Particles in turbulent separated flow over a bump: effect of the Stokes number and lift force
Jean-Paul Mollicone, Mahshid Sharifi, Francesco Battista, Paolo Gualtieri, Carlo Massimo Casciola
PParticles in turbulent separated flow over a bump
Particles in turbulent separated flow over a bump: effect of the Stokesnumber and lift force
J.-P. Mollicone, a) M. Sharifi, F. Battista, P. Gualtieri, and C.M. Casciola Department of Civil and Environmental Engineering, Imperial College London, United Kingdom Faculty of Sciences and Engineering, Sorbonne University, Paris, France ENEA C.R. Casaccia, S.M. di Galeria, Rome, Italy Department of Mechanical and Aerospace Engineering,Sapienza University of Rome, Italy (Dated: 5 July 2019)
Particle-laden turbulent flow that separates due to a bump inside a channel is simulated to analyse the effects of theStokes number and the lift force on the particle spatial distribution. The fluid friction Reynolds number is approximately900 over the bump, the highest achieved for similar computational domains. A range of particle Stokes numbers areconsidered, each simulated with and without the lift force in the particle dynamic equation. When the lift force isincluded a significant difference in the particle concentration, in the order of thousands, is observed in comparison withcases where the lift force is omitted. The greatest deviation is in regions of high vorticity, particularly at the walls andin the shear layer but results show that the concentration also changes in the bulk of the flow away from the walls. Theparticle behaviour changes drastically when the Stokes number is varied. As the Stokes number increases, particlesbypass the recirculating region that is formed after the bump and their redistribution is mostly affected by the strongshear layer. Particles segregate at the walls and particularly accumulate in secondary recirculating regions behind thebump. At higher Stokes numbers, the particles create reflection layers of high concentration due to their inertia as theyare diverted by the bump. The fluid flow is less influential and this enables the particles to enter the recirculating regionby rebounding off walls and create a focus of high particle concentration.
I. INTRODUCTION
Turbulent flows laden with particles are common in natu-ral phenomena and engineering applications. The understand-ing of turbulence and multiphase flows is considered a chal-lenge in both experiments and numerical simulations . Asfor single-phase flow simulations, different formulations forthe carrier phase in multiphase flows can be used such as di-rect numerical simulation (DNS), the method used here, large-eddy simulation (LES), , or Reynolds-averaged Navier-Stokes equations (RANS), .To these possible descriptions of turbulent flow, Lagrangianmethods can be used to couple particles and fluid at differentlevels, i.e. one-way, two-way or four-way coupling regime, .In the one-way coupling the particle volume fraction and massloading are low, the particles are transported by the carrierphase which is not modified. For higher mass loading andsmall volume fraction, two-way coupling considers the flowmodulation due to the particles. In four-way coupling, colli-sions and hydrodynamic interaction between particles are sig-nificant.In simulations involving a large number of particle smallerthan the Kolmogorov scale, a suitable numerical approach isthe Lagrangian point-particle method, . In such mixedEulerian-Lagrangian method, the continuous phase is de-scribed in a Eulerian framework whilst the dispersed phase bya Lagrangian approach solving the dynamic equation for eachparticle. Particles may be considered to be spherical or, re-quiring more complex modelling, non-spherical particles maybe considered . a) Electronic mail: [email protected]
The study of particle-laden turbulent flow has been of in-terest for decades by considering isotropic turbulence, ,and homogeneous shear turbulence, , and cases where theflow is confined by solid boundaries to investigate, for ex-ample, particle behaviour in turbulent boundary layers, or in Couette flow . Various studies are conducted in wall-bounded flows, , and they include confined flows such aspipe flows, and channels, , sometimes with the ad-dition of roughness on the surface of the channel’s walls tomodulate the flow and hence the particle dynamics, . Thestudy of particle-laden fluid jets is also active due to their vastuse in engineering and their occurrence in nature, . Somestudies focus, for example, on the effect of the Stokes num-ber on particle behaviour in turbulent jets, both experimentally and numerically .In many applications though, the geometry involved is morecomplex than these standard domains. For example, mi-croparticles are used in inhalable drug delivery systems sincethey provide a non-invasive treatment and localised deliverymethod. Some authors discuss their use in the treatment oflung cancer whilst other show how microparticles can beused to deliver embedded nanocrystals to the lungs. Theseapplications call for numerical simulations to study micropar-ticle behaviour in realistic models of human airways to-gether with the effect of different breathing conditions . An-other health related example is the obstruction in blood vesselsdue to atherosclerosis which is nowadays also studied with theaid of computational modelling .A geometrical change may be intentional, for example, toenhance mixing of fluid and particles or to separate particlesfrom fluid for filtration. Specific geometries can be used forthe preferential separation of particles when populations ofparticles with different characteristics are present in a carrier a r X i v : . [ phy s i c s . f l u - dyn ] J u l articles in turbulent separated flow over a bump 2phase. In Refs. the authors study particulate dispersionand mixing through DNS in a serpentine channel by consider-ing a large range of particle Stokes numbers. The authors ob-serve high concentrations of particles near the surface of theouter wall and show how the heaviest particles reflect from thewall to form reflection layers whilst the lighter particles con-centrate in the streaks at the wall. A recent study also ob-serve particle reflections in the outer bend of turbulent curvedpipe flow laden with micro-sized inertial particles. The au-thors document the modification of particle axial and wall-normal velocities and the increase in particle turbulent kineticenergy. Other studies show the importance of geometry forparticle-capture mechanisms in branching junctions by com-paring experiments and numerical simulations. The authorsshow that the capture is dependent on vortex breakdown, a re-sult of the creation and evolution of recirculating regions inthe system and a crucial factor that determines whether parti-cle accumulation is maximised or eliminated.The aim of the present paper is to study the microparticlesbehaviour in a turbulent channel flow with bump at one of thewalls by means of a Lagrangian point-particle method in anincompressible flow simulated using DNS. The bump makesthe flow separate, creating a strong shear layer and recirculat-ing region, . The configuration is nonetheless still acces-sible to classical statistical tools and turbulence theory for thedetailed study of turbulence dynamics, . To the best of ourknowledge, this is the first simulation of such a configurationladen with particles at a friction Reynolds number of 900 overthe bump. A wide range of populations ranging from almosttracers to ballistic particles are addressed. Additionally, dueto the high vorticity present in some regions of the flow, weinvestigate if and where the lift force significantly influencesthe particle dynamics.The paper is divided as follows: the simulation setup is de-scribed in section II, the results are discussed in the III and thefinal remarks are in section IV. II. SIMULATION SETUPA. Fluid phase
The computational domain has dimensions ( L x × L y × L z ) =( × × π ) × h , where x , y , and z are the streamwise, wall-normal and spanwise coordinates respectively and h is halfthe nominal channel height, see figure 1. Periodic boundaryconditions are enforced in both x and z directions, whilst at thewalls no-slip conditions are applied. The bottom wall containsa bump that is described by the easily reproducible and differ-entiable equation (especially important for particle rebound) y = a ( + cos (( π / c )( x − b ))) where a = . b = c = x ranges from 2 to 4. The periodicity in the streamwisedirection avoids artificial inflow/outflow boundary conditionsand the period is chosen as large as possible, within computa-tional limitations, to allow the analysis of an almost isolatedbump, with definite flow reattachment and negligible stream-wise correlation. The incoming flow accelerates at the channelrestriction and a recirculating region forms behind the bump, starting downstream of the bump tip. An intense shear layerseparates the recirculating region from the outer flow. Down-stream of the bump, the flow re-attaches completely. The tur-bulence dynamics for such flows over a bump is discussed indetail in .Direct numerical simulation (DNS) is used to solve the in-compressible Navier-Stokes equations, ∂ u ∂ t + u · ∇ u = − ∇ p + ν ∇ u ∇ · u = , (1)where u is the fluid velocity, t is the time, p is the hydrody-namic pressure and ν is the kinematic viscosity. Nek5000 ,which is based on the spectral element method (SEM) , isused to solve both the flow domain and the dispersed phase.The implemented algorithm for the computation of the par-ticle dynamics is deeply described in subsection II B. Thesimulations are carried out at bulk Reynolds number Re = h U b / ν = U b is the bulk velocity. All lengthscales are made dimensionless with the nominal channel half-height h , time with h / U b and pressure with ρ U b . The max-imum friction Reynolds number, achieved close to the bumptip, is Re τ = τ = u τ h / ν , where the fric-tion velocity is u τ = (cid:112) τ w / ρ , τ w is the local mean shear stressand ρ is the constant fluid density. The simulation has beenperformed with about 400 million grid point on 32768 coresusing approximately 30 million core hours, using spectral el-ements of order of N =
11. In this case the grid spacing, ∆ x + = . ∆ z + = . ∆ y + max / min = . / .
9, is adeguatefor the high fidelity description of all the flow scales. For de-tailed discussion about the resolution the reader can refer tosection 2 of Ref. Mollicone et al. . Approximately 500 sta-tistically uncorrelated fields, separated by a time interval of ∆ t stat =
6, have been collected for each simulation in orderto obtain properly converging statistics. Defining the ’flow-through time’, t ft , as the time needed for a turbulent structureto travel all along the channel length, the simulation time is T tot = (cid:39) t ft , which makes sure that statistics con-verge. B. Solid phase
The solid phase is composed of spherical particles with ra-dius smaller than the dissipative scale, the wall unit in ourcase. In a dilute suspension at low mass loading, the turbu-lence modulation due to the particles, the inter-particle col-lisions and the hydrodynamic interactions can be neglected .Under these conditions, the one-way coupling regime can beassumed and the Newton equation is forced by the Stokes dragtogether with the lift force which we intentionally include oromit to investigate its effect, namely d x p dt = v p d v p dt = τ p (cid:16) u (cid:12)(cid:12) p − v p (cid:17)(cid:124) (cid:123)(cid:122) (cid:125) Stokes Drag + β (cid:16) u (cid:12)(cid:12) p − v p (cid:17) × ζ (cid:12)(cid:12) p (cid:124) (cid:123)(cid:122) (cid:125) Lift (2)where x p is the particle position, v p is the particle veloc-ity, u p is the fluid velocity at the particle position and τ p = articles in turbulent separated flow over a bump 3 (cid:0) ρ p / ρ f (cid:1) (cid:0) d p / ν (cid:1) is the particle relaxation time, ρ p / ρ f = d p is the particle di-ameter, the coefficient of the lift force is β = / (cid:0) (cid:0) ρ p / ρ f (cid:1)(cid:1) and ζ is the vorticity. The particles elastically bounce at thewall, where in correspondence of the bump the exact bounc-ing direction is evaluated by the analytical function of the wallprofile.The dynamics of the particle in the one-way couplingregime is well described by the Stokes number, i.e. the ratiobetween the particle relaxation time and the fluid characteris-tic time scale. Two different Stokes numbers are defined, thereference one, St = τ p U b / h , and the viscous Stokes num-ber, St + = τ p u τ / ν . The two are related through the Reynoldsnumber, St = St + Re / Re τ .Equations (2) are evolved for each single particle with afourth order Adams-Bashforth method in time. A spectralinterpolation, intrinsic to the Nek5000 code, is employed toevaluate the fluid variables at the particle position. Twentydifferent populations are evolved: ten different Stokes num-bers St + = [ . , , , , , , , , , ] each withand without the lift term. Ten million of particle are con-sidered in the simulation. The huge number of particles isemployed to obtain an adeguate statistical convergence, sinceeach single particle does not feel the presence of the otherand does not modify the carrier flow. The effect of grav-ity is negligible for the particle population considered in thepresent study. In fact, the dimensionless terminal velocity, V t = τ p g , can be expressed in terms of the control param-eters as V t / U b = St + Re / ( Fr Re τ ) where Fr = U b / √ g h is the bulk Froude number. For the heaviest population atSt + = V t / U b is of the order of10 − . III. RESULTS
Figure 1 shows instantaneous snapshots of the particlescoloured with their instantaneous streamwise velocity for St + = [ , , ] . The whole computational domain is shownin the left panels whilst the area around the bump, in a vieworthogonal to the x - y plane, is shown in the right panels.Unless otherwise stated, figures show the particles that havebeen evolved with the lift term included in equation (2). Atlow Stokes number, the particles distribute themselves evenlythroughout the domain and have velocities comparable to thefluid velocity. The low particle velocity inside the recirculat-ing region behind the bump contrasts the high particle velocityat the centre of the channel. On the other hand, at St + = y = . C ( x , y ) = (cid:104) ( n ( x , y , z , t ) / V ( x , y , z )) / ( N T / V T ) (cid:105) , where n ( x , y , z , t ) is (a) (b)(c) (d)(e) (f) FIG. 1. Instantaneous snapshot of particles coloured by stream-wisevelocity for St + = St + =
50 in panels (c) and(d), St + =
600 in panels (e) and (f). Particles are not to scale. the instantaneous local number of particles in a cell at ( x , y , z ) position, V ( x , y , z ) is the corresponding cell volume, N T is thetotal number of particles and V T is the total domain volume.The angular brackets denote averaging in the homogeneousspanwise direction, z , and in time. The normalisation by N T / V T represents the homogeneous particle concentration,i.e. the concentration value that would be obtained at anypoint if all the particles were equally distributed in all thedomain. The data sets are taken after the system is statisticallystationary and at time intervals larger than the correlationtime. Six out of the total ten particle populations that havebeen simulated, St + = [ , , , , , ] , will be presentedsince the populations that exhibit similar behaviour are omit-ted. The population at St + = . St + =
1, some particles segregate towards the channelwalls whilst a homogenous concentration, indicated by thegreen colour, is present in most of the domain. The concentra-tion increases at the bump wall, particularly in three locations:before the bump, just after the tip of the bump and after thebump. These locations coincide with three small recirculatingbubbles that form in the fluid, see the zero velocity isoline infigure 5(a), that capture the particles. Figure 4 shows line plotsof mean particle concentration at these locations, specificallyat x = [ . , . , . ] . C = x positions, confirming the homogeneous distribution. The con-centration increases to the order of 10 at the top wall and order100 at the bottom wall, both before and after the bump, panels(a) and (c). In the latter, the particles manage to enter the recir-culating region and the concentration only decreases slightlyarticles in turbulent separated flow over a bump 4 (a)(b)(c)(d)(e)(f) FIG. 2. Mean particle concentration in the ( x , y ) plane for St + =[ , , , , , ] in panels (a) to (f) respectively. with respect to C = y ≈ .
6. A more pronounced de-crease, even though the particles are still clearly present, canbe seen further behind the bump, but still in the recirculatingregion, shown by the cyan colour in figure 3(a). Just afterthe tip of the bump, the particles are affected by the shearlayer forming in the fluid. Figure 5(b) shows the turbulent ki-netic energy production Π = (cid:104) u (cid:48) i u (cid:48) j (cid:105) ( ∂ (cid:104) u i (cid:105) / ∂ x j ) , where angu-lar brackets and apices indicate the mean and the fluctuations,respectively. The high value of Π indicates the location andintensity of the shear layer. Figure 4 reports the wall normalprofiles of the particle concentration at different streamwiseposition, in particularpanel (b) refers to a streamwise positionthat traverses the shear layer. The plot shows the accumulationof particles at the bump wall, y ≈ .
49, up to y ≈ .
53. Theseare the particles that move up the bump wall and towards thetip, since they follow the fluid which is recirculating under theshear layer behind the bump, see the negative particle velocityin figure 1(b) and the region of negative fluid velocity in fig-ure 5(a). As the shear layer is encountered, the concentrationslightly drops below C = (a) (b)(c) (d)(e) (f) FIG. 3. Mean particle concentration around the bump in the ( x , y ) plane for St + = [ , , , , , ] in panels (a) to (f) respectively. y C St = 1no liftwith lift (a) y C St = 1no liftwith lift (b) y C St = 1no liftwith lift (c)
FIG. 4. Mean particle concentration for particles having St + = x = [ . , . , . ] in panels (a) to (c) respectively. Panel (b) shows alimited y range to zoom in the area just after the tip of the bump. Bluecircles and red squares show the particles evolved with and withoutthe lift force respectively. ther away from the shear layer and towards the center of thechannel. For particles having this Stokes number, the effect ofthe lift is negligible.At St + =
2, see figure 2(b) and the zoom in figure 3(b), theparticle behaviour is similar to the one for St + =
1, except forthe significant decrease in concentration away from the wallswhich is compensated by an increase at the walls. The con-centration also decreases in the stream above the shear layerarticles in turbulent separated flow over a bump 5 (a)(b)
FIG. 5. Panel (a): Mean streamwise velocity (cid:104) u x (cid:105) . Panel (b): Meanturbulent kinetic energy production Π . Black solid isoline shows (cid:104) u x (cid:105) = y C St = 5no liftwith lift (a) y CSt = 5no liftwith lift (b) y C St = 5no liftwith lift (c)
FIG. 6. Mean particle concentration for particles having St + = x = [ . , . , . ] in panels (a) to (c) respectively. Blue circles andred squares show the particles evolved with and without the lift forcerespectively. with respect to the previous Stokes number. The inertia of theparticles increases with Stokes number and consequently lessparticles are capable of entering the recirculating region. Theincreased particle segregation towards the wall persists down-stream of the bump where C ≈ .
5, and it is more intense thanthe lower Stokes number case.When the Stokes number is increased to St + =
5, figures2(c) and 3(c), the particles segregate more towards the wallsand have a low concentration (dark blue) in the rest of thedomain. The main recirculating region behind the bump con-tains no particles (purple colour) except for concentrations ofparticles in the two, small, secondary recirculating bubbles atthe walls (discussed for St + = x ≈ x = [ . , . , . ] . In general, a slight difference in concen-tration appears when the lift term is included or not, except for an evident deviation from the two values in the shear layer,see panel (b) at y ≈ .
55. The strong vorticity present in theshear layer, which influences the lift, is a direct responsiblefor this discrepancy. Panel (c) shows that the concentration islow in the recirculating region, y ≤ .
6, but nonetheless highat both walls, even if the bottom wall is underneath a regioncontaining no particles. The particles must therefore reachthis region from the edge of the recirculation where the flowreattaches and they are transported towards the left along thewall by reverse (upstream) flow.At higher Stokes numbers, the particles’ inertia becomesdominant and therefore their response to the flow is minimal.Figures 2(d), (e) and (f), together with the corresponding pan-els in figure 3, show the concentration for St + = [ , , ] .After hitting the bump and being projected upwards, the parti-cles proceed towards the upper wall. At St + =
50, the streamof particles still bends downwards towards the centre of thechannel, since the mean flow has some effect on them sincethe characteristic time of the mean flow is comparable with theparticle relaxation time which in turn is sensibly larger thanthe fluctuation characteristic time. At St + = x ≈ .
1, theflow re-directs them in the streamwise direction and the parti-cles do not enter the recirculating region from above. This isnot the case for the highest Stokes number, St + = x ≈ , which resembles a shockwave sincethe population of particles act as a compressible phase. Theparticles segregate towards the upper part of the channel forthese higher Stokes numbers. Apart from the particles thatare projected upwards by the bump, another contribution tothis segregation is due to the particles that travel at y > . St + =
200 and St + =
600 re-spectively with the colour contour representing the mean par-ticle concentration for the fully turbulent DNS simulation asin panels (e) and (f) in figure 3. The white iso-surface showsthe particle concentration of a separate simulation when thereis no coupling, i.e. the particle acceleration is zero and onlythe rebound from the solid walls is considered. Note that thewhite iso-contours are identical in both panels since the dy-namics is independent of the Stokes number (no fluid inter-action). The particles are given an initial streamwise velocityarticles in turbulent separated flow over a bump 6 (a) (b)
FIG. 7. Mean particle concentration in the fully turbulent flow ascoloured contour. The white iso-surface represents the particle con-centration with no fluid interaction. The black iso-surface representsthe particle concentration when interacting only with the mean flow. St + =
200 in panel (a) and St + =
600 in panel (b). Both iso-surfacesare set at C = and the ones travelling at y < . x ≈ . x ≈ . < x < .
0) and focus in this small re-gion. The black iso-surface shows the particle concentrationwhen they are coupled with only the mean flow (no fluctua-tions) obtained from the turbulent simulation. At St + = St + = x ≈
4, superimposing the white and coloured con-tours. The turbulent fluctuations therefore play an importantrole by deviating the particle stream or dispersing it. Nonethe-less, in high Stokes number cases, considering only the meanflow for particle transport, the qualitative behaviour is wellpredicted.The lift force plays a crucial role in determining the particleconcentration at St + =
50, see figure 8. Away from the walls,the concentration for the particles without the lift is approxi-mately half the one for particles with the lift and therefore thelift cannot be considered to be negligible, see panels (a) and(b) for plots taken at x = . x = . x = . y C St = 50no liftwith lift (a) y C St = 50no liftwith lift (b) y CSt = 50no liftwith lift (c)
FIG. 8. Mean particle concentration for particles having St + =
50 at x = . x = . x = .
5. Blue circles and red squares showthe particles evolved with and without the lift force respectively lift force. This difference can be again attributed to vorticitywhich is strong in this region due to the presence of the bound-ing wall. The particles migrating towards the walls experiencea negative slip velocity with respect to the fluid, u | p − v p < x ≈
4) also decreases significantly comparedto the lower Stokes numbers, see figure 9(a). The reverse flowon the right of the recirculating region is not able to capture asmany particles as previously seen for the lower Stokes num-bers since the particles’ inertia is now higher. When the Stokesincreases to St + = y < . St + = IV. FINAL REMARKS
The effects of Stokes number and lift force have been anal-ysed for particle-laden turbulent flow that separates due to thepresence of a bump in a channel-like domain. A strong shearlayer and a recirculating region are formed behind the bump,both affecting the particle dynamics. The Reynolds numberis relatively high when considering similar geometries andthe present multi-phase configuration has, to the best of ourknowledge, never been simulated before.A vast range of Stokes numbers are considered, simulatedboth with and without the lift force in the particle dynamicequation. The conclusion is that the lift force must not be ne-glected, since there are drastic changes in particle concentra-articles in turbulent separated flow over a bump 7 y C St = 50no liftwith lift (a) y CSt = 200no liftwith lift (b) y CSt = 600no liftwith lift (c)
FIG. 9. Mean particle concentration at x = St + = [ , , ] in panels (a) to (c) respectively. Blue circles andred squares show the particles evolved with and without the lift forcerespectively. tion for some Stokes numbers with respect to results obtainedwithout the lift force. Regions of high vorticity, particularlyat the walls and in the shear layer, exhibit the greatest differ-ences. However, for some intermediate Stokes numbers, thedifference is also evident in the bulk of the flow throughoutthe whole domain.The particles behave as tracers at low Stokes numbers andclosely follow the fluid phase. As the Stokes number is in-creased, the particles tend to segregate at the walls and do notenter the recirculating region behind the bump. Some parti-cles are captured by the recirculation, forced upstream as theymove close to the wall and transported back downstream asthey encounter the strong shear layer formed by the bump.Secondary recirculating regions, one before the bump and twoinside the primary recirculating region, manage to capture theparticles. At the highest Stokes numbers, the particles’ inertiais high and their ballistic nature makes them bounce off thebump, top and bottom walls, creating reflection layers as theyare only slightly affected by the fluid flow. This enables theparticles to enter the recirculating region by bouncing off theback of the bump and creating a focused spot of high particleconcentration.Understanding particle-laden flows in presence of featuressuch as a bump is important for engineering applications thatconcern geometries relatively more complex than classicalflows such as straight pipes or planar channels. Such fea-tures may be intentional (such as for filtration or separation ofparticles) or unintentional (such as defects) and it is essentialto comprehend how particles, that may have different Stokesnumbers, behave in such particle-laden flows. ACKNOWLEDGMENTS
The research has received funding from the European Re-search Council under the ERC Grant Agreement no. 339446.We acknowledge CINECA for awarding us access to su-percomputing resource MARCONI based in Bologna, Italythrough ISCRA project no. HP10BLVPKA. S. Balachandar and J. K. Eaton, “Turbulent dispersed multiphase flow,” An-nual Review of Fluid Mechanics , 111–133 (2010). S. Elghobashi, “Direct numerical simulation of turbulent flows laden withdroplets or bubbles,” Annual Review of Fluid Mechanics , 217–244(2019). C. Marchioli, “Large-eddy simulation of turbulent dispersed flows: a reviewof modelling approaches,” Acta Mechanica , 741–771 (2017). A. Innocenti, C. Marchioli, and S. Chibbaro, “Lagrangian filtered den-sity function for les-based stochastic modelling of turbulent particle-ladenflows,” Physics of Fluids , 115106 (2016). G. I. Park, M. Bassenne, J. Urzay, and P. Moin, “A simple dynamic subgrid-scale model for les of particle-laden turbulence,” Physical Review Fluids ,044301 (2017). J.-P. Minier, S. Chibbaro, and S. B. Pope, “Guidelines for the formulationof lagrangian stochastic models for particle simulations of single-phase anddispersed two-phase turbulent flows,” Physics of Fluids , 113303 (2014). H. Sajjadi, M. Salmanzadeh, G. Ahmadi, and S. Jafari, “Lattice boltzmannmethod and rans approach for simulation of turbulent flows and particletransport and deposition,” Particuology , 62–72 (2017). S. Vahidifar, M. R. Saffarian, and E. Hajidavalloo, “Introducing the theoryof successful settling in order to evaluate and optimize the sedimentationtanks,” Meccanica (2018), 10.1007/s11012-018-0907-2. S. Elghobashi, “On predicting particle-laden turbulent flows,” Applied Sci-entific Research , 309–329 (1994). F. Toschi and E. Bodenschatz, “Lagrangian properties of particles in turbu-lence,” Annual review of fluid mechanics , 375–404 (2009). J. G. M. Kuerten, “Point-particle dns and les of particle-laden turbulentflow-a state-of-the-art review,” Flow, turbulence and combustion , 689–713 (2016). K. Gustavsson, J. Jucha, A. Naso, E. Lévêque, A. Pumir, and B. Mehlig,“Statistical model for the orientation of nonspherical particles settling inturbulence,” Physical review letters , 254501 (2017). G. A. Voth and A. Soldati, “Anisotropic particles in turbulence,” AnnualReview of Fluid Mechanics , 249–276 (2017). K. D. Squires and J. K. Eaton, “Preferential concentration of particles byturbulence,” Physics of Fluids A: Fluid Dynamics , 1169–1178 (1991). A. D. Bragg, P. J. Ireland, and L. R. Collins, “Mechanisms for the clusteringof inertial particles in the inertial range of isotropic turbulence,” PhysicalReview E , 023029 (2015). C. Nicolai, B. Jacob, P. Gualtieri, and R. Piva, “Inertial particles in homo-geneous shear turbulence: experiments and direct numerical simulation,”Flow, turbulence and combustion , 65–82 (2014). F. Battista, P. Gualtieri, J.-P. Mollicone, and C. M. Casciola, “Applicationof the exact regularized point particle method (erpp) to particle laden turbu-lent shear flows in the two-way coupling regime,” International Journal ofMultiphase Flow , 113–124 (2018). G. Sardina, F. Picano, P. Schlatter, L. Brandt, and C. M. Casciola, “Largescale accumulation patterns of inertial particles in wall-bounded turbulentflow,” Flow, turbulence and combustion , 519–532 (2011). G. Sardina, P. Schlatter, F. Picano, C. M. Casciola, L. Brandt, andD. S. Henningson, “Self-similar transport of inertial particles in a turbulentboundary layer,” Journal of Fluid Mechanics , 584–596 (2012). D. Li, A. Wei, K. Luo, and J. Fan, “Direct numerical simulation of aparticle-laden flow in a flat plate boundary layer,” International Journal ofMultiphase Flow , 124–143 (2016). M. Bernardini, S. Pirozzoli, and P. Orlandi, “The effect of large-scale turbu-lent structures on particle dispersion in wall-bounded flows,” InternationalJournal of Multiphase Flow , 55–64 (2013). C. Marchioli, A. Soldati, J. G. M. Kuerten, B. Arcen, A. Taniere, G. Gold-ensoph, K. D. Squires, M. F. Cargnelutti, and L. M. Portela, “Statisticsof particle dispersion in direct numerical simulations of wall-bounded tur-bulence: results of an international collaborative benchmark test,” Interna-tional Journal of Multiphase Flow , 879–893 (2008). C. Marchioli, A. Giusti, M. V. Salvetti, and A. Soldati, “Direct numericalsimulation of particle wall transfer and deposition in upward turbulent pipeflow,” International journal of Multiphase flow , 1017–1038 (2003). F. Picano, G. Sardina, and C. M. Casciola, “Spatial development ofparticle-laden turbulent pipe flow,” Physics of Fluids , 093305 (2009). G. Sardina, P. Schlatter, L. Brandt, F. Picano, and C. M. Casciola, “Wallaccumulation and spatial localization in particle-laden wall flows,” Journal articles in turbulent separated flow over a bump 8 of Fluid Mechanics , 50–78 (2012). J. D. Kulick, J. R. Fessler, and J. K. Eaton, “Particle response and turbu-lence modification in fully developed channel flow,” Journal of Fluid Me-chanics , 109–134 (1994). X. Liu, K. Luo, and J. Fan, “Turbulence modulation in a particle-ladenflow over a hemisphere-roughened wall,” International Journal of Multi-phase Flow , 250–262 (2016). M. De Marchis, B. Milici, G. Sardina, and E. Napoli, “Interaction betweenturbulent structures and particles in roughened channel,” International Jour-nal of Multiphase Flow , 117–131 (2016). A. W. Vreman, “Turbulence attenuation in particle-laden flow in smoothand rough channels,” Journal of Fluid Mechanics , 103 (2015). F. Picano, F. Battista, G. Troiani, and C. M. Casciola, “Dynamics of pivseeding particles in turbulent premixed flames,” Experiments in Fluids ,75–88 (2011). P. Gualtieri, F. Battista, and C. Casciola, “Turbulence modulation in heavy-loaded suspensions of tiny particles,” Physical Review Fluids , 034304(2017). F. Battista, F. Picano, G. Troiani, and C. M. Casciola, “Intermittent featuresof inertial particle distributions in turbulent premixed flames,” Physics ofFluids , 123304 (2011). W. Wu, G. G. Soligo, C. Marchioli, A. Soldati, and U. Piomelli, “Parti-cle resuspension by a periodically forced impinging jet,” Journal of FluidMechanics , 284–311 (2017). T. C. W. Lau and G. J. Nathan, “Influence of stokes number on the veloc-ity and concentration distributions in particle-laden jets,” Journal of FluidMechanics , 432–457 (2014). T. C. W. Lau and G. J. Nathan, “The effect of stokes number on particlevelocity and concentration distributions in a well-characterised, turbulent,co-flowing two-phase jet,” Journal of Fluid Mechanics , 72–110 (2016). X. Wang, X. Zheng, and P. Wang, “Direct numerical simulation of particle-laden plane turbulent wall jet and the influence of stokes number,” Interna-tional Journal of Multiphase Flow , 82–92 (2017). H. M. Abdelaziz, M. Gaber, M. M. Abd-Elwakil, M. T. Mabrouk, M. M.Elgohary, N. M. Kamel, D. M. Kabary, M. S. Freag, M. W. Samaha, S. M.Mortada, et al. , “Inhalable particulate drug delivery systems for lung can-cer therapy: Nanoparticles, microparticles, nanocomposites and nanoaggre-gates,” Journal of Controlled Release (2017). R. Ni, J. Zhao, Q. Liu, Z. Liang, U. Muenster, and S. Mao, “Nanocrys-tals embedded in chitosan-based respirable swellable microparticles as drypowder for sustained pulmonary drug delivery,” European Journal of Phar-maceutical Sciences , 137–146 (2017). E. Ghahramani, O. Abouali, H. Emdad, and G. Ahmadi, “Numerical in-vestigation of turbulent airflow and microparticle deposition in a realisticmodel of human upper airway using les,” Computers & Fluids , 43–54(2017). F. S. Stylianou, J. Sznitman, and S. C. Kassinos, “Direct numerical simu-lation of particle laden flow in a human airway bifurcation model,” Interna-tional Journal of Heat and Fluid Flow , 677–710 (2016). M. Rahimi-Gorji, O. Pourmehran, M. Gorji-Bandpy, and T. B. Gorji, “Cfd simulation of airflow behavior and particle transport and deposition in dif-ferent breathing conditions through the realistic model of human airways,”Journal of Molecular Liquids , 121–133 (2015). V. Thondapu, C. V. Bourantas, N. Foin, I.-K. Jang, P. W. Serruys, andP. Barlis, “Biomechanical stress in coronary atherosclerosis: emerging in-sights from computational modelling,” European heart journal , 81–92(2016). W. Choi, J. H. Park, H. Byeon, and S. J. Lee, “Flow characteristics arounda deformable stenosis under pulsatile flow condition,” Physics of Fluids ,011902 (2018). X. Huang and P. Durbin, “Particulate dispersion in a turbulent serpentinechannel,” Flow, turbulence and combustion , 333–344 (2010). X. Huang and P. A. Durbin, “Particulate mixing in a turbulent serpentineduct,” Physics of Fluids , 013301 (2012). A. Noorani, G. Sardina, L. Brandt, and P. Schlatter, “Particle velocity andacceleration in turbulent bent pipe flows,” Flow, Turbulence and Combus-tion , 539–559 (2015). J. T. Ault, A. Fani, K. K. Chen, S. Shin, F. Gallaire, and H. A. Stone,“Vortex-breakdown-induced particle capture in branching junctions,” Phys-ical review letters , 084501 (2016). F. Stella, N. Mazellier, and A. Kourta, “Scaling of separated shear lay-ers: an investigation of mass entrainment,” Journal of Fluid Mechanics ,851–887 (2017). B. Krank, M. Kronbichler, and W. A. Wall, “Direct numerical simulationof flow over periodic hills up to reh=10595,” Flow, Turbulence and Com-bustion , 1–31 (2017). L. A. C. A. Schiavo, W. R. Wolf, and J. L. F. Azevedo, “Turbulent kineticenergy budgets in wall bounded flows with pressure gradients and separa-tion,” Physics of Fluids , 115108 (2017). J.-P. Mollicone, F. Battista, P. Gualtieri, and C. M. Casciola, “Turbulencedynamics in separated flows: the generalised kolmogorov equation forinhomogeneous anisotropic conditions,” Journal of Fluid Mechanics ,1012–1039 (2018). J.-P. Mollicone, F. Battista, P. Gualtieri, and C. M. Casciola, “Effect ofgeometry and reynolds number on the turbulent separated flow behind abulge in a channel,” Journal of Fluid Mechanics , 100–133 (2017). P.-Y. Passaggia and U. Ehrenstein, “Optimal control of a separatedboundary-layer flow over a bump,” Journal of Fluid Mechanics , 238–265 (2018). C. Kähler, S. Scharnowski, and C. Cierpka, “Highly resolved experimentalresults of the separated flow in a channel with streamwise periodic constric-tions,” Journal of Fluid Mechanics , 257–284 (2016). P. Fischer, J. W. Lottes, and S. G. Kerkemeier, “Nek5000 -Open source spectral element CFD solver. Argonne National Labora-tory, Mathematics and Computer Science Division, Argonne, IL, seehttp://nek5000.mcs.anl.gov,” (2008). A. T. Patera, “A spectral element method for fluid dynamics,” Journal ofComputational Physics54