Reduced Sediment Settling in Turbulent Flows Due To Basset History and Virtual Mass Effects

The behavior of suspended particles in turbulent flows is a recalcitrant problem spanning wide‐ranging fields including geomorphology, hydrology, and dispersion of particulate matter in the atmosphere. One key mechanism underlying particle suspension is the difference between particle settling velocity (ws) in turbulence and its still water counterpart (wso). This difference is explored here for a range of particle‐to‐fluid densities (1–10) and particle diameter to Kolmogorov micro‐eddy sizes (0.1–10). Conventional models of particle fluxes that equate ws to wso result in eddy diffusivities and turbulent Schmidt numbers contradictory to laboratory experiments. Incorporating virtual mass and Basset history forces resolves these inconsistencies, providing clarity as to why ws/wso is sub‐unity for the aforementioned conditions. The proposed formulation can be imminently used to model particle settling in turbulence, especially when sediment distribution outcomes over extended time scales far surpassing turbulence time scales are sought.


Plain Language Summary
In rivers and streams, mixed-up dirt called "suspended sediments" is known to influence water quality and its concomitant effects on many physical, chemical, and biological processes.Suspended sediments can be harmful to aquatic ecosystem productivity because they reduce light penetration.They can clog gills of fish and other aquatic organism, and they can impact reservoir operation and their capacity necessitating frequent dredging.There is debate about how these sediments interact with swirling motions ("turbulent flows") in moving water.Traditional mathematical models overlook how individual grains are affected by these swirls ("turbulent eddies") and how they displace water ("virtual mass") when sediments move in a fluid.For large grains, these overlooked details can generate new terms in the force balance that are associated with complex eddying motion lagging behind the grain motion ("Basset history term").Including the Basset history and virtual mass factors reconciles controversies between recent laboratory experiments and traditional theories about how grains settle in turbulent flows.LI ET AL. vortex trapping, (c) non-linear drag, and (d) loitering.The latter three mechanisms all result in w s /w so < 1 whereas the first leads to w s /w so > 1 (Fornari et al., 2016;Good et al., 2014).Preferential sweeping appears to dominate for d p /η ≤ 1 but for heavy inertial particles (s ≪ 1) that tend to be swept into downdraft regions characterized by low vorticity and high stain rate (Squires & Eaton, 1991;Stout et al., 1995).Vortex trapping also occurs for small particles (d p /η ≤ 1) but for R * ≪ 1.When the difference between fluid and sediment velocity (or slippage velocity) fluctuates in time, a non-linear drag favors higher drag values over lower drag values resulting in a net increase in the effective drag coefficient sensed by the settling grain.Thus, averaging over the settling time yields a larger drag force resulting in w s /w so < 1 (Stout et al., 1995).Loitering can occur when sediments with finite inertia spend more time moving upward (which is assumed to take longer) than moving downward (which is assumed to take less time), resulting in an asymmetric behavior (Akutina et al., 2020).This behavior, partly connected to ejections dominating over sweeps outside the roughness sublayer of boundary layers (Raupach, 1981), leads to a reduction in w s compared to w so .Values of w s /w so for conditions where d p /η > 1, s ≤ O(1), and when sediments remain in suspension (i.e., with R * not deviating appreciably from unity) are not frequently considered despite their ubiquity in streams (Vanoni, 1984) and the atmosphere as summarized in Figure 1.
Compared to the aforementioned mechanisms, the unsteady vortex generation (Basset history forces) due to flow memory and the displacement of water with lower density than the grain density (virtual mass term) have received less attention in conventional suspended settling models such as Rouse's mean concentration profile and its associated R * .As illustrated in Figure 1, the virtual mass term and Basset history forces can be separately significant across a wide range of particle sizes (10 −6 < d p /η < 10 2 ) and relative densities (1 < ρ p /ρ < 10 4 ) when combining turbulent stream and atmospheric flows.This range covers clay, some micro-plastics, and various sands in water, as well as volcanic ash, snowflakes, and aerosols in the atmosphere and is thus pertinent to a plethora of geographical and atmospheric applications.This work investigates particle settling velocity in turbulent flows taking into account virtual mass and Basset history terms where they may be jointly and dynamically more significant compared to other processes.This range is 1 < ρ p /ρ < 3 (for virtual mass) and 0.1 < d p /η < 10 (for the Basset history) in Figure 1.Incorporating the effects of these two terms in a simplified mean sediment concentration budget provides a fresh perspective on ongoing controversies related to whether w s /w so exceeds unity or not and whether Sc = ν t /D S , a turbulent Schmidt number defined by the ratio of a turbulent viscosity (ν t ) and a particle turbulent diffusivity (D s ) also exceeds unity or not (Bombardelli & Moreno, 2012;van Rijn, 1984).Estimates of Sc inferred from measured mean sediment concentration profiles by fitting Rouse's solution often yield Sc < 1 whereas estimates of Sc from measured turbulent fluxes (sediments and momentum) and mean gradients yield Sc > 1 (Chauchat et al., 2022).Plausible connections between Sc, w s /w so , and the simultaneous role of the Basset history and virtual mass effects are the main novelty offered here for the regimes dominated by 1 < ρ p /ρ < 3 and 0.1 < d p /η < 10.These connections are explored in the inertial sublayer of boundary layers where the log-law reasonably describes the mean velocity profile and the second-order flow statistics do not vary appreciably with wall-normal distance.

Theory
Assuming a continuum representation for suspended sediment concentration or SSC (but see Bragg et al. (2021) for alternative approaches when this is not valid), the instantaneous sediment mass balance yields (Richter & Chamecki, 2018) where t is time, C is the sediment concentration at position x (or x i ) and time t, D m is the molecular diffusivity, V i is the instantaneous advective velocity of the sediment grains, and i = 1, 2, 3 is a live index.In this notation, x 1 (or x), x 2 (or y), and x 3 (or z) represent the longitudinal, transverse, and vertical directions respectively.Upon using Reynolds averaging (indicated by overline), assuming stationary (i.e.,  (⋅)∕ = 0 ) and planar homogeneous (i.e.,  (⋅)∕ = (⋅)∕ = 0 ) turbulent flow conditions with V 3 = w p , Equation 1 simplifies to where transport by the molecular diffusion term  ( ∕ ) is ignored relative to the turbulent flux   ′   ′ after averaging, w p = w − w s is the instantaneous net vertical advection velocity of the sediment particles and w s is a terminal settling velocity of a single sediment particle and frames the scope of the work.Integration of Equation 2 with respect to z results in   +  ′   ′ =  , where F T is an integration constant that can be interpreted as the total sediment flux into or out of the control volume and is therefore determined by the boundary conditions.To evaluate F T , the free water surface boundary is considered.At this boundary, it is assumed that no sediment additions occur,   = 0 , and ν t = 0 resulting in F T = 0.This budget equation with F T = 0 is the starting point of all operational models including Rouse's suspended sediment concentration (SSC) profile.
The sediment vertical velocity w p is governed by a local force balance (Figure 1), and when the sediment grain size satisfies d p /η ≪ 1 and the particle Reynolds number   = |((), ) − |∕ is small (where ν is the fluid viscosity, and x(t) is the sediment position at time t), this force balance may be approximated by formulations discussed in Duman et al. (2016) and Maxey and Riley (1983) leading to where g o = 9.8 m s −2 is the gravitational acceleration, and C d is a particle dimensionless drag coefficient.For s = ρ/ρ p → 0, this budget simplifies to dw p /dt = −g o as expected.However, as s → 1, the contribution of g o can be ignored and the remaining terms including the flow acceleration, drag force, virtual mass, and the Basset history terms all become significant.To proceed further, an estimate of C d is required.For smooth spherical grains of diameter d p , C d may be approximated as (Cheng, 2009) In Equation 4, the C d = 24/Re p is the Stokes drag coefficient valid for small Re p , and the remaining terms in Equation 4 are additive and multiplicative corrections to the Stokes coefficient.When the Stokes contribution in Equation 4 is dominant, the drag force in Equation 3 scales as The Basset history term, which arises due to a temporal delay and unsteadiness of the boundary layer developing on the moving grain, is much more difficult to represent in sign and magnitude.However, inspection of Equation 3 is suggestive that the magnitude of the Basset history scales as   −1  .Hence, the ratio of Basset history to drag force scales with d p , and the Basset term can only be ignored if d p /η is very small, where η is presumed to be the smallest scale of motion where turbulence remains significant.Direct numerical simulation (DNS) studies have shown that the behavior of Basset history term may be approximated as being proportional to the drag force according to Daitche (2015), Daitche and Tél (2011), and Guseva et al. ( 2016) where α η is a similarity coefficient of order unity (Daitche, 2015;Guseva et al., 2016).The sign of the Basset history is not obvious but evidence also suggests that (at least on average) it reduces the slip velocity between fluid and particle grains implying Re p is reduced and C d is enhanced by this term (Daitche, 2015;Guseva et al., 2016).Accordingly, the particle acceleration in Equation 3 simplifies to Equation 6 can be expressed as (Druzhinin, 1995;Good et al., 2014;Kawanisi & Shiozaki, 2008), where the characteristic particle timescale τ p , the modified gravitational acceleration and a density modification coefficient λ are now defined as The τ p now includes the effect of the Basset history term through a finite α η and the added virtual mass in λ.When d p /η is sufficiently small, the Basset history term can be ignored.The Basset history and virtual mass jointly act to increase the effective C d but those terms are associated with different mechanisms when compared to averaging a non-linear form of C d over variations in  |((), ) − | .Moreover, the g is also reduced from g o by (2 − 2s)/ (2 + s) due to s < 1 as is the case for sedimentation.
To link w p to w in Equation 7, a small Stokes number (St + = τ p u * /H, H is a large length scale) perturbation analysis (Druzhinin, 1995;Ferry & Balachandar, 2001;Maxey & Riley, 1983) is used and shown in Supporting Information S1.The suitability of the small St + assumption when the target variable is  () within the log-layer will be discussed later.As a result, the sediment settling velocity (dimensional) can be derived as a function of the fluid vertical velocity and particle time scale using, The new characteristic sediment settling velocity w s emerges as a zeroth order estimation of the sediment settling velocity in turbulent flow and is now presented as , where  = 1

− 4𝑠𝑠 3𝑠𝑠 The outcome in Equation 10 suggests that w s /w so → 1 when d p /η → 0.Moreover, w s is reduced with increasing d p /η.The virtual mass and the Basset history terms appear to act in concert to reduce w/w so from unity.Substituting Equation 9into Equation 2 and performing the Reynolds averaging operation on each term, a sediment mass budget can be obtained as where   = √  ′2 is the fluid vertical velocity standard deviation.The terms on the right-hand side of Equation 11 are respectively produced from the turbophoretic effect and finite particle inertia as discussed in Supporting Information S1.This latter effect indirectly accounts for preferential sweeping (especially with increasing St + ).The term  ( − ) on the left-hand side of Equation 11is the missing flux in Rouse's analysis correcting the still-water gravitational term attributed here to the Basset history and virtual mass.A result similar to Equation 11 was previously derived (Richter & Chamecki, 2018); however, this prior analysis did not include the Basset history, virtual mass and the fluid acceleration terms.
To recap, when using Rouse's analysis, the following approximations are invoked: w so = w s and all the terms on the right-hand side of Equation 11are ignored (τ p → 0).The Basset history term is expected to be significant when d p /η is not very small, and the effects of virtual mass are large when s is close but below unity.Equation 11suggests that this residual flux arising from the difference between the turbulent settling velocity and still water settling velocity is given as where θ cd = C d (w s )/C d (w so ) is an adjustment coefficient due to the variations of the differences in drag coefficients in still water and turbulence.For inhomogeneous turbulent flows where   2  ∕ can be large (e.g., turbulent boundary layers near the wall or in the outer region), it may be argued that the turbophoretic effect on the righthand side of Equation 11 must be included given that it scales with   .However, in the inertial region (i.e., where the log layer for the mean velocity is expected to hold), the turbophoretic effect is minor because σ w /u * is roughly constant in z.The last two terms in Equation 11 representing turbulent sweeping effects are rarely quantitatively scaled in the literature due to their intractability.Thus, as a logical starting point, they are temporarily dropped here and the focus is given to only the Basset history and virtual mass terms.The budget in Equation 11 that ignores the right-hand side but maintains the residual flux R f is now used to examine the outcome of a number of laboratory experiments that report w s /w so < 1 for d p /η > 1 and ρ p /ρ ∈ [1, 3] within the inertial sublayer.

Experiments and Model Comparison
Three different laboratory experiments are used and interpreted using Equation 11.The first experiment is presented in Revil-Baudard et al. (2015) and is labeled as RB15 while the other two experiments are presented in Shen and Lemmin (1999) and are labeled as SLR1 and SLR2.These experiments are selected because flow and sediment concentration statistics were publicly available.Briefly, the experiments were conducted in rectangular channels with width B and bed slope S o for steady and uniform flow characterized by a constant flow rate Q.Sidewall friction was presumed to be small compared to bed friction so that the water level H reasonably approximates the hydraulic radius R H .A partial justification for setting R H = H is not based on geometry because H/B is not an order of magnitude smaller than unity in all experiments.It is based on the fact that the bed is covered with sediments and is much rougher than the side channels (usually smooth glass or plastic to permit optical access).Thus, ] 1∕4 .Near the channel bed and in the inertial (or log-layer), ϵ(z)/ϵ b > 1 so that η(z)/η b < 1.The η b must be viewed as an upper limit on the Kolmogorov microscale and d p /η b as a lower limit for the experiment at hand when evaluating the Basset history in the target region.The bulk conditions across experiments are compared and summarized in Table S2 in Supporting Information S1.All three experiments have commensurate U b , u * , and η b .The computed R * = w o /(κu * ) > 2.5 using the reported measured settling velocity in still water (labeled as w o ) for RB15, where κ = 0.4 is the von Kármán constant.However, the computed R * for SLR1 and SLR2 were consistent with a 100% suspended sediment and wash-load classification (i.e., R * ∈ [0.5, 1.2]).This outcome alone hints that adjustments to R * (i.e., reduced settling velocity) must be invoked to maintain suspension in RB15.It is also known that the mean settling velocity of sediments may be reduced by hindrance effects, caused by obstacles or other particles in the fluid that impede their motion.However, recent analysis by Chauchat et al. (2022) using their RB15 data showed that such hindrance mechanism may not be sufficient to account for the observed retardation effects.Therefore, particle-particle hindrance effects are not considered.
Common issues in turbulent flux measurements are short sampling duration and small sampling frequency f s .To explore these two effects on the data here, especially RB15, an idealized co-spectrum of the turbulent sediment flux was assumed and featured in Supporting Information S1.Key findings from this supplementary are that the combined effects of these two corrections appear to be large (as high as a factor of 2) but their leading order effect can be accommodated in the comparisons presented next, at least for RB15 where all the necessary data are available.The corrected (where possible) and uncorrected turbulent fluxes are presented in the figures for RB15. ) in all three studies, the computed settling velocity w so , and the characteristic turbulent settling velocity w s are highlighted in Figure 2. The w o /w so < 1 when w so is computed for spherical smooth grains as noted in previous studies (Cheng, 1997).
When α η = 0.7 predetermined from DNS (Dai et al., 2016;Daitche, 2015;Guseva et al., 2016), good agreement (a maximal deviation of 25%) between the settling flux and turbulent flux can indeed be achieved except for RB15.The RB15 data set can also be matched if the turbulent flux is corrected (e.g., with   ′  ′  or α η is fitted as 2.0 without correction using   ′  ′ ).It is noted that the co-spectral correction to the turbulent flux in RB15 are large (see Supporting Information S1) and may be an overestimate of the correction magnitude.Furthermore, the measured turbulent flux, the corrected turbulent flux, and the gravitational settling flux in Equation 11 are separately examined (see Supporting Information S1) implying that the inclusion of the Basset history term offers a plausible explanation to reduced sediment settling velocity in turbulent flows for all these experiments.However, for α η = 0.7, the flux budget is indeed matched after co-spectral corrections are applied to the turbulent flux for RB15.Likewise, an α η = 0.7 reasonably matches the other data sets (original fluxes) though co-spectral corrections could not be determined for SLR1 and SLR2.The more interesting feature here is that the computed w s from Equation 10 with α η = 0.7 remains significantly smaller compared to w o .This finding supports the conjecture that turbulence indeed reduces the sediment settling with the highest retardation occurring for the largest d p .Equation 10 attributes this reduction to the Basset history and the added virtual mass terms.For z/H < 0.18, inferred w s appears to be not constant in two experiments (SLR1 and SLR2), which may be indicative that other effects are significant near the wall such as turbophoresis or particle-particle interaction due to the high concentration near the bed.

Revisiting the Turbulent Schmidt Number
The turbulent Schmidt number can be expressed as The inferred Sc directly from measurements is shown in Figure 2 in the inertial layer.As shown in panel (i) of Figure 2, the Sc = 3.5 appears constant for all experiments in the inertial layer.An Sc > 1 was suggested to counter Rouse's original model interpretation (Rouse, 1937;van Driest, 1956).The original outcome that the turbulent Sc < 1 (traditionally denoted as β −1 in many studies) from Rouse's mean concentration profile (Rouse, 1937;van Driest, 1956) can be explored using R f .Adopting van Rijin's formula (van Rijn, 1984), a definition of a "modified" Schmidt number represented by β −1 was introduced and may be expressed as where R f = 1 − w s /w so is the residual flux when the Basset history and virtual mass are included in the particle force balance.As shown from Equation 14, the turbulent Schmidt number β −1 used in Rouse's formula can now be viewed as a product of two terms: a "true" Schmidt number modified by a factor (1 − R f ) to accommodate the missing forces (here, the virtual mass and Basset history).The plot of β −1 is shown in panel (j) of Figure 2 where the RB15 data are smaller than unity (R f > 0) while the SLR1 and SLR2 are not (discussed later).The "controversy" about Sc > 1 can be addressed when noting that the reported β −1 is not the actual turbulent Schmidt number as may be inferred from Equation 13.The β −1 must be interpreted as a characteristic turbulent Schmidt number "pinned" to a settling flux based on the still water settling velocity instead of a turbulence-reduced settling velocity.and Runs 1 (green color, SLR1) and 2 (red color, SLR2) from Shen and Lemmin (1999).Different data sets are marked in different colors in a consistent manner throughout all the figures.The inertial layers are also highlighted in light gray (z/H > 0.12 but <0.3 for RB15) and light blue (z/H > 0.18 but <0.3 for SLR1 and SLR2).
Middle panel (f-h): The three subplots show the computed and reported settling velocities (w) corresponding to RB15, SLR1, and SLR2 respectively.In each subplot, the four settling velocities are (i) computed turbulent settling velocity w s from Equation 10with α η = 0.7 from Direct numerical simulation (the error bar shows a 25% deviation off the mean), (ii) the settling velocity w so from Equation 10 in black dot-dash lines, (iii) the experimentally reported still water settling velocity w o in solid line and (iv) the settling velocity from measurements using raw It is noted that an Sc > 1 (as is the case here) qualitatively agrees with the "crossing-trajectories" argument developed for heavy particles in turbulent flows (Csanady, 1963;Stout et al., 1995;Wells & Stock, 1983).The crossing trajectories argument suggests that heavy particles will move from highly correlated regions in the flow to less correlated regions when their trajectories intersect those of fluid elements, typically due to the influence of gravity.This phenomenon leads to an increase in flow diffusivity and a decrease in sediment diffusivity, resulting in Sc > 1.This behavior is analogous to the Basset history effect, which causes the generation of unsteady vortices and added mass due to the displacement of particles and water within the local region bounded by the particle volumes.

A Re-Examination of Rouse's Budget
The improved skill of accommodating the Basset history and virtual mass in the SSC budget is now evaluated in terms of predicting the  () with z/H.Numerical integration of Equation 15 is used to derive the mean SSC.The measured flow statistics profiles were digitized and used in the computation of   (𝑧𝑧) .The depth integration to derive the SSC for the two models is given by, To optimize Rouse's original concentration profile solution (i.e., R f = 0), β −1 was set as a "free parameter" in Equation 15.For the revised Rouse budget, no fitting was necessary provided Sc was externally set.The R f was computed (and varied with z) using the reported s for the virtual mass and the modeled Basset history term with α η = 0.7 (see Figure 2) and Sc = 3.5 determined from the experiments shown in Figure 2. The predicted SSC profiles using the original Rouse solution and the present method are shown in Figure 2. The comparison features the optimized β −1 summarized in Table S2 in Supporting Information S1.
Figure 2 demonstrates that predicted  () profiles agree with observed values for both formulations though minor improvements are noted for the proposed formulation that includes the z dependency of R f .Free-fitting β −1 makes the modeled SSC profiles match measurements even though the assumption that the turbulent flux is balanced by the still water settling flux is incomplete.This agreement between data and the original Rouse model may be due to the fact that d p /η is insensitive to small variations in z within the log-layer as η scales only as z 1/4 (i.e., sub-unity exponent).Hence, different grain sizes or flow conditions simply increase the variability in β −1 when fitting the original Rouse model to measured  () but the  () dependency on z is not significantly altered.Table S2 in Supporting Information S1 makes clear why prior studies reported an Sc = β −1 < 1. Computing β −1 by fitting Rouse's formula to measured  () and then setting Sc = β −1 yields an apparent Sc < 1.The inferred β −1 from fitting Rouse's profile to concentration data all result in β −1 < 1 whereas computed β −1 from data in Figure 2 exceeds unity for two experiments (SLR1 and SLR2).Accounting for R f variations in z and using Sc = 3.5 yields good agreement between measured and modeled  () for the same data sets.

Model Limitation
The model assumptions warrant further scrutiny.First, the particle equation of motion (i.e., Equation 3) was derived from a force balance on a single grain assuming the grain is finite in size but small in diameter (i.e., d p /η ≤ 1), spherical and rigid, and experiencing a small particle Reynolds number.Moreover, a solution to the particle equation of motion was derived assuming that the particle Stokes number St + = τ p u * /H is small.The data in Table S2 in Supporting Information S1 shows that the particles in all three experiments have values St + = O(10 −3 ) and therefore the assumption is clearly validated.This assertion remains true even when H is replaced by z in the St + expression for z within the log-layer (order 0.1 H).
Returning to the case where d p /η b > 1, it is more crucial that d p be much smaller than the flow scales influencing  () .Thus, it is the large scales (i.e., z), not the Kolmogorov scales, that predominantly affect the behavior of  () .That is, Equation 3 still applies for modeling  () in the log-layer, especially when d p /z ≪ 1.This finding is supported by Costa et al. (2020) who showed that away from the wall (z/H > 0.2), finite-size particles' behavior agree with point-particle results from Equation 3. In the experiments here, d p /z remains much smaller than unity within the log-layer suggestive that Equation 3 can be used for predicting  () in that layer.Finally, the model emphasizes the significance of the Basset history in the experiments here due to the non-negligible value of d p /η.It could be argued that the Faxen correction terms in the particle equation of motion (i.e., Equation 3) must also be employed when d p /η is finite (Maxey & Riley, 1983).The relevance of those corrections was explored in a number of studies (Dai et al., 2016;Daitche, 2015) and they were deemed minor for the parameter range here.The inclusion of the Basset history and added mass terms suffice to explain the reduction in the settling velocity relative to its still value hinting that the omission of the Faxen corrections is acceptable in the log-region.More evaluations are also presented in the Supporting Information S1.

Conclusion
The work here focused on the particle setting velocity in a turbulent flow for the cases where (a) the diameter of sediment particles is of the same order as or exceeds the Kolmogorov micro-scale, and (b) the particle density is not substantially different from (but exceeds) that of the fluid.It was shown that for such ubiquitous conditions in many geophysical flows, an "upgrade" to the current formulation of the sediment settling velocity is necessary.The proposed upgrade incorporates the simultaneous effects of the Basset history term and virtual mass and leads to particle settling velocities smaller than their still water counterparts.These two effects also elucidate the observed variability in the turbulent Schmidt number, which is commonly inferred by fitting measured mean sediment concentration profiles to Rouse's formula.The turbulent Schmidt number derived from fitting Rouse's profile to measurements a priori assumes that w s /w so = 1.Hence, deviations from such an assumption can be erroneously attributed to a turbulent Schmidt effect.
Future effort seeks to expand this model to include (a) inhomogeneous flow conditions where turbophertic effects can be significant (e.g., near boundaries or free water interfaces), (b) other phoretic effects such as thermophoresis needed in stratified atmospheric flow studies, and (c) finite particle inertia where preferential sweeping effects can become significant.Moreover, for many other applications such as dispersion of micro-plastics, relaxing the rigid spherical assumption is also needed.To accomplish these revisions requires dedicated (and expensive) direct numerical simulations, a task which must be kept for the future.

Figure 1 .
Figure 1.Top: Geophysical applications of particle settling in water (left) and air (right) organized by d p /η and relative densities ρ p /ρ, with ρ = ρ w for water and ρ = ρ a for air.The shaded regions indicate when the virtual mass (orange-red shaded) and Basset history (green shaded) terms can be significant.The applications include suspended sediment transport in rivers and conveyance structures, snow, aerosols, dust and ashes, and certain types of aggregate plastics in the atmosphere.Bottom: A sketch of an open channel flow with suspended sediments entrained from the channel bed (left) and a single grain (right) interacts with the micro-scale eddy (η) commensurate with d p .The inclusion of virtual mass and Basset history terms in the mean sediment concentration budget become necessary when d p /η > 0.1 and ρ p /ρ is of order unity.

Figure 2 .
Figure 2. Top panel (a-e):The reported profiles of turbulent flow and sediment statistics fromRevil-Baudard et al. (2015),Chauchat et al. (2022) (blue color, RB15), and Runs 1 (green color, SLR1) and 2 (red color, SLR2) fromShen and Lemmin (1999).Different data sets are marked in different colors in a consistent manner throughout all the figures.The inertial layers are also highlighted in light gray (z/H > 0.12 but <0.3 for RB15) and light blue (z/H > 0.18 but <0.3 for SLR1 and SLR2).Middle panel (f-h): The three subplots show the computed and reported settling velocities (w) corresponding to RB15, SLR1, and SLR2 respectively.In each subplot, the four settling velocities are (i) computed turbulent settling velocity w s from Equation10with α η = 0.7 from Direct numerical simulation (the error bar shows a 25% deviation off the mean), (ii) the settling velocity w so from Equation 10 in black dot-dash lines, (iii) the experimentally reported still water settling velocity w o in solid line and (iv) the settling velocity from measurements using raw  (  ′  ′ ∕ ) and co-spectrally corrected turbulent flux values (for RB15 only)  (  ′  ′  ∕ ) .Bottom panel Chauchat et al. (2022)hen & Lemmin, 1999)and summarized byChauchat et al. (2022)are shown in Figure2.The shaded area in these figures indicates the inertial (or log-law) region.This region is characterized by measured γ = σ w /u * not varying appreciably with z but different across experiments due to H/B restrictions as noted earlier.Hence, turbophoresis may be ignored in this layer.When H/B → 0 (e.g., a near-neutral atmospheric surface layer flows), γ = 1.2 − 1.3 and is larger than the reported γ in the experiments here.Only RB15 reported a γ of order unity.The measured still water settling velocity (i.e., w o Comparison of different turbulent Schmidt numbers (Sc) computed from Equation 13 and using Equation14.For the overall estimates of Sc and β −1 reported in TableS2in Supporting Information S1, fitting measured (symbols) to modeled (lines) C(z) is used.In panels (k-l) the dashed lines are the predictions of the SSC.(k) Shows the revised Rouse model that accommodates the Basset history and virtual mass terms with no fitting parameters, while (l) shows the best-fit of the original Rouse equation to data.