Abstract A three-dimensional mixture theory model (SedMix3D) was used to simulate the flow and sediment transport over vortex ripples for scaled laboratory conditions. SedMix3D treats the fluid-sediment mixture as a continuum of varying density and viscosity with the concentration of sediment and velocity of the mixture calculated using a sediment flux equation coupled to the Navier-Stokes equations for the mixture. Mixture theory allows the model to simulate the three-dimensional flow and sediment concentration within and above an evolving sediment bed. Grid spacing was on the order of a sediment grain diameter and time steps were O(10 -5 s). The simulation was forced with a time series of free-stream velocity measured in a free-surface laboratory flume. Spatial variations in the simulated suspended sediment concentration were primarily associated with the non-uniform generation of vortex structures over the ripple flanks. The suspended sediment was initially picked up in regions of high vorticity, and then caused a damping of the vorticity while being advected through the water column

A M Penko

J Calantoni

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Marine and River Dune Dynamics – MARID IV – 15 & 16 April 2013 - Bruges, Belgium 
 209 
Three-dimensional spatial variations of suspended sediment 
concentration over vortex ripples 
A. M. Penko(1) and  J. Calantoni(1) 
 
1. Naval Research Laboratory, Stennis Space Center, MS, USA – Code7434@nrlssc.navy.mil 
 
 
Abstract 
A three-dimensional mixture theory model (SedMix3D) was used to simulate the flow and sediment transport over 
vortex ripples for scaled laboratory conditions. SedMix3D treats the fluid-sediment mixture as a continuum of 
varying density and viscosity with the concentration of sediment and velocity of the mixture calculated using a 
sediment flux equation coupled to the Navier-Stokes equations for the mixture. Mixture theory allows the model to 
simulate the three-dimensional flow and sediment cocentration within and above an evolving sediment bed. Grid 
spacing was on the order of a sediment grain diameter and time steps were O(10-5 s). The simulation was forced 
with a time series of free-stream velocity measured in a free-surface laboratory flume. Spatial variations in the 
simulated suspended sediment concentration were primarily associated with the non-uniform generation of vortex 
structures over the ripple flanks. The suspended sediment was initially picked up in regions of high vorticity, and 
then caused a damping of the vorticity while being advected through the water column. 
 
1. INTRODUCTION 
Bedforms on the seafloor in the neashore coastal 
and continental shelf regions impact erosion and 
deposition, waves and currents, and coastal 
infrastructure. Ripples are ubiquitous in sandy 
coastal regions, generally covering the seafloor in 
depths up to 20 m during temperate weather (i.e, 
during non-extreme wave events). Ripples 
drastically affect the total sediment transport and 
wave energy dissipation; however, it is difficult to 
quantify their total effect due to the complexity 
that ripples bring to the hydrodynamics of bottom 
boundary layer flow.   
 
The hydrodynamics above sand ripples are 
dominated by the coherent vortices formed on the 
ripple flanks and ejected into the water column at 
flow reversal.  This process erodes sediment off 
the surface of the ripple and suspends it into the 
water column where it is then advected with the 
flow (Thorne et al., 2003, van der Werf et al., 
2006).  
 
 
 
 
Recent advances in high performance computing 
hardware and software have allowed us to 
implement three-dimensional, high-resolution 
simulations to study small-scale bedform 
dynamics. Experimental (laboratory and field) and 
complex numerical models are all necessary to 
better understand and quantify the effect of ripples 
on sediment transport and the turbulent dynamics 
of the wave boundary layer. However, most 
research examining sediment transport over three-
dimensional dynamic beds has been limited to 
laboratory and field studies. Recent advances have 
allowed for the measurement of vortex dynamics 
and sediment transport in the laboratory (van der 
Werf, 2006) and field (Traykovski, 1999; Hurther 
and Thorne, 2011).  Few three-dimensional, high-
resolution models exist to study small-scale 
sediment dynamics in detail.  Here we use a 
mixture theory model to study the dynamic 
coupling of vorticity and suspended sediment and 
their spatial variability over dynamic sandy rippled 
beds.  
 
 
Marine and River Dune Dynamics – MARID IV – 15 & 16 April 2013 - Bruges, Belgium 
 210 
2. METHODOLOGY 
2.1. Numerical model 
Using the three-dimensional Navier-Stokes solver, 
SedMix3D, we performed a high spatial and 
temporal resolution simulation of oscillatory flow 
over a sandy rippled bed for 18 wave periods.  
 
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 
(Penko et al., 2011). The mixture is treated as a 
continuum of varying density and viscosity with 
the concentration of sediment and velocity of the 
mixture calculated with a sediment flux equation 
coupled with the Navier-Stokes equations for the 
mixture. The mixture continuity equation was 
derived by combining the fluid and sediment phase 
continuity equations, 
∂ρ
∂t
+ ∇ ⋅ (ρu) = 0
 (1) 
where u is the mixture velocity and ρ is the 
mixture density, 
ρ = φρs + (1−φ)ρ f  (2) 
where  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, 
∂ρu
∂t
+ ρu⋅ ∇u = −∇P+ ∇ ⋅ (µ∇u)+ F − ρg− Sbu
  (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/s2), and Sb is the particle pressure 
parameterization. 
 
SedMix3D employs a modified Eilers equation 
(Eilers, 1941) to represent effective viscosity, µ, 
here scaled by the pure water viscosity, µf, 
 
µ
µ f
= 1+ 0.5[µ]φ
1−φ φm






2
 (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 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,  
∂φ
∂t
+ u ⋅ ∇φ = D∇2φ − ∂φWt
∂z  (5) 
where Wt is the concentration dependent settling 
velocity (Richardson and Zaki, 1954) and Rep is 
the particle Reynolds number. The shear-induced 
diffusion of sediment, D, is a function of grain 
size, volumetric concentration, and mixture 
stresses (Leighton and Acrivos, 1986), 
D = 1
4
d2β(φ) ∇u = 0
 (6) 
and where d is the grain size diameter, α is an 
empirical constant, and β is an empirical function 
of concentration. Previously, SedMix3D 
demonstrated good agreement with detailed 
laboratory observations using particle image 
velocimetry to measure vortex dynamics over 
rippled beds with uniform grain size (Penko et al., 
2013). Here, we initialized the numerical 
simulation with the measured bed profile and 
imposed hydrodynamic conditions. 
 
2.2. Simulation set-up 
The simulation had a domain size of 19.0 cm x 2.4 
cm x 14.2 cm with approximately 1.5 million grid 
points and ran on 96 processers on a Department 
of Defense high performance computer. The 36 s 
real-time simulation took approximately 170 wall-
clock hours to complete.  The laboratory data used 
to initialize and force the model simulation were 
collected using Particle Image Velocimetry (PIV) 
in a free-surface wave flume with a wave generator 
that produced regular sinusoidal waves 5 cm in 
height with 2 s periods over an artificial sediment 
bed.  
 
Marine and River Dune Dynamics – MARID IV – 15 & 16 April 2013 - Bruges, Belgium 
 211 
 
 
Figure 1. Free stream velocity time series recorded by 
an ADV in a free surface flume that was used as forcing 
in the simulations. 
 
 
Two–dimensional ripples persisted uring the 
experiment with wavelengths of ~5 cm and heights 
of ~2 cm.  The simulation was forced with a time 
series of free stream velocity recorded with an 
ADV (Figure 1) for a total of 18 wave periods (36 
s).  The first 9 wave periods of the simulation were 
discarded for model spin-up and not included in 
the data analysis.   
 
The initial bed consisted of four ripples (height~2 
cm, length~5 cm) extracted from profiles 
measured in the laboratory (Figure 2).  The initial 
bed profile was determined from the measured 
laboratory bed profile at the beginning of the 
experiment and then evolved with the oscillatory 
flow.  
 
 
 
Figure 2. The initial simulation bed profile in the 
simulation coordinate system.  
 
The bed profile reached equilibrium in the 
simulation by the seventh wave period and the 
ripples remained in equilibrium for the remainder 
of the simulation. Due to a lengthy model spin-up 
time, only the last 9 wave periods were used in the 
analysis. The model output of velocity and 
concentration was ensemble-averaged over the 9 
wave periods and the results are presented in the 
following section.  
 
 
3. RESULTS 
SedMix3D outputs the three-component velocity 
vector and volumetric concentration of sediment (0 
< ϕ < 0.63) at every grid point within the domain. 
The mixture theory approach is advantageous for 
examining sand ripple dynamics because the 
model simulates the velocity and concentration 
everywhere, from the top of the boundary layer to 
the packed sand bed, including the highly 
concentrated bedload layer that is very difficult to 
measure in situ and arbitrary in thickness. One can 
also directly examine the dynamically coupled 
effect of the suspended sediment on the vorticity 
and conversely, the vorticity on the transport of 
sediment. 
 
A plan view (x-y) of the vorticity magnitude and 
the suspended sediment concentration contour 
levels for ϕ = 0.01 (blue solid line) and ϕ = 0.1 
(magenta dashed line) at an elevation of z = 2.86 
cm is plotted in Figure 3. The insets in the upper 
left corners denote the phase of the wave for each 
subfigure. Note the significant variability of the 
vorticity and bursts of suspended sediment in the 
cross-flow (y) direction (Figure 3a). Immediately 
before flow reversal, sediment was picked up in 
the regions of strong vorticity (Figure 3a). Once 
suspended in the water column, the sediment 
began to damp the vorticity, causing pockets of 
decreased vorticity and consequently causing a 
spatial variability in the vortex dissipation (Figure 
3b). As the flow began to accelerate, the model 
produced strong and spatially uniform vortex 
structures over ripple crests (Figure 3c). However, 
over the troughs, where the flow decelerated, 
vortices broke-up and dissipated at random, 
producing more three-dimensional variability than 
in the accelerated flow over the crests (Figure 
Marine and River Dune Dynamics – MARID IV – 15 & 16 April 2013 - Bruges, Belgium 
 212 
3a,b). Also just after flow reversal, the suspended 
sediment concentrated in the leading (left) edge of 
the ejected vortex, damping out most of the 
vorticity before maximum flow (Figure 3b,c). 
 
 
 
 
| Ω
 |  (1/s)
y 
(c
m
)
x (cm)
a
2 4 6 8 10 12 14 16 18
0.5
1
1.5
2
0
10
20
30
40
φ = 0.01
φ = 0.1
 
 
 
| Ω
 |  (1/s)
y 
(c
m
)
x (cm)
b
2 4 6 8 10 12 14 16 18
0.5
1
1.5
2
0
10
20
30
40
φ = 0.01
φ = 0.1
 
 
 
| Ω
 |  (1/s)
y 
(c
m
)
x (cm)
c
2 4 6 8 10 12 14 16 18
0.5
1
1.5
2
0
10
20
30
40
φ = 0.01
φ = 0.1
 
 
 
| Ω
 |  (1/s)
y 
(c
m
)
x (cm)
d
2 4 6 8 10 12 14 16 18
0.5
1
1.5
2
0
10
20
30
40
φ = 0.01
φ = 0.1
 
 
Figure 3. Plan view of the ensemble-averaged vorticity 
magnitude (filled black and white contours) and 
suspended sediment concentration (blue solid and 
magenta dashed contour lines) at z = 2.86 cm. Insets 
show the ensemble-averaged free stream velocity (black 
line) and the position of the wave for each figure (r d 
dot).  
 
 
The simulation also showed a suspended sediment 
burst on the flank of the ripples after maximum 
flow, despite relatively low vorticity, which is a 
behavior we have observed in laboratory 
experiments of oscillatory flow over sand ripples 
(Figure 3d).  
 
The cross-flow variability is also evident in a plot 
of the three-dimensional swirling strength (Figure 
4). Swirling strength is a scalar quantity of the 
vorticity that identifies the closed rotational vortex 
structures in turbulent flow (Zhou et al., 1999). 
The method is effective in determining the location 
of vortex cores by excluding the voriticity due to 
boundary-generated shear, but does not identify 
the rotational direction. At flow reversal (Figure 
4a), the vortices over the troughs were very three-
dimensional with high spatial variability, 
especially in the cross-flow direction. However, 
where the flow accelerated over the ripple crests, 
the vortices were more uniform, with very little 
variability in the cross-flow direction. After 
maximum flow (Figure 4b), the vorticies are 
stretched and dissipated by the strong flow. The 
vorticies dissipate randomly, especially over the 
ripple troughs.  
 
  
4. CONCLUSIONS 
To the authors’ knowledge, no one previously has 
been able to simulate the three-dimensionality of 
small-scale boundary layer processes in this 
amount of detail. SedMix3D allows for detailed 
examinations of the three-dimensional boundary 
layer processes occurring due to oscillatory flow 
over sand ripples. Here, we present three-
dimensional model results of vorticity and 
suspended sediment concentrations throughout the 
phase of a wave. Over ripple crests and during the 
accelerated portion of the flow the model produced 
strong uniform vortex structures. However, over 
the troughs, where the flow decelerated, vortices 
broke-up and dissipated at random, producing 
more three-dimensional variability than in the 
accelerated flow over the crests. Spatial variations 
in the simulated suspended sediment concentration 
were primarily associated with the non-uniform 
generation of vortex structures over the ripple 
flanks and non-uniform vortex dissipation as flow 
accelerated. The suspended sediment was initially 
picked up in regions of high vorticity, and then 
damped the vorticity while being advected through 
the water column. The simulation also showed 
bursts of suspended sediment concentration when 
vorticity was low, similar to observations of 
laboratory experiments. 
 
 
5. ACKNOWLEDGMENT 
AMP was supported by the Jerome and Isabella 
Karle Distinguished Scholar Fellowship Program 
at the Naval Research Laboratory. JC was 
supported under base funding to the Naval 
Research Laboratory from the Office of Naval 
Marine and River Dune Dynamics – MARID IV – 15 & 16 April 2013 - Bruges, Belgium 
 213 
Research. 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. 
 
 
6. REFERENCES 
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viscous substances as function of concentration. 
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bed load sediment transport processes above a 
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Leighton, D. & Acrivos, A. 1986. Viscous 
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Nir, A. & Acrivos A. 1990. Sedimentation and sediment 
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Figure 4. The plot shows the ensemble-averaged isosurface of the three-dimensional swirling strength at two phases 
of the wave. 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.  
 
 
 
 


Three-dimensional spatial variations of suspended sediment concentration over vortex ripples

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Three-dimensional spatial variations of suspended sediment concentration over vortex ripples

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