Abstract—The formation of spatially ordered structures in a suspended sediment under the action oftwo-dimensional standing surface gravity waves is studied experimentally for the first time in a rect-angular vessel oscillating in the vertical direction. The parameters of the structured regions in vesselswith individual vortex ripples and groups of ripples are found for the first and second wave modes.Isolated structured regions of the suspended sediment appear over the bottom topography and graduallyreach the free surface. The corresponding spatial horizontal scales are determined by the sand rippledimensions, while the vertical scale of the clouds increases with time. In all experiments, the structuresformed remained unchanged during the whole interval of the fluid wave motion and disappeared whenthe parametric excitation of the waves stopped.
Investigation of the mechanisms of formation of suspended sand clouds due to the interaction of periodic-wave fluid flows with the sediment is a basic problem of fundamental and applied fluid dynamics [1–3]. Theresults of natural and large-scale laboratory experiments indicate a high spatial and temporal variabilityof the particle concentration over a rippled bottom [4–6]. In this connection, it is interesting to study thesediment resuspension as a dynamic process of the entrainment of bottom particles in the fluid wave flow.Since the coastal oceanic zone and inland water basins are usually characterized by a shaped sandy bottom,the resuspension of sediments is determined by a complex interaction of periodic wave flows with granulardeposits.
In the existing models, the spatial and temporal variability of the particle concentration in the wave flowis described on the basis of either the turbulent-diffusion mechanism [7] or on the convective transport withaccount of vortex structures [3]. Among the few experimental studies, one can distinguish the results of[8, 9] demonstrating a sudden jump of the suspension concentration at a fairly large distance from the vortexripple surface.
The behavior of suspensions and suspended sediments in the bulk flow, entrained in periodic wave mo-tion, is not well studied. Mathematical models of wave propagation in inhomogeneous media, in additionto regularly perturbed functions, contain a rich family of functions characterizing the fine-structure flowcomponents [10], which affect the dissipation and localization of vorticity and transport of the admixture.The experiments also revealed the process of structuring the dispersed phase in initially homogeneous sus-pensions entrained in wave motion over a profiled sandy bottom [11, 12]. However, the question remainsopen about the influence of the particle shape on the effects observed. Aluminum powder particles with theshape of thin plates were used in [11, 12].
So far, the process of formation of a cloud of sand particles due to the interaction of periodic-wave flowswith bottom sediments have not been studied. In this paper, for the first time we investigate the formation
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of spatially ordered regions in the suspended sediment consisting of spherical particles in a field of standingwaves in a rectangular laboratory vessel.
1. FLOW PARAMETERS
The working medium is water or an aqueous suspension, the disperse phase of which consists of particleswith diameter dp and density ρp. The medium is characterized by the density ρs, the kinematic viscosity νs,and the volume concentration of suspended particles C. For small C, the density and viscosity of the sus-pension is close to the corresponding values in the pure liquid: ρs ∼ ρ f , νs ∼ ν f . The governing parametersof the suspension are also the diffusion coefficient κ and the particle settling velocity (hydraulic size) W .
Steady-state two-dimensional standing waves in a narrow rectangular vessel on the free surface of aliquid layer of depth h are characterized by the frequency ω and the magnitude of horizontal oscillationsnear the bottom A = H/sinhkh (L is the vessel length, k = πn/L is the wave number, n is the mode number,and H is the wave height). The wave Reynolds number for the suspension is Res = UbA/νs, where Ub =gHk/(2ω cosh kh), and g is the gravity force acceleration.
The flow is characterized by several length scales, including the thickness of the constant-concentrationlayer, which is a consequence of zero balance between the downward sedimentation and the upward dif-fusion: ly ∼ D/W , where W = gd2(s − 1)/(18ν f ), s = ρs/ρ f . The fluid fluctuations determine two other
scales, namely, the Stoksean viscous scale δ (ν)ω ∼√ν/ω and the diffusion scale δ (κ)
ω ∼√κ/ω [10].The quantitative characteristics of the ripples are their length lr and height hr, as well as the dimension-
less parameters: the steepness sr = hr/lr and the dimensionless amplitude of fluid fluctuations r = A/lr.In describing the ripples, the parameters which take into account the characteristics of the flow and the sed-iment are also used. These are the sediment mobility Ψ = (Aω)2/[4(s − 1)gd] and the Schields coefficientΘ = u2∗/[(s − 1)gd] = 0.5 fwΨ [3]. Here, s = ρs/ρ f and d are the relative density and the average diameterof the bottom particles, respectively; u∗ is the dynamic velocity, and fw is the drag coefficient, which is afunction of A and the roughness scale Z0. The latter is equal to 2.5d for a horizontal layer of sand particleswith diameter d and 3hr − 4hr for the ripples of height hr; for a smooth bottom, Z0 = 0. In the case of lam-inar flow over a rough sandy bottom, we have Θ = Ub
√ων/[(s− 1)gd]. With account of the compression
of streamlines over the vortex ripple crests, we have [3]: Θr = Θ/(1−πhr/lr)2.Resulting from the wave nonlinearity and the viscous effects, there forms a system of slow steady circu-
lation flows with a horizontal scale of the order of one quarter the wave length L/2n.
2. EXPERIMENTAL TECHNIQUE
The generation of a wave flow over the bottom sediment layer was investigated in the regime of paramet-ric excitation of the first and second modes (n = 1, 2) of standing gravitational waves on the free surface ofa fluid in a rectangular vessel measuring 50×4×40 cm3, made of organic glass. The vessel was placed ona platform of a precision vibration table, which performs harmonic oscillations in the vertical direction (theangular deviations from the vertical did not exceed 8′) [11–13].
We studied the flows in the main Faraday resonance regime for the first mode n = 1, when the vesseloscillation frequency Ω is twice the excited wave frequency ω , or in the harmonic-resonance regime ω ∼Ωfor the second wave mode n = 2. At a fixed oscillation amplitude of the vessel s = 2 or 2.5 cm (n = 1 or 2,respectively ), the variation of Ω ensured the variation of the wave steepness Γ = kH/2π within 0.08–0.12.
To create a layer of loose sediments on the vessel bottom, we used aluminosilicate proppant (parti-cle diameter d = 0.063± 0.007 cm with roundness of 0.8–0.9 and material density ρ = 2.65 g/cm3) orpolydisperse sand (d = 0.18–0.25 mm with the same density). The mechanism of suspending the bottomparticles was studied using spherical polystyrene beads with diameter d = 0.060± 0.008 cm and densityρ = 1.05 g/cm3. Optical microphotographs of the particles obtained using a Motic-BA410T microscope areshown in Fig. 1.
The experiments were conducted mainly with water and an aqueous suspension (pure fluid: ρ = 1 g/cm3,ν f = 1 cSt; h = 7–15 cm), in which the dispersed phase consisted of degreased particles of PAP-2 pigmentaluminum powder (with material density 2.7 g/cm3). The dispersed particles were plate-shaped, measuring30×30×0.5 μm3, with equivalent diameter dp = 10 μm. The fall velocity of the particles W did not exceed10−2 cm/s. The particle diffusion coefficient in water at room temperature was κ = 10−10 cm2/s (accordingto the Einstein relation [14]). The particle Reynolds number was estimated as Re ∼ 10−3 based on the fallvelocity and Re∼ 2 based on the near-bottom velocity Ub.
The amplitude of fluid-particle oscillations near the bottom at the frequency of the first wave modeω = 5.19–5.45 s−1 was A = 4.5–10.8 cm, the magnitude of the bottom velocity was Ub = 13.1–30.0 cm/s.The corresponding wave Reynolds number Rew ≈ 104. For ω = 5.36 s−1, the fluid wave flow in the vesselwith a horizontal bed (0.15 cm) of monodisperse sand (d = 0.06 cm) was accompanied by the formation ofthree stationary ripples in the central part of the vessel.
The parameters of periodic flow in the case of the first and second wave modes and vortex ripples usedin the analysis of the observed structures of suspended sediment are presented in the table in dimensionaland dimensionless forms.
In the experiments with suspensions, the concentration of aluminum powder was 300–400 particles per1 cm3 and made almost no effect on the density and effective viscosity of the medium.
The flow patterns were recorded using a DIMAGE Z2 digital video camera (frame frequency24–30 frame/s, F/2.8–3.6) and a VS FAST high-speed video camera (100–500 frame/s) in the moving ref-erence frame fitted to the vessel. The video record of the motion was then processed using a ImageJ 1.43utool. The image resolution was 0.15 mm/pixel.
To estimate the concentration of suspended particles using ImageJ 1.43, we constructed the dimension-less profiles of C(x, y, t) as functions of the brightness of each pixel on a scale with 256 shades of grayalong the selected control lines. On the images processed, the regions of high particle concentration werelighter, and low particle concentration regions were darker.
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Table
n h, cm ω , s−1 H, cm Ub, cm/s lr, cm hr, cm W , cm/s θ θr
Spherical polystyrene particles
1 7 4.81 3.6 21.01 5.0 0.9 0.98 1.57 8.29
2 15 10.78 7.5 12.74 2.2 0.4 0.98 1.42 7.72
Proppant
1 7 4.81 3.6 21.01 5.0 0.9 32.39 0.05 0.25
2 15 10.78 7.5 12.74 2.2 0.4 23.32 0.04 0.23
3. RESULTS AND DISCUSSION
Earlier, in the experiments with a periodic wave flow over sand ripples the structuring effect was detectedin an initially homogeneous suspension of suspended plate-shaped particles. This effect is manifested in theappearance of bands with high and low concentrations in the entire liquid layer [11, 12]. In these experi-ments, the formation of spatially ordered regions of suspended sediments of 3-D particles was investigatedfor a pure fluid and a suspension of plate-shaped particles in the field of 2-D standing surface gravity wavesin a rectangular vessel oscillating in the vertical direction.
Formation of spatially ordered regions of suspended sedimentin a vessel with a topography
The dynamics of entrainment of the bottom sediment, initially being at rest, into the bulk flow under theaction of stationary harmonic oscillations in a vessel with a profiled bottom is illustrated in Fig. 2.
In the preparation of the experiment, on the vessel bottom we placed a layer of aluminosilicate proppantparticles (d = 0.06 cm) with a thickness equal to 2–3 particle diameter. After the excitation of the first wavemode (ω = 5.39 s−1 H = 4.2 cm) for 20 min, all the particles were accumulated in the central part of thevessel (18 < x < 33 cm) and formed three stable crests with the length lr = 5 cm and height hr = 0.9 cm.The shapes of the bottom structures, which retain when the first mode is excited repeatedly, are traditionallyreferred to stationary bottom structures of loose sand particles and are termed ‘vortex ripples’ [1, 3].
The transition of the bottom sediment into the suspended state was studied using spherical polystyrenebeads (d = 0.060±0.008 cm ρ = 1.05 g/cm3) with the sinking velocity W = 0.98 cm/s, significantly lowerthan for the proppant particles W = 32.39 cm/s (see Table).
The lighter fraction of the bottom particles was added into the oscillating vessel after the formation ofvortex ripples. The wave flows prompted the uniform distribution of the particles in the valleys between theripple crests.
We studied the process of entrainment of the bottom granules into motion after the excitation of wavemodes from the state of rest.
In establishing the oscillations with the first wave mode, at first we observed a smooth phase of particlemotion: the particles traveled along the riffle surface without transition into suspension. Small groups ofparticles resuspended in the valleys of the ripples started to form at wave height H = 2.5 cm (Fig. 2, image 1).As the motion develops and the wave height increases, both the number of particles involved in the periodicmotion and the dimensions of regions occupied by the particles, the number of which is determined by thegeometry of the vortex ripples (regions I and II in Fig. 2, image 2), also increase.
In the resuspension phase, the further increase in the wave height and near-bottom velocity results ina vortex flow around the ripples (Fig. 2, images 3 and 4), with the horizontal and vertical scales of thestructures continuing to grow with time, even at a steady wave height H .
For a quantitative description of the dynamics of formation of the large-scale areas of suspended bottomparticles on times 0 < t/T ≤ 10 (T is the wave period), we use the following parameters: the maximum
Fig. 2. Evolution of clouds I and II of suspended bottom particles over the vortex ripples: (1–4) τ = t/T = 6, 7, 9 , 10;T = 1.31 s, H = 2.5, 2.8, 3.4, 3.5 cm.
horizontal and vertical dimensions X , Y , the area S, the coordinates of the center of mass Xc and Yc, and therelative number of suspended particles n (similar to [11, 12]).
In Fig. 3a, the relative linear dimensions X , Y (scaled to h) of the cloud are described by an exponentialfunction aτn (here, τ = t/T ). The interpolation of the experimental data (curve 3) gives the values a = 0.32,n = 0.5. The form of the dependence makes it possible to classify the cloud growth process as ‘diffusion’ andestimate the coefficient of entrainment of the bottom particles or the effective induced-diffusion coefficientas De f f = 3.69 cm2/s, which is consistent with the linear increase in the cloud area in Fig. 3b. The value of
the coefficient De f f determines the effective scale of particle entrainment δ (D)ω =
√2D f/ω = 1.2 cm, which
corresponds to twice the particle diameter d.The time dependences of the coordinates of the center of mass of cloud II (Fig. 2) are shown in Fig. 3a,c.
The vertical coordinate of the center of mass Yc(t) (Fig. 3a) monotonously increases with time, and theexperimental dots 4 are grouped with a small scatter around the curve Yc(t) = aY τn, aY = 0.11, n = 0.5.
In contrast to the vertical coordinate, for the dependence Xc(t) the regular alternating of oscillations withlarge and small amplitudes is typical (Fig. 3c). The character of the curve reflects the complexity of the flowpattern over the topography, in which both wave components and rapidly varying spatially inhomogeneousvortex components are present. The vortices directly above the ripples are visualized in a number of studies;however, the flow structure as a whole has not been studied.
STRUCTURING OF SUSPENDED SEDIMENTS IN PERIODIC VORTEX FLOW 227
Fig. 3. Evolution of the parameters of the particle structures: (a, b) the size and area of the clouds; (c) the horizontalcoordinates of the center of mass; (d) the relative number of suspended particles of cloud II (Fig. 2); the linear characteristicsare scaled to h = 7 cm and the areas to hL = 350 cm2.
Processing the photographs in Fig. 2 using the ImageJ tool made it possible to estimate the number ofbottom particles which are suspended during the interval 0 < t/T < 10. Figure 3d shows the results ofautomatic counting of the number of particles in the cloud, which indicate a linear time dependence of thenumber of particles entrained in motion.
Formation of spatially ordered structures in a vessel withtwo groups of vortex ripples
The second series of experiments on resuspension of a bottom sediment was carried out in a vessel witha sandy deformable bottom covered by a 1 cm layer of polydisperse sand particles with d = 0.18–0.25 mm.The wave flow was formed at a frequency of vertical oscillations Ω = 21.08 s−1 with amplitude s = 2.5 cm,when in the liquid layer 15 cm deep the second wave mode n = 2 is excited (two fixed points of the freesurface with coordinates x = 12.5 and 37.5 cm).
Under the nodes of a steady wave of height H = 5.2 cm on the originally horizontal surface of the sandlayer two groups of stationary vortex ripples were formed (Fig. 4). Each group consisted of seven elements2.2 cm long and 0.4 cm high. In this configuration the ripples had a stable shape, which retained after theshutdown of the setup, damping all fluid motions, and a subsequent independent excitation of waves in thenew experiment.
The transition of the sediment into the suspension state was studied using spherical polystyrene parti-cles (d = 0.060± 0.008 cm, ρ = 1.05 g/cm3). In addition to the sediment resuspension, we recorded therestructuring of initially homogeneous suspension of aluminum pigment powder consisted of plate-shapedparticles.
In the regime of the second wave mode ω = 10.78 s−1, the breaking of the uniformity of the suspension
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228 KALINICHENKO, CHASHECHKIN
Fig. 4. Structurization of a homogeneous suspension and resuspension of bottom particles for excitation of the second wavemode in a vessel with two groups of vortex ripples: (1–4) τ = 11, 17, 22, 34 and H = 4, 4.9, 7.1, 7.3 cm; T = 0.586 s.
concentration started simultaneously both under the wave nodes over the bottom ripples and at the antinodesunder the free surface (at the center and on the vessel walls).
In Fig. 4, the dynamics of the development of disperse structures is shown as a series of images (1–4).The process of establishing the oscillations is accompanied by the formation of localized zones with
a high and low particle concentration, simultaneously near the free surface and near the bottom, i.e. theprocess of restructuring the suspension due to the effects of bottom topography and mass transfer.
The resuspension of the bottom sediment is manifested in growing clouds of suspended particles, startingwith the image 1 in Fig. 5. At first, there are only two clouds above each group of ripples (image 1), thenthe number of clouds grows: four in image 2, and eight in image 3. Next, the clouds lose their individuality,and in image 4 the merged clouds form a sheet of suspended bottom particles.
For a quantitative description of the dynamics of forming large-scale regions of suspended bottom parti-cles, from the instant of appearance of individual clouds to the formation of a particle sheet, we also usedthe maximum horizontal and vertical dimensions of these regions X , Y and the area and coordinates of thecenter of mass Xc, Yc (similar to the quantitative characteristics used to describe the clouds of suspendedparticles over individual vortex ripples, Fig. 3).
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Fig. 5. Evolution of the size (a), area (b), vertical (c) and horizontal (d) coordinates of the center of mass of the cloud ofsuspended particles (Fig. 4) in the case of the second wave mode; the linear characteristics are scaled to h = 15 cm and theareas to hL = 750 cm2.
In Fig. 5a, the relative linear dimensions of the cloud X , Y (scaled to h) are described by an exponentialfunction aτn. The interpolation of the experimental data (curve 3) gives the values a = 0.03, n = 0.5. Asin the case of individual vortex ripples, the form of the dependence makes it possible to classify the cloudgrowth process as ‘diffusion’ and estimate the coefficient of entrainment of bottom particles or the effectiveinduced-diffusion coefficient as De f f = 0.32 cm2/s, which is consistent with a linear increase in the cloudarea in Fig. 5b.
The time dependences of the coordinates of the center of mass of one cloud (Fig. 4) are shown inFig. 5c, 5d. The vertical coordinate of the center of mass Yc(t) (Fig. 5c) monotonically increases withtime, the experimental points with a small scatter are grouped around the curve Yc(t) = aY τn, aY = 0.01,n = 0.5. The horizontal coordinate of the center of mass Xc(t) (Fig. 5d) is characterized by fluctuations withan increasing amplitude.
Segregating properties of wave flows with a bottom topography
Earlier, in the experiments on dynamics of initially homogeneous aqueous suspensions of small thinplates in the vessels with a topography the restructuring effect was detected, which is manifested in theformation of isolated dark areas over large ripples, that have been classified as zones with low particleconcentration (‘voids’) [11, 12 ]. However, a side lighting technique used in the experiments made it possibleto give an alternative interpretation of the observed darkening, due to a coherent variation in the orientationof the scattering surface of flat particles in the flow; in the experiments [11, 12] aluminum flakes were used.To clarify the structure formation mechanism, we conducted control experiments using particles of differentshapes and sizes.
The images in Fig. 6 indicate a close interaction between the formed disperse structures and the sus-pended bottom particles: the motion of the latter is restricted by the regions of reduced concentration of the
Fig. 6. Disperse structures and suspended bottom particles over the vortex ripples for excitation of the second wave modein a vessel with two groups of vortex ripples: (1–8) τ = 0.5, 1.5, 2.5 , 3.5, and H = 3.6 cm; T = 0.6 s.
dispersed phase (‘voids’). The suspended particles travel along very complex trajectories. Simultaneouslywith the appearance of dark spots in Fig. 6, image 1, near the edges of the bottom topography the suspendedbottom particles are accumulated (white dots) in the same selected regions. The voids grow slowly with time,and simultaneously the dimensions of the particle circulation zones increase (images 2–4, the compared re-gions are shown by the arrow). In the circulation regions, the number of particles changes very slowly.
In general, a fine structure is formed in the suspension distribution, which is manifested in the formationof regions with high (light bands) and low (dark spots) concentration. The first group of bands over thevoids follows the shape of the boundary of the low particle concentration regions.
The above data for the first two wave modes make it possible to formulate the following general conclu-sions about the process of resuspension of bottom particles in the wave flow over vortex ripples. As in thecase of structuring a homogeneous suspension, over each ripple a large-scale disperse structure (or cloud) isformed by the suspended bottom particles. The horizontal and vertical dimensions of the near-bottom cloudsgradually increase, the amount of suspended particles forming these clouds also increases. The experimentson structure formation in suspensions and resuspension of bottom particles in a vessel with vortex rippleswere repeated several times after long time intervals required for the recovery of a homogeneous particleconcentration. These experiments demonstrated the high reproducibility of the results presented above ingeneral and in detail.
Summary. A dynamic process of the formation of spatially ordered regions of a suspended sediment inthe field of two-dimensional standing surface gravity waves is first studied experimentally in a rectangularvessel oscillating in the vertical direction.
In vessels with individual vortex ripples and groups of ripples, the parameters of the structured regionsare determined for the first and second wave modes. It has been established that isolated structured regionsof suspended sediments are formed near the bottom and gradually move towards the free surface. It is shown
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that the corresponding spatial horizontal scales are determined by the length of sand ripples, and the verticalscale increases with time. In all the experiments, the structures formed in the process of resuspensionmaintained during the entire time of the fluid wave motion and disappeared after ceasing the parametricexcitation of the waves.
The experiments were performed on the PR-2M USU setup of the RAS Institute for Problems in Me-chanics with partial financial support of the Russian Foundation for Basic Research (projects 12-01-00128and 12-08-00067).
REFERENCES
1. Dynamic Processes of Coastal Zone (Eds. R.D. Kos’yan, I.S. Podymov, and N.V. Pykhov) [in Russian] (ScientificWorld, Moscow, 2003).
2. F. Charru, B. Andreotti, and P. Claudin, “Sand Ripples and Dunes,” Annu. Rev. Fluid Mech. 45, 469–493 (2013).3. P. Nielsen, Coastal Bottom Boundary Layers and Sediment Transport (Word Scientific Publ., Singapore, 1992).4. M. Canals and G. Pawlak, “Three-Dimensional Vortex Dynamics in Oscillatory Flow Separation,” J. Fluid Mech.
674, 408–432 (2011).5. R.B. O’Hara Murray, P.D. Thorne, and D.M. Hodgson, “Intrawave Observations of Sediment Entrainment Pro-
cesses above Sand Ripples under Irregular Waves,” J. Geophys. Res. 116, C01001 (25 P.)(2011).6. A.G. Davies and P.D. Thorne, “Modeling and Measurement of Sediment Transport by Waves in the Vortex Ripple
Regime,” J. Geophys. Res. 110, C05017 (25 P.) (2005).7. H. Kim, “Temporal Variation of Suspended Sediment Concentration for Waves over Ripples,” Int. J. Sediment
Research 18 (3), 248–265 (2003).8. P.D. Thorne, A.G. Davies, and P.S. Bell, “Observations and Analysis of Sediment Diffusivity Profiles over Sandy
Rippled Beds under Waves,” J. Geophys. Res. 114, C02023 (16 P.) (2009). DOI: 10.1029/2008JC004944.9. J.J. Williams, N. Metje, L.E. Coates, and P.R. Atkins, “Sand Suspension by Vortex Pairing,” Geophys. Res. Lett.
34 (15) (2007). DOI: 10.1029/2007GL030761.10. Y.D. Chashechkin, “Hierarchy of Models of Classical Mechanics of Inhomogeneous Fluids,” Morskoi Gidrofiz.
Zh. (5), 3–10 (2010).11. V.A. Kalinichenko and Y.D. Chashechkin, “Structurization and Restructurization of a Uniform Suspension in
a Field of Standing Waves,” Fluid Dynamics 47 (6), 776–788 (2012).12. Y.D. Chashechkin and V.A. Kalinichenko, “Topography Patterns in the Suspension Structure in Standing Waves,”
