Turbulent particle flux and momentum flux statistics in suspension flow

Transcription

1 WATER RESOURCES RESEARCH, VOL. 39, NO. 5, 1139, doi: /2001wr001113, 2003 Turbulent particle flux and momentum flux statistics in suspension flow D. Hurther Woods Hole Oceanographic Institution, Woods Hole, Massachusetts, USA U. Lemmin Laboratoire d Hydraulique Environnementale, Ecole Polytechnique Federale Lausanne, Lausanne, Switzerland Received 10 December 2001; revised 10 January 2003; accepted 18 March 2003; published 31 May [1] The particle entrainment ability of coherent flow structures is investigated by comparing statistical properties of momentum flux u 0 w 0 and of turbulent mass fluxes c 0 u 0 and c 0 w 0 in suspension, open-channel flow under capacity charge conditions. The quadrant repartitions of these quantities as a function of the corresponding threshold levels are estimated. A cumulant discard probability density distribution is used to calculate the theoretical quadrant dynamics. Good agreement between the third-order model and the experimental results is found for all investigated quantities in the wall and intermediate flow regions. In the free surface domain, the increase of intermittency of the momentum and mass transport processes leads to small discrepancies between the model and the experimental results. The quadrant distributions of the horizontal and vertical turbulent mass fluxes are dominated by the same two quadrants as the momentum flux u 0 w 0. Ascendent mass flux events are found to correlate with ejections over the entire water depth. A dynamical equilibrium between the shear stress production term and the turbulent energy dissipation term is found in the intermediate flow region where the value of the normalized vertical flux of turbulent kinetic energy in suspension flow corresponds well with the one observed in clear water flows. This points toward a universality of the normalized vertical flux of turbulent kinetic in highly turbulent boundary layers. The suspended particle transport capacity of coherent structures is directly quantified from the estimation of the conditionally sampled terms of the particle diffusion equation. Coherent structures are found to play a dominant role in the mass transport mechanism under highly turbulent flow conditions in open-channel flows. INDEX TERMS: 1815 Hydrology: Erosion and sedimentation; 4558 Oceanography: Physical: Sediment transport; 4568 Oceanography: Physical: Turbulence, diffusion, and mixing processes; 1894 Hydrology: Instruments and techniques; KEYWORDS: sediment transport, coherent structures, particleturbulence interaction, suspension flow, quadrant method Citation: Hurther, D., and U. Lemmin, Turbulent particle flux and momentum flux statistics in suspension flow, Water Resour. Res., 39(5), 1139, doi: /2001wr001113, Introduction [2] In recent years, the dynamics of coherent structures in open-channel flow have been studied in great detail in order to understand their generation mechanism and their role in different turbulence-related processes such as mixing, transport, and gas transfer at the air-water interface. A large number of the investigations have been concerned with the quantification of shear stress dynamics in boundary layers over smooth and rough beds. They have revealed the presence of dominant turbulence-generating events identified as ejections and sweeps. These events, which have been well documented, are chacaterized as highly intermittent with large amplitudes and turbulent timescales. However, no consensus has been found yet concerning the scaling laws of these structures. Models are based on inner region, Copyright 2003 by the American Geophysical Union /03/2001WR ESG 2-1 outer region, or mixed flow quantities, and most rely on some empirical assumptions. [3] Several studies have tried to determine the link between these coherent flow structures and particle transport in different turbulent flows. Particles are found to respond to different flow structures in suspension transport and in bed load transport. Bed load transport may arise from pressure fluctuations induced by sweeps at the bed, while ejections drive the particle movement in suspension flows. Soulsby et al. [1994] observed in a tidal current that the vertical sand flux is dominated by the dynamics of largescale turbulent structures and that the damping of the turbulent kinetic energy is due to the presence of suspension [Barenblatt, 1955; Smith and McLean, 1977]. Niño and Garcia [1996] and Garcia et al. [1997] showed in their open-channel flow study over smooth and transitionally rough beds that ejections are responsible for particle entrainment into suspension if the particles are completely immersed in the viscous sublayer. They concluded that this

2 ESG 2-2 HURTHER AND LEMMIN: PARTICLE FLUX AND SHEAR STRESS STATISTICS mechanism is independent of the relative roughness. This result coincides with the earlier conclusions obtained by Grass [1971] and Grass et al. [1993], who observed that ejections and sweeps occur in a similar manner in shear flows over smooth and rough beds. While the last two papers have examined the importance of ejections in lifting up particles into suspension, the studies of Heathershaw and Thorne [1985], Hogg et al. [1995], Séchet and LeGuennec [1999] investigated the role of sweeps in the eventdriven bed load transport of open-channel flows and tidal flows. [4] The results of these studies confirm the importance of ejections and sweeps and their effects near the bed when particle suspension transport in highly turbulent boundary layers is considered. Assessing particle transport only from shear stress ru 0 w 0 obviously lacks insight into the underlying physical processes. In the following we will omit the term r when we refer to shear stress. [5] The aim of the present study is to provide quantitative information on the role of coherent structures in the particle transport mechanism. For this purpose, we have applied a quadrant analysis to u 0 w 0 and particle flux simultaneously measured over the entire height of the boundary layer. Our motivations for the use of this well-known conditional sampling method are the following: [6] We observe from instantaneous flow field measurements that flow patterns delimited by high-threshold-level values are quasi-coherent flow structures surrounded by a more randomly distributed fluctuating velocity field. (An example of this observation is shown in Figure 10, to be discussed in detail later, where the contours of high u 0 w 0 structures are represented by the solid lines.) The corresponding superimposed instantaneous fluctuating velocity regions persist in terms of their direction throughout the turbulent event. [7] The present study is concerned with a statistical analysis of the turbulence-particle interactions in openchannel suspension flow. We will not characterize the interaction forces between the two phases but will focus on the statistical properties of conditionally sampled shear stress and particle fluxes. Since the momentum and particle fluxes are measured simultaneously, the correlations between the two fluxes will be evaluated. The statistical quantities calculated using the quadrant method (experimental approach) will be compared with the results obtained from a statistical model (theoretical approach). The theoretical expressions of the probability density functions allow us to identify the relevant velocity moments dominating the observed dynamics. [8] Particular attention will be given to the particle fluxes in the intermediate and free surface flow regions where outer flow parameters are found to be more appropriate for the scaling [Nezu and Nakagawa, 1993; Cao, 1999]. For this purpose, an extension of a statistical approach (Nakagawa and Nezu [1977] (hereinafter referred to as NN77), Katul et al. [1994], and Chu et al. [1996]) for the investigation of the shear stress statistics to mass fluxes is proposed. In a previous clear water, open-channel flow study we have applied this statistical model to determine the theoretical values of the quadrant dynamics for covariance terms u 0 w 0 /u 0 w 0 and v 0 w 0 /v 0 w 0 using instantaneous three-dimensional (3-D) velocity profile measurements [Hurther and Lemmin, 2000]. The model results were in good agreement with experimental results obtained with the quadrant threshold technique developed by Lu and Willmarth [1973]. From the theoretical expressions of the probability density functions, the relevant cumulants for the quadrant repartition were obtained. [9] Here this conditional statistical model will be applied to the variable u 0 w 0 and to the turbulent particle fluxes c 0 u 0 and c 0 w 0 in order to calculate their relative quadrant repartition in the corresponding planes. By comparing the statistical properties of shear stress and turbulent particle fluxes for specific quadrants, the contribution by ejections, sweeps, and inward and outward interactions to the mean particle entrainment c 0 w 0 can be evaluated. On the basis of these results, we will validate the existence of an extended equilibrium region (0.25 < z/h < 0.75) over which the normalized vertical flux of turbulent kinetic energy is constant. The obtained constant is in good agreement with values found in clear water flows [Hurther and Lemmin, 2000]. As a result, a wall similarity concept valid in both clear water and suspension flows is discussed. [10] Conditional sampling will then be applied to the terms of the advection-diffusion equation in order to quantify the transport capacity of coherent structures. Thus, combined with flow visualization, the contribution to the transport in the particle concentration profile will be estimated as a function of the threshold level. This is usually considered as the parameter delimiting coherent structures in the flow field. 2. Experimental Details [11] The data which will be analyzed here were obtained by Cellino [1998] in a laboratory study on suspension flow. The experiments were carried out in a recirculating tilting open-channel, 16.8 m long and 0.6 m wide [Cellino et al., 1996]. The channel bed was rough, with a mean roughness of 4.8 mm. Special care was taken to ensure steady and uniform flow conditions. [12] The acoustic particle flux profiler (APFP) [Shen and Lemmin, 1997; Shen and Lemmin, 1999] was employed to measure quasi-instantaneous particle flux profiles with a sampling frequency of 25 Hz and a record length of 180 s. The sampling volume is smaller than mm, which permits us to resolve small-scale turbulence structures in the studied flow conditions. Conventional nonintrusive acoustic backscattering systems are limited to a maximal concentration range of 5 kg m 3 by mass or 0.2% by volume over a 1 m measuring range. Using an acoustic attenuation compensation algorithm, the two-transducer system proposed by Shen and Lemmin [1997] is capable of profiling high concentration as high as 120 kg m 3 over a measuring range of 40 cm. This is a versatile tool for simultaneously measuring the profiles of the instantaneous velocity and the concentration over the entire water depth of a suspension flow. The APFP instrument was located 13 m from the entrance at the centerline of the channel where turbulence is well developed. [13] Figure 1 is an example of an APFP measurement sequence used in the present work. It shows the instantaneous isocontour lines of the conditionally sampled u 0 w 0 patterns (see section 4.1 for the conditional sampling

3 HURTHER AND LEMMIN: PARTICLE FLUX AND SHEAR STRESS STATISTICS ESG 2-3 Figure 1. Isocontours of momentum flux patterns u 0 w 0 and vertical turbulent particle fluxes at the centerline of the open-channel flow [Shen and Lemmin, 1997]. technique) over the whole water depth and the corresponding vertical turbulent particle flux (drawn in gray scales). The horizontal turbulent particle flux which is not shown will be treated in the same way in order to quantify the correlation of turbulent momentum flux u 0 w 0 to c 0 u 0. [14] The hydraulic parameters given in Table 1 characterize a fully developed turbulent subcritical flow of depth h = 12 cm with a bed friction velocity of 3.9 cm s 1. Quartz particles of d 50 = 135 mm and specific density of 2.65 were gradually added to the flow until a layer of particles, remaining stable during the experiments, appeared on the bed of the channel completely covering the bottom roughness. No more particles were added from this moment onward because the capacity charge equilibrium condition was reached. In that way the maximum suspension transport capacity of the flow is achieved. Any further supply of particles will only increase the thickness of the deposition layer on the bed. The reference concentration C a was measured by a suction device under isokinetical conditions at the water depth z/h = 0.05 [Cellino, 1998]. [15] An important aspect in the interpretation of the velocity measurements is the relative motion between the particles and the water. In the present flow conditions where the volumetric particle concentration is lower than 2%, the ratio between the turbulent microscale and the particle diameter exceeds a value of 100 [Shen and Lemmin, 1999; Cellino, 1998]. Fortier [1967] and Hinze [1975] have shown that this value is high enough to assure that the relaxation time of the suspended particles is negligible compared to the temporal turbulent microscale. Hence the turbulent motion of particles is following the turbulent flow with negligible inertial lag. 3. Theoretical Aspects [16] In this section we briefly present the theoretical expressions of the probability density functions used to calculate the quadrant contributions of the three investigated quantities. [17] We define the following variables: u 0, w 0, c 0 are the zero mean fluctuating longitudinal, vertical velocity, and concentration p components, respectively; _ u, w _, _ c are equal to u 0 = ffiffiffiffiffi p u 02, w 0 = ffiffiffiffiffiffi p w 02, c 0 = ffiffiffiffiffi c 02, respectively. We shall quantify the contributions of the relative shear stress for the cross-product term e 1 = u 0 w 0 /u 0 w 0 and the contribution of the relative mass fluxes for e 2 = c 0 u 0 /c 0 u 0 and e 3 = c 0 w 0 /c 0 w 0. Figure 2 shows the orientation of the quadrants in the respective planes. Three joint probability density functions, p 1 ð _ u ; w _ Þ, p 2 ð _ c ; _ u Þ, and p 3 ð _ c ; w _ Þ, given as the inverse Fourier transforms of the characteristic functions q 1 ð _ u ; w _ Þ, q 2 ð _ c ; _ u Þ, and q 3 ð _ c ; w _ Þ, respectively, can be expressed as functions of the moment and cumulant generating functions in which m 1; jk ¼ _ u j w _k, m 2; jk ¼ _ c j_k u and m 3; jk ¼ _ c j w _k denote the moments of ( j + k)order and k 1, jk, k 2, jk and k 3, jk correspond to the cumulants of ( j + k)th order. By limiting these cumulant expansion series to an order of three, NN77 determined the conditionally sampled probability densities of covariance events e 1 over the four quadrants from a high-order cumulant discard Gram-Charlier probability density function. The mathematical manipulation is described in detail by NN77. The following general equations are given: p i;2 ðe i Þ ¼ p in ðe i Þþj i ðe i Þ p i;1 ðe i Þ ¼ p in ðe i Þþj þ i ðe i Þ p i;4 ðe i Þ ¼ p in ðe i Þ j i ðe i Þ ðe i > 0Þ i ¼ 1; 2 ð1þ ðe i < 0Þ p i;3 ðe i Þ ¼ p in ðe i Þ j þ i ðe i Þ where the index q in p i,q denotes the quadrant index (1 to 4) in the ithplane with planes 1, 2, and 3 corresponding to the (u 0, w 0 ), (c 0, u 0 ), and (c 0, w 0 ) planes, respectively. The probability density p in (e i ) is directly developed from the Table 1. Hydraulic Parameters for the Experiment Parameter Value Q, m 3 s h, cm 12 U, cms u *,cms S, Re h, Fr h d 50, mm r S, kg m w 0,mms 1 12 C a,kg m k s +

4 ESG 2-4 HURTHER AND LEMMIN: PARTICLE FLUX AND SHEAR STRESS STATISTICS Figure 2. Division of events: (a) Momentum fluxes. (b) Horizontal turbulent particle flux. (c) Vertical turbulent particle flux. corresponding bivariate normal distribution (second-order function). The nonconditionally sampled probability function of shear stress is p i (e i )=p i,1 (e i )+p i,2 (e i )+p i,3 (e i )+ p i,4 (e i )=2p in with i = 1, 2, 3 and 8 p in ðe i Þ ¼ r i 2p exp r K 0 ðjt i jþ ð it i Þ ð1 ri 2 Þ 1=2 j þ i ðe i Þ ¼ r i 2p exp r jt i j 1=2 ð it i ÞK 1=2 ðjt i jþ ð1 ri 2 Þ ð1 þ r i Þ Sþ i 3 þ Dþ i jt i j 2 r i >< Si þ þ D þ i 3 j i ðe i Þ ¼ r i 2p exp ð r jt i j 1=2 it i ÞK 1=2 ðjt i jþ ð1 þ ri 2 Þ ð1 r i Þ S i 3 þ D i jt i j 2 þ r i Si þ D i 3 t i ¼ r ie i ð1 ri 2 Þ ; S i ¼ 1 2 k i;03 k i;30 ; D i ¼ 1 2 k i;21 k >: i;12 where r i are the corresponding correlation coefficients and K 0 (t) is the zeroth-order modified Bessel function of the second kind. [18] The standard deviation, skewness, and flatness factors of the e i terms can be approximated by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s i ¼ ðe i e i Þ 2 S ei ð ¼ e i e i Þ 3 s 3 ¼ i ¼ 1 m i;11 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m i;22 m 2 i;11 1 3=2 hm i;30 m i;03 þ 9m i;12 m i;21 m i;22 m 2 i;11 i þ 3 m i;31 þ m i;13 2mi;11 5m 2 i;11 3m i;22 þ 9 ð2þ ð3þ From equation (1) we will calculate the first-order moment of each conditional probability density distribution as a function of the threshold level H i. By increasing the level of H i, progressively stronger fractional e i events will be selected. Their distribution over the different quadrants can be investigated using the following expressions: R i;q ðh i Þ ¼ R i;q ðh i Þ ¼ Z 1 H i Z Hi 1 e i p i;q ðe i Þde i H i 0 i ¼ 1; 2 q ¼ 2; 4 e i p i;q ðe i Þde i H i 0 i ¼ 1; 2 q ¼ 1; 3 This method is known as the u-w quadrant threshold technique [Lu and Willmarth, 1973] applied in this case to the model results. The parameters R i,q, evaluated from the probability densities, will be compared to those from experimental results in order to obtain information on the quadrant distribution of the relative covariances. The time fractions T i,q of these conditionally sampled events are estimated by T i;q ðh i Þ ¼ T i;q ðh i Þ ¼ Z 1 H i Z Hi 1 p i;q ðe i Þde i H i 0 i ¼ 1; 2 q ¼ 2; 4 p i;q ðe i Þde i H i 0 i ¼ 1; 2 q ¼ 1; 3 Consequently, those terms of the events having relative shear e i lower than the defined thresholds are called the hole event terms (Figure 2). R i,5 and T i,5 and are given by R i;5 ðh i Þ ¼ 1 X4 R i;q ðh i Þ T i;5 ðh i Þ ¼ 1 X4 q¼1 q¼1 ð4þ ð5þ T i;q ðh i Þ ð6þ F ei ð ¼ e i e i Þ 4 s 2 ¼ i 1 2 m i;40 þ m i;04 þ 16m i;31 m i;13 m i;22 m 2 i;11 h i þ 24 m i;21 m i;03 þ m i;12 m i;30 þ 18 m 2 i;22 þ 2 m 2 i;21 þ m2 i;12 30m 2 i;11 m2 i;22 3 m4 i;11 þ 24m2 i;11 þ 6 12r i m i;31 þ m i;13 9m i;21 m i;12 m i;30 m i;03 Þ 4. Statistical Properties of Momentum and Mass Fluxes 4.1. Probability Density Functions, Skewness, and Flatness Factors [19] The lines in Figure 3 represent the probability density functions p i (e i ), evaluated for the relative shear stress, as well as the horizontal and vertical turbulent mass fluxes. As mentioned in the previous section, all variables e i were treated independently. This point will be further

5 HURTHER AND LEMMIN: PARTICLE FLUX AND SHEAR STRESS STATISTICS ESG 2-5 Figure 3. Theoretical and experimental density probabilities of momentum flux and horizontal and vertical turbulent particle fluxes at different flow depths. discussed later on. The symbols in Figure 3 show the probability density functions of the same variables estimated from the corresponding experimental histograms. For each of them, 64 cells are taken over a range of H i values varying from 8 to 8. The results are shown at four flow depths corresponding to the inner, intermediate, and free surface flow regions. [20] Good agreement between the model results and the experimental results can be seen for the u 0 w 0 and the turbulent mass fluxes at all depths. The range of variation of the variables is of the same order of magnitude for all investigated relative quantities. Furthermore, only small discrepancies exist between the shape of the different probability density functions, indicating indentical statistical properties for u 0 w 0 and the turbulent mass fluxes. [21] This is confirmed when profiles of skewness approximated by equation (3) are compared with experimental ones (Figure 4a). Again, the experimental results are fairly well described by the model results. In this case the model results are not dependent on the choice of the analytical probability density function (i.e., a Gram-Charlier distribution), but rather depend on the relations between cumulants and moments [Antonia and Atkinson, 1973]. All three quantities are positively skewed with almost the same constant value of 4 (compared with a value of 0 in the case of a Gaussian distribution) over the whole water depth. Large positive values of these random variables are more frequent than negative ones. As for the shear stress dynamics, there is a dominant contribution of two quadrants (known as ejections and sweeps for the shear stress) compared with the remaining two. Since the mean horizon- Figure 4. (a) Skewness and (b) flatness profiles for shear stress, horizontal, and vertical mass fluxes.

6 ESG 2-6 HURTHER AND LEMMIN: PARTICLE FLUX AND SHEAR STRESS STATISTICS Figure 5. Theoretical and experimental quadrant repartition of relative momentum flux e 1 as a function of threshold H 1. tal mass flux is negative [Shen and Lemmin, 1999], the two dominant quadrants are quadrants 2 and 4, the same as for the shear stress. The mean vertical mass flux, however, is positive, and therefore quadrants 1 and 3 are the main contributors. We will calculate the exact relative contribution of all quadrants for each quantity and comment on their physical meaning in the next section. [22] The flatness factors are much higher than those for a Gaussian distribution, i.e., equal to 3 (Figure 4b). This emphasizes the highly intermittent character of both shear stress and mass fluxes. A certain difference of the flatness factors can be distinguished for z/h < 0.6, which may indicate that the mass flux events are not responding exactly to u 0 w 0 events. The detailed quadrant repartition of the relative contributions and time fractions as functions of the corresponding threshold levels discussed in the next section will clarify that observation. [23] It can be concluded that the nonconditionally sampled probability density functions can well be modeled by the first terms of equation (1), which only take into account second-order variables and therefore the corresponding bivariate normal distribution. This has been shown by Antonia and Atkinson [1973], who compared the second-order probability density function of u 0 w 0 to a fourth-order cumulant discard Gram-Charlier probability function. The difference between the two was found to be negligible. From the results shown here the same observation is also valid for the horizontal and vertical mass flux events, which occur with small values, but occasionally an event with larger amplitude occurs. The importance of these high-amplitude structures in the particle transport mechanism is quantified in section Quadrant Contribution of Shear Stress and Mass Fluxes [24] The quadrant repartition introduced in section 3 was separately analyzed for each quantity. Figures 5 7 show the relative magnitude of u 0 w 0 and horizontal and vertical mass fluxes, respectively, as functions of the corresponding selection criteria H i. The theoretical (from equation (2)) and experimental results are given for three different flow depths in the wall, intermediate, and free surface flow regions. The time fraction of the hole events at those three depths are also presented in each figure. The agreement between the theoretical and experimental results is good for all quantities at each depth in the inner and the intermediate Figure 6. Theoretical and experimental quadrant repartition of relative momentum flux e 2 as a function of threshold H 2.

7 HURTHER AND LEMMIN: PARTICLE FLUX AND SHEAR STRESS STATISTICS ESG 2-7 Figure 7. Theoretical and experimental quadrant repartition of relative momentum flux e 3 as a function of threshold H 3. flow regions. In the free surface flow region, small discrepancies are observed between the model and the experimental results. NN77 had attributed these discrepancies to the limitation of the cumulant expansion series to an order of 3. This order appears to be too low in the free surface region where the process becomes more intermittent, as indicated above. [25] The quadrant dynamics of the three investigated quantities is obviously dominated by the contribution of two quadrants. In Figure 5, ejections (quadrant 2) and sweeps (quadrant 4) dominate, in accordance with results given in the literature. For example, the experimental contributions of quadrants 2 and 4 in Figure 5 at z/h = 0.78 for H i = 0 are equal to 1.1 and 0.76, respectively, which are in agreement with the values of 1 and 0.65 at z/h = given by NN77. The evolution of the curves with respect to H 1 is also found to be identical to results of Raupach [1981]. They show high quadrant 2 and 4 contributions for large H 1 values while the interaction contributions (quadrants 1 and 3) vanish earlier. This trend can be explained by the larger tails of the conditional probability density functions (not shown here) of the ejection and sweep events. It implies that these occur with higher intermittency than the interaction events. [26] Another well-documented characteristic of the coherent large-scale structures is the distribution of the hole event time fraction of the relative contribution. For example, in Figure 5 at z/h = 0.42 and H 1 = 5, the hole event time fraction is equal to 85%, meaning that only 15% of the events still contribute to 40% of the shear stress u 0 w 0 in quadrant 2 and 35% of the shear in quadrant 4, revealing the short lifetime and large amplitudes of the turbulence producing events. These observations are in agreement with several experimental studies [Nakagawa and Nezu, 1981; Luchik and Tiederman, 1987] in clear water turbulent boundary layers. Here we have demonstrated that these characteristics of u 0 w 0 dynamics in suspension flow under capacity charge condition are very similar to those in clear water flow. [27] From the observed probability density functions, the important contributors to the horizontal mass fluxes are identified as quadrant 2 and 4 events (Figure 6). The quadrant 4 contribution is more important than the quadrant 2 contribution for any H 2, which indicates that the entrainment mass fluxes from ejections are higher than the downward particle fluxes from sweep events. An association of quadrant 4 horizontal mass flux structures with ejections can be suggested. When an ejection occurs (i.e., with u 0 <0 and w 0 > 0), it will lift up particles from a region of higher mean concentration to one of lower mean concentration. The same reasoning holds for the combination of quadrant 2 horizontal mass fluxes with sweeps. This hypothesis will be confirmed in the next section when the transport capacity of different event types will be estimated. [28] The quadrant contributions of the vertical mass fluxes are presented in Figure 7. Here quadrant 3 and 1 events are associated with ejections and sweeps, respectively. The experimental curves are fairly well described by the theoretical third-order model. The orders of magnitude of the relative contributions of vertical and horizontal mass fluxes versus H 2 and H 3 correspond to the relative u 0 w 0 contributions. Therefore the general quadrant dynamics, including the hole event time fraction, of the mass fluxes and u 0 w 0 are in good agreement. This indicates the importance of the dynamics of coherent large-scale structures in the particle transport dynamics Wall Similarity Concept in Suspension Flow [29] No significant differences between the quadrant distribution of u 0 w 0 in clear water and suspension flows have appeared in the results presented above. Therefore the validity of the concept of wall similarity which is directly related to the coherent structure dynamics will be investigated in suspension flows. The concept states that a dynamical equilibrium exists between the production term of turbulent kinetic energy and its dissipation rate over an extended flow region roughly corresponding to the intermediate flow domain (0.2 < z/h < 0.65). As a result, the vertical diffusion terms of turbulent kinetic energy and pressure fluctuations must compensate each other or be negligible in that flow region. The concept has recently been validated in clear water, uniform open-channel flows in the region of 0.25 < z/h < 0.75 where the normalized (with the bed friction velocity to the power of 3) flux remains constant with a value of ffi0.33, independent of the flow conditions [López and Garcia, 1999; Hurther and Lemmin, 2000].

8 ESG 2-8 HURTHER AND LEMMIN: PARTICLE FLUX AND SHEAR STRESS STATISTICS the measure volume is weak. This is a great simplification combined with a gain in accuracy in the determination of u *, particularly suited for field measurements. Figure 8. Profiles of third-order moments for different flow conditions. Experiment A: suspension flow, Re = 271,000, k + s =7,Fr = Experiment B: clear water flow, Re = 27,000, k + s = 34, Fr = Experiment C: clear water flow, Re = 49,600, k + s = 45, Fr = Re is Reynolds number, k + s is relative roughness, and Fr is Froude number. [30] The relation between the shear stress dynamics and the normalized vertical flux of turbulent kinetic energy for a uniform 2-D mean flow, is given as F k ¼ 1 2u 3 ðu 02 þ v 02 þ w 02 Þw 0 * pffiffiffiffiffiffi w ¼ 02 2u 3 u 02 m 1;21 þ v 02 m t 1;21 þ w02 m 1;03 * t where m i,jk are the moments defined in section 3, m 1,21 = v_ 2 w _ is the transverse crossed third-order moment, and u * is the mean bed friction velocity. The moments m 1,03, m 1,21, and m t 1,21 measured in the suspension flow are compared with clear water flow results (Figure 8) investigated by Hurther and Lemmin [2000]. The hydraulic parameters of the clear water experiments are given in the figure. It can be seen that the different moments are almost identical in the domain 0.25 z/h 0.75, independent of the flow conditions. In the present study the transverse crossed third-order moment has not been measured since 2-D velocity measurements were undertaken. Instead, the approximation v 02 w 0 ffi w 03 (also assumed by NN77), verified previously through 3-D velocity profile measurements [Hurther and Lemmin, 2000], is made here for the estimation of the normalized vertical flux of turbulent kinetic energy. [31] The energy flux for the different experiments is shown in Figure 9. In the flow region 0.25 z/h 0.75, a value of ffi0.3 is found. This confirms experimentally the existence of the wall similarity in the case of the suspension flow and points toward a universality of this concept. [32] This result provides for a new method to determine the bed friction velocity u * [López and Garcia, 1999]. Methods used so far to determine u * suffer from the problem that the reference depth of the profile had to be fixed precisely, which often introduces important errors especially in flows over rough beds. Using the wall similarity concept, u * can be determined by measurements made at a single depth far from the bed (anywhere in 0.25 z/h 0.75) where the gradient of the mean longitudinal velocity is low and the effect of spatial averaging over ð7þ 5. Transport Capacity Coherent Structures 5.1. Method [33] In this section we will evaluate the contribution of coherent structures to the concentration profile by considering the conditionally sampled diffusion equation for particles which can be expressed in its general formulation hi c H1 ¼ w hcu i hcvi H hcwi H where hi H1 denotes an averaging over the structures delimited by the selection criteria H 1 and w 0 represents the mean settling velocity of the particles in pure, still, clear water. The Reynolds decomposition of the variables is not applied in order to avoid the ambiguous definition of the mean velocity when conditional sampling is undertaken. The mass fluxes are measured directly and do not have to be approximated by a phenomenological law such as the Fick law Coherent Structure Selection Method and Validation of the Mass Flux Equilibrium State [34] The importance of the different terms in equation (8) as functions of the selection criteria H 1 will be determined in order to quantify the contribution of coherent structures to the concentration profile. The identification and selection of the flow structures is achieved with the classical uwquadrant threshold technique. However, the value of H 1 defining a coherent structure depends on the selection method used, and no consensus can be found in the literature about the exact value. Therefore the quadrant threshold technique will be combined with flow visualization available from the measurement technique. [35] Figure 10 presents a typical example of the flow over the whole flow depth taken in the middle of the channel. The fluctuating 2-D velocity field V(u 0, w 0 ) and the u 0 w 0 isocontour lines (with H 1 equal to 4) are superimposed. The structure of the flow features delimited by the isocontour lines is similar to that observed by Grass et al. [1993] in clear water conditions. Several flow patterns such as vortex cross sections associated with an underlying high shear layer (at t = 14.2 s, z/h = 0.225), Figure 9. Profile of normalized flux of turbulent kinetic energy for different flow conditions (see Figure 8 for legend). ð8þ

9 HURTHER AND LEMMIN: PARTICLE FLUX AND SHEAR STRESS STATISTICS ESG 2-9 Figure 10. Isocontours of high momentum flux u 0 w 0 and fluctuating velocity field V(u 0,w 0 ) at the centerline of the channel. Figure 11. Conditionally sampled terms of the sediment diffusion equation: (a) Vertical particle flux gradient. (b) Horizontal particle flux gradient. shear layers elongated over the outer flow region (at t = s,z/h= ),anddownwelling(layeratt=13.4s, z/h = ) can be observed. The present observations give support to the existence of common flow features in both clear water and turbulent open-channel suspension flows. As seen in Figure 10, the u 0 w 0 threshold selection technique accounts for coherent structures appearing instanteanously in the flow field. [36] The different terms of equation (8) for different values of H 1 are given in Figure 11. For H 1 = 0, no particular flow structures are selected and the longitudinal gradient of the mean horizontal mass flux is obviously negligible compared to the vertical gradient of the mean vertical mass flux. The equilibrium between the mean ascending particle mass flux caused by the entrainment capacity of coherent structures and the deposition flux due to the effect of gravity is evident. The validity of this equilibrium state is not a priori obvious since horizontal nonuniformity could appear in the structures. However, the equilibrium condition can still be assumed for H 1 5. From the flow visualization (Figure 10), coherent structures are clearly distinguished from the background flow field for H 1 = Results and Discussion [37] Figure 12 shows the concentration profiles relative to the reference concentration (taken at z/h = 0.05) for several threshold levels H 1 and the quadrant repartition diagrams of the relative vertical mass flux sampled as a function of H 1 and H 3 at z/h = 0.16, z/h = 0.5, z/h = From this figure, the dynamics of the relation of the instantaneous mass flux c 0 w 0 to the instantaneous u 0 w 0 events can be investigated in more detail. [38] Along the whole water column, the transport capacity of the coherent structures decreases proportionally Figure 12. Relative particle concentration versus threshold level H 1 and quadrant distribution of c 0 w 0 versus H 3 (dotted lines) and H 1 (solid lines) at three different flow depths.

10 ESG 2-10 HURTHER AND LEMMIN: PARTICLE FLUX AND SHEAR STRESS STATISTICS with increasing H. For values of H 1 =3andH 1 = 5, which represent strong structures, the transport capacity of coherent structures is still equal to 49% and 31%, respectively, of the total transport. Combining this information with the hole events time fraction given in the quadrant repartition diagrams, the time fraction of these coherent structures for the same two H 1 values are found to be only 30% and 10%, respectively. From this example, the importance of coherent structures in the particle entrainment mechanism becomes quantitatively evident, even if their lifetimes are relatively short. [39] In quadrant 1, corresponding to ejection events (Figure 7), good agreement is found between the vertical mass fluxes sampled as functions of H 1 and H 3 for all three depths. This indicates that the upward mass flux is directly correlated with ejection events for all values of H 1 and H 3 throughout the whole water column. Particle entrainment from the bed is strongly controlled by ejections even though their time fraction quickly becomes relatively small with increasing H. In Figure 1 the conditionally sampled Reynolds stress superimposed on the conditional sampled vertical mass fluxes for H 1 = 4 has been presented in order to indicate the role of the coherent flow structures in particle transport. The quantitative results presented in Figure 12 confirm the hypothesis that the contours delimiting important ejections coincide with the regions of high positive vertical mass fluxes. [40] In quadrant 3, which corresponds to sweeps, good agreement is again found for the functions of H 1 and H 3. However, at all depths but more so when approaching the channel bed, the falloff with increasing H is more rapid than for the ejection events. Sweeps are obviously predominantly organized in low H events. [41] Ejections and sweeps also influence the particle flux in quadrants 2 and 4, evident from the contributions below the solid lines in those quadrants. The contribution of interaction events of quadrants 2 and 4 is represented by the difference between the solid line and the dotted line in Figure 12. It is obvious that interactions are hardly correlated with the vertical mass fluxes. Therefore interaction events can be ignored in the particle transport and do not contribute to transport capacity curves of Figure 12. [42] It has previously been observed by Soulsby et al. [1994] that saltating particle movement in bed load transport may arise from local pressure gradients resulting from the impinging of sweep events on the bed. The initial phase of particle resuspension will then be related to sweeps. Further investigations may help to understand more details of the dynamical interactions between coherent structures and the particle flux through the boundary layer. [43] The results in Figure 12 reveal the strong interplay between the important u 0 w 0 structures with high particle flux events. However, a nonnegligible part of the suspended particle concentration is decorrelated with coherent structures, as can be seen in Figure 12 for z/h = Only 50% of the relative concentration is directly correlated with u 0 w 0 patterns for H 1 =3. 6. Summary and Conclusions [44] A third-order cumulant discard Gram-Charlier probability density function has been applied to the velocity cross-product term u 0 w 0, as well as to the horizontal and vertical turbulent mass fluxes in order to quantify their quadrant dynamics. Good agreement was found between the model results and the experimental estimations in the wall and intermediate flow regions. In the free surface flow region, for all investigated quantities, the limitation of the model to the order of 3 leads to small discrepancies due to an increase in intermittency. [45] The quadrant dynamics of u 0 w 0 correspond to results found in the literature with a clear domination of quadrant 2 (ejections) and quadrant 4 (sweeps) events. Thus the presence in the flow of particles of the size investigated here, even at capacity charge, does not influence the flow dynamics on the scales of coherent structures. Instead, a quadrant repartition similar to the shear stress distribution is observed for the turbulent mass fluxes, and the effect of the hole size parameter H on c 0 u 0 and c 0 w 0 is comparable to that of the shear stress. This shows that the turbulent mass fluxes are also strongly organized in coherent structures. The quadrant repartition obtained for the turbulent mass fluxes can be interpretated through the dynamics of ejections and sweeps. [46] On the basis of the conditionally sampled sediment continuity equation, the suspended particle transport capacity of coherent structures has been quantified. The proportion of the relative particle concentration profile (relative to the near-bed reference concentration taken at z/h = 0.05) and the time fraction were estimated as functions of the u 0 w 0 threshold level (delimiting the coherent structures in the instantaneous flow field). It has been shown quantitatively that coherent structures are important contributors to suspended particle transport. Strong structures which are only present for 30% of the time carry nearly 50% of the vertical particle flux. This indicates that particle transport is highly intermittent and that particle concentration in the water column varies strongly. On the basis of these results, we have discussed [see Hurther and Lemmin, 2001] a formulation given by Cao [1999] of the near-bed equilibrium concentration dependent on scales of coherent structures. The validation of the particle entrainment function in suspension flows with different particle diameters and varying Shields parameters has been investigated. [47] The results of Hurther and Lemmin [2000] have demonstrated that the third-order velocity moments determine the shear stress difference between ejection and sweep events in the outer domain of the boundary layer. In the present investigation of suspension flow, these moments are found to be in good agreement with those obtained in clear water conditions (see Figure 8). This analogy allows us to validate the wall similarity concept in highly turbulent suspension flows. This result points toward a universality of this concept. It provides for a simplification combined with increased accuracy in the determination of the effective wall shear velocity particularly in flows with strongly varying bottom roughness height where classical mean profile methods fail [López and Garcia, 1999; Hurther and Lemmin, 2000]. Further investigations on the range of validity of this concept in suspension flow with different grain sizes and under field conditions will be carried out. Also, a more objective technique similar to the one proposed by Szilagyi et al. [1999] will be applied to compare the dominant scales of the momentum flux events to the scales of mass flux events in suspension flows.

11 HURTHER AND LEMMIN: PARTICLE FLUX AND SHEAR STRESS STATISTICS ESG 2-11 Notation x, y, z longitudinal, transverse, and vertical variables in the Cartesian coordinate system, respectively. u, v, w, c local instantaneous longitudinal, transverse, vertical velocity, and mass concentration components, respectively. u; v; w; c local time mean of longitudinal, transverse, vertical velocity, and mass concentration components, respectively. u 0, v 0, w 0, c 0 local fluctuating longitudinal, transverse, vertical velocity, and mass concentration _ u ; _ v ; w _ ; _ c t m 1,21 components, respectively. root mean square normalized local fluctuating longitudinal, transverse, vertical velocity, and mass concentration components, respectively. C a near-bed equilibrium particle concentration (mass concentration taken at z/h = 0.05). d 50 particle diameter from granulometric curve at 50% F k normalized vertical flux of turbulent kinetic energy h water depth. H i sampling threshold i. k i,jk cumulant of order j + k of bivariate velocity probability density function. K 0, K 1/2 modified Bessel function of second kind of order 0 and 1/2, respectively. m i,jk moment of order j + k of bivariate velocity probability density function. = _2 v w _ crossed third-order moment in the transverse flow section. p i probability density function of random variable e i in plane i. p i, q probability density function of random variable e i in quadrant q of plane i. p in probability density function of random variable e i in plane i developed from the corresponding bivariate normal distribution (taking into account cumulants of second order). r i correlation coefficient i. s i, S ei, F ei R i,q first-order moment of probability density function p i,q. T i,q time fraction of conditionally sampled e i events of quadrant q in plane i. u * mean bed friction velocity. w 0 settling velocity of particle in still water. e i random variable i. i characteristic function i. ± j i additional high-order probability density function of e i (taking into account cumulants of order higher than 2). variance, skewness, and flatness factors of probability density function p i. [48] Acknowledgments. The present study was funded by the Swiss National Science Foundation, grant We are grateful for the support. Comments by two anonymous reviewers helped to improve the presentation. References Antonia, R. A., and J. D. Atkinson, High-order moments of Reynolds shear stress fluctuations in a turbulent boundary layer, J. Fluid Mech., 58, , Barenblatt, G. I., On the motions of suspended particles in a turbulent flow occupying a half-space or a plane open channel of finite depth, Prikl. Mat. 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Hurther, Woods Hole Oceanographic Institution, Woods Hole, MA 02543, USA. (david.hurther@hmp.inpg.fr) U. Lemmin, Laboratoire d Hydraulique Environnementale, Ecole Polytechnique Federale Lausanne, CH-1015 Lausanne, Switzerland. (ulrich.lemmin@epfl.ch)