Internal microstructure driven turbulence enhancement of fluids
IInternal microstructure driven turbulence enhancement of fluids
G. Sofiadis and I. E. Sarris ∗ Department of Mechanical Engineering, University of Thessaly, 38334, Volos, Greece. Department of Mechanical Engineering, University of West Attica, Athens, Greece. (Dated: November 11, 2020)Fluids with internal microstructure like dense suspensions, biological and polymer added fluids,are commonly found in the turbulent regime in many applications. Their flow is extremely difficultto be studied as microstructure complexity and Reynolds number increase, even nowadays. Thisbottleneck is novelty and efficiently treated here by the micropolar theory. Our findings support thatwhen denser microstructure occurs turbulence is intensified by the chaotic rotation of internal fluid’selements. Unlike in Newtonian fluids, shear stress is now decreased, and viscous and microstructureinterrotational stresses increase near the walls and mark the turbulence intensification.
In the usual Newtonian fluid flow, shear is the domi-nant mechanism for turbulence production, however, asthe fluid departs from the Newtonian regime, its mi-crostructure is found to alter the usual turbulent charac-teristics i.e. to result in drag increase [1] or decrease [2].Turbulence intensification or reduction is not reportedto be only due to viscosity increase is such cases, butrather than due to the presence of internal elements likeits microstructure, added polymer strings or microparti-cles. Phenomena like the increase of effective viscosity,turbophoresis, shear thickening or thinning are associ-ated with the existence of fluid’s internal microstructure[3–6]. In the case of suspensions, it is reported that asthe volume fraction of additives increases, their inducedstress is increased and results in shear stress reduction.The usual turbulence production mechanism is affectedor even dominated that way by the internal elements in-duced turbulence. As particle volume fraction increasestheir turbulent interaction increases and sustains hightotal drag. This is found not to be related to higherfriction velocity, but to the increased viscosity of densesuspensions [1]. More specific, turbulence mechanism incase of dense suspensions does not only vary in termsof production but location as well. It has recently beenproved [7] that when particles dominate the flow (becomelarger than the smaller turbulent scales), near wall tur-bulence production becomes significant. This near-wallturbulence mechanism has been attributed to an effectivesuspension viscosity, but is still not fully understood. Asimilar near-wall mechanism has been found to arise ingeneral, when fluid microstructure is taken into consider-ation. This peculiar turbulence enhancement mechanismseems to be directly connected to a microstructure stresstensor.Polymer and viscoelastic fluids may also contain de-formable micro-structures. The major advantage of suchfluid turbulent flows is the reduced drag due to poly-meric additions. Toms [8] firstly observed this pressuredrop decrease when polymers are added in water or othersolvents. Later in an experimental study of polymer di-lute pipe flows, Giles and Pettit [9] agreed that in similartypes of flows, like Toms [8], polymer additives in low concentrations decrease friction. Experiments in tubeswere also conducted by White and McEligot [10] for dif-ferent types of polymers in deionized water solutions.Their results showed a polymer-concentration dependentReynolds number for transition to turbulence. Moreover,in a recent paper, Rosti and Brandt [11] measured withthe same methodology that polymer additives in densesuspensions may lead to drag decrease.Dense suspensions or other fluids with internal mi-crostructure need special numerical attention of their mo-tion and interaction of rigid internal additives, whereasin significant high Reynolds numbers it is extremely diffi-cult to be used for simulations. This bottleneck is treatedhere by the micropolar theory which is able to mathemat-ically describe a wide range of fluids spanning from densesuspensions, liquid crystals, blood and other polymeric-like fluids consisting of internal microstructure [12]. Theclass of equations that describes the micropolar theory isattractive due to its capability to include an asymmetricand couple stress tensor in a rather simple way [12, 13].The symmetric stress tensor arises in macroscopic contin-uum theory when certain degrees of freedom are assignedto the fluid, such as internal spin, leading to a general-ization of the incompressible, ∇ · u = 0, Navier-Stokesequation [12–15], as: ∂u∂t + ( u · ∇ ) u = −∇ P + 1 Re ∇ u + mRe ∇ × ω (1) JNm (cid:20) ∂ω∂t + ( u · ∇ ) ω (cid:21) = 1 Re ∇ ω + NRe ∇ × u − NRe ω (2)where, u and ω are the linear and angular velocity vec-tors, respectively, and t and P stand for time and pres-sure, respectively.In this Letter, Eqs. (1)-(2) are solved for the usualturbulent channel flow test-bed cases of [16–18]. Weadopt the set of constitutive equations that have beenoriginally proposed by Eringen [12], as an extension ofthe classical Navier-Stokes equation for fluids with mi-crostructure. Direct numerical simulations (DNS) of theturbulent channel flow at various Re and m values areperformed to demonstrate the key role of the micropolareffect at turbulence generation. Turbulent channel flow a r X i v : . [ phy s i c s . f l u - dyn ] N ov is a well documented test-bed for this type of flows andlarge databases of turbulent statistics are available forNewtonian and non-Newtonian fluids [1, 2, 16–18]. Theimmediate advantage of the present model is that a meshof about 50 times smaller is needed to perform the densesuspensions simulations at the same Reynolds number asin ref. [1]. Furthermore, both Eqs. (1)-(2) can be uni-formly solved in a Eulerian frame, with no need to treata discrete phase.The linear and angular velocities, time and pressure atthe above equations are made dimensionless by selectingthe characteristic quantities U , U δ , δU , and ρU , respec-tively, where U is the constant mean velocity, 2 δ is theheight of the channel and ρ is the density of the fluidwhich is considered to be homogenous. At Eqs. (1)-(2)four non-dimensional quantities are recovered, i.e. themodified bulk Reynolds number, Re = ρ U δµ + κ , with µ be-ing the molecular and κ the micropolar fluid viscosities,the so-called vortex viscosity parameter, m = κµ + κ , thedimensionless microrotation parameter, J = jδ , where j is the microinertia of the fluid, and the so-called spingradient viscosity parameter, N = κδ γ , where γ is thematerial coefficient of the fluid [19, 20]. Throughoutthis study, the spin gradient viscosity and the microro-tation parameters are kept constant at N = 8 . × and J = 10 − that are usual magnitudes for biolog-ical flows [21], while m is varied between 0 and 0.9.This is chosen because of the straightforward connec-tion between the linear and angular momentum equa-tions through m . Fluids near m = 0 behave similar tothe Newtonian one, while when m = 0 . m inthe present simulations, under constant bulk Reynoldsnumbers, imply the simultaneously reduction of molec-ular viscosity so that the total viscosity of the fluid toremain the same.Initially, successful experiments are made in the New-tonian regime to compare our numerical facility againstothers [18]. As can be seen in Figure 1 for the temporaland wall-normal plane averaged velocity U + along thewall-normal direction y + , both normalized by the shearvelocity u τ = (cid:112) τ w /ρ , where τ w is the shear stress and + indicates inner (wall) units. Cases in the Newtonianrange of Re = 3300 − Re τ = 110 − Re τ = ρu τ δµ + κ . In all these Newtonian cases,the channel size and grid arrangement is kept equal asin the DNS study of Moser et al. [18]. Moreover, theaccuracy of the micropolar part of the model is alreadyverified in recent works [22, 23] against other numericaland experimental results.As can be seen in Fig. 1 for example at Re τ = 180,the increase of the vortex viscosity parameter from zeroin the Newtonian case to m = 0 . y + for variousNewtonian cases and the case of m = 0 . Re τ = 180 based on the Newtonian case.of U + and consequently, pressure drop increase and tur-bulence intensification due to the strongly interrotatingmicrostructure. The m increase is also connected to theincrease of Re τ by 8% at most in the Re τ = 395 case dueto increased u τ as the microstructure effect is more signif-icant near the walls, as will be discussed later. The rateof turbulence enhancement as connected by the presenceof microstructure can be efficiently presented in the fric-tion factor, C f = τ w / ρU , schema of Fig. 2. There, thefriction factor variation of various non-Newtonian casesis compared against the present micropolar results as m increases. In the case of flows containing polymer addi-tives [2] or small heavy solid particles [24] friction is foundto decrease in lower values than in the Newtonian one,however, in the case of dense suspensions [1] as well as inthe present case of internal microstructure, friction factoris increased, and thus, turbulence is enhanced with vol-ume fraction and vortex viscosity increase, respectively.Turbulence enhancement due to microstructure is notonly found to exist quantitatively in the above figures,but more importantly qualitatively in the different formsbetween the flow structures. This is made visual in thestreamwise velocity contour snapshots aligned with the x − y and x − z planes of Fig. 3. It is observed thatfiner structures are encountered for m = 0 . m = 0 in the x − y plane that is a strong indicationof shear thickening and the tendency for more isotropicturbulence. Furthermore, the stronger turbulent mixingis evident in the x − z plane, for example at y + = 1, wherefiner and isotropic structures are recorded for m = 0 . y + = 1 for a Newtonian fluid at Re τ = 180(left) and as m = 0 . Re τ = 180. As m increases,the peak value of the streamwise fluctuation componentof the velocity, U + rms , is reduced and approaches the wall.Thus, the peak from y + ≈
20 at m = 0 goes at about y + ≈
10 for m = 0 .
9. Simultaneously, the reduction ofpeak value indicates the turbulent homogenization trendof internal microstructure due to increasingly turbulentfluctuations of the angular velocity, ω + z,rms , as m in-creases, see Fig. 4b. The micropolar fluctuation ω + z,rms that mostly interacts with U + rms is found to approach itspeak value at about y + ≈
10 for m = 0 .
9. As com-pared to the polymer additive case [2], an opposite trendis found due to its shear thinning nature.Moreover, the present results are in alignment with thedense suspension case of ref. [1] in two major points, al-though the visual differences. Firstly, the peak values of U + rms are reduced due to shear thickening and this reduc-tion is proportional to particles volume fraction increase,and secondly, despite that the peak value of U + rms of thefluid phase at ref. [1] departs from the walls with volumefraction increase, its particulate counterpart approachesthe walls like in the present results. The peak value ofthe particulate root-mean-squared fluctuation velocity ofref. [1] is found to be of the same order of magnitude asits fluid part, thus, turbulence intensification is fed bythe particulate induced turbulence in dense suspensions.Similarly here, the internal microstructure is respon-sible for the turbulence enhancement as can be seen inthe stress balance of the micropolar fluids flow equations.In turbulent wall-bounded fluid flows with internal mi-crostructure, the total stress is balanced by Reynoldsshear, viscous and microstructure stresses as: τ t = τ s + τ v + τ m (3)where, τ s = − u + (cid:48) v + (cid:48) and τ v = dU + dy + represent the shearand viscous stresses, respectively, while the micropolarstress due to microstructure is τ m = mω + z , u (cid:48) and v (cid:48) are the streamwise and wall-normal velocity fluctuations, FIG. 4: Root-mean-square velocity fluctuationsnormalized by the wall-shear velocity: a) the linearstreamwise component U + rms , and b) the angularspanwise component ω + z,rms .respectively, and bar indicates temporal and wall-normalplane average.Reynolds stress is the usual mechanism of turbulenceproduction, but here is peculiarly reduced with m in-crease as can be observed in Fig. 5. Similar shear stressdecrease is encountered in dense suspensions, polymerflows and flows under the effect of external forces, likeexternal magnetic fields [25]. Shear reduction is replacedby the simultaneous increase of both, the viscous and mi-cropolar stresses near the walls at y + (cid:46)
40. This newturbulence mechanism ignited by the internal microstruc-ture is reported for the first time here. Thus now, themicropolar stress alters the dominance of viscous stressnear the walls and, as m increases, the magnitude of τ m is getting comparable to that of the viscous one. The mi-cropolar stress is found to be of the same nature as theviscous one, i.e. rises only very close to the wall, while farfrom it drops to zero and mostly affected by the total vis-cosity. The peak value of the micropolar stress is foundto be located inside the viscous sublayer at y + ≈ − Re b = 3300 almost in-dependently of m , and with an tendency to departs fromthe wall as Reynolds number increases. This decrease ofshear stress is responsible also for the reduction of totalstress, similarly to [25], due to the homogenization effectof the internal microstructure.The near walls new induced turbulence mechanismthrough microstructure is of similar nature as the usualvorticity induced turbulence mechanism of the Newto-nian fluids (now act together) that excite streaks struc-tures near the walls [26]. In fact, this is evident fromFig. 5 where the profiles of the mean angular velocity,i.e. mω + z , is of similar shape to the fluid spanwise fric-tion vorticity profile, i.e. Ω z = dU + dy + . It is found easilyfrom the present statistics, and also proved mathemati-cally, that 2 ω z = Ω z = dU + dy + . The difference however ofthe micropolar stress production is that it is an activeterm capable to sustain and enhance turbulence, and nota passive one as the usual vorticity is.In summary, the micropolar set of equations has beenapplied to investigate the internal microstructure effect inFIG. 5: Stress profiles for various m along y + at Re τ = 180 for m = 0: a) total and shear stresses, b)viscous and micropolar stresses, and c) locations of thepeak values of all stresses. Arrows indicate the increaseof m , Lines with (cid:78) at Re b = 3300, with (cid:4) at Re b = 5600, and with (cid:72) at Re b = 13800.a turbulent fluid flow. Computational experiments havebeen conducted at relative low Re channel flow and re-sults for different vortex viscosity ratios have been com-pared against other non-Newtonian fluids flows. Littleis known so far for this type of turbulent flows, whichranks this study among pioneer ones. Results exhibitshear thickening, drag increase and enhanced turbulenceas the vortex viscosity increases. By further analysis,it has been shown that the main mechanism for turbu-lence generation in this case is the micropolar stress, asit increases close to the walls when vortex viscosity, andthus microstructure density, increases. The concept ofmicropolar turbulence formation could lead to a betterunderstanding of turbulence mechanics in more generaltype of fluids.The authors are grateful for the support of the GreekResearch and Technology Network (GRNET) for thecomputational time granted in the National HPC facilityARIS. ∗ Electronic address: [email protected] [1] F. Picano, W.P. Breugem and L. Brandt, J. Fluid Mech. , 463–487 (2015).[2] V. Dallas, J.C. Vassilicos and G.F. Hewitt, Phys. Rev. E , (2010).[3] N.J. Wagner and J.F. Brady, Phys. Today , 27–32(2009).[4] J.J. Stickel and R.L. Powell, Annu. Rev. Fluid Mech. ,129–149 (2005).[5] M. Reeks, J. Aerosol Sci. , 729–739 (1983).[6] F. Picano, W.-P. Breugen, D. Mitra and L. Brandt, Phys.Rev. Lett. , 098302 (2013).[7] P. Costa, F. Picano, L. Brandt and W.P. Breugem, Phys.Rev. Lett. 117, 134501 (2016).[8] B.A. Toms, Proceedings of the International Congress onRheology , 135, North-Holland (1949).[9] W.B. Giles and W.T. Pettit, Nature , 470–472(1967).[10] W.D. White and D.M. McEligot, J. of Basic Engineering , 411–418 (1970).[11] M.E. Rosti and L. Brandt, Phys. Rev. Fluids , 041301(2020).[12] A.C. Eringen, Technical report , Purdue University(1965).[13] N. Mitarai, H. Hayakawa and H. Nakanishi, Phys. Rev.Lett. , 174301-1 (2002).[14] G. Lukaszewicz, Springer Science and Business MediaNew York (1999).[15] D.W. Condiff and J.S. Dahler, Phys. of fluids , 842(1964).[16] T. Tsukahara, Y. Seki, H. Kawamura and D. Tochio, 4thInternational Symposium on Turbulence and Shear FlowPhenomena, Williamsburg, VA, USA, 935-940 (2005).[17] J. Kim, P. Moin and R. Moser, J. Fluid Mech. , 133–166 (1986).[18] R.D. Moser, J. Kim and N.N. Mansour, Phys. of fluids , 943 (1999).[19] A.C. Eringen, J. Math. Mech. , 1-16 (1966).[20] M.I. Cheikh, J. Chen and M. Wei, Phys. Rev. Fluids ,104610 (2019).[21] T. Ariman, M.A. Turk and N.D. Sylvester,J. Appl. Mech.Trans. ASME , 1-7 (1974).[22] E. Karvelas, G. Sofiadis, T. Papathanasiou and I.E. Sar-ris, Fluids , 125 (2020).[23] E.G. Karvelas, A. Tsiantis and T.D. Papathanasiou,Comp. Meth. and Prog. in Biomed. , 105135 (2020).[24] C.D. Dritselis and N.S. Vlachos, Phys. of Fluids ,055103 (2008).[25] D. Lee and H. Choi, J. Fluid Mech. , 367-394 (2001).[26] W. Schoppa and F. Hussain, J. Fluid Mech.453