Turbulence modulations and drag reduction by inertialess spheroids in turbulent channel flow
11 Turbulence modulations and drag reductionby inertialess spheroids in turbulent channelflow
Ze Wang Chun-Xiao Xu and Lihao Zhao † , AML, Department of Engineering Mechanics, Tsinghua University, 100084 Beijing, China(Received xx; revised xx; accepted xx)
Previous studies on nonspherical particle-fluid interaction were mostly confined to elon-gated fiber-like particles, which were observed to induce turbulence drag reduction.However, with the presence of tiny disk-like particles how wall turbulence is modulatedand whether drag reduction occurs are still unknown. Motivated by those open questions,we performed two-way coupled direct numerical simulations of inertialess spheroids in tur-bulent channel flow by an Eulerian-Lagrangian approach. The additional stress accountsfor the feedback from inertialess spheroids on the fluid phase. The results demonstratethat both rigid elongated fibers (prolate spheroids) and thin disks (oblate spheroids) canlead to significant turbulence modulations and drag reduction. However, the disk-induceddrag reduction is less pronounced than that of rigid fibers with the same volume fraction.Typical features of drag-reduced flows by additives are observed in both flow statisticsand turbulence coherent structures. Moreover, in contrast to one-way simulations, thetwo-way coupled results of spheroidal particles exhibit stronger preferential alignmentsand lower rotation rates. At the end we propose a drag reduction mechanism by inertialessspheroids and explain the different performance for drag reduction by fibers and disks. Wefind that the spheroidal particles weaken the quasistreamwise vortices through negativework and, therefore, the Reynolds shear stress is reduced. However, the mean shear stressgenerated by particles, which is shape-dependent, partly compensates for the reductionof Reynolds shear stress and thus affects the efficiency of drag reduction. The presentstudy implies that tiny disk-like particles can be an alternative drag reduction agent inwall turbulence.
1. Introduction
Drag reduction in wall-bounded turbulence is of importance in industrial applications(Kim 2011). Toms (Toms 1949) observed a dramatic drag reduction in turbulent flowsinduced by adding a small amount of long-chain flexible polymer and followed by Toms’swork the polymer induced by drag reduction has been widely investigated (White &Mungal 2008; Benzi & Ching 2018). Compared with polymer additives, tiny rigid fiber-like particles are also found to produce drag reduction in wall-bounded turbulence butless efficient (Radin 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 Z. Wang, C.-X. Xu and L.-H. Zhao (2008)). The spheroids with small size influence the surrounding flow via additional stressand the feedback forces and torques are often neglected with negligibly small particleinertia, i.e. inertialess (Guazzelli et al. a , b , 1971) advanced the understanding of particlesuspensions. He firstly derived the expression of bulk stresses in a dilute suspensionof force-free ellipsoidal particles (Batchelor 1970 b ). The particle stress generated byelongated prolate spheroids in a dilute or semi-dilute suspensions is also described basedon slender-body theory (Batchelor 1970 a , 1971). Brenner (1974) developed a generalrheology theory for a dilute suspension of rigid axisymmetric particles. Several studies(Doi & Edwards 1986; Hinch & Leal 1972, 1976; Shaqfeh & Fredrickson 1990) furtherinvestigated the particle stress in a suspension of elongated spheroids theoretically. On theother hand, direct numerical simulations (DNSs) of suspensions of inertialess spheroidshave not been performed until 1990s to computational limitations. With an aligned-particle approximation (Lipscomb et al. et al. (1997) showed that elongated spheroids lead to a significantdrag reduction and the modulated turbulence statistics qualitatively agreed with theexperimental measurements in turbulent pipe flows. Paschkewitz et al. (2004) performeda two-way coupled DNS of suspensions of rigid fibers in a minimal channel flow (Jim´enez& Moin 1991). They demonstrated that the fluctuations of particle stress in inter-vortexextensional regions weaken the near-wall vortex structures and induce turbulent dragreduction. Gillissen et al. (2008) carried out coupled Eulerian simulations and observedthat tiny elongated spheroids lead to a reduction of the frictional drag in wall turbu-lence at various Reynolds numbers. A two-way coupled Eulerian-Lagrangian approachis adopted by Moosaie & Manhart (2013) to investigate drag reduction caused by rigidfibers. This Lagrangian point-particle tracking method is computationally expensive,because the particle orientation is solved directly without using any closure model orartificial diffusivity (Gillissen et al. et al. et al. (2013) observed a very high intrinsicviscosity in a dilute suspension of graphene oxide. Recent studies (Wang & Zhao 2020;Moosaie et al. rag reduction by inertialess spheroids
2. Methodology
Governing equations
Eulerian fluid phase
The turbulent channel flow of a dilute suspension is computed by DNS in an Eulerianframework. The carrier fluid is incompressible, isothermal, and Newtonian. The fluidmotion is governed by mass continuity and momentum equations: ∂u i ∂x i = 0 , (2.1) ρ (cid:18) ∂u i ∂t + u j ∂u i ∂x j (cid:19) = − ∂p∂x i + µ ∂ u i ∂x j ∂x j + ∂τ pij ∂x j . (2.2)where p is the pressure, ρ and µ are the fluid density and dynamic viscosity, respectively.The particle stress tensor τ pij represents the influence of inertialess spheroids on the fluid.A friction Reynolds number is defined as Re τ = u τ h/ν , based on the channel half-height h and the nominal friction velocity u τ = (cid:115) − hρ d (cid:104) p w (cid:105) d x = (cid:114) τ w ρ , (2.3)where d (cid:104) p w (cid:105) / d x is the mean pressure gradient in the streamwise direction and τ w = − h d (cid:104) p w (cid:105) / d x is the wall shear stress. Hereafter, the superscript (cid:48) + (cid:48) denotes the normal-ization by viscous scales for velocity ( u τ ), length ( ν/u τ ), and time ( ν/u τ ).2.1.2. Lagrangian particle phase
Lagrangian point-particle approach is adopted to describe the dynamics of smallinertialess particles. These non-Brownian tracers passively follow the translational motionof local fluid, but the rotational motion is determined by the following equations (Jeffery1922) : ω x (cid:48) = − κE y (cid:48) z (cid:48) + R x (cid:48) ,ω y (cid:48) = κE x (cid:48) z (cid:48) + R y (cid:48) ,ω z (cid:48) = R z (cid:48) . (2.4)Here, ω i (cid:48) is the particle rotation vector, E i (cid:48) j (cid:48) is the fluid rate-of-strain tensor and R i (cid:48) is thefluid rate-of-rotation tensor, defined in a particle frame of reference x i (cid:48) = (cid:104) x (cid:48) , y (cid:48) , z (cid:48) (cid:105) . Thetransformation between the inertial and the particle frame is obtained by an orthogonalmatrix (Goldstein 1980) . The shape parameter κ = ( λ − / ( λ + 1) denotes theeccentricity, where λ is the aspect ratio defined as the ratio between the length of thesymmetry axis (in the z (cid:48) direction) and that of the two equal axes. Therefore, prolatespheroids have λ > κ >
0, while oblate ones have λ < κ < Z. Wang, C.-X. Xu and L.-H. Zhao
Particle stress
The spheroids considered in the present study are inertialess and, therefore, force-freeand torque-free, and particles affect the fluid flow via particle stress. This additionalstress in a dilute suspension corresponds to the local volume average of stresslets,which represent the resistance of rigid particles to a straining motion (Batchelor 1970 b ;Guazzelli et al. S ij ) for a spheroidal particle: S ij = 5 µV p [2 Q E ij + Q δ ij E kl n k n l + 2 Q ( n i n l E lj + E ik n k n j ) + Q n i n j n k n l E kl ] . (2.5)Here, V p is particle volume. The material constant Q α ( α = 1 , , ,
4) is a function ofaspect ratio λ . A direction cosine n i ( i = x, y, z ) is defined as the cosine of the anglebetween the particle symmetry axis and the x i -direction. The particle stress producedby N p particles within a given volume (cid:52) (e.g. a grid cell) is expressed as, τ pij = 1 (cid:52) N p (cid:88) β =1 S βij , (2.6)where S βij is the stresslet of the β th particle. The inertialess spheroids are uniformlydistributed and a sufficiently large amount of particles is needed to reach the givenvolume fraction in the whole channel. It is clear that the volume fraction plays a crucialrole concerning the interaction between particles and turbulence. The two-way coupledsimulations are implemented by substituting the stress tensor τ pij into Eq.(2.2). Similarnumerical approaches have been employed in earlier studies (Terrapon 2005; Moosaie &Manhart 2013). 2.2. Simulation setup
As described in Section 2.1, we use an Eulerian-Lagrangian point-particle approach tosimulate the suspensions. The two-way coupling scheme accounted for the feedback frominertialess spheroids on fluid phase via particle stress.A fully developed turbulent channel flow at friction Reynolds number Re τ = 180is considered. Flow is driven by a constant mean pressure gradient in the streamwisedirection. Simulations are performed on a 10 h × h × h channel domain using 128 × ×
128 grid points in the streameise x , spanwise y and the wall-normal z directions,respectively. The grids are uniformly distributed in the homogeneous directions with ∆x + = 14 . ∆y + = 7 .
0. The mesh size ∆z + varies between 1 . . x and y directionsand no-slip and impermeability conditions at the channel walls. Spatial derivativesare computed by a pseudo-spectral method in the two homogeneous directions and asecond-order central finite-difference method in the wall-normal direction. The governingequations are integrated forward in time by using an explicit second-order Adams-Bashforth scheme with a time step ∆t + = 0 . λ = 100, 0 .
01 and 0 .
002 are consideredto explore the effect of particle shape on turbulence modulations. In two-way coupledsimulations, 40 million particles of each type are released randomly into fully developedturbulent channel flows. The volume fraction is chosen as 0 . et al. rag reduction by inertialess spheroids Aspect ratio Volume fraction Bulk velocity enhancement Drag reduction λ = 100 0.75% 8.42% 14.93% λ = 0 .
01 0.75% 0.98% 1.92% λ = 0 .
002 0.75% 3.78% 7.15%
Table 1.
Drag reduction of channel flows laden with spheroids of different shapes. a marginal influence on tiny spheroids with small Stokes number (Shin & Koch 2005;Parsa & Voth 2014; Ravnik et al. Re τ = 180 as the initial field in all simulations.Statistics are gathered over a temporal sampling interval ∆T + = 3600 after the bulk flowreaches a steady state. One-way coupled simulations is also conducted for comparisonpurposes. We follow the same approach as that adopted in our earlier work (Challabotla et al. et al. < t + <
3. Results and discussion
Fluid statistics
The turbulent flows for all cases are driven by the same mean pressure gradient in thestreamwise direction and the increase in bulk velocity U b is indicative of drag reduction.The drag coefficient is defined as C f = τ w / . ρU b . As showing in Table 1, the degree ofdrag reduction with the same particle volume fraction (0 . λ = 100 lead to an enhancement of8 .
42% of the bulk velocity and 14 .
93% reduction of the drag coefficient. The results for thesuspension of prolate spheroids are in agreement with previous observations (Paschkewitz et al. λ = 0 .
01 only lead to a modest drag reduction, which becomes significant for theflattest disks with λ = 0 . µ d U d z − ρ (cid:104) u (cid:48) w (cid:48) (cid:105) + (cid:104) τ pxz (cid:105) = τ total = − h d (cid:104) p w (cid:105) d x (1 − zh ) . (3.1)The distribution of the viscous, Reynolds, and particle shear stress in the wall-normaldirection is plotted in Figure 2. The linear profile of total shear stress confirms that Z. Wang, C.-X. Xu and L.-H. Zhao (a) (b)
Figure 1.
Mean fluid velocity profiles in the streamwise direction. (a) linear plot; (b) semi-logplot. (a) s h e a r s t r e ss z + (b) z + (c) z + Figure 2.
Stress balance for laden and unladen flows (solid lines with circles). (a) λ = 100; (b) λ = 0 .
01; (c) λ = 0 . the flow has reached a statistically steady state and satisfies the integral momentumbalance. In comparison with the unladen flow, Reynolds shear stress for particle-ladenflow decreases throughout the whole half channel, while viscous shear stress is attenuatedin the near-wall region and augmented from the buffer layer. The flattest disks with λ = 0 .
002 induce the maximum particle shear stress and have the greatest impact onshear stress balance.As shown in Figure 3, turbulent kinetic energy k and turbulence intensities rms( u (cid:48) i ) arealso altered by the presence of inertialess spheroids. The k + and rms( u (cid:48) + ) are enhancedexcept in the near-wall region, while turbulence intensities in the spanwise and wall-normal directions are attenuated. Additionally, all peaks are shifted further away from thewall compared with that in the unladen flow. The results of intensities imply the modifiedanisotropy of turbulence in particle-laden flows presented in Figure 4. The flows withspheroids are less isotropic in the channel center and more prolate axisymmetric in theregion away from the wall. The addition of particles also makes the right tip approach theone-component limit. Interestingly, the wall value for the suspension of disk-like particlestends towards the state of isotropic two-component turbulence, contrary to that for thesuspension of rod-like particles. Frohnapfel et al. (2007) concluded that drag-reducedchannel flow is commonly accompanied with the increased anisotropy of turbulence inthe near-wall region. The modulations of the Lumley anisotropy map indicate that disksare less effective than rods in drag reduction.The statistical results of the vorticity field are depicted in Figure 5. The meanspanwise vorticity corresponds to the mean velocity gradient and resembles the profileof the viscous shear stress in Figure 2. In the near-wall region, the decrease of mean rag reduction by inertialess spheroids (a) (b)(c) (d) Figure 3. (a) Turbulent kinetic energy and (b-d) turbulence intensities in the streamwise,spanwise, and wall-normal directions, respectively.
Figure 4.
Lumley anisotropy map (Lumley & Newman 1977) of the unladen flow andparticle-laden flows. The second and third invariants for turbulence are defined as II = a ij a ji ,III = a ij a jk a ki , respectively, where a ij = (cid:104) u (cid:48) i u (cid:48) j (cid:105) / k − δ ij /
3. The dotted line represents theLumley triangle, which shows the limiting states of turbulence. The states of the wall, right tip,and channel centerline are highlighted by symbols. spanwise vorticity caused by spheroids is due to the attenuation of local mean velocity.As shown in Figure 5(b-d), the vorticity fluctuations are damped in the particle-ladenchannel, indicating the suppression of vortical structures. The distance between the localmaximum and minimum of the streamwise fluctuations represents the average radius of
Z. Wang, C.-X. Xu and L.-H. Zhao (a) (b)(c) (d)
Figure 5. (a) Mean spanwise vorticity and (b-d) root-mean-square of the streamwise,spanwise and wall-normal fluctuating fluid vorticity components, respectively. the streamwise vortices, while the magnitude of the local maximum reflects the averagestrength (Kim et al. ∂u (cid:48) /∂z in the near-wall region, the reduced rms( Ω (cid:48) + y ) also indicates that the mean streak spacing is increaseddue to the presence of spheroids.3.2. Particle statistics
From the previous results, it is known that inertialess spheroids could affect the fluidflows. On the other hand, the modulated fluid field will be influential in the particledynamics, in return. Since the massless spheroids passively translate along with the localfluid, the focus in this section is on orientational and rotational behavior.Figure 6 shows the mean absolute values of the direction cosines in particle-laden flowscompared with that in one-way coupled simulations. We observe that disks tend to alignin the wall-normal direction, while rods preferentially align with the wall. The shape-dependence of particle alignment becomes marginal in the core region of the channel. Thisorientational tendency is in accordance with earlier studies (Challabotla et al. et al. rag reduction by inertialess spheroids (a) < | n i | > z + (b) z + (c) z + Figure 6.
Mean absolute values of the direction cosines for spheroidal particles. (a) (cid:104)| n x |(cid:105) ; (b) (cid:104)| n y |(cid:105) ; (c) (cid:104)| n z |(cid:105) . The solid lines with symbols represent particle statistics of one-way coupledsimulations. (a) (b)(c) (d) Figure 7. (a) Mean spanwise angular velocity; (b-d) Root-mean-square of the fluctuatingangular velocity of particles in the streamwise, spanwise, and wall-normal direction, respectively.The solid lines with symbols represent particle statistics of one-way coupled simulations. orientation and fluid gradients (including mean shear and turbulent vorticity) (Yang et al. et al. (cid:104) ω + y (cid:105) and rms( ω (cid:48) + i ) of spheroids in the particle-ladenflows. Again, discrepancies between the rotational behavior of disks with λ = 0 .
01 and λ = 0 .
002 are enhanced due to particle-turbulence interactions.0
Z. Wang, C.-X. Xu and L.-H. Zhao (a) < τ p + i j > z + (b) z + (c) z + Figure 8.
Mean particle stresses in two-way coupled simulations for spheroids with (a) λ = 100, (b) λ = 0 .
01, and (c) λ = 0 . The variation of the mean particle stresses from the wall to the channel center isplotted in Figure 8. The components (cid:104) τ pxy (cid:105) and (cid:104) τ pyz (cid:105) vanish due to the symmetry of thechannel flow. Different from the fluid viscous stresses, the appearance of mean particlenormal stresses reveals that spheroids result in the non-Newtonian contribution to thecarrier fluid. The mean particle stresses attain peaks in the buffer layer and vanish atthe channel center. The normal stress (cid:104) τ pxx (cid:105) is slightly larger than other components forrods in the near-wall region, while the shear stress (cid:104) τ pxz (cid:105) is the dominant one for disks.The present results show the same shape-dependence of particle stress with previousfindings (Manhart 2003; Wang & Zhao 2020; Moosaie et al. τ pxz causedby disks is accompanied with pronounced turbulence modulations, e.g. the fluid vorticityand turbulence intensities, but the degree of drag reduction is less than expected. Thedistribution of mean particle stress for disks is qualitatively different from that for rods.Further details about the effect of particle stress on the fluid phase will be explored inthe following section. 3.3. Drag reduction mechanism
In Section 3.1 and 3.2, we have presented the fluid statistics and the particle statisticsin the suspensions of spheroids with high asphericity, compared with results in one-waycoupled simulations. However, two questions remain unanswered. Firstly, how do iner-tialess spheroids modulate the turbulence field, especially coherent structures? Secondly,why rods are more effective than disks in drag reduction? Motivated by those questions,in this section, we will explore the mechanisms for drag reduction by inertialess spheroidsin turbulent channel flow.The near-wall coherent structures formed in the self-sustaining process are responsiblefor high skin-friction drag in wall-bounded turbulence (Kravchenko et al. et al. λ -criterion (Jeong & Hussain 1995) and colored by the fluctuating streamwise velocity. Inthe near-wall region, the dominant structures are the quasistreamwise vortices, slightlytilted away from the wall. Compared with the unladen flow case in Figure 9(a), theaddition of spheroids results in fewer vortices with larger sizes. Disks with λ = 0 . λ = 100, and finallydisks with λ = 0 .
01. These phenomena are consistent with the suppression of the rag reduction by inertialess spheroids (a) u (cid:48) + (b) u (cid:48) + (c) u (cid:48) + (d) u (cid:48) + Figure 9.
Instantaneous isosurfaces of λ +2 = − .
007 colored by streamwise velocity u (cid:48) + in thelower half channel for (a) unladen flow and particle-laden flows with (b) λ = 100; (c) λ = 0 . λ = 0 . fluctuating fluid vorticity and partly explain why the turbulence intensities in thespanwise and wall-normal directions are attenuated. Kim (2011) pointed out that theattenuation of the quasistreamwise vortices is a common feature of drag-reduced wall-bounded flows.Produced by the quasistreamwise vortices, the near-wall streaky structures shouldrespond to the addition of spheroids. From Figure 10, we observe that streaks becomemore regular and the spanwise spacing is increased in particle suspensions. The reductionof waviness and small scales means that the low-speed streaks are more stable, hencethe generation of the quasistreamwise vortices is suppressed (Schoppa & Hussain 2002).The mean streak spacing is further investigated through velocity autocorrelation in thespanwise direction. The separation of minimum correlation corresponds to half of themean streak spacing (Kim et al. z + <
40, the spacingfor particle suspensions is larger than that in unladen flow, but the differences graduallydiminish as away from the wall (see Figure 11). Since the near-wall streak spacing isindependent of Reynolds number (Smits et al.
Z. Wang, C.-X. Xu and L.-H. Zhao (a) y/ h u (cid:48) + (b) u (cid:48) + (c) y/ h x/ hu (cid:48) + (d) x/ hu (cid:48) + Figure 10.
Instantaneous contours of streamwise velocity fluctuations at z + = 15 for (a)unladen flow and particle-laden flows with (b) λ = 100; (c) λ = 0 .
01; (d) λ = 0 . Figure 11.
Variation of mean spanwise streak spacing in near-wall region.
A, the transport equation of the turbulent kinetic energy k in particle-laden flow can bewritten as ∂k∂t + (cid:104) u i (cid:105) ∂k∂x i = −(cid:104) u (cid:48) i u (cid:48) j (cid:105) ∂ (cid:104) u i (cid:105) ∂x j − ∂∂x j (cid:18) (cid:104) p (cid:48) u (cid:48) j (cid:105) ρ + (cid:104) u (cid:48) i u (cid:48) i u (cid:48) j (cid:105) − ν ∂k∂x j (cid:19) − ν (cid:28) ∂u (cid:48) i ∂x j ∂u (cid:48) i ∂x j (cid:29) + 1 ρ (cid:104) f (cid:48) i u (cid:48) i (cid:105) . (3.2)Here, f i = ∂τ pij /∂x j is particle body force (Paschkewitz et al. γ ( γ = x, y, z ) direction is thendefined as W γ = (cid:104) f (cid:48) γ u (cid:48) γ (cid:105) . Note that the usual summation convention is not adoptedfor the repeated Greek indices. The work done by particles on the fluid has also beenanalyzed in earlier studies (Paschkewitz et al. et al. et al. rag reduction by inertialess spheroids Figure 12.
Probability density functions of the work W + y + z done by particles on the fluidconditionally sampled with λ < − λ , rms in 20 < z + <
60. The solid lines with symbolsrepresent negative values. which mainly consist of velocities in the spanwise and wall-normal directions, and thework W y + z = W y + W z is taken into account. The negative (positive) W y + z indicatesthat particles weaken (strengthen) the quasistreamwise vortices. The sign of W y + z isdetermined by the angle between two vectors of v (cid:48) + w (cid:48) and f (cid:48) y + f (cid:48) z .Figure 12 plots the probability density functions (PDFs) of W + y + z conditionally sampledin the region 20 < z + <
60 where the self-sustaining process happens. The vortices areextracted from background fluctuations with the condition λ < − λ , rms . We observethat the probability of negative work is higher than that of positive work, regardless ofthe particle shape. This observation reveals that the work done by spheroids contributeto the attenuation of velocity fluctuations in the y and z directions in the vortex regions.However, rods and the flattest disks, with the less sharp profiles of PDFs, are more likelyto produce large work than disks with λ = 0 . λ in half channeland slices of W + y + z in the y-z and x-z planes. It is clear that spheroids tend to producenegative work on the vortex structures. From slices of W + y + z , the large negative work isassociated with the regions near the quasistreamwise vortices. The magnitude of W + y + z done by disks with λ = 0 .
01 is mostly smaller than that by other spheroids, consistentwith PDFs in Figure 12. The instantaneous results manifest that W y + z done by spheroidsweakens the quasistreamwise vortices.The conditional ensemble average flow fields and the work W y + z are presented tofurther validate the above findings from a statistical standpoint. The instantaneousquasistreamwise vortices with typical features, such as λ < − λ , rms , a streamwise lengthat least 150 wall units, and positive streamwise vorticity, are sampled conditionally in therange 0 < z + <
60. Then the flow fields and the work W y + z around these structures areensemble-averaged after proper alignment. It should be noted that λ for this flow field isobtained from the ensemble-averaged fluid velocity. The detailed procedures are referredto Jeong et al. (1997); Dritselis & Vlachos (2008). As shown in the right column panels ofFigure 13, spheroids generally produce negative W + y + z in the regions of quasistreamwisevortex (see the isosurfaces). The large negative work appears in the upper left and lowerright of the vortex (see the slices), where the velocity fluctuations in the spanwise andwall-normal directions are intense. Comparisons between Figure 13 (b), (d) and (f) revealthe dependence of the magnitude of W + y + z on particle shape.4 Z. Wang, C.-X. Xu and L.-H. Zhao (a) W + y + z (b) W + y + z (c) W + y + z (d) W + y + z (e) W + y + z (f ) W + y + z Figure 13.
The relationship between vortex structures and the work W + y + z . Left column panelsare instantaneous isosurfaces of λ +2 = − .
007 colored by the work W + y + z in half channel andslices of W + y + z in the y-z and x-z planes for particle-laden flows with (a) λ = 100; (c) λ = 0 .
01; (e) λ = 0 . λ +2 = − .
002 for ensemble-averaged coherentstructures with positive streamwise vorticity colored by W + y + z and slices of W + y + z in y-z planefor particle-laden flows with (b) λ = 100; (d) λ = 0 .
01; (f) λ = 0 . Based on these findings, we demonstrate that inertialess spheroids weaken the qua-sistreamwise vortices through the work W + y + z whose magnitude depends on particleshape. The Reynolds shear stress in particle-laden flow is then reduced. However, thedamping of the Reynolds shear stress −(cid:104) u (cid:48) w (cid:48) (cid:105) cannot directly lead to drag reductionof wall turbulence (Eshghinejadfard et al. C f can be expressed as: rag reduction by inertialess spheroids (a) (b) Figure 14. (a) Reynolds shear stress and particle shear stress; (b) the weighted shear stress. C f = τ w ρU b = ν u τ h [ u τ − ρ (cid:82) (1 − η )( − ρ (cid:104) u (cid:48) w (cid:48) (cid:105) + (cid:104) τ pxz (cid:105) )d η ] , (3.3)where η = z/h . Because the present flows are driven by a constant mean pressuregradient, the differences in drag coefficients among all cases are only dependent onReynolds shear stress and particle induced shear stress. The weighted integration (cid:0)(cid:82) (1 − η )( − ρ (cid:104) u (cid:48) w (cid:48) (cid:105) + (cid:104) τ pxz (cid:105) )d η (cid:1) is positively correlated with the drag coefficient, while negativelywith the efficiency of drag reduction. This integration in particle suspensions determineswhether drag reduction happens. Figure 14(a) illustrates −(cid:104) u (cid:48) w (cid:48) (cid:105) and (cid:104) τ pxz (cid:105) as functionsof the wall distance z/h . The particle shear stress compensates for the reduction ofReynolds shear stress, especially in the near-wall region, where the weight factor (1 − η )is relatively higher. The weighted shear stress (1 − η )( − ρ (cid:104) u (cid:48) w (cid:48) (cid:105) + (cid:104) τ pxz (cid:105) ) is depicted inFigure 14(b). The integrations, i.e. areas under profiles of the weighted shear stresses, arein good agreement with the degree of drag reduction. Compared with the results of rods,the large particle shear stress (cid:104) τ pxz (cid:105) generated by disks in the near-wall region increasesthe integration and reduces the efficiency of drag reduction. Therefore, it is reasonablethat disks are less effective than rods in drag reduction, even though the former also hasa significant impact on the turbulence modulations.
4. Concluding remarks
The present study has examined the interaction between spheroidal particles and wallturbulence in two-way coupled direct numerical simulations. We focused on how fiber-like and disk-like particles modulate the near-wall turbulence and the mechanism ofdrag reduction induced by non-spherical particles. Three types of spheroids with aspectratios of λ = 100, 0 .
01 and 0 .
002 were considered and 40 million particles of each type,corresponding to a volume fraction of 0 . Re τ = 180. These tiny inertialess spheroids were tracked in a Lagrangianframework and affected fluid phase via particle stress. One-way coupled simulations,where the feedback from particles onto the fluid was ignored, were also performed forcomparison.Our results show that rods with λ = 100 lead to the most pronounced drag reduction(14 . λ = 0 .
002 (7 . λ = 0 . . et al. et al. Z. Wang, C.-X. Xu and L.-H. Zhao
Compared with the unladen flow case, the shear stress balance was modulated andReynolds shear stress was attenuated. The velocity fluctuations were also altered by thepresence of inertialess spheroids that the streamwise velocity fluctuation was enhancedexcept in the near-wall region, whereas the fluctuations in the other directions weredamped. Moreover, the modulated vorticity field indicated that the addition of spheroidsresulted in fewer vortices with larger sizes, which was confirmed by the instantaneousvortex structures (Figure 9). Consequently, near-wall streaks became more regular andthe mean streak spacing was increased in particle suspensions. On the other hand, theparticle dynamics were affected by the modulated fluid field. Similar to the observations inone-way coupled simulations, disk-like particles tended to align norm to the wall, whilerod-like particles preferentially aligned parallel to the wall. However, such alignmenttendency was strengthened in the two-way coupled simulations. Under the influences ofthe damped fluid vorticities and the enhanced alignment tendency, mean angular velocityand spin fluctuations of spheroids were reduced considerably.The physical mechanism responsible for drag reduction by inertialess spheroids can bedepicted as follows. Spheroidal particles weaken the near-wall quasistreamwise vorticesthrough the work W + y + z whose magnitude depends on particle shape, which reveals that,in the regions of quasistreamwise vortex structures, particle body force tends to suppressthe fluid fluctuating motions in the y and z directions. Therefore, the Reynolds shearstress in particle-laden flow is damped, which contributes to the turbulent drag reduction.According to equation (3.3), the degree of drag reduction depends on the profiles ofReynolds shear stress and particle shear stress (cid:104) τ pxz (cid:105) . As shown in Figure 14, the particleshear stress partly compensates for the reduction of Reynolds shear stress and limitsthe efficiency of drag reduction. Because the stress (cid:104) τ pxz (cid:105) induced by disks is relativelylarge, especially in the near-wall region, disks induce less pronounced drag reduction thanrods. The present results suggest that tiny disk-like particles can lead to a noticeable dragreduction in wall turbulence and can be an alternative drag reduction agent. Acknowledgments
The work was supported by the Natural Science Foundation of China (grant Nos.11911530141 and 91752205).
Declaration of Interests
The authors report no conflict of interest.
Appendix A. Transport equation of turbulent kinetic energy
By subtracting the mean flow equation from the Navier-Stokes equation (2.2), thefluctuating momentum equation of fluid can be written as: ρ (cid:18) ∂u (cid:48) i ∂t + (cid:104) u j (cid:105) ∂u (cid:48) i ∂x j + u (cid:48) j ∂ (cid:104) u i (cid:105) ∂x j (cid:19) = − ∂p (cid:48) ∂x i + µ ∂ u (cid:48) i ∂x j ∂x j + ρ ∂ (cid:10) u (cid:48) i u (cid:48) j (cid:11) ∂x j − ρ ∂u (cid:48) i u (cid:48) j ∂x j + ∂τ p (cid:48) ij ∂x j . (A.1)Multiplying by u (cid:48) i and taking the mean of that equation, we obtain rag reduction by inertialess spheroids ∂ (cid:104) u (cid:48) i u (cid:48) i (cid:105) ∂t + 12 (cid:104) u j (cid:105) ∂ (cid:104) u (cid:48) i u (cid:48) i (cid:105) ∂x j = −(cid:104) u (cid:48) i u (cid:48) j (cid:105) ∂ (cid:104) u i (cid:105) ∂x j − ∂∂x j (cid:18) (cid:104) p (cid:48) u (cid:48) j (cid:105) ρ + 12 (cid:104) u (cid:48) i u (cid:48) i u (cid:48) j (cid:105) − ν ∂ (cid:104) u (cid:48) i u (cid:48) i (cid:105) ∂x j (cid:19) − ν (cid:28) ∂u (cid:48) i ∂x j ∂u (cid:48) i ∂x j (cid:29) + 1 ρ (cid:42) u (cid:48) i ∂τ p (cid:48) ij ∂x j (cid:43) . (A.2)Then the transport equation of turbulent kinetic energy is given as: ∂k∂t + (cid:104) u i (cid:105) ∂k∂x i = −(cid:104) u (cid:48) i u (cid:48) j (cid:105) ∂ (cid:104) u i (cid:105) ∂x j − ∂∂x j (cid:18) (cid:104) p (cid:48) u (cid:48) j (cid:105) ρ + (cid:104) u (cid:48) i u (cid:48) i u (cid:48) j (cid:105) − ν ∂k∂x j (cid:19) − ν (cid:28) ∂u (cid:48) i ∂x j ∂u (cid:48) i ∂x j (cid:29) + 1 ρ (cid:104) u (cid:48) i f (cid:48) i (cid:105) , (A.3)where k = (cid:104) u (cid:48) i u (cid:48) i (cid:105) / f i = ∂τ pij /∂x j is defined as particlebody force. The last term on the right-hand side is the rate of work done by particles tofluid per mass. Appendix B. Drag coefficient of spheroid-laden channel flow
In order to derive the drag coefficient in the present flow configuration, we start withthe mean momentum equation in the streamwise direction:0 = − ρ d (cid:104) p w (cid:105) d x + ν d U d z + 1 ρ d (cid:104) τ pxz (cid:105) d z − d (cid:104) u (cid:48) w (cid:48) (cid:105) d z , (B.1)where d (cid:104) p w (cid:105) / d x is the mean pressure gradient at channel walls in the streamwise directionand U is the mean streamwise velocity. Integrating the above equation from z to h (thehalf-channel height) in the wall-normal direction, we obtain the shear stress balanceequation: µ d U d z − ρ (cid:104) u (cid:48) w (cid:48) (cid:105) + (cid:104) τ pxz (cid:105) = τ w (cid:16) − zh (cid:17) . (B.2)Here, τ w = − h d (cid:104) p w (cid:105) / d x = ρu τ is the wall shear stress. A double integral of this equationfrom 0 to z and from 0 to h in the wall-normal direction gives hµU b + (cid:90) h (cid:90) z ( − ρ (cid:104) u (cid:48) w (cid:48) (cid:105) + (cid:104) τ pxz (cid:105) ) d z d z = h τ w , (B.3)8 Z. Wang, C.-X. Xu and L.-H. Zhao where the bulk velocity is defined as U b = (cid:82) h U d z/h . The second term on the right-handside can be simplified with the application of partial integration: (cid:90) h (cid:90) z ( − ρ (cid:104) u (cid:48) w (cid:48) (cid:105) + (cid:104) τ pxz (cid:105) ) d z d z = z (cid:90) z ( − ρ (cid:104) u (cid:48) w (cid:48) (cid:105) + (cid:104) τ pxz (cid:105) ) d z (cid:12)(cid:12)(cid:12)(cid:12) h − (cid:90) h z ( − ρ (cid:104) u (cid:48) w (cid:48) (cid:105) + (cid:104) τ pxz (cid:105) ) d z = (cid:90) h ( h − z ) ( − ρ (cid:104) u (cid:48) w (cid:48) (cid:105) + (cid:104) τ pxz (cid:105) ) d z = h (cid:90) (1 − η ) ( − ρ (cid:104) u (cid:48) w (cid:48) (cid:105) + (cid:104) τ pxz (cid:105) ) d η, (B.4)in which η = z/h is the dimensionless height from the lower channel wall. The bulkvelocity U b is then expressed as: U b = hν (cid:18) u τ − ρ (cid:90) (1 − η ) ( − ρ (cid:104) u (cid:48) w (cid:48) (cid:105) + (cid:104) τ pxz (cid:105) ) d η (cid:19) . (B.5)Finally, the drag coefficient C f of the present laden flows is given as follows C f = τ w ρU b = ν u τ h (cid:16) u r − ρ (cid:82) (1 − η ) ( − ρ (cid:104) u (cid:48) w (cid:48) (cid:105) + (cid:104) τ pxz (cid:105) ) d η (cid:17) . (B.6) REFERENCESBatchelor, G. K. a Slender-body theory for particles of arbitrary cross-section in Stokesflow.
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