Near-wall turbulence modulation by small inertial particles
TThis draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics Near-wall turbulence modulation by smallinertial particles
Pedro Costa † , Luca Brandt and Francesco Picano Faculty of Industrial Engineering, Mechanical Engineering and Computer Science, Universityof Iceland, Hjardarhagi 2-6, 107 Reykjavik, Iceland Linn´e FLOW Centre and SeRC (Swedish e-Science Research Centre), Department ofEngineering Mechanics, KTH, SE-100 44 Stockholm, Sweden Department of Industrial Engineering, University of Padova, Italy(Received xx; revised xx; accepted xx)
We use interface-resolved simulations to study near-wall turbulence modulation by smallinertial particles in dilute/semi-dilute conditions. We considered three bulk solid massfractions, Ψ = 0 . . . Ψ = 3 .
1. Introduction
The turbulent channel flow laden with small, gravity-free inertial spheres is a paradig-matic multiphase flow, yet not fully understood and correctly modeled. This systemhas been widely studied using direct numerical simulations (DNS) of the Navier-Stokesequations using the point-particle approximation (Balachandar & Eaton 2010), whichassumes that interphase coupling is localized at a single point and the particle dynamicsdriven by an undisturbed flow field, evaluated at the particle position. This approximationallows us to study the flow dynamics beyond the particle scale without solving theactual flow around each particle. Despite the many studies, the validity of these particle-fluid coupling models has not been fully assessed for canonical turbulent wall flows, inparticular when the underlying turbulence is altered by the dispersed phase.When the particle feedback on the flow becomes important – the two-way couplingregime – care should be taken in the estimation of the undisturbed velocity. The challengeis conciliating the estimation of an undisturbed velocity sampled by the particle with theneed of forcing the local velocity field. Indeed, approaches for accurate and realistictwo-way coupling methods have been pursued since the work of Crowe et al. (1977),and are still object of active research (see, e.g., Gualtieri et al. et al. † Email address for correspondence: [email protected] a r X i v : . [ phy s i c s . f l u - dyn ] F e b P. Costa, L. Brandt and F. Picanoet al. (2010) have observed a drag- reducing behavior in two-way coupled particle-ladenturbulent channel flow, other studies have measured a drag increase (Battista et al. et al. (2019), which compares one-to-one two-way coupled point-particle DNS with experimental measurements for verticalturbulent channel flow at different volume loads, and the interface-resolved (also denotedparticle-resolved) DNS of spherical particles near the point-particle limit in homogeneousisotropic turbulence (Schneiders et al. et al. et al. et al. et al. (2020) and add results at higher volume fraction. Three bulk mass fractions Ψ = 0 . . .
7% at fixed particle to fluid density ratio ( Π ρ = 100) arehence considered, where the latter two show non-negligible two-way coupling effects.Our results reveal two distinct mechanisms for turbulence modulation. At lower volumefractions, the flow is only dense very close to the wall, and the higher flow inertiain this region results in a drag-increased flow resembling single-phase turbulence atslightly higher Reynolds number. At higher volume fractions, the dispersed phase isdynamically important over the entire channel. Here the drag increasing effect is amplifiedby correlated particle velocity fluctuations, but counterbalanced by a substantial fluidturbulence drag reduction. This results in a milder increase in drag for the denser case.These observations may be explained in light of the streamwise momentum balance forvanishing volume fraction, but with non-negligible mass fraction.
2. Methods and Computational Setup
Since we use here the tools and setup in Costa et al. (2020), with one additional case atthe largest volume fraction, we briefly summarize the numerical method and refer to theprevious work for more details. We solve the continuity and Navier-Stokes equations foran incompressible Newtonian fluid, together with the Newton-Euler equations driving themotion of the solid spherical particles. These two sets of equations are coupled directlyusing the immersed-boundary method developed by Breugem (2012), built on a standardsecond-order finite-difference method on a three-dimensional, staggered Cartesian grid,using a fast-Fourier-transform-based pressure-projection method (Kim & Moin 1985).Short-range particle-particle/particle-wall interactions are modeled using the method ofCosta et al. (2015) as in Costa et al. (2020).Turbulent channel flow is simulated in a domain periodic in the streamwise ( x ) andspanwise ( z ) directions, with no-slip and no-penetration boundary conditions imposedat the walls ( y = h ∓ h ), where h is the channel half height. The flow is driven bya uniform pressure gradient that ensures a constant bulk velocity. The bulk Reynoldsnumber is equal to Re b = U b (2 h ) /ν = 5 600, which corresponds to an unladen frictionReynolds number Re sphτ = u τ h/ν ≈ U b is the bulk flow velocity and u τ the wall friction velocity. The particle properties are chosen to yield a particle Reynoldsnumber Re p = Du τ /ν = D + = 3, and Stokes number St p = Π ρ Re p /
18 = 50, where D ear wall turbulence modulation by small inertial particles Case Φ ( N p ) Ψ Notes VD . . D . . SD . .
67% interface-resolved (semi dilute)
Table 1.
Computational parameters. Φ / Ψ denote the bulk solid volume/mass fraction, and N p the total number of particles. Common to all cases: Bulk Reynolds number Re b = 5 600(i.e. friction Reynolds number in the single-phase limit Re sphτ ≈ D/ (2 h ) = 1 / Π ρ = 100. These correspond to aparticle Reynolds number based on the unladen reference values of D + = 3 and Stokesnumber St = 50. The fluid domain is discretized on a regular Cartesian grid with( L x /N x ) × ( L y /N y ) × ( L z /N z ) = (6 h/ × (2 h/ × (3 h/ D/∆x = 12 grid points over the particle diameter (in total 420 Lagrangian gridpoints) (same as Costa et al. is the particle diameter and Π ρ the particle-to-fluid mass density ratio. Three values ofsolid volume fraction are considered, increasing by factors of 10: Φ (cid:39) · − , denoted very dilute ( VD ), Φ (cid:39) · − , dilute ( D ), and Φ (cid:39) · − , semi dilute ( SD ), with thedata pertaining to the first two cases also used in the recent study by Costa et al. (2020).Table 1 shows all relevant physical and computational parameters.Finally, unless otherwise stated, the mesoscale-averaged profiles reported in thismanuscript correspond to intrinsic averages in time and along the two homogeneousdirections for each phase (Costa et al.
3. Results
A three-dimensional visualization of the flow pertaining to the different cases isreported in figure 1, where iso-surfaces of constant Q -criterion (colored by the local wall-normal velocity) and the particles are displayed. As expected, no qualitative differencescan be seen between the single phase and the very dilute case VD , apart from the verysmall number of dispersed particles in the latter case. Qualitative differences betweenthese cases and case D are also small; however, an increased number of high vorticityspots, footprint of the presence of the particles, is obvious over the entire domain. Finally,a strong modulation of the flow dynamics by the particles is seen when inspecting case SD , where the disruption of the flow coherent structures is evident. One of the mainmessages of the present work is that, although cases D and SD have macroscopic flowvariations and, therefore, are formally in a two-way coupling regime, the mechanism ofturbulence modulation is definitively different.Figure 2( a ) presents the friction Reynolds number as a function of the bulk volumefraction, while the change in drag relative to the unladen flow is quantified in panel( b ). The friction Reynolds number increases significantly (5%) from case VD to case D , despite the relatively low mass fraction Ψ = O (10 − ). As discussed in Costa et al. (2020), this increase is attributed to the higher inertia induced by the large local massfraction near the wall; the particles are driven by turbophoresis towards the wall, wherethey experience a large apparent slip velocity. Further increasing the mass fraction byan order of magnitude (case SD ) results in a mild increase in drag, only about 3%with respect to case D . This milder increase indicates that a competing drag-reducingmechanism comes into play at higher mass loading. The blue symbols in figure 2 displayquantities computed using a wall friction velocity determined from the centerline slope P. Costa, L. Brandt and F. Picano single phase case VD case D case SD Figure 1.
Three-dimensional flow visualizations of the different cases under consideration. Wedisplay surfaces of constant second-invariant of the velocity gradient tensor Q = 20( U b /h ) ,colored by the local wall-normal velocity (white – high and blue – low). The particles are shownto scale, in orange color. See table 1 for a description of the different cases. − − − Ψ R e τ Re τ Re tτ − − − Ψ − . − . . . . D C DCDC t ( a ) ( b ) Figure 2. ( a ) Friction Reynolds number Re τ versus the bulk volume fraction Ψ ; Re tτ is based ona velocity scale computed form the slope of the outer-scaled Reynolds shear stresses evaluatedat the channel centreline u tτ = (cid:112) ∂ y/h (cid:104) u (cid:48) v (cid:48) (cid:105) | y/h =1 . ( b ) Drag modulation DC = 1 − Re τ / Re ,sphτ ,computed with Re τ and Re tτ (DC t ). of the Reynolds stress, so to quantify the decrease in turbulent momentum transfer, asdiscussed in detail later.Figure 3(a) shows the mass fraction normalized by the corresponding bulk value versusthe wall-normal distance in particle diameters. All the cases show a peak at the wall,corresponding to a particle layer. Clearly, the fraction of particles near the wall decreaseswith increasing mass loading. This decrease in wall accumulation is attributed to theincreased shear rate near the boundary, which enhances lift forces (Costa et al. SD shows a disproportionally strong decrease of the relative particleconcentration, considering the relatively low increase in wall shear with respect to case S . This is partly caused by two-way coupling effects, which dampen the intensity ofthe wall-normal fluid velocity fluctuations responsible – to first approximation – for theturbophoretic drift (Marchioli & Soldati 2002). In addition, particle-particle interactionsmay become significant. The inset of figure 3( a ) shows the mass fraction profile as afunction of the outer-scaled wall-normal distance. While the mass fraction profiles becomeuniform far from the wall for cases VD and D , the distribution shows a mild monotonicincrease with the wall distance in case SD . This indicates that, for this densest case, ear wall turbulence modulation by small inertial particles y/D ψ / Ψ − − − y/h ψ VDDSD z/h -0.2500.250.50.751 R z uu sph, y/h = 0 . y/h = 0 . y/h = 0 . y/h = 0 .
067 sph, y/h = 0 . y/h = 0 . y/h = 0 . y/h = 0 . ( a ) ( b ) Figure 3. ( a ) Local solid mass fraction ψ , normalized by the bulk value Ψ as a function of thewall-normal distance in particle diameters; the inset shows the nominal local mass fraction ψ asa function of the outer-scaled wall-normal distance. ( b ) Outer-scaled spanwise autocorrelationof stramwise velocity at y/h = 0 .
067 and y/h = 0 . y/δ sphν ≈
12 and 30); ‘sph’ correspondsto the unladen flow. non-negligible particle-particle interactions may be driving particles towards regions oflow shear. The mechanism underlying this drift is most likely similar to that reportedin Fornari et al. (2016) – particle inertia and local high shear promote inter-particleinteractions, causing a net migration towards low shear regions.The inset of figure 3( a ) also illustrates where two-way coupling effects are expectedto be important. These can be envisaged near the wall for the less dilute cases, as thesolid mass fraction increases to 20% for case D , and to about 80% for case SD . Awayfrom the wall, instead, the solid mass fraction retains high values only for case SD ,25 − D to case SD . In the latercase, we observe a significant turbulence modulation by the solid particles: the spacingbetween low and high-speed streaks is larger, the spanwise velocity variation smoother,and the maximum streak amplitude occurs at a much larger wall-normal distance, whichis reflected in the spanwise autocorrelation of the streamwise velocity in wall-parallelplanes (figure 3( b )). Case SD also presents non-negligible four-way coupling effects. Inparticular, we estimated the particle collision frequency near the wall following Sundaram& Collins (1997) and found it to be O (10) per unit volume L x L z D and bulk eddy turnovertime ( O ( h/u τ )), about one order of magnitude larger than in case D .The inner-scaled mean velocity profiles, see figure 4(a), resemble those of the single-phase flow, with a downward shift that indicates drag increase (the difference is lessapparent in the corresponding outer-scaled profiles shown in panel (b)). The inset ofthe figure shows the particle-to-fluid apparent slip velocity, defined as the differencebetween the mean velocity profiles of the fluid and solid phase. The negative minimumin the near-wall region is due to the higher particle velocity where the fluid velocity isvanishing. The local maximum of positive slip velocity occurs in the buffer region, and isthe footprint of the well-known tendency of near-wall inertial particles to reside in regionsof low streamwise fluid velocity (Rouson & Eaton 2001). The higher the mass loading,the weaker this slip velocity is. Case SD , in particular, shows a deeper minimum of theslip velocity, and virtually no slip between the two phases beyond y > δ ν . Interestingly,the fluid-to-particle apparent slip velocity does not attain positive values for case SD ,meaning that particles do not oversample low-speed regions. Since the low (and high)speed regions are located further away from the wall, particles departing from the wallmay not reside for long enough in these regions. Finally, we should note that wider streakspacing is often a feature of turbulent drag reduction (see e.g. Tiederman et al. P. Costa, L. Brandt and F. Picano y/δ ν h u i / u τ SphVDDSD y/δ ν -2-101 ∆ h u i / u τ y/h h u i / U b SphVD DSD y/h -0.100.1 ∆ h u i / U b ( a ) ( b ) Figure 4.
Inner- ( a ) and outer-scaled ( b ) mean streamwise velocity profiles. Lines – fluidvelocity; symbols – particle velocity. The insets show the apparent fluid-to-particle slip velocity,defined as the difference between the intrinsic mean velocity of each phase. y/δ ν u r / u τ SphVDDSD 0 50 100 150 200 y/δ ν v r / u τ ( a ) ( b ) y/δ ν w r / u τ y/h h u v i / U b ( c ) ( d ) Figure 5.
Second-order moments of mean fluid (lines) and particle (symbols) velocity. ( a )inner-scaled streamwise, ( b ) wall-normal and ( c ) spanwise velocity r.m.s., and ( d ) outer-scaledReynolds stresses profile. The legend in ( a ) holds for all panels. Figure 5 presents the second-order moments of the fluid and particle velocity. Particlefluctuations are usually higher in the near wall region and smaller or similar to that ofthe fluid in the bulk. While modifications with respect to the unladen case are relativelysmall for cases VD and D , case SD shows three significant differences: a strong reductionof the streamwise velocity fluctuations near the wall and enhancement in the bulk, aremarkable decrease of the wall-normal and spanwise velocity r.m.s. across the channel,and a reduction in Reynolds shear stresses. Interestingly, the same trends have beenobserved in the point-particle simulations of drag-reducing turbulent channel flow byZhao et al. (2010). Focusing on the fluid Reynolds shear stress, case D features a slightincrease of the peak value and slope in the outer region, while the opposite applies tocase SD . Note that the particle Reynolds stresses are always larger than the fluid ones.Despite the overall increase in drag, the significant decrease of the Reynolds shear ear wall turbulence modulation by small inertial particles SD is consistent with the reduced contribution to the overall dragfrom the fluid turbulence shown in figure 2 by the friction Reynolds number basedon the velocity based on the slope of the Reynolds stresses, u tτ = (cid:112) ∂ y/h (cid:104) u (cid:48) v (cid:48) (cid:105) | y/h =1 .This quantity initially increases with the mass loading (case VD to D ), denoting anenhancement of turbulence and its induced drag. Further increasing the solid massfraction, (case SD ), Re tτ decreases – even below the single phase flow – while theconventional friction Reynolds number increases. In other words, while turbulence isattenuated, a fundamentally different mechanism acts to anyway increase the drag. Also,despite showing drag increase, case D features statistics that highly resemble those ofthe single-phase flow. Hence, only case SD shows a more intricate two-way couplingmechanism, at play over the whole channel region.To gain further insight, we examine the streamwise momentum balance of the fluid, τ = ρu τ (1 − (cid:104) φ (cid:105) ) (cid:18) ν d (cid:104) u (cid:105) d y − (cid:104) u (cid:48) v (cid:48) (cid:105) (cid:19) + (cid:104) φ (cid:105) τ p = ρu τ (cid:16) − yh (cid:17) , (3.1)where the first and second terms denote the fluid viscous and Reynolds stresses, whilethe last term the total particle stress τ p = τ p,ν + ρ p (cid:10) u (cid:48) p v (cid:48) p (cid:11) , with τ p,ν including theviscous and collisional particle stresses. The data are shown in figure 6( a - c ), where theinsets depict the local relative contribution of each term to the total momentum transfer,while panel ( d ) reports the wall normal integral of the different terms normalized by thetotal stress, ∫ h τ i d y/ ∫ h τ d y , with τ i the different contributions. Expectedly, the stressbudget pertaining to case VD resembles that of the single-phase flow, with negligiblecontribution from the particles. Case D displays a noticeable, though small, contributionof the particle stresses which have a peak of about 5% close to the wall and are of the orderof the viscous stresses in the bulk. Finally, the total particle stresses show a significantrelative contribution to the total in case SD , of about 20%. Examining panel ( d ) of thefigure, we note that the sum of the viscous and turbulent stress corresponds to the single-phase drag in case VD , the particle stress adding a negligible effect. In case D , the sumof the viscous and turbulent stress contributions is higher than the overall single-phasedrag, which is further increased by the particle stress. This indicates that the overall dragis increased by two different mechanisms: a direct one – the particles induce an extrastress, localised near the wall where the average slip velocity is not negligible; and anindirect effect – the solid phase enhances the Reynolds stress and hence the turbulence.Surprisingly, the sum of the viscous and turbulent stress contributions is smaller thanthe overall single-phase drag in case SD ; the particle stress, however, counterbalancesthis reduction and the overall drag is still higher than the corresponding unladen case.This confirms that we have turbulent drag reduction and a strong contribution of thetotal particle stress, which combine to a net drag increase.Let us take the limit of vanishing volume fraction, φ →
0, but finite mass fraction ψ = ρ p φ = ψ , of equation eq. (3.1), τ two − way = ρ (cid:18) ν d (cid:104) u (cid:105) d y − (cid:10) u (cid:48) f v (cid:48) f (cid:11)(cid:19) − (cid:104) ψ (cid:105) (cid:10) u (cid:48) p v (cid:48) p (cid:11) ≈ ρu τ (1 − y/h ) . (3.2)This limit corresponds to negligible particle excluded volume, but finite effects of theparticle mass, i.e. two-way coupling conditions. In this limit, the contribution from thecorrelated particle fluctuations corresponds to a particle direct contribution to the drag.When plotting the terms in equations (3.2), see figure 7, we note that the differencebetween the total stress and the sum of the different terms is very small, confirmingthat the contribution of the particle inertial shear stress to the budget is dynamicallysignificant and increases with the mass fraction. Moreover, the “two-way coupling” P. Costa, L. Brandt and F. Picano y/h h τ i i / ( ρ u τ ) totalturbulent particleviscous y/h h τ i i / τ y/h h τ i i / ( ρ u τ ) y/h h τ i i / τ ( VD ) ( D ) y/h h τ i i / ( ρ u τ ) y/h h τ i i / τ VD D SD0 . . . . . c o n t r i bu t i o n t o t o t a l s t r e ss viscous turbulent particle ( SD ) Figure 6.
Budget of streamwise momentum as a function of the outer-scaled wall-normaldistance, for the different cases. The inset shows the same budget, but normalized by the localvalue of the total stresses. The bottom-right panel shows the relative contribution of each termin the stress budget to the integral of the total stress, i.e. to the mean wall friction, normalizedby the corresponding values of the unladen case. y/h h τ i i / ( ρ u τ ) τψ D u p v p ED u f v f E ν d h u i / d y residual y/h h τ i i / ( ρ u τ ) τψ D u p v p ED u f v f E ν d h u i / d y residual ( D ) ( SD ) Figure 7.
Budget of streamwise momentum in the two-way coupling limit, for cases D and VD ,(see equation (3.2)). The dashed gray line corresponds to the difference between the sum of theterms in equation (3.2), and the total stress, to assess the validity of the approximation. budget explains the true nature of the turbulence attenuation for case SD . As the massloading is increased, the fluid Reynolds stresses progressively give in to particle Reynoldsstresses, which alter the nature of the flow. A significant portion of the total stress isspent to accelerate the inertial particles, which in turn lowers the fluid Reynolds stresses.Consequently, the flow shows fluid turbulence attenuation. Unlike correlated fluid velocityfluctuations, particle correlated motions cannot sustain near-wall turbulence, since thedispersed phase dynamics differ with respect to a Newtonian fluid (e.g. there is noincompressibility condition dictated by a pressure field).Since the net effect of the particle Reynolds stresses is an increased drag for case SD , we speculate that at higher mass fractions the drag may eventually decrease – thecontribution of Reynolds stresses to the momentum transport would decrease and the ear wall turbulence modulation by small inertial particles particle velocity fluctuations wouldalso decrease, drastically reducing the overall flow drag. New experiments and simulationsare needed to confirm this hypothesis.
4. Conclusions
We have used particle-resolved direct numerical simulations to study the near-wallturbulence modulation by small inertial particles. Three cases have been consideredwith volume fractions progressively increased by one order of magnitude, chosen in theone-way and two-way coupling regimes. The two densest cases show a non-negligibleturbulence modulation, however of fundamentally different nature. The case with Ψ = 3 .
4% ( Φ = 0 . DILPART . PC acknowledges fundingfrom the University of Iceland Recruitment Fund grant no. 1515-151341, TURBBLY.Declaration of Interests. The authors report no conflict of interest.
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