Frohnapfel, B.

Hasegawa, Y.

Kasagi, N.

English

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Effect of Near-Wall Componental Modificationof Turbulence on its Statistical PropertiesBettina Frohnapfel, Yosuke Hasegawa and Nobuhide KasagiAbstract Turbulence control techniques are of great economical and ecological in-terest. In the present work a fundamental study is carried out in which body forcesare introduced in the near-wall region of a turbulent channel flow and thus mod-ify the near-wall behavior. It is investigated how these forces, which selectively acton one of the velocity components, modify near-wall turbulence and its statisticalproperties with the goal to extract properties that can directly be linked to the skinfriction drag. The alignment between the principal axis of the Reynolds stress tensorand the mean flow direction is identified as an interesting quantity in this respect.1 ProcedureWe carry out direct numerical simulations of a fully developed channel flow witha constant flow rate. The bulk Reynolds number based on the channel height isReb = 4460 and the corresponding friction Reynolds number for the uncontrolledcase is given by Reτ = 150. As a control input for modification of the near-wallvelocity fluctuation we consider a feedback body force, b f j, which is introducedinto the Navier-Stokes equations. The modified momentum equations ( j = 1,2,3)thus read:∂u+j∂ t++u+i∂u+j∂x+i= − ∂ p+∂x+j+∂ 2u+j∂x+i ∂ x+i+f (x+2 )Φ+(u+j −U+j)︸ ︷︷ ︸b f +j, (1)Bettina FrohnapfelCenter of Smart Interfaces, Technical University of Darmstadt, Petersenstr. 32, 64287 Darmstadt,Germany, e-mail: frohnapfel@csi.tu-darmstadt.deYosuke Hasegawa, Nobuhide KasagiDepartment of Mechanical Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku,Tokyo 113-8656, Japan, e-mail: hasegawa / kasagi@thtlab.t.u-tokyo.ac.jp1Notes on Numerical Fluid Mechanics and Multidisciplinary Design, Springer, Vol. 110: 127-133 presented at: Second International Conference on Turbulence and Interaction - TI 2009 31 May - 5 June 2009 Sainte-Luce, Martinique2 Bettina Frohnapfel, Yosuke Hasegawa and Nobuhide Kasagiwhere the subscript ()+ indicates normalization with the kinematic viscosity, ν , andthe wall shear velocity, uτ , of the uncontrolled channel flow. The velocity compo-nents in streamwise, wall-normal and spanwise direction are given by u1,u2,u3 andu,v,w, respectively and the decomposition into mean and fluctuating components isdefined as: u j = U j +u′j. The last term on the right hand side of equation (1) repre-sents the body force which is proportional to the corresponding velocity fluctuation.The relaxation time constant,Φ+, determines the strength and type of the forcing,and f (x+2 ) is a step function which determines the region where the forcing is ap-plied on top and bottom wall: f (x+2 ≤ y+f ) = 1, f (x+2 > y+f ) = 0. The control inputcan thus be varied by modifying the relaxation time constant,Φ+, and the forcinglayer thickness, y+f . For positive values of Φ+ the introduced body force acts to en-hance the velocity fluctuations in the corresponding direction; for negative valuesof Φ+ the introduced body force acts as a damping force. In the present study thefocus is placed on the influence of the forcing orientation. Therefore, the forcinglayer thickness is set to a constant value of y+f = 10.2 Results and DiscussionTable 1 shows the obtained drag reduction for different exemplary test cases inwhich a body force according to equation (1) is introduced within the forcing layer.The results show that damping of the wall-normal velocity fluctuation, v′, and thespanwise fluctuation, w′, leads to drag reduction (DR). The DR obtained with afixed value of Φ+ is higher for w-forcing. This result is not surprising since thebody force depends on the strength of the velocity fluctuation itself and similar re-sults have been reported by [5] who employed an exponentially decaying dampingforce in the near-wall region. With increasing strength of the w′-damping force, thetheoretical maximum of DR (Iwamoto et al., 2005) is realized for (Φ+)−1 <−1 [1].Table 1 Drag reduction (DR) obtained for channel flows in which different forcing is appliedwithin a forcing layer thickness of y+f = 10. The forcing strength is given by (Φ+)−1.forcing b f j (Φ+)−1 DRu’-damping b f1 < 0 -1/15 −7%v’-damping b f2 < 0 -10/15 20%w’-damping b f3 < 0 -10/15 42%u’-increase(weak)b f1 > 0 2/150 2%u’-increase(strong)b f1 > 0 5/150 −8%v’-increase b f2 > 0 10/15 −10%w’-increase b f3 > 0 5/150 −25%Second International Conference on Turbulence and Interaction - TI 2009 31 May - 5 June 2009 Sainte-Luce, MartiniqueNear-Wall Componental Modification of Turbulence 3101w -damping‘uj,rms**x210-410-2 102w -increase‘u -damping‘v -damping‘v -increase‘ u -increase‘Fig. 1 Near-wall behavior of the rms values (u∗rms,v∗rms,w∗rms from top to bottom in each plot) forthe controlled flow (open symbols) in comparison to the uncontrolled channel flow (solid sym-bols). The subscript ()∗ indicates that all values are normalized with the inner variables of thecorresponding flow. The plot for u’-increase contains results for both, weak (black open symbols)and strong (grey solid symbols) forcing.This amount of DR can not be achieved with increasing the strength of v’-dampingup to (Φ+)−1 = −1000/15, which corresponds to the maximum forcing strengthconsidered in the present investigation.For positive values of b f2 and b f3, which increase v′ and w′, respectively, dragincrease is observed. Again, a body force acting on the w-component (b f3) has amore pronounced effect than b f2 with similar values of (Φ+)−1. The general simi-larity between modifications of the v- and w-component can be explained as a resultof streamwise vortices in the near-wall region which are damped or enhanced dueto the introduced body forces.For body forces acting in the streamwise direction, b f1, drag increase is observedfor damping of the streamwise fluctuations, u′. This is in contrast to the results of[5] who reported slight DR for an exponentially decaying damping force in stream-wise direction. In the present investigation, weak enhancement of the streamwisevelocity fluctuations leads to a slight DR, while stronger forcing enhances the skinfriction drag. Figure 2 shows the near-wall behavior of the velocity fluctuations ofthe controlled flow in comparison to the uncontrolled channel flow. It can be seenthat strong body forces in spanwise or wall-normal direction significantly modifythe other near-wall components, too.Second International Conference on Turbulence and Interaction - TI 2009 31 May - 5 June 2009 Sainte-Luce, Martinique4 Bettina Frohnapfel, Yosuke Hasegawa and Nobuhide KasagiFig. 2 Anisotropy-invariantmapping for the uncontrolledflow and one of the controlled(w’-damping, DR=42%)flows. The second invariant,IIa, captures the magnitude ofanisotropy and arrows markthe value of IIa at the wall.0.15 0.250.050.5 uncontrolledflowcontrolledflow-0.05IIaIIIaIIa=aijajiIIIa=aijajkakiIn [1] it was demonstrated that, for a variety of different flow control techniques,DR is associated with increased anisotropy of turbulence in the near-wall region.When the near-wall behavior of the present controlled flows is analyzed in theanisotropy-invariant map (see figure 2), an increase of the second invariant, IIa, ofthe anisotropy tensor, ai j [6]:ai j =u′iu′jq2− 13δi j, , IIa = ai ja ji , (2)in the near-wall region is observed for all drag reduced cases. This result is in agree-ment with the above mentioned prior findings. It is interesting to check whethera reversed trend in the near-wall turbulence anisotropy can be found for drag in-creased flows. In the present investigation, the anisotropy in the near-wall regionis decreased for all drag increased flows, expect for the case of strong streamwiseforcing for which a slight increase of IIa is observed.In general, the magnitude changes of the near-wall anisotropy cannot be relatedquantitatively to the Reynolds shear stress,−u′v′ , which forms the turbulent contri-bution to the skin friction drag in a fully developed turbulent channel flow [3]:c f =12Reb+12∫ 102(1− y)(−u′v′)dy (3)Further information about −u′v′ can be found when considering the transformationof the Reynolds stress tensor to its diagonal form which forms the basis for theinvariant analysis:Second International Conference on Turbulence and Interaction - TI 2009 31 May - 5 June 2009 Sainte-Luce, MartiniqueNear-Wall Componental Modification of Turbulence 50123456780 50 100 1500.000.050.100.150.200.250.300.350.400 50 100 150( )22sin12''aEVEVvu --=-channel, Re =150channel, Re =100w’-damping:(?+)-1= -1/15(?+)-1= -10/15x2* x2*ttFig. 3 Contributions to the Reynolds shear stress in terms of the eigenvalues of the Reynolds stresstensor and the misalignment of the principle axis with the channel coordinate system. Normaliza-tion is based on the inner variable of each flow.cosα sinα 0−sinα cosα 00 0 1u′u′ u′v′ 0u′v′ v′v′ 00 0 w′w′cosα −sinα 0sinα cosα 00 0 1 =EV 1 0 00 EV 2 00 0 EV 3(4)It is possible to express −u′v′ in terms of the angle α of this rotation and the differ-ence of the first and second eigenvalues, EV , of the Reynolds tensor:−u′v′ =−(EV 2−EV 1) sin2α2. (5)Figure 3 shows the change in both terms of this identity for two w′-damping casesin comparison with the uncontrolled channel flow. In order to distinguish betweenReynolds number effects (for the controlled flows Reτ = 130 and Reτ = 115, re-spectively) and the influence of the applied control, channel flow data of Reτ = 100is also included. It is obvious that the reduction of the angle α , i.e. the reducedmisalignment between the principal axis and mean flow direction, in the near-wallregion is the main reason for the reduction of the Reynolds shear stress and thusDR, while the reduction of α in the outer flow region can be attributed to the de-crease in Reynolds number. The same trends are found for v’-damping. For weaku’-enhancement, only DR = 2.2% is achieved and the corresponding changes inthe quantities of equation (5) are small but they suggest the same conclusion: thereduction of α can be identified as the source for the reduced Reynolds shear stress.The drag increase for u’-damping, v’-increase and w’-increase is related to ageneral increase of α while (EV 2−EV 1) is reduced. Especially for u′-damping aSecond International Conference on Turbulence and Interaction - TI 2009 31 May - 5 June 2009 Sainte-Luce, Martinique6 Bettina Frohnapfel, Yosuke Hasegawa and Nobuhide Kasagidrastic increase of 0.5sin2α is observed in the forcing layer. It is accompanied bya strong increase of the correlation coefficient for −u′v′ in this region. This factexplains the increase of the Reynolds shear stress in spite of damping of the stream-wise velocity fluctuations u′.For strong enhancement of the streamwise velocity fluctuations, an increase ofIIa is observed while the skin friction drag is increased. This result suggests that thecommonly observed trend of increased near-wall anisotropy for drag reduced flowscannot be identified as a sufficient condition for DR. The analysis in terms of eigen-values and alignment can shed some light into this point of discussion. Figure 4shows the split-up of the corresponding Reynolds shear stress into its contributionsaccording to equation (5). In the near-wall region a reduction of the α-term in com-parison to the uncontrolled channel flow can be observed. Although the Reynoldsnumber effect would correspond to an increase of α in the entire flow domain, itremains identical to the one for an uncontrolled channel flow in the region of ap-proximately 30 < y+ < 100 before shifting to higher values around the channelcenter. In spite of the α-reduction in the near-wall region −u′v′ increases due toa large increase of (EV 2−EV 1). This term can further be split up into the turbu-lent kinetic energy, k = 0.5q2, of the flow field and the difference between first andsecond eigenvalue of the anisotropy tensor, ai j, according to:(EV 2−EV 1) = q2(EV 2(ai j)−EV 1(ai j)) (6)The split up reveals that the large increase in the eigenvalue difference (EV 2−EV 1)and thus the increase of −u′v′ can be related to the increase of turbulent kinetic en-ergy in the flow field due to the enhancement of u′. This behavior suggests that thehigh turbulent kinetic energy, induced by the u′-enhancing body force, suppressesthe drag reducing potential of the improved alignment in the near-wall region. In thisrespect it should be noted that the energy input required to change v′ and w′ is gen-erally small compared to the pumping power of the channel flow while significantlyhigher energy input is needed for changes of u′.In summary, the results of the present investigation suggest that flow control thatacts on the w’-component is most effective. Modifications of the v’-component yieldsimilar results but tend to be less effective resulting in lower gain, i.e. energy savingsper energy input for the DR cases. Direct modifications of the streamwise velocityfluctuation yield the surprising result that drag increase is found for a reduction ofthe turbulent kinetic energy in the flow field and vice versa (for small increase ofu′). Since these modifications require high energy inputs, however, they do not seema smart target for flow control. The misalignment angle, α , between principle axisof the Reynolds stress tensor and the mean flow direction of the channel is intro-duced as an interesting quantity for flow control. At this point the present resultssuggest that a reduction of α in the near-wall region which is achieved with littleenergy input will result in drag reduction. Much further work will be needed to clar-ify whether this fact can be used in the design of practical flow control techniques.Simulations of w′-damping at Reτ = 300 and 450 show that the drag reduction rateachievable with near-wall componental modifications exhibits almost no ReynoldsSecond International Conference on Turbulence and Interaction - TI 2009 31 May - 5 June 2009 Sainte-Luce, MartiniqueNear-Wall Componental Modification of Turbulence 7051015000.20.40 50 100 150channel, Ret=150u -increase(F+)-1=5/150 50 100’( )22sin12''aEVEVvu --=-x2* x2*Fig. 4 Contributions to the Reynolds shear stress in terms of the eigenvalues of the Reynolds stresstensor and the misalignment of the principle axis with the channel coordinate system. Normaliza-tion is based on the inner variable of each flow.number dependency, at least in the investigated regime, and confirm that the reduc-tion of the misalignment angle α is a key quantity for the reduction of the Reynoldsshear stress, −u′v′.References1. Frohnapfel B, Hasegawa Y, Kasagi N (2008) Flow Control by Turbulence State Modificationsin the Near-Wall Region. In: XXII Int Cong Theoretical and Applied Mechanics, Adelaide,Australia2. Frohnapfel B, Lammers P, Jovanović J, Durst F (2007) Interpretation of the mechanism asso-ciated with turbulent drag reduction in terms of anisotropy invariants. In: J Fluid Mech 577:457–4663. Fukagata K, Iwamoto K, Kasagi N (2002) Contribution of Reynolds stress distribution to theskin friction in wall-bounded flows. In: Phys. Fluids 14: L73–L764. Iwamoto K, Fukagata K, Kasagi N, Suzuki Y (2005) Friction drag reduction achievable bynear-wall turbulence manipulation at high Reynolds number. In: Phys Fluids 17: 0117025. Lee C, Kim J (2002) Control of the viscous sublayer for drag reduction. In: Phys Fluids 14:2523–25296. Lumley JL, Newman GR (1977) The return to isotropy of homogeneous turbulence. In: JFluid Mech 82: 161–178Second International Conference on Turbulence and Interaction - TI 2009 31 May - 5 June 2009 Sainte-Luce, Martinique

Effect of near-wall componental modification of turbulence on its statistical properties

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Effect of near-wall componental modification of turbulence on its statistical properties

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