment particles used in these experiments varies from 0.09 to 0.27 mm, and the amplitude of the free-stream velocity varies from 0.2 to 0.75 m s−1. As a result, the scale of the vortex ripples covers a wide range. In addition, Steetzel’s (1984) experiments include four cases, that is, cases T226, T235, T260, and T264, of which the amplitude of the free-stream velocity differs while other conditions are fixed. It may also be necessary to mention that cases T226 and T264 of Steetzel’s (1984) experiments cannot be regarded as vortex ripple cases based on the criterion of Thorne et al. (2009). However, the sediment concentration is still reasonably predicted by the present model as long as the ripples are stable and the ripple profiles used in the computations are accurate enough.
Comparison of computed, mean, sediment volumetric concentration with experiments.
Citation: Journal of Physical Oceanography 45, 1; 10.1175/JPO-D-14-0031.1
4. Conclusions
This study uses a two-phase turbulent flow model to study the sediment motion over equilibrium sandy ripples with regular geometry. The numerical results show good agreement with the data of the intrawave fluid and sediment motion obtained by different investigators either in oscillatory flow tunnels or in wave flumes. For both the clear water flow over fixed solid ripples and sediment-laden flow over sandy ripples, the wavy variation of the horizontal velocity in the vertical direction, which has an equivalent scale to the ripple height and is related to the presence of the vortex that always accompanies the oscillatory flow over a ripple, has been confirmed. The dynamic process of the vortex generation, growth, ejection, and finally being swept over a sandy ripple bed can also be well represented by the numerical model. The sediment trapping–lifting process associated with the formation–ejection process of vortices has also been demonstrated. The temporal variation of the sediment volumetric concentration at a fixed point is shown to have four peaks within a period of the oscillatory flow, of which two correspond to the ejection of a highly concentrated sediment cloud and the others correspond to sweeping of the ejected sediment cloud as the flow reverses.
Acknowledgments
This research is supported by Natural Science Foundation of China (NSFC) under Grant 11472156 and by State Key Laboratory of Hydroscience and Engineering, China, under Grant 2014-KY-02.
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