Numerical Simulation of Sediment Particles Released at the. Edge of the Viscous Sublayer in Steady and Oscillating. Turbulent Boundary Layers

Transcription

1 Numerical Simulation of Sediment Particles Released at the Edge of the Viscous Sublayer in Steady and Oscillating Turbulent Boundary Layers Yeon S. Chang a) and Alberto Scotti b) Deartment of Marine Sciences, University of North Carolina, Chael Hill, North Carolina Submitted to Physics of Fluids ABSTRACT In this reort e studied numerically the movement of susended sediments in a turbulent boundary layer over a flat bed. We used LES to generate the velocity field, hile the motion of individual articles as calculated using a modified version of the Maxey and Riley equation. We considered three kinds of flos -- steady, oscillating and ulsating, and articles of different sizes ranging from silt to fine sand. In each exeriment, 4096 articles ere released at the uer edge of the viscous sublayer. The susension rate, defined as the ercentage of articles still afloat after the initial shake a) b) yschang@ .unc.edu ascotti@ .unc.edu 1

2 don deended strongly on the ratio of vertical rms velocity fluctuation to settling velocity in all tyes of flos. In the unsteady case e also found a non trivial relation ith the hase of the flo. In oscillating and ulsating flos e found that the mean elevation of the susended articles increased more raidly around the time of flo reversal. While the susension rate in steady and oscillating layers as comarable, it as significantly larger in the ulsating layer. We also looked at the effect of coherent structures in lifting the articles aay from the all. We found ositive evidence for that, though e did not observe any significant traing. Finally, e discussed the imlication of our study to the modeling of bottom sediment fluxes (icku rates) in unsteady flos in an aendix. I. INTRODUCTION Sediment susension events and the movement of susended sediments are still oorly understood henomena, desite the fact that susended sediments can contribute significantly to the total sediment transort. In coastal areas the roblem is further comlicated by the unsteady nature of the flo due to surface gravity aves. In general, near shore it is believed that aves are mainly resonsible for sediment susension hile currents carry the entrained sediments aay. Thus, in order to investigate the sediment susension in this regime, an understanding of turbulent oscillating flos is necessary. Exerimenting ith turbulent oscillatory flos over rough beds, Sleath 1 demonstrated that turbulent intensities significantly fluctuate during cycle ith to eaks er cycle. Jensen et al. 2 also resented some exerimental results on urely oscillating turbulent 2

3 boundary layers; esecially imortant is the observation that, ithout a mean current, oscillating flos cycle beteen laminar and turbulent state, ith the transition usually occurring just before near-boundary flo reversal. Turbulent fluctuations due to the oscillatory free stream are usually confined ithin a thin oscillating boundary layer 3,4. Because of this, detailed measurements are often difficult to erform, and sometimes the interretation of data is controversial. At lo concentration, the resence of sediments does not alter significantly the roerties of the flo, and the fluid hase can be modeled searately from the articulate hase. For the numerical comutation of turbulent flos over the seabed, Reynolds-averaged Navier-Stokes (RANS) equations are often emloyed 5-8. RANS models are the choice tools for coastal engineering alication for their robustness and because of the relatively small comutational cost. Hoever, since fluctuations at all scales need to be modeled, the models are sensitive to the large scale driving conditions. For examle, recent evidence suggests that commonly used RANS models misreresent key turbulent quantities in oscillating ulsating boundary layers 9. At the oosite end of the modeling sectrum, Direct Numerical Simulation (DNS) has been successfully emloyed in many studies of oscillating boundary layers Hoever, DNS simulations are severely limited by grid size and time ste requirements. For this reason DNS is still confined to relatively lo Reynolds number flos. Large Eddy Simulation (LES) steers a middle course beteen DNS and RANS. In LES the large scale eddies that are considered to be imortant in energy transfer are resolved, and the smallest "subgrid-scale" eddies are modeled. 13,14 Hence, the LES aroach lies beteen the extremes of DNS in hich all fluctuations are resolved, and RANS in hich 3

4 only mean values are calculated and all fluctuations are modeled. For this reason LES results are usually less sensitive to modeling errors than RANS method. For the modeling of the SGS stress, eddy-viscosity based models have been commonly emloyed, and one of the most idely used exression for the eddy-viscosity is the dynamic Smagorinsky model, 15,16 hich has successfully been alied to both high Reynolds number flos 17 and ulsating flos. 4 As for the dynamics of the articulate hase, the aroach that is most idely used is to consider the evolution of the Susended Sediment Concentration (SSC) using an advection-diffusion equation, in hich the time variation of the volumetric concentration of a control volume is balanced by flo advection, turbulent diffusion, and settling due to gravity. 18,19 By combining the flo fields calculated by RANS models or LES ith an aroriate SSC equation, it is ossible to comute the susended concentration field at each time ste. Andersen 20 used a K ω tye RANS model to calculate the SSC and to investigate the rile dynamics. Chang and Hanes 21 use same model for the comarison ith the field data measured in a near-shore region, shoing that turbulent eddies are formed even by lo-amlitude riles ith steeness 1/15 and the sediment susension events are affected by these eddies. Zedler and Street 22 used LES for a three-dimensional calculation of the SSC in channel flo over riles of size 0.25 cm in height and 5 cm in length. Though the calculation of the volume concentration from sediment advectiondiffusion equation is efficient for ractical uroses, it has several shortcomings: It is very sensitive to the bottom boundary conditions; 21 lack of knoledge about the transort mechanism has resulted in using the eddy viscosity of the flo as eddy diffusivity in most models, hich may result in under (or over)-estimation of the SSC redictions. To 4

5 address these issues, LES or DNS have been couled to an equation for the articulate hase. The Maxey and Riley equation 23, hich includes the effects of ambient ressure gradient, added mass, Stokes drag, Basset force, and buoyancy on a single article, has been idely used for the investigation of the motion of small articles in turbulent flos couled to either a DNS or a LES In this aer e couled a LES ith a modified Maxey and Riley equation for articulate matter to study 1) the settling and susending rocesses in relation to article size and flo energy, 2) the effects of coherent structures on the motion of sediments and 3) the imortance of unsteadiness over the susension rocess. We simulated both steady and unsteady flo conditions, ith arameters as close as ossible to the ones found in the environment, and different article sizes ranging from silt to small sand. To isolate the effect of coherent structures from large-scale vortical motion, e considered a flat, hydrodynamically smooth surface, even though in the field this condition is rarely encountered. II. PROBLEM FORMULATION We assume that the concentration of sediments at any time is too lo to affect the dynamics of the fluid. Thus, the equations for the fluid art can be solved indeendently from the articulate hase. A. Fluid Equations 5

6 The governing equations emloyed in the resent study for the fluid hase are the filtered Navier-Stokes equations u t i + x j ( u u ) i j P = x i 2 ui + ν x x i j τ x ij j, (1) u i x i = 0, (2) here the subgrid scale stress is ij ( u u u u ) τ =. (3) i j i j In the above equations, a bar is used to indicate filtered variables. We used the standard geohysical convention here (x,y,z) or (1,2,3) stand for the streamise, sanise, and vertical directions, and (u,v,) or (u 1,u 2,u 3 ) for the flo velocities in the resective directions. The comutational domain is a rectangular channel ith height 2 H =0.2 m, and idth L x =0.6 m and the viscosity ν = 10 6 m 2 /s. L y =0.2 m, resectively (Figure 1) and The subgrid scale stress in Eq. (3) is modeled using the dynamic eddy viscosity model. 15,16 The flo is forced by an external ressure gradient that is the sum of a steady and an oscillating art P f [ α + β cos( ω t) ]. ( x, t) = x. (4) 6

7 _ The total ressure, P ( x, y, z, t), is the sum of the forcing ressure, P f ( x, t), and the mechanical ressure, ( x, y, z, t), the latter necessary to satisfy the zero divergence condition of the flo. Equations (1) (3) are solved using an Adams-Bashforth fractional-ste method. Both advective and diffusive terms are treated exlicitly. All satial derivatives are aroximated by second-order central differences on a staggered grid. 33 The boundary conditions are eriodic in x and y, and no-sli at the loer and uer boundary. The grid sacing is uniform in the sanise and streamise direction. In the vertical direction, a hyer tangential grid generator is emloyed to create dense grid sacing near the loer all. B. Sediment Particle Motion We use the equation derived by Wiberg and Smith 30 to describe the force balance acting on sediment articles in ater. Buoyancy, ressure gradient of ambient flo, added mass, drag, and lift forces are included in the force balance that reads & dv ρ V dt (5) = ( ρ 1 + CL ρa 2 & Du f ρ) Vg n + ρv Dt + C & Du f ρv Dt & & 2 & & 2 [ u V u V ] n, f to f m bottom & dv dt + 1 C 2 D & ρau f & V & ( u V ) f & here V & is sediment article velocity vector, u & f is fluid velocity interolated at the article osition, 7

8 ρ and ρ are the density of article and fluid resectively, V and A are the volume and cross sectional area of the article, C m is the coefficient of added mass, C D and C L are drag and lift coefficients, n a vector of magnitude one ointing in the vertical. The subscrition to and bottom mean that the relative velocities are calculated at the to and bottom of the sediment article and the derivative D / Dt denote the time D derivative folloing a arcel of fluid such that = + u f. In the above j Dt t x & equation, the drag force is relaced by the Stokes drag, π r µ ( ) j & 6 u f V, hen the article Reynolds number Re P D = & u f ν & V is less than 1. Here µ is the dynamic viscosity of ater and r the radius of the article. The coefficient of added mass, sherical article and the lift coefficient, drag coefficient C [ Re ] D C m, is 0.5 for a C L, is set to 0.2. The emirical relation for the = P is used folloing Wang and Squires 29. The Re P lift force induced by the velocity difference beteen to and bottom of the article is usually considered negligible hen the size of the article is small. Inside the viscous sublayer layer, hoever, the velocity gradient is large, so e have decided to retain it. Other forces, such as the Magnus force due to the article rotation and the Basset history force that accounts for changes in fluid drag due to changes in the flo structure around the article are ignored in the resent study. 8

9 The article velocities are calculated by integrating Eq. (5) in time using a semi-imlicit method. Since the article ositions at any given time do not in general corresond ith the oints at hich the Eulerian fluid velocity field is calculated, the velocity field and its derivatives need to be interolated onto the article osition. The interolation is carried out by sixth-order Lagrangian olynomials in three directions 34. Once the article velocities are comuted, the article dislacement can be calculated from * d X dt P here & = V (6) X & P, P is the osition vector of the sediment article. At the horizontal boundaries, articles leaving one side are injected at the oosite side. In the vertical direction, the articles are removed if they reach the bottom or the middle of the channel. C. Parameters: Continuum and articulate hase In the resent study e considered three flo regimes: A steady flo, α 0, β = 0 (Case 1); a urely oscillating floα = 0, β 0 (Case 2); a ulsating flo, α 0, β 0, (Case 3), here ( α, β ) have been defined in eq.(4). The choice of the forcing arameters reresents a comromise beteen achieving the largest ossible Reynolds number hile keeing the execution time ithin reasonable bounds. In the steady case α = 10 3 m/s 2 gave a friction velocity ut = 10 2 m/s and a Reynolds number based on the friction velocity and height of the channel Re H ν = The hysical domain as t u t discretized using 98 oints in the vertical and 130 oints in each horizontal direction. In 9

10 the vertical, the grid as stretched to accommodate 10 oints ithin the first 10 all units (1.u. ν u = 0. 1mm), hile in the horizontal the resolution as 4.7 mm in the t streamise direction and 1.6 mm in the sanise direction + + ( x = xu ν = 47, y = yu ν = 16 ), values that are considered aroriate for LES t t of channel flos 17. In the urely oscillating case e set β = m/s 2 and T=8 s, resulting in a free stream velocity U = 0. 4 m/s, a realistic value in shallo aters. Purely oscillating flos over a flat surface can be characterized in terms of a Reynolds number Re s = U l s ν based on the thickness of the laminar boundary layer l = 2ν / ω s and the free stream velocity. In the resent case Re s = 640 laces the flo in the transitional regime according to Hino et al. 35. The flo conditions are listed in Table 1. We have considered three grain tyes, varying in diameter from 0.05 to 0.02 mm (fine sand to silt), ith a density ratio s = 2.65 (quartz). Table 2 summarizes their roerties. D. Initial conditions: Continuum hase The solver as run ith β = 0 until a statistically steady state as reached. At this oint, the statistical roerties of the solution ere checked against standard benchmarks 36 and found in good agreement. This field as then used as an initial condition for the steady flos simulations. To generate the initial condition for the unsteady simulation e emloyed a similar strategy, running the code until a statistically steady (in the sense of hase averages) as reached. 10

11 E. Initial condition: Particulate hase. The assumtions under hich eq.(5) is derived break don in the immediate vicinity of the loer boundary, at a distance of the order of the diameter of the articles. In this region, one cannot ignore the finite extent of the articles and the comlexities due to article-article interaction and collision ith articles on the bottom. 37 Ideally, in the viscous sublayer, one should solve the fully couled laminar roblem treating the articles as sheres of finite extent. Hoever, couling a turbulent channel solver to a 3D couled laminar solver is comutationally too exensive (a 2D simulation of a laminar flo over nearly buoyant articles has been recently ublished 38, hich illustrates ell the difficulties involved). Once detached from the bottom, articles can be susended if, at the zenith of their trajectories, the articles encounter turbulent fluctuations of large enough magnitude, and it is on the latter roblem that e focus in this aer. Thus, in the resent study, e have decided to ignore the dynamics in the viscous layer, and e have concentrated on the dynamics of the articles outside the viscous layer, here eq. (5) is a valid aroximation. Hence, e released the articles in the flo at the edge of the viscous sublayer (z=0.95 mm). In the steady case, the ratio / varied from 1 to rms s The goal as to determine ho the susension rate deends on rms / s, as ell as to investigate the role of coherent structures in lifting sediments aay from the boundary. For each run, e released 4096 articles and folloed them until a seudo steady state as reached. Particles that hit the bottom or that reached the middle of the channel ere removed from the simulation. 11

12 III. RESULTS A. Particles under uniform flos (Case 1) Figure 2 shos the turbulent intensity of the fluctuations in the steady case. In the uer anel the vertical distances are normalized by the channel height, and in the loer anel the same roerties are exressed in all units. The maximum values of the intensities occurs around z + = 13 in the streamise direction, z + = 59 in the sanise, and about z + = 104 in the vertical direction in good agreement ith DNS results 36. Once the flo reaches a statistically stable state, the sediment articles are released at the initial elevation. Figure 3 shos the numbers of surviving articles (i.e. articles that have not hit the bottom) as a function of time. More than 95% of the 0.05 mm articles settle don ithin the first second, hile more than 60% of the 0.02 mm articles still survive even after 18 seconds. This indicates that the vertical uard fluctuations near the initial elevation are not strong enough to suort the 0.05 mm articles, even though 15% of the articles at t=0 exerience an uard velocity larger than the settling velocity. In figure 4, the mean vertical article velocity,, mean vertical flo velocity at article ositions, f, and mean vertical article osition, z, of the 0.05 mm articles are comared ith those of the 0.02mm articles during the first 0.6 seconds ithin hich most of the 0.05mm articles settle don. In anel (a) f and, normalized by s, are shon for the 0.05mm articles. In this case the mean is taken only for the articles that have ositive (uard) velocities at t=0 (otherise the mean ould be close 12

13 to zero due to the articles ith the negative (donard) velocities). The vertical flo velocities are also calculated at the article ositions through a three-dimensional Lagrangian interolation. Although is smaller than s (only 0.4 times of For the 0.02mm articles, hoever, rms is comarable to s at the initial elevation, s ) and the articles move donard (anel (a)). f is about 3 times greater than s, so f is also greater than zero resulting in the uard motion of a large fraction of 0.02 mm articles (anel (b)). Therefore, as shon in anel (c), the center of mass of the 0.05 mm articles dros to the bottom, hile the 0.02mm articles are susended into the flos. The mean values of the article velocities ( u, v, ) and ositions ( x, y, z ) of the 0.02mm and 0.035mm sediment articles are shon in figure 5. From no on, the averages are taken including all the susended articles, regardless on their vertical velocity. The 0.05 mm articles are not included because most of the articles fail to be susended as shon in figure 3. In the streamise direction, u converges to 0.15m/s after 10 second.. This is a little bit smaller than the bulk mean velocity 0.178m/s. Hoever, anel (e) shos that the mean article osition is beteen m above the bottom. If a bulk mean velocity is calculated in this vertical range only, the value is m/s and it is no closer to u. The streamise article osition, x, increases uniformly due to the uniform flo velocity in this direction and shos no significant difference beteen the to article sizes (anel (a)). In the sanise direction y and v is exected to be zero because no forces are acting on this direction. Panel (c) shos that y is not zero but small enough to be neglected (5 mm or less). v is also close to zero (anel (d)). In the vertical direction one may exect that becomes zero and z 13

14 remains constant after a stable condition is reached. Panel (e) and (f) sho that e are close to a stable condition though there is still some increase in z as is significantly different from zero. In this exeriment, an exactly stable condition cannot be reached regardless of the rocessing time because the number of articles reaching the to of the channel or hitting the bottom (here they are removed from the calculation) does not tend to zero. In figure 6 e sho the evolution of the concentration field over time. The vertical rofiles are lotted by dividing the vertical range into 30 segments from 0 to 0.06m and by counting the number of articles contained ithin each segment at three different times, 3,10, and 18 sec after the articles are released. The articles that have settled to the bottom are not included for counting. At t = 3 most of the articles are still confined to lo elevations (z<0.02m). At t = 10 the rofile becomes more stable ith the concentration decreasing both belo 0.5 cm and above 4 cm. At t = 18 the number of articles decreases gradually ith elevation and a simle exonential model for the mean concentration rofile of the susended sediments can be alied to the rofile if it is exressed as C( z) / C0 = ex( α z) here α is a coefficient due to turbulent mixing. B. Effect of near-all coherent structures The near-all region in a turbulent boundary layer is characterized by regions of relatively ell organized flo knon as coherent structures. 39 In general, they aear as airise vortexes ith cores aligned ith the direction of the flo. 14

15 Figure 7 shos a distribution of the vertical comonents of the velocity on a x-y (streamise-sanise) horizontal lane at the initial elevation, z = 0.95 mm, and time t = 0 sec. As can be seen in the uer anel, regions here the vertical velocity gradient is strong along the horizontal direction (dark sots) are found in many laces. One of the structures, denoted S, is magnified in the loer anel. Strong donard and uard velocity comonents are found together in this small area, hich has the size of 4.0cm x 2.0cm. For a better understanding of the flo structure in the rectangle S, it is helful to look at the vertical distribution of the velocity, shon in Figure 8. To counter rotating eddies can be clearly seen, searated by a strong dondraft area. In order to investigate articulate motions of sediments near this kind of eddies, four sediment articles are chosen and the trajectories of each article are traced in time. Figure 8 shos the locations of the four selected articles, P1, P2, P3, and P4, that are initially located on the line A-B. P1 is located here the local flo is strong and directed to the uer-left of the frame. P2 is located near the edge of the eddy and the direction of the local flo is donard. The local flo around P3 is directed to the right of the domain due to the eddy in the right side of frame. P4 is also under the influence of the same eddy as P3 but the flo is much eaker. Once the four articles are released, the trajectories of each article are traced as they move ith the local flos. Figure 9 and 10 sho the trajectories of the four articles as ell as the local flos around the articles at four time stes. Though the articles movements initially start at the same donstream locations along the line A-B, they subsequently move at different seed. For this reason, the x-ositions of each article are 15

16 calculated searately based on each article velocity and the y-z lanes are lotted at the x-locations of each article. The history of the movements of the article P1 is shon in the left four anels of figure 9. At t = 0.23 sec P1 moves about 5 mm in the negative y- direction and 5 mm in the ositive z-direction. The trajectory is consistent ith the direction of the local flo hich surrounds P1 at t = 0 (Figure 8). In addition, it can be seen that the article trajectory at t = 1.91 sec has formed a small circle in resonse to the small eddy hich is about to form around P1 at t = 0.95 sec. The article sitches to the ositive y-direction at t = 2.39 sec hich is also to be exected from the local flo attern at t = 1.91 sec. In summary, the article aears to track the flo closely. Though the direction of the local flo is donard at t = 0 for P2, it becomes soon horizontal (ositive y-direction) and so does the article s movement at t = P2 kees moving in the ositive y-direction until the flo reverses its direction at t = 0.91 sec. The flo around P2 at t = 1.67 sec changes the direction uard and the resulting article s movement is uard as shon in the anel at t = 2.39 sec. P3 kees moving horizontally (ositive y) along the flo direction at t= 0 sec (figure 10), and no changes is found in its direction until t = 2.86 sec. Once the local flo changes the direction at t = 2.86 sec, the article moves uard folloing the flo. The movement of P4 is not as interesting as the other articles. It is found that the flo around P4 is eak and not enough to susend the article. This article finally settles don to the bottom about 3 sec after it is released. As can be seen from figure 9 and 10, the eddies found in figure 8 are short lived. Thus, e do not exect that article remain traed ithin these eddies for long time. Hoever, the turbulent structures certainly lay a significant role in the initial susension of the articles as can be seen in case of P1 and P4. 16

17 In figure 11 the vertical comonents of the article and the flo velocities at the same location are comared. As can be seen from the figure, the flo velocities that are interolated at the article ositions are almost identical to the article velocity but differences are still found here the flo velocity changes raidly. For examle, the flo velocity around P1 raidly decreases after a eak at t ~ 1 sec but the article velocity decreases more sloly, hich revents a raid dro of the article. Similarly, the flo velocity sharly increases at t ~3.5 sec from the negative values. In case of P2 the article raidly move uard at t ~ 2 3 sec due to a shar eak of the velocity. After that the article osition shos little changes because the velocity has small fluctuations near zero for t ~ 3 5 sec. Folloing a shar increase at t ~ 9 sec, the velocity raidly dros belo zero, hich results in the settling of the article. For article P3, the velocity remains close to zero for the first 2 sec and the article hovers above the bottom during this time. After a eak at t ~ 7 the velocity sharly decreases but still kees the ositive values, so the article also kees moving uard until t ~9 sec. As already shon in figure 10 the vertical range of P4 is smallest and it settles don to the bottom at t ~ 3 due to negative velocity after t ~ 1. C. Particle motions under ure oscillating flo (Case 2) and combined oscillating and uniform flo (Case 3). The motion of articles under ure oscillating flo (Case 2) as ell as combined oscillating and uniform flo (Case 3) is investigated on the same comutational grid as Case 1. As already mentioned, Re s for Case 2 is set to 640 ith U = 0.4 m/s and T = 8 17

18 sec, under hich condition the flos oscillates beteen turbulent and laminar conditions during one eriod. For Case 3 the ressure gradient has the same mean comonent as Case 1 and the same oscillating comonent of Case 2. The angle beteen the uniform and the oscillating flos is zero, so ossible effects caused by the various angles beteen the to flos are not considered here. In both flos, the oscillation in the amlitude of the turbulent fluctuations is confined to a layer extending u to 5 cm from the bottom, the oscillating turbulent boundary layer. For a urely oscillating driving force, the flo is laminar outside the oscillating boundary layer, hile in the case of ulsating flos the turbulent quantities converge to their steady state values outside of it 4,40. In figure 12 the hase-averaged streamise velocities in the middle of the channel, elevation, U T, and at the initial U I, are shon in the uer anel (a) for one oscillating eriod for Case 2. The hase difference beteen U T and U I agrees ith revious studies 40. Panel (a) also shos the temoral variation of the Shield arameter, the non-dimensional bottom shear τ b ( t) u stress, exressed as θ ( t) = here the bottom shear stress is τ b ( t ) = µ z= 0 ρ ( s 1) gd z and D = 0.05mm. The bottom stress leads U T, ith a hase difference of about 1 sec., or π / 4, the latter being the value exected for ure laminar flos 40. In the loer anel (b) in figure 12, the time develoment of shon. The mm). Both rms and the turbulent kinetic energy (TKE) are rms and TKE distributions are evaluated at the initial elevation (z = 0.95 rms and TKE are eriodic ith to eaks during one oscillating eriod. eaks at the time of flo reversal in stream velocity. The vertical distributions of U T hile the TKE eaks in hase ith the free rms rms and TKE are shon in figure 14 here 18

19 time lags of the eak values are found in the vertical direction for TKE. The time lag shos that the fluctuations, generated near the all, roagate uard as attenuated aves. It can be seen that strong TKE fluctuations are confined at lo elevation hile the fluctuations in rms extend to higher elevation. This reflects the fact that vertical fluctuations are induced by eddies hose size is limited by the distance from the all. In case of ulsating flos (Case 3), UT can be ell reresented as a sinusoidal oscillation at the driving frequency around a non-zero mean. The maximum value of UT is 0.6 m/s and the minimum is 0.2 m/s, as shon in Figure 13. Close to the all, the resonse of the system becomes non-linear, and higher harmonics are required to describe θ. The non-linear resonse of the system results in very different atterns in the time develoments of magnitude both rms and TKE for Case 3. Instead of having to eaks of equivalent rms and TKE have one dominant eak near the time of flo reversal (maximum deceleration) (t = 5~6 sec). Another interesting feature is that the hase difference beteen the eaks of rms and TKE is much smaller than Case 2. In Case 2 the eak of TKE occurs in hase ith the free stream, hile the eak in Case 3 occurs some time later and the time lag beteen the to eaks is smaller. Therefore, the turbulent energy as ell as the energy of the vertical fluctuations concentrates near the time of the flo reversal. In figure 14 the vertical distribution of rms and TKE shos that the dominant eak is concentrated near the time of flo reversal (t = 5~6 sec) for Case 3. In order to investigate the sediment article motions for Case 2 and 3, five different times (t1, t2, t3, t4, and t5) ere chosen as the initial times for the articles to be released as 19

20 shon in figure 12 and 13. These initial times here chosen in relation to the time variation of θ. t1 is chosen hen θ just exceeds the critical value of the Shields arameter, θ c hich is the criterion for the movement of the sediment article and it is usually taken to be 0.05(Nielsen 19 ). Similarly, t5 is chosen just before θ becomes smaller than θ c. t3 is the time hen θ is maximum during the oscillating eriod, and t2 (or t4) are chosen beteen t1 and t3 (t3 and t5) hen the increase (decrease) of θ is maximum. Once the initial times from t1 to t5 are reached, the sediment articles of three different sizes are released into the flos at the initial elevation just as in Case 1. Figure 15 shos the number of articles that remain in the flo after they are released at each time stes under ure oscillating flos (Case2), here the time axis is same as in the revious figures 12, 13, and 14. More than 95 % of the largest articles settle to the bottom ithin 1 sec after they are released. The settling time is slightly larger for articles released hen θ is maximum, even though at that time both TKE and vertical fluctuations are near their minimum level. At any rate, the flo is not active enough to suort the article of this size as in Case 1. For D = 0.035mm more articles survive, though a large amount of them settle shortly after they are released. With the smallest articles (D=0.02mm), more than 50% articles survive, even after to oscillating cycles. So, as already seen in Case 1, the susension rate is very sensitive ith the size of articles. The largest survival rate is obtained by releasing the articles at t3. This relation can be seen even clearly for Case 3 in figure 16. Overall, the survival rate in Case 1 and Case 2 is roughly the same. Not so in Case 3, here the results deends on the release time (figure 16). Particles released at t1 and t2 have the same chance of being susended 20

21 as in the steady case, hile articles released at t3, t4 and t4 have a much higher survival rate. The time develoment of the mean article osition in the streamise direction, x, for Case 2 and 3 is shon in figure 17. The mean is taken only for the 0.02mm size articles that are still afloat after to oscillating cycles. Figure 18 shos the corresonding mean article velocity in the same direction, u. No significant differences are found in the mean x osition ith regard to the release time. Desite the non-linear nature of the flo, no drift is observed in Case 2. The influence of the temoral variation of TKE and vertical velocity fluctuations is thus small on the article motions in the streamise direction. The mean horizontal velocity relaxes to the mean velocity of the flo after a cycle. The mean article osition, z, and velocity,, in the vertical direction are also comared in figure 19 and 20. The vertical elevation z, monotonically increases ith time. The sloe though is not uniform, clear evidence that the susension rate is deendent on the hase. That can be seen by the eaks at t ~ 2, 6, 10, and 14 sec in figure 20 in the time evolution of. Since these times are corresondent ith the time of flo reversal and time of eaks in rms, it can be argued that the vertical movement of the susended articles is greatly affected by the vertical turbulent fluctuations. The articles resonses to the vertical flos can be seen also clearly in Case 3. It shos that z raidly increases near t ~ 6 sec (figure 19). At the same time dislays a eak (figure 20), hich is corresondent to the eak in distribution (figure 13). After one rms 21

22 cycle, the effect is reduced, since the number of articles in the oscillating layer is reduced. The individual article motions near the time of flo reversal are examined in connection ith the local flos in figure 21. The trajectories of five 0.02 mm articles are shon at six times from t = 4.9 ~ 7.0 sec for Case 2 flos. The trajectories and flo vectors are lotted in the x-z lane here the ranges are 1.0 ~ 1.25 m in x-direction and 0 ~ m in z-direction at y = 0.15 m. The comutational domain has size of only 0.6 m in x- direction, but since the eriodic condition are alied in the lateral boundaries the ositions can be extended beyond this range. The figure shos that the flo at t = 4.86 sec is directed uniformly toard the ositive x-direction and the five articles are sread near the bottom (< m). The articles move horizontally ith the flo as shon in the frame at t = 5.57 until the flo near the bottom is close to reversal. At t = 6.05 and 6.29 sec it can be seen that the vertical fluctuations become rather active. The articles resond by moving. Once the flo changes its direction the articles kee moving in the negative x-direction as shon in the frames of t = 6.77 and 7.01 sec. The result is that 4 out of five articles are ejected into the core of the flo during flo reversal. The ejection rate near flo reversal seems to suggest the existence of organized structures that form during this time. This hyothesis is further confirmed by the insection of the coherent structures at the time of release, shon in figure 22. A thorough analysis of these structures is beyond the scoe of this aer. Here e limit to comment them in relation to the observed settling rate. 22

23 At t1 and t2, in both Case 2 and Case 3, the coherent structures (streaks) are disorganized, resulting in a large settling rate. In Case 2, the flo becomes more organized as time rogresses, hich is reflected in a larger survival rate. In Case 3, the flo has a more comlex develoment. At t3, e can see relatively long and ell-organized structures, hich disaear at t4, hen the flo actually becomes very smooth. Hoever, instabilities soon develo in the flo, leading at t5 to the formation of regions of highly turbulent flo searated by still laminar regions. This, in our vie, exlains hy the survival rate is large for articles released at t3 and t5, but not at t4. In addition to the toology of the flo, at t3 the mean shear is maximum, resulting in a mean lift force of the order of 10% of the donard gravitational acceleration, hile it as usually negligible at other times. The vertical concentration rofiles calculated in the same ay as Case 1 are shon in figure 23 for 0.02mm articles for Case 2 and 3. Four different times are chosen during the second oscillating eriod corresonding to maximum deceleration (t = 14 sec for Case 2 and 14.5 sec for Case 3), minimum U T (t = 12 sec), maximum U T, maximum acceleration (t=18 sec for Case 2 and t=17.5 sec for Case 3). Also, mean rofiles are dran ith the average taken over the second oscillating eriod. The differences beteen Case 2 and 3 are clear since the number of sediment articles decreases raidly at elevations higher than 0.02 m for Case 2, hile the decreasing rate is exonentially uniform for the hole range of the elevations for Case 3. The differences are small at lo elevations (z < 0.02m) here the numbers of articles are comarable beteen the to cases, though the sloe is different. In the absence of a mean flo (Case 2), the turbulent fluctuations are confined ithin the oscillating boundary layer (figure 24); hence, the 23

24 sediment cannot escae into the laminar core. On the contrary, the fluctuations extend to the center of channel in Case 3. It must be remarked that the rofiles in Case 2 are not stable yet, since the number of articles slightly increases at high elevations as time increases. The mean rofiles during the second oscillating eriod are comared in figure 25 using articles released at different initial times, t1, t3, and t5 for Case 3. A field measurement of SSC rofile under ave and current conditions 41 is also comared. The rofiles in figure 25 are normalized by the values at the loest elevations, here the loest elevation of the field rofile is 1.38cm. The exerimental conditions ( u t =1.1 cm/s, U 0 =0.28 m/s) did not match exactly our values. Desite that, the rofiles calculated from our exeriment are comarable ith the measurements. The sloes are in reasonable agreement, esecially for the articles released at t1 that had more time to relax toards seudo equilibrium. IV. CONCLUSIONS The motion of sediment articles under turbulent oscillating and steady flos is imortant in assessing the large-scale transort of sediments in coastal areas. In the resent aer, e have studied numerically the movement of sediments released at the edge of the viscous sublayer in a turbulent channel flo. The turbulent fields ere generated numerically using LES. Three flo conditions ere studied, alying a steady and unsteady ressure gradient to a fluid contained in a channel hose size as 0.6m x 0.2m x 0.2m. The Reynolds number of the steady flo based on the friction velocity and 24

25 height of the channel as While this value does not come close to the values observed in the field, it is knon that the hysics of the near all turbulence, of interest here, is ell reroduced. The driving conditions for the ure oscillating case generated a flo ith a maximum velocity in the middle of the channel of 0.4 m/s, having a eriod of 8 seconds. These values are reresentative of conditions found in shallo aters under a steady sell. The third flo as obtained by forcing the fluid ith the sum of the driving conditions of the to revious cases. The amlitude of the free stream oscillation as tice its mean value, a common feature of the coastal environment, leading to flo reversal near the bottom. We calculated the motion of individual sediment grains using a variant of the Maxey and Riley equation, neglecting article-article interactions and assuming that the flo is not affected by the susended sediments. This is justified if the concentration of sediment is small. We varied the size of the articles (ranging from 0.05 to 0.02 mm) keeing the density constant to 2650 kg/m 3, the latter being the density of quartz. The articles ere released at the edge of the viscous sublayer, 0.95 mm from the flat bottom, regardless of the size of the articles. At that height, under steady driving condition, the ratio beteen the rms vertical-velocity fluctuations and the settling velocity varies from 1 for the largest articles to 6.25 for the smallest. Under steady driving conditions, the articles ere folloed for 18 seconds, until a seudo steady state as reached. It as found that most of the articles ith size 0.05mm droed to the bottom ithin one second hile about 17% and 60% of the mm and 0.02 mm articles resectively ere still in susension after 18 seconds. The susended articles moved ith the local mean velocity of the flo. We thus conclude that the to settling velocity ratio necessary to achieve significant resusension should exceed 2. At the end rms 25

26 of the exeriment, the concentration of articles ith height decreased exonentially ith height, suggesting that the effect of turbulence on average can be catured ith a simle eddy viscosity model. We also looked at the interaction of coherent structures near the all ith the susended sediments. We gathered evidence that ulifting of articles across the buffer layer is correlated to coherent structures. Hoever, our results did not suort the idea that eddies efficiently tras articles. Overall, the article velocities track the flo velocities ell, ith the excetions of regions of large gradients. The article movements ere also investigated under oscillating and combined flo conditions. During one oscillating eriod, to eaks are found in rms and TKE distributions in the ure oscillating case, hile only one eak is found in the ulsating case, near the time of maximum deceleration. Significant susension rates ere observed only for the smallest articles. For the urely oscillating case, the cloud of susended sediments remains confined ithin the oscillating turbulent boundary layer, hile in the ulsating case the sediments ended u occuying the entire column. The ulift of articles correlates ell ith the eaks in rms fluctuations. The survival rate as strongly correlated to the toology of the coherent structures at the time of release. Long, relatively smooth coherent structures ere associated to the highest susension rate. Thus e can consider the susension mechanism as a to-ste rocess: at the beginning, the organized motion of the coherent structures moves the articles aay from the all into the buffer layer; there, the articles are further lifted by the turbulent eddies. Overall, the susension rate as much larger in the ulsating case than in either the steady or oscillating case. 26

27 ACKNOWLEDGMENTS The numerical code used to simulate the turbulent flo is a arallel version of a serial code originally develoed by Dr. E. Balaras, hom e thank for sharing it ith us. We ould like to thank Dr. A. Provenzale, Dr. G. Passoni and Dr. J. von Harderberg for useful discussions. This research as suorted by NSF under grant OCE Comutations ere suorted by NSF grant APPENDIX: Some comments regarding icku functions in unsteady flos In this aendix, e discuss some general imlications of the results discussed here ith regard to icku rates 19. The icku rate is defined as the uard flux of sediment from the bottom due to the flo over it.. In the regime under study, the drag over the articles forces them to tumble along the bottom in a rather comlex motion. Because of the collisions, a certain number N of articles er unit area er unit time are ushed uards, u to a height z 0 that deends, inter alia, on the magnitude of the bottom stress. Once detached from the bottom, the articles can be icked u by the turbulent flo and brought into susension ith a robability P that ill deend on the intensity of the turbulent fluctuations at z 0. The flux can be ritten as F = NP. 27

28 In a steady flo, our data indicates that P deends on ) /. Since in a steady + rms ( z 0 s turbulent boundary layer, the distribution of the rms vertical velocity fluctuations follos a universal rofile hen exressed in all units, the flux can be ritten as F + = N( θ, D ) P( θ, D + ), τ θ =, (A1) ρ ( s 1)gD + + here e have assumed that z + = z ( θ, D ). The idely used van Rijn s formula 42 is of 0 0 this kind. When the flo is unteady, the common ractice is to relace the stress in (A1) ith the instantaneous stress. This requires that both the roerties of the tumbling layer ( N, z0 ) and the entrainment robability P adjust immediately to the driving conditions. The former assumtion is justifiable if the flight time of the articles is small relative to characteristic time of the unsteadiness, but the latter is clearly violated in the flos considered here, esecially in Case 3, here a strong asymmetry as observed in P. REFERENCES 1 J. F. A. Sleath, Turbulent oscillatory flo over rough beds, J. Fluid Mech. 182, (1987). 2 B. L. Jensen, B. M. Sumer, and J. Fredsoe, Turbulent oscillatory boundary layers at high Reynolds numbers, J. Fluid Mech. 206, (1989). 3 S. Tardu, G. Binder and R. F. Balckelder, Turbulent channel flo ith largeamlitude velocity oscillations, J. Fluid Mech. 267, 109 (1994). 4 A. Scotti and U. Piomelli, Numerical simulation of ulsating turbulent channel flo, Phys. Fluids 13(5), (2001). 28

29 5 P. G. Saffman, A model for inhomogeneous turbulent flo, Proc. R. Soc. Lond. A317, (1970). 6 B. E. Launder and B. I. Sharma, Alication of the energy dissiation model of turbulence to the calculation of flo near a sinning disc, Letters in Heat and Mass Transfer 1(2), (1974). 7 S. Tjerry, Morhological calculations of dunes in alluvial rivers, Ph.D. thesis. Technical University of Denmark (1995). 8 D. C. Wilcox, Turbulence modeling for CFD, second edition, DWC industries INC. (1998). 9 A. Scotti and U. Piomelli, Turbulence models in ulsating flos, AIAA Journal 40(3), 537 (2002). 10 G. Vittori and R. Verzicco, Direct simulation of transition in an oscillatory boundary layer, J. Fluid Mech. 371, (1998). 11 G. Scandura, G. Vittori, and P. Blondeaux, Three-dimensional oscillatory flo over stee riles, J. Fluid Mech. 412, (2000). 12 P. Moin and K. Mahesh, Direct numerical simulation - A tool in turbulence research, Annu. Rev. Fluid Mech. 30, (1998). 13 P. Moin and J. Kim, Numerical investigation of turbulent channel flo, J. Fluid Mech. 118, (1982) 14 R. S. Rogallo and P. Moin, Numerical simulation of turbulent flos, Ann. Rev. Fluid Mech. 16, (1984) 15 M. Germano, U. Piomelli, P. Moin, and W. H. Cabot, A dynamic subgrid-scale eddy viscosity model, Phys. Fluids A3(7), (1991). 29

30 16 D. K. Lilly, A roosed modification of the Germano subgrid-scale closure method, Phys. Fluids A4(3), (1992). 17 U. Piomelli, High Reynolds number calculations using the dynamic subgrid-scale stress model, Phys. Fluids A5(6), (1993). 18 J. Fredoe and R. Deigaard, Mechanics of coastal sediment transort, Advanced series on Ocean engineering. 3, World Scientific, Sigaore (1992). 19 P. Nielsen, Coastal bottom boundary layers and sediment transort, Advanced series on Ocean engineering, 4, World Scientific, Sigaore (1992). 20 K. H. Andersen, The dynamics of riles beneath surface aves and Toics in shell models of turbulence, Ph.D. thesis. Technical University of Denmark (1999). 21 Y. S. Chang and D. M. Hanes, Field observation and model investigation of the susended sediment distribution over riled seabeds, Submitted to Continental Shelf Research (2002). 22 E. A. Zedler and R. L. Street, Large-eddy simulation of sediment transort - currents over riles, J. Hydraulic Engineering 127(6), (2001). 23 M. R. Maxey and J. Riley, Equation of motion for a small rigid shere in a nonuniform flo, Phys. Fluids 26(4), (1983). 24 S. Pedinotti, G. Mariotti, and S. Banerjee, Direct simulation of article behavior in the all region of turbulent flos in horizontal channels, Int. J. Multihase Flo 18(6), (1992). 25 A. M. Ahmed and S. Elghobashi, Direct numerical simulation of article disersion in homogeneous turbulent shear flos, Phys. Fluids 13(11), (2001). 30

31 26 V. Armenio and V. Fiorotto, The imortance of the forces acting on articles in turbulent flos, Phys. Fluids 13(8), (2001). 27 Q. Wang and K. D. Squires, K.D. Large eddy simulation of article-laden turbulent channel flo, Phys. Fluids 8(5), (1996). 28 V. Armenio, U. Piomelli, and V. Fiorotto, Effect of the subgrid scales on article motion, Phys. Fluids 11(10), (1999). 29 Q. Wang and K. D. Squires, Large eddy simulation of article deosition in a vertical turbulent channel flo, Int. J. Multihase flo 22(4), (1996). 30 P. L. Wiberg and J. D. Smith, A theoretical model for saltating grains in ater, J. Geohys. Res. 90(4), (1985). 31 T. G. Drake and J. Calantoni, Discrete article model for sheet flo sediment transort in the nearshore, J. Geohys. Res. 106(C9), (2001). 32 O. S. Madsen, Mechanics of cohesionless sediment transort in coastal aters, Coastal Sediments'91, (1991). 33 E. Balaras, Finite-difference comutations of high Reynolds-number flos using the dynamic subgrid-scale model, Theor. Com. Fluid Dyn. 7(3), (1995). 34 S. Balachandar and M. R. Maxey, Methods for evaluating fluid velocities in sectral simulations of turbulence, J. Com. Phys. 83, (1989). 35 M. Hino, M. Kashiayanagi, A. Nakayama, and T. Hara, Exeriments on the turbulent statistics and the structure of a recirocating oscillatory flo, J. Fluid Mech. 131, (1983). 36 J. Kim, P. Moin, and R. Moser, Turbulence statistics in fully develoed channel flo at lo Reynolds number, J. Fluid Mech. 177, (1987). 31

32 37 R. A. Bagnold, The flo of cohesionless grains in fluids, Phil. Trans. Roy. Soc. Lond. 249, 235 (1954). 38 N.A. Patankar, T. Ko, H.G. Choi, and D.D. Joseh, A correlation for the lift-off of many articles in lane Poiseuille flos of Netonian fluids, J. Fluid Mech. 445, (2001). 39 S. K. Robinson, Coherent motions in the turbulent boundary layer, Ann. Rev. Fluid Mech. 23, (1991). 40 P. R. Salart and B. S. Baldin, Direct simulation of a turbulent oscillating boundary layer, In Turbulent Shear Flos 6 (Sringer, Berlin) (1987). 41 T. H. Lee and D. M. Hanes, Comarison of field observations of the vertical distribution of susended sand and its rediction by model, J. Geohys. Res. 101(c2), (1995). 42 L. C. Van Rijn, Sediment ick-u functions, J. Hydraulic Engineering 110(10), (1984). 32

33 List of Tables Table 1: Flo conditions Table 2: Sediment arameters 33

34 u t (m/s) U (m/s) Re t Re T (sec) Mash s Case x130x98 Case x130x98 Case x130x98 Table 1 34

35 D (mm) s 2 = ( s 1) gd /18ν (m/s) Re D / ν D = u t Table 2 35

36 List of Figures Figure 1: Schematic of the comutational domain. Figure 2: Root-mean-square value of velocity fluctuations (Case 1), circles: streamise, asterisks: sanise, squares: vertical, uer anel: normalized by channel height, loer anel: in all units. Figure 3: Number of surviving articles that remain in the flo, Solid: D=0.02mm, dashed: 0.035mm, dotted: 0.05mm. Figure 4: (a): mean vertical flo velocity at article ositions,, (solid) and mean article velocity,, (dashed) normalized by settling velocity,, 0.05mm articles. (b): f s f (solid) and (dashed) normalized by s, 0.02mm articles. (c): mean vertical osition, z, solid: 0.05mm, dashed: 0.02mm. Figure 5: Mean article osition and velocities. Solid: 0.02mm, dashed: 0.035mm. Figure 6: Vertical rofiles of sediment concentration, squares: t = 3 sec, asterisks: t = 10 sec, circles: t = 18 sec. Figure 7: Horizontal distribution of the vertical flo velocity comonents on x-y lane at z = 0.95mm and at t=0, Case 1. Loer anel: magnified vie of rectangle S in uer anel. Figure 8: Velocity distribution on the y-z lane along A-B in figure 10 at t = 0 sec, and the initial locations of the four articles selected to be traced. Arros: velocity vector, circle: P1, square: P2, triangle: P3, diamond: P4. Figure 9: Particle trajectories of P1 and P2 ith the flo on y-z lane at four different times. P1: left column, P2: right column. 36

37 Figure 10: Particle trajectories of P3 and P4 ith the flos on y-z lane at four different times. P3: left four anels, P4: right four anels. Figure 11: Comarison of the vertical comonents of the flo velocity, article velocities, and the article ositions of P1, P2, P3, and P4. solid: flo velocity at the article osition, Dashed (thinner): article velocity, Dashed (thicker): article osition. Figure 12: (a) Time develoment of U T : solid, 0.95 mm: solid, TKE at z = 0.95 mm: dashed, Case 2. Figure 13: (a) Time develoment of U T : solid, 0.95 mm: solid, TKE at z = 0.95 mm: dashed, Case 3. Figure 14: Vertical distribution of rms and TKE, Case 2. U I : dotted, θ : dashed; (b) U I : dotted, θ : dashed; (b) rms at z = rms at z = Figure 15: Number of articles that remain in the flo for D=0.02mm (a), for D = 0.035mm (b), and for D = 0.05mm (c), Case 2. Figure 16: Number of articles that remain in the flo for D=0.02mm (a), for D = 0.035mm (b), and for D = 0.05mm (c), Case 3. Figure 17: Time develoment of Figure 18: Time develoment of x for Case 2 and 3, D=0.02mm. u for Case 2 and 3, D=0.02mm. Figure 19: Time develoment of z for Case 2 and 3, D=0.02mm. Figure 20: Time develoment of for Case 2 and 3, D=0.02mm. Figure 21: Examle of the article trajectories near the time of flo reversal, Case 2, D = 0.02 mm. Figure 22: Contour lots of vertical velocity fluctuations ( ) in the horizontal lane at initial elevation at t = 0 sec (Case 1) and t = t1, t2, t3, t4, and t5 (Case 2 and 3). 37

38 Figure 23: Vertical rofiles of the sediment articles for Case 2 and 3, squares: t =12sec, asterisks: t=14sec for Case 2, 14.5sec for Case 3, circles: t=16sec, triangles: t=18sec for Case 2, 17.5sec for Case 3, solid: averaged over the oscillating eriod. Figure 24: Root-mean-square value of velocity fluctuations (Case 2), circles: streamise, stars: sanise, squares: vertical. Figure 25: Comarison of the sediment article rofiles ith measured SSC rofile, normalized by the values at the loest elevations. squares: released at t1, asterisks: released at t3, circles: released at t5, solid: measurements. 38

39 Figure 1: Yeon S. Chang and Alberto Scotti. Physics of Fluids. 39

40 Figure 2: Yeon S. Chang and Alberto Scotti. Physics of Fluids. 40

41 Figure 3: Yeon S. Chang and Alberto Scotti. Physics of Fluids. 41

42 Figure 4: Yeon S. Chang and Alberto Scotti. Physics of Fluids. 42

43 Figure 5: Yeon S. Chang and Alberto Scotti. Physics of Fluids. 43

44 Figure 6: Yeon S. Chang and Alberto Scotti. Physics of Fluids. 44

45 Figure 7: Yeon S. Chang and Alberto Scotti. Physics of Fluids. 45

46 Figure 8: Yeon S. Chang and Alberto Scotti. Physics of Fluids. 46

47 Figure 9: Yeon S. Chang and Alberto Scotti. Physics of Fluids. 47

48 Figure 10: Yeon S. Chang and Alberto Scotti. Physics of Fluids. 48

49 Figure 11: Yeon S. Chang and Alberto Scotti. Physics of Fluids. 49

50 Figure 12: Yeon S. Chang and Alberto Scotti. Physics of Fluids. 50

51 Figure 13: Yeon S. Chang and Alberto Scotti. Physics of Fluids. 51

52 Figure 14: Yeon S. Chang and Alberto Scotti. Physics of Fluids. 52

53 Figure 15: Yeon S. Chang and Alberto Scotti. Physics of Fluids. 53

54 Figure 16: Yeon S. Chang and Alberto Scotti. Physics of Fluids. 54

55 Figure 17: Yeon S. Chang and Alberto Scotti. Physics of Fluids. 55

56 Figure 18: Yeon S. Chang and Alberto Scotti. Physics of Fluids. 56

57 Figure 19: Yeon S. Chang and Alberto Scotti. Physics of Fluids. 57

58 Figure 20: Yeon S. Chang and Alberto Scotti. Physics of Fluids. 58

59 Figure 21: Yeon S. Chang and Alberto Scotti. Physics of Fluids. 59

60 Figure 22: Yeon S. Chang and Alberto Scotti. Physics of Fluids. 60