Christopher S. Thaxton1 and Joseph Calantoni2 ... The DPM (Drake and Calantoni ... tangential stiffness and a frictional force (Drake and Walton 1995),.
VERTICAL SORTING AND PREFERENTIAL TRANSPORT IN SHEET FLOW WITH BIMODAL SIZE DISTRIBUTIONS OF SEDIMENT Christopher S. Thaxton 1 and Joseph Calantoni2 The vertical sorting of grains that occurs during the process of sheet flow largely has been ignored in parameterized models for sediment transport. Here we performed computer simulations of sheet flow transport with bimodal distributions of coarse grains under forcing from idealized nearshore waves. The results demonstrate that the mobile bed rapidly experiences vertically segregation of grains by size, where the larger grains migrate to the top of the mobile bed, while the smaller grains percolate to the bottom. Consequently, the larger grains transport at a rate disproportionately higher than their mass fraction compared to the smaller grains.
INTRODUCTION
The sorting of grains by size has been observed in nature, in the laboratory (e.g., Rosato et al. 1987; Savage and Lun 1988), and in numerical simulations (e.g. Thaxton et al. 2001). When vibrations, oscillations, or steady currents drive a bed of mixed grain sizes, the larger grains tend to migrate to the top of the bed (e.g., the “Brazil Nut Effect” – Rosato et al. 1987). The behavior leads to a preferential transport of the larger grains when driven by a net shearing force, such as nearshore ocean waves. The spatial distribution of sediments, bedform evolution, and ultimately large-scale bathymetric changes (e.g., sandbar migration) may be affected by the processes of granular sorting. METHODS
Using an existing discrete particle model (DPM), we simulate the physics of sheet flow sediment transport in the wave bottom boundary layer (WBBL). The simulations attempt to emulate the physics of the sea floor, at the fluid-sediment interface, in shallow water under wave forcing. The DPM (Drake and Calantoni 2001) is a rigorous two-phase flow solution for the WBBL that couples a threedimensional Lagrangian particle model for sediments to a simple onedimensional Eulerian fluid model. Two-way coupling along the flow direction attempts to strictly enforce Newton’s Third Law, so that fluid exerts force on sediment particles, and sediment particles collectively exert force back on the fluid. Sediment Phase
Individual sand grains are modeled with spherical particles. Arguably, the sediment phase of the model performs a direct numerical simulation. We
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1Department of Physics and Astronomy, Appalachian State University, Boone NC, 28608, USA 2 Code 7440.3, Naval Research Laboratory, Stennis Space Center MS, 39525, USA
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2 compute the three-dimensional deterministic motions of every single grain in the model while resolving the fundamental length scale, that of an individual particle. Particles interact with other particles through forces generated at point contacts between spheres. The contact force model is essentially a modified Walton model (Walton and Braun 1986) where normal and tangential forces generated at the contact point are modeled with springs and friction, respectively. For the normal loading of two particles the repulsive force is modeled as,
FN = k1a ,
(1)
where FN is the magnitude of the normal force at the contact point, k1 is the loading spring constant, and a is the overlap distance at the contact point. For unloading the force is given as,
(
)
FN = max k2 a a0 , k3 a ,
(2)
where k2 is the unloading spring constant, a0 is the value of a where the loading curve goes to zero, and k3 is a third spring whose value is small ( 0.01 k2 ). The addition of the spring, k3 , prevents permanent deformation in a static pile of grains that may occur since the unloading curve may go to zero while two particles are still in contact. An effective coefficient of restitution is computed as, 1
k1 2 e= . k2
(3)
For tangential forces at the contact point we use the minimum magnitude of a tangential stiffness and a frictional force (Drake and Walton 1995),
FT = min kt ds , µ FN ,
(4)
where kt is the tangential spring constant, ds is the tangential displacement at the contact point, and µ is the coefficient of friction. No distinction is made here between static and kinetic friction. Instead, grains resist tangential motion at the contact point with the tangential stiffness, until the force from friction is exceeded and then the contact is allowed to break free and slide with friction. The fluid forces acting on the particles are found in the equation for translational motion of a single spherical grain (e.g., Madsen 1991),
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sV
dus 1 = s Vg + AC D* u us u us 2 dt , Du dus Du + VC M V + F + dt
Dt z= Dt
(
)
(
)
(5)
where s and are the sediment and fluid density, respectively, us and u are the sediment and fluid velocity, respectively, V is the volume of a spherical particle, A is the cross-sectional area, and g is the acceleration due to gravity. All derivatives are evaluated at the particle center unless otherwise specified. The first term on the right-hand-side (RHS) represents the particle buoyancy, which is effectively treated as a reduced mass of the particle. The second term on the RHS is the fluid drag force determined from the empirical drag law for a sphere. Here the drag coefficient is represented with the following expression,
(
1
)
C D* = c* 24Res1 + 4Res 2 + 0.4 ,
(6)
where Res is the relative particle Reynolds number and c* is a correction based on local particle concentration (by volume), c (e.g., Richardson and Zaki 1954), 5
1 2 c = 1 c c2 . 3 *
(6)
The third term on the RHS represents the added mass force with the coefficient of added mass, C M = 0.5 (Batchelor 1967). The fourth term on the RHS is the force from the horizontal pressure gradient, which is assumed equal to the free stream fluid acceleration and acts directly on the particles. The fifth term on the RHS represents the sum of intergranular forces, described above. Note particles do exhibit rotational motion that results solely from tangential forces generated at contact points between particle pairs. Fluid Phase
The fluid phase of the DPM is modeled on a one-dimensional Eulerian grid that produces a velocity profile for the flow in the WBBL. Fluid motion is constrained parallel to the bed. The turbulence closure uses a simple eddy viscosity that is determined from a Prandtl mixing length, such that the eddy viscosity between adjacent grid cells is given by,
T = 2 l 2
u , z
(6)
where = 0.4 is the von Karman constant, and l is the distance from the origin of the mixing length profile. Choosing the vertical position for the origin of the mixing length profile is problematic. We select the origin for the mixing length
4 profile to coincide with the location in the bed where the particle concentration, c = 0.4 . As a result, the location will shift downward as sediment is mobilized and then back upward as sediment motion ceases. The eddy viscosity defined in (6) is used to calculate the Reynolds shear stress between adjacent grid cells with
m = T
u . z
(7)
The driving force for both fluid and particles in the simulations comes from the applied horizontal pressure gradient. The forcing from waves passing above is assumed to exert a time varying but spatially constant horizontal pressure gradient across the entire simulation domain that exerts body forces on both fluid and the embedded particles. It is assumed that the forcing from the horizontal pressure gradient may be quantified by, and is equivalent to the free stream fluid acceleration. The precise form for the time series of the free stream fluid acceleration used in this study is given in the next section. RESULTS
Simulations using a bimodal size distribution of coarse grains were performed with the DPM. From here on, we will refer to the two size fractions as large and small, respectively. The large particles of the bimodal distribution had a fixed diameter, DL = 0.0015 m , and the small particles had a fixed diameter, DS = 0.001 m . All particles had a fixed density, s = 2,650 kg m -3 , equivalent to quartz. The bed composition, C = M L M S , is defined as the ratio of the mass of large grains, M L , to the mass of the small grains, M S , contained in the simulation domain. The bed composition, C , was varied systematically over 9 values for M L M S from 10 90 , 20 80 , 30 70 , up to 90 10 , where the numbers in the ratio represent the percentage of mass contained in each size fraction. The total mass of sediment was held approximately constant so that the top of the at-rest bed for all 9 configurations was typically around z = 0.01 m , where z = 0 is the location of the hard bottom in the simulation coordinate system. Care was taken when constructing the initial bed configurations so that, as much as possible, the grains were well mixed prior to the initiation of the simulations (Figure 1). To simulate waves, the time varying forcing from the horizontal pressure gradient acting on the simulation volume (along the x direction) was given the functional form, 4
F(t) ua i=0
(
)
(
)
1 i + 1 sin i + 1 t + i , 2i
(6)
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Figure 1. Snapshots of the initial bed configuration with C = 1. The grains are initially well mixed. The upper image shows a two-dimensional plan view from the top of the bed. The lower image shows a three-dimensional view from the side of the domain. Periodic boundary conditions are used in the x and y directions. The hard bottom at z = 0 has several fixed particles embedded in the plane to prevent bulk sliding of the entire granular assemblage. The domain dimensions are approximately 0.02 m in x and 0.01 m in y.
where = 2 T with T = 6 s . The value of ua was varied to produce monochromatic waves with three different maximum free-stream velocity magnitudes of u = 0.85 m s-1 , u = 1.1 m s-1 , and u = 1.35 m s-1 . Likewise,
6 the value of was varied from 0, 4 , and 2 , to produce waves with three different values of the velocity skewness (e.g., Elgar et al. 1988) equal to 1.2, 0.8, and 0, respectively, for a total of 9 different idealized wave conditions. Each of the 9 unique bed configurations was used to simulate sheet flow transport with the 9 idealized nearshore wave conditions for a total of 81 different simulations. Each simulation condition was run out for a minimum of six consecutive wave periods or 36 seconds. Previous work has demonstrated the ability of the model used here to accurately predict time-averaged transport rates for grain size distributions and maximum free stream velocities in the range considered in this study (Drake and Calantoni, 2001; Calantoni et al., 2004; Calantoni and Puleo, 2006).
Figure 2. Vertical profiles of particle concentration by volume for the bed configuration with C = 1, under forcing from the wave condition with u = 1.1 m s-1 and = 2 . The heavy black line denotes large particle concentration, the thin black line denotes small particle concentration, and the gray line is the total particle concentration.
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Figure 3. Time averaged flux per wave period for the bed configuration with C = 1, under forcing from the wave condition having u = 1.1 m s-1 with (a, d) = 0 , (b, e)
= 4 , and (c, f) = 2 . Time averaged flux is calculated both along the flow direction, x (a-c), and in the vertical direction, z (d-f). The open circles represent large particle flux, the closed circles represent small particle flux, and the squares represent the total flux.
Results from the majority of simulations demonstrate vertical sorting and preferential transport of grains by size. Initially mixed beds of sediment sort such that the large grains migrate to the top of the active layer and the small grains move to the bottom of the active layer. After just one to two consecutive wave periods the two size fractions of grains become fairly well sorted as illustrated by plots of depth versus particle concentration (Figure 2). These
8 results are robust for the full range of wave intensities and wave shapes simulated. The vertical sorting that occurs between large and small grains impacts the transport rates of the different size fractions in the bimodal distribution. Shown in Figure 3 are both horizontal (along the flow direction) and vertical transport rates for same initial bed configuration from Figure 1 under forcing from three different wave conditions. In all cases, the large grain transport along the flow direction increase as the bed sorts vertically, while the small grain transport tends to decrease. Also, the rate of vertical sorting is shown to be most rapid during the first wave period, with large grains migrating to the top of the bed and small grains percolating down into the mobile sediment layer, evidenced by positive vertical transport for large grains, and negative vertical transport for small grains. DISCUSSION
The vertical sorting of grains in the bimodal distributions is rapid for all cases simulated. We observe large grains migrating to the top of the mobile bed layer and smaller grains percolating down to the bottom of the mobile layer. Consider the initial snapshot from the side of the simulation domain shown in Figure 1. Here the bed composition, C = 1 , which has equal percentages of large and small grains by mass. The vertical profile at t = 0 s shown in Figure 2 is the quantification of the visual images shown in Figure 1. Notice that at t = 0 s , there is a thin veneer of smaller particles on the top of the bed. At t = 6 s , after just one complete wave period, the condition has completely reversed, and the bed is covered with larger particles. The rate of sorting is most rapid during the first wave period when the bed is initially well mixed and decreases subsequently. The rate of vertical sorting can be quantified by looking at the time averaged vertical transport rates for large and small grains shown in the right column of Figure 3 (d-f). The upper two sets of panels (a, d, and b, e) exhibit very similar behavior. There is rapid initial sorting and then they appear to be approaching a steady state by the end of 6 wave periods. However, the bottom panels (c, f) trend differently. For this wave condition, there is a strong spike in the applied horizontal pressure gradient. We see that while the vertical sorting rates of the smaller grains has trended towards zero, the larger grains are still being squeezed from the bottom of the bed to the surface. The wave condition in these lower panels (c, f) is the same condition shown in Figure 2. In Figure 2, it is fairly clear that the bottom of the mobile bed layer is located at approximately z = 0.004 m throughout the 6 wave periods. However, notice that from t = 12 s to t = 36 s there appears to be measurable sorting occurring beneath the bottom of the mobile layer. The contribution to the net transport from grains located below the level z = 0.004 m is nominally zero, but at flow reversal, when the forcing from the horizontal pressure gradient is strongest the lower immobile bed is agitated just enough to squeeze out larger grains and capture smaller ones.
9 The processes operating beneath the mobile bed layer are analogous to granular sorting that occurs in a packed, vibrated bed. CONCLUSIONS
We demonstrate with DPM simulations of sheet flow transport using bimodal distributions of coarse sized sediments under waves that the processes of vertical sorting of grains by size are critical to understanding and quantifying the partial transport rates of each size fraction. An initially well mixed bed sorts very rapidly when subjected to strong shear stresses that mobilize layers at least a couple particle diameters deep. Larger grains migrate to the top of the mobile bed, while the smaller grains percolate to the bottom. As a result of the vertical dynamics the horizontal transport is influenced in a way that is typically not addressed by parameterized models for sediment transport. ACKNOWLEDGMENTS
Calantoni was supported under base funding to the Naval Research Laboratory from the Office of Naval Research (PE# 61153N). This work was supported in part by a grant of computer time from the DoD High Performance Computing Modernization Program at the ERDC MSRC. REFERENCES
Batchelor, G.K. 1967. An introduction to fluid dynamics, Cambridge University Press, Cambridge, 615 pp. Calantoni, J., Holland, K.T., and T.G. Drake. 2004. Modelling sheet-flow sediment transport in wave-bottom boundary layers using discrete-element modelling, Philosophical Transactions of the Royal Society of London A, 362, 1987–2001. Calantoni, J., and J.A. Puleo. 2006. Role of pressure gradients in sheet flow of coarse sediments under sawtooth waves, Journal of Geophysical Research, 111, C01010, doi:10.1029/2005JC002875. Drake, T.G., and O.R. Walton. 1995. Comparison of experimental and simulated grain flows, Journal of Applied Mechanics, 62(1), 131-135. Drake, T.G., and J. Calantoni. 2001. Discrete particle model for sheet flow sediment transport in the nearshore, Journal of Geophysical Research, 106(C9), 19,859-19,868. Elgar, S., R.T. Guza, and M.H. Freilich (1988), Eulerian Measurements of Horizontal Accelerations in Shoaling Gravity-Waves, Journal of Geophysical Research, 93(C8), 9261-9269. Madsen, O.S. 1991. Mechanics of cohesionless sediment transport in coastal waters, Coastal Sediments, 1, 15-27. Richardson, J.F., and W.N. Zaki. 1954. Sedimentation and fluidisation, Transactions of IChE, 32, 35-53. Rosato, A., K.J. Strandburg, F. Prinz, and R.H. Swendsen. 1987. Why the Brazil nuts are on top: size segregation of particulate matter by shaking, Physical Review Letters, 58(10), 1038-1040.
10 Savage, S.B. and C.K.K. Lun, 1988. Particle size segregation in inclined chute flow of dry cohesionless granular solids, Journal of Fluid Mechanics, 189, 311-335. Thaxton, C.S., J. Calantoni, and T.G. Drake. 2001. Can a single grain size represent bedload transport in the surf zone?, EOS Transactions AGU, 82(47), Fall Meeting Supplement, F587. Walton, O.R., and R.L. Braun. 1986. Viscosity, granular-temperature, and stress calculations for shearing assemblies of inelastic, frictional disks, Journal of Rheology, 30, 949-980.
11 KEYWORDS – ICCE 2006 VERTICAL SORTING AND PREFERENTIAL TRANSPORT IN SHEET FLOW WITH BIMODAL SIZE DISTRIBUTIONS OF SEDIMENT Christopher S. Thaxton and Joseph Calantoni 1041 Sheet flow Bimodal size distribution Vertical sorting Discrete particle model Segregation Two-phase flow Wave bottom boundary layer Particles
