Fundamental Physics Underlying Polymer Drag Reduction, from Homogeneous DNS Turbulence with the FENE-P Model

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1 nd International Symosium on Seawater Drag Reduction Busan, Korea, 3-6 May 005 Fundamental Physics Underlying Polymer Drag Reduction, from Homogeneous DNS Turbulence with the FENE-P Model J.G. Brasseur, A. Robert (The Pennsylvania State University, U.S.A.) L.R. Collins, T. Vaithianathan (Cornell University, U.S.A.) ABSTRACT Reduction in surface shear stress in turbulent boundary layers imlies suression of turbulent momentum flux, a large-eddy henomenon. The hysics by which dilute concentrations of long-chain molecules alter large-eddy structure and momentum flux is not well understood. Exeriment, however, indicates an essential mix of turbulent velocity fluctuations, olymer molecules, and mean shear. We study the consequences of this essential mix through direct numerical simulation (DNS) of homogeneous shear-driven turbulence with olymer-turbulence interactions modeled using the FENE-P (Finite Extensible Nonlinear Elastic with the Peterlin aroximation) reresentation for olymer stress, and the conformation equation redicted with an advanced hyerbolic solver. Progressive increases in nondimensional olymer relaxation time (Weissenberg number) roduce rogressive reductions in Reynolds stress concurrent with increasing olymer stretch. The redicted -D sectra change with olymer addition consistent with exeriment. Polymer-turbulence energy exchange is confined to the large scales, indicating a direct, rather than indirect, effect on the flux-carrying eddies. Polymer-turbulence energy exchange, ressure-strainrate correlations, and olymer-stretch lay major roles in the dynamics that underlies the suression of momentum flux and drag reduction. THE BASIS FOR THE STUDY That olymer added at low concentration to wallbounded turbulent shear flows dramatically reduces wall shear stress is well established exerimentally (Lumley 973, Virk 975). The secific olymerturbulence interactions that lead to drag reduction also alter turbulence structure, velocity correlations, and momentum flux within the large eddies (Oldaker et al. 968, Donohue et al. 97, Luchik & Tiederman 97, Wei et al. 99, McComb 977, Dimitrooulos et al. 998, 00, van Doorn et al. 999, Warholic et al. 999). Even with recent advances in simulation and measurement of olymer-turbulence drag reduction, the detailed hysical mechanisms by which large-eddy turbulent fluxes are suressed by molecules orders of magnitude smaller in scale are not well defined. The current analysis is intended as a ste forward in this direction. A few key exerimental and numerical observations underlie our study. Firstly, drag reduction initiates only when the boundary layer is turbulent (Virk 975), so that to deciher the underlying mechanisms is to deciher the interactions between olymer molecules and turbulent fluctuations that initiate and maintain alteration of the large-scale turbulent eddies, where momentum flux is concentrated. Indeed, drag reduction occurs with dramatic reductions in near-wall Reynolds shear stress (Virk 97, Luchik & Tiederman 988, Warholic et al. 999, Dimitrooulos et al. 00), imlying suression of normal momentum flux and skin friction. However, drag reduction occurs equally over both rough and smooth surfaces (McNally 968, Sangler 969, Virk 97), so that the essential hysics of drag reduction does not involve a viscous sublayer. Furthermore, drag reduction initiates as olymer enters lower inertial wall layer from above or below (Wells & Sangler 967, McComb & Rabie 979, 98), suggesting that the source of the underlying olymer-turbulence dynamics lies within the lower inertial layer, which can be aroximated as

2 quasi homogeneous turbulence modulated by shear (Khanna et al. 99). We conclude, from these exerimental observations, that the existence of a surface is not essential to the hysics underlying olymer drag reduction, and that the essential dynamics that alters large-eddy inertial-layer momentum flux and suresses drag can studied, at its most basic level, with direct numerical simulation (DNS) of olymerladen shear-driven homogenous turbulence. We have carried out a series of numerical studies using the FENE-P model of continuum-level olymer stress within DNS of isotroic and homogeneous shear turbulence. We focus, in this aer, on shear-driven turbulence and statistical mechanisms underlying olymer-induced modulation of the Reynolds stress tensor and the suression of momentum flux. Polymer relaxation time scale (Weissenberg number) is systematically increased, and nondimensional shear rate adjusted to corresond to the transition between the lower inertial layer and uer buffer layer of the high Reynolds number turbulent boundary layer. THE FENE-P MODEL SYSTEM FOR POLYMER-LADEN TURBULENT FLOWS The FENE-P reresentation (Bird et al. 987, Peterlin 96) of the interaction between dilute olymer and turbulent flow is given by the following dynamical system: i = 0, () x i s i τ τ ρ = ρ ( uu i j) + +, () t x x x x j i j j f( R) C δ s where τ = β ( µ S ), τ = ( β ) µ, λ L 3 f ( R) =, C ˆˆ / ˆ = RR i j L0, R C. (3) ii L R Newton s law, eq. (), suorts a stress contribution from the olymer, τ, in addition to the Newtonian solvent stress, τ s. µ = µ s + µ is the zero shear viscosity of the mixture, β = µ s /µ is the ratio of solvent to mixture viscosity, and ( - β) = µ /µ the ratio of zero shear olymer to mixture viscosity. Thus, β and ( - β) are concentration arameters for solvent and olymer, resectively. Polymer stress is roortional to f( R) C, where the olymer conformation tensor C is a local nondimensionalized ensemble average of the dyadic roduct of end-to-end olymer vector, R ˆi. R is the local nondimensional average olymer length, and L the maximum nondimensional olymer length. All length variables (C, R, L) are nondimensionalized by L 0, where L ˆ ˆ 0 3 RR is a characteristic olymer i i equil length in the equilibrium no-flow state. f(r) is the socalled Peterlin function, constructed to diverge as R L. In this limit olymer stress becomes infinite and the finite extensibility of the olymer is forced, by the FENE dumbbell model, to remain below L. The FENE dumbbell models single olymer molecules with end-to-end vector length R i. The extension of the model to the continuum level, eq. (), introduces the Peterlin re-averaging aroximation (Peterlin 96) and leads to the following FENE-P dynamical equation for C : C j i = ( uc k ) + Cik + Cjk τ (4) t xk xk xk µ Conformation is generated by the stretching of olymer by local strain-rate field. The restoration term in eq. (4) contains the Peterlin function f(r) and enforces finite extensibility of olymer. Given the redominance of strain-rate fluctuations at the smallest turbulence scales, the imortance of small-scale turbulence-olymer interactions for the stretching and modulation of olymer conformation is clear in eq. (4). The mechanisms by which the largest eddies are altered by olymer, however, is unclear. Given the imortance of mean olymer stretching and Reynolds stress to drag reduction hysics, we consider the Reynolds stress budgets, suitably restricted to homogeneous shear turbulence with mean shear rate, S = du/dy: d u d v d w = Suv + φ ε Γ, (5a) = + φ ε Γ, (5b) = + φ ε Γ, (5c)

3 d d uv q = Suv ε Γ, (5d) = Sv + φ ε Γ. (5e) q = uu i i and φ = s ρ are ressure-strain-rate correlations resonsible for intercomonent energy i j transfer, and ε = ρµ s and ε = ε ii are k k dissiation rates. Γ and Γ= Γ ii are the rates of energy exchange between the turbulence and the olymer, where Γ = + j i τik τ jk ρ k k (6) has dynamics similar to the stretching term in eq. (4). When olymer is stretched energy asses, on average, from turbulent fluctuations to the olymer, where it is stored as elastic energy. When olymer relaxes elastic energy is assed from the olymer to the turbulence. Thus, unlike viscous dissiation, Γ describes a reversible transfer of energy. We shall find that Γ acts as an additional dissiative mechanism, on average, although locally Γ is of both signs. On average, olymer conformation evolves according to the following equations in homogeneous shear-driven turbulence: d C = SC +Λ τ (7a) µ d C d C d R d C τ µ = +Λ (7b) τ 33 µ = +Λ (7c) = SC +Λkk τ kk (7d) µ SC τ µ = +Λ (7e) In both the Reynolds stress and mean conformation equations, the source terms in the shear comonents are roortional to the comonent in the direction of mean gradient. The source terms in the mean flow directional comonent, in contrast, are roortional to the shear comonents. All source terms are 3 roortional to mean shear rate. In articular, mean olymer stretch has a contribution from the interaction between mean shear and mean conformation as well as from the correlation between fluctuating strain-rate and fluctuating conformation: Λ = C + C i ik jk xk x j k (8) Each of the hysical-sace balances have Fouriersectral counterarts from which the scales involved in olymer-turbulence interactions can be studied. In articular, the comonent energy equations can be written: Eαα =Ρ αα + Ta α Dαα Γαα, (9) t where Ea α, Ρ aα, Dαα and Γαα are the 3- D sectral equivalents to comonent turbulent kinetic energy (TKE), roduction-rate, dissiation-rate, and olymer-turbulence energy exchange-rate. Ta α is the rate of energy transfer between wavenumber shell k and all other sectral shells resulting from the advective nonlinearity in Newton s law. The corresonding equation for the 3-D energy sectrum E(k) = /E ii (k) is a contraction of (7). A HIGH-ACCURACY NUMERICAL ALGO- RITHM FOR POLYMER CONFORMATION For numerical stability, the solution to eq. (4) must maintain olymer stretch R between 0 and L, and the conformation tensor should remain ositive semidefinite. The lack of natural diffusion in the conformation equation has lead to the introduction of numerical diffusion to artificially stabilize the advancement of the conformation tensor (Sureshkumar et al. 995, Min et al. 003, Dubief et al. 004). Vaithianathan & Collins (003) develoed an algorithm based on the formal decomosition of C into a roduct of eigenvalue and eigenvector matrices in a form that necessarily maintains C as ositive definite. The algorithm was unconditionally stable without the need for artificial diffusion. However, whereas olymer stretch was maintained ositive definite, the algorithm did not guarantee ositive definiteness of individual eigenvalues, leading to an imbalance in the advective derivative. The current simulations of shear turbulence were carried out with a different algorithm that maintains both the ositive definiteness of the conformation tensor as a whole and the individual eigenvalues at all oints in sace-time (Vaithianathan et al., 004).

4 Recognizing that the C equation is hyerbolic in nature, the Kurganov-Tadmor (000) algorithm was adated to the solution of eq. (4). The Kurganov- Tadmor method is a second-order advanced centraldifference scheme develoed to cature shocks with minimal numerical diffusion of shock strength. The scheme successfully treats the sensitivity in the equations to olymer stretch by turbulent strain-rate fluctuations at the smallest resolved scales. Individual eigenvalues are maintained absolutely ositive definite, and the conformation and Reynolds stress equations are within % balance at all times. An imortant consequence of roer treatment of gradients in olymer stress by this scheme is a higher sensitivity to olymer in the modulation of turbulence structure than with other, more classical, algorithms that lose olymer effect to a smearing of small-scale olymer stress gradients. SHEAR-MODULATED POLYMER TURBULENCE SIMULATIONS Direct numerical simulations (DNS) of homogeneous turbulent shear flow with FENE-P olymer dynamics were carried out seudosectrally for the solution of eqs. () and (), and in hysical sace using the Kurganov-Tadmor algorithm for the solution of eq. (4). Both solutions were carried out using the Rogallo (98) transformation to a frame of reference deforming with mean shear. In this way, the equations for fluctuating velocity and for the conformation tensor (4) were solved with eriodic boundary conditions. Newtonian homogeneous shear flow turbulence is inherently nonstationary with growing length scales and with roduction rate exceeding dissiation rate. The box is twice as long in the streamwise direction as in the cross-stream directions to accommodate growing streamwise integral scales. The objective is to initiate the turbulence such that a nonstationary equilibrium growth state is attained before the integral scales grow large enough for significant influence from the eriodic boundary conditions. Simulations were initiated with an initial velocity sectrum chosen to maintain high resolution at both integral and Kolmogorov scales. In all simulations at least ten integral scales. sanned the box in the streamwise direction, and the Kolmogorov scales were always well resolved (k max η ~.5.50). The conformation tensor was secified initially as uniform with C ii determined by scaling the olymer to viscous force terms in eq. (): R init =.5βλ SL ( β ) L + 3βλ. (0) S The constant.5 was chosen from exerimentation. Relevant arameters in the simulations are: Sq λ uv S*, We τ, α = ( - β)l. () ε ν s S* is shear rate nondimensionalized by an integral time scale, ε/q. We τ (a Weissenberg number) is olymer time scale nondimensionlaized by τ + ν s /u τ, the nearwall time scale in the viscous sublayer of a corresonding Newtonian boundary layer with viscosity ν s and surface stress τ w ρuv in the corresonding Newtonian flow. We τ is consistent with homogeneous shear flow as a model for the lower (a) (b) Figure. Evolution of (a) Reynolds stress, ρuv and (b) olymer stretch, R = C ii for the Newtonian (black) and olymer-laden shear flow, over We τ from 5 to 09. 4

5 Figure. Reynolds stress normalized by q = uu, i i showing that at St 8, the turbulence aears to be roughly in equilibrium. Polymer stretch has reached a eak at aroximately the same St. inertial layer in a high Reynolds number turbulent boundary layer where Re λ qλ, /ν s ~ α is an effective concentration arameter roortional to the extensional viscosity, or Trouton ratio. SUPPRESSION OF TURBULENT MOMENTUM FLUX BY POLYMER IN SHEAR FLOW In Fig. the evolution of Reynolds stress and mean olymer stretch are lotted as a function of total shear St, for Newtonian turbulence, and with We τ from 5 to 09. Weissenberg number (eq. ) is based on Reynolds stress at St = 8 which, from figure, is take to be the best aroximation of an equilibrium state when uv scales aroximately on q. As Weissenberg number increases, the influence of the olymer strengthens as a result of a rogressively higher olymer stretch (figure b; note that R eaks near St = 8 in all cases). Figure a shows a rogressively stronger suression of turbulent momentum flux (Reynolds stress) with increasing Weissenberg number, consistent with exeriment in olymer-laden boundary layers. Figure 3 indicates that the reduction in Reynolds stress comes about largely from a decorrelation between u (streamwise) and v (vertical) velocity fluctuations, consistent with exerimental data from the near-wall Figure 3. Correlation coefficient between streamwise and vertical velocity fluctuations with increasing Weissenberg number. In the simulations reorted in this study, β = 0.95, L = 00, and α = 500. At the times analyzed (next section), S* varied from ~ and Re λ from ~ 0-0, corresonding to the lower-inertial/uer-buffer layer of a corresonding high Reynolds number boundary layer. In this study we vary the influence of olymer through Weissenberg number, from Newtonian to We τ ~ 0. 5 Figure 4. -D energy sectra in the inertial region measured in a Newtonian channel flow and with increasing olymer concentrations. Measurements were made using a laser Doler velocimeter at 0.3 the half channel wih. From Warholic et al. (999). boundary layer (Warholic et al. 999). SUPPORT OF HOMOGENEOUS SHEAR FLOW AS A MODEL FOR DRAG REDUCTION Figure 4 shows -D comonent streamwise energy sectra measured exerimentally within the inertial

6 layer of olymer laden boundary layers by Warholic et al. (999). A rogressive suression of small scale energy at high wavenumbers is observed with increasing olymer concentration, then a reduction in energy overall at higher concentrations. through interactions between molecules orders of magnitude smaller in scale and strain-rate fluctuations, as described by the stretching terms in eq. (4). One manifestation of this olymer-turbulence interaction is energy exchange Γ between turbulent kinetic energy and olymer elastic energy given in eqs. (5) and (9). In figure 6 we lot the 3-D sectrum of the total kinetic energy transfer, -Γ(k) = -/ Γ ii (k) (-Γ(k) < 0 imlies drain of TKE). Polymer is observed to strictly drain energy, on average, at each scale, and that energy drain is confined to the larger scales. The 3-D energy sectra (not shown) for all but the highest Weissenberg number eak at k ~ 6-7, coinciding with the eak energy drain. Figure 5. -D steamwise energy sectra redicted in olymer-laden homogeneous turbulence shear flow with increasing olymer effect (Weissenberg number). The Newtonian case is given by the black curve. Corresonding -D streamwise comonent energy sectra for the homogeneous shear flow simulations are given in figure 5. Whereas increasing influence of the olymer in the exeriment is obtained by increasing olymer concentration, in the simulations a similar effect is obtained by increasing Weissenberg number. The simulations show the same modulation of the -D sectra as exeriment a suression of energy at high wavenumbers, then, at the highest Weissenberg numbers suression also at low wavenumbers. Thus, FENE-P olymer turbulence in homogeneous shear flow with high-order numerics redicts both the suression of Reynolds shear stress, and the internal adjustments to the energy sectrum, measured in the inertial layer of wall-bounded olymer-laden shear flows. These results rovide confidence for more detailed analysis of the fundamental hysics and mechanisms underlying the suression of Reynolds stress and consequent reductions in total wall shear stress. DIRECT INTERACTION BETWEEN POLYMER MOLECULES AND LARGE EDDIES The suression of Reynolds stress comes about through the modulation of the largest eddies, ultimately Figure 6. 3-D sectra of total olymer turbulence energy exchange, -Γ(k), where Γ = Γ. This result (and a similar result found in isotroic turbulence) is inconsistent with the cascade theory of Tabor & DeGenne (986), who roosed as a mechanism for large-eddy modulation and drag reduction, an interrution to the energy cascade by olymer-turbulence energy exchange with consequent indirect influence on the energy-containing turbulent motions. We find, instead, that the influence of olymer molecules on the turbulence is directly on the large scales. Somehow, in a manner as yet not fully understood, olymer molecules orders of magnitude smaller than the integral scales, directly modulate the flux-carrying eddies. In the resence of shear this direct interaction leads to a decorrelation of streamwise and vertical velocity fluctuations, a reduction in vertical turbulent momentum flux, and a reduction in skin friction on an underlying surface. ii 6

7 We conclude that the addition of olymer adds an additional energy drain mechanism, but energy drain directly from the energy-containing eddies through a mechanism very different from viscous dissiation, which remains concentrated at the Kolmogorov scales. Figure 7. Comonent olymer-turbulence energy exchange sectra. See Γ αα in eq. (9). The anisotroic behavior of olymer-turbulence energy exchange may lay a role in olymer-induced restructuring of the large scales. In figure 7 we show comonent energy exchange sectra corresonding to -Γ αα (k) in eq. (9). All three comonent energy exchange rates are confined to the energy-containing eddies. However the rate of energy drain is much Figure 8. Comarison of relative contributions of friction (viscous dissiation rate, ε ) and olymer energy drain (γ ) with increasing Weissenberg number (St = 8). greater from the streamwise TKE comonent than the other two, with energy drain from the sanwise comonent higher than the vertical comonent. 7 THE NEW DISSIPATIVE MECHANISM Figures 6 and 7 show that, whereas olymer-turbulence energy exchange Γ may be of both signs locally, on average Γ αα is dissiative in nature. The changes in relative contribution to total dissiation of olymer vs. viscous dissiation with increasing Weissenberg number are shown in figure 8. At low We τ viscous dissiation-rate dominates olymer dissiation in all comonents. However, with increasing olymer affect, viscous dissiation is rogressively more suressed and olymer dissiation more enhanced until a crossover Weissenberg number is reached above which olymer dissiation exceeds viscous dissiation. This cross-over We τ occurs first in vertical kinetic energy drain, then sanwise, then streamwise TKE, over the range We τ ~ However, at higher Weissenberg numbers, the dominance of olymer dissiation over viscous decreases. It is clear that in the resence of a significant olymer effect, both viscous dissiation and energy drain to the olymer at the large scales contribute to the dynamics of turbulence evolution. That the direct drain of turbulent kinetic energy into olymer stretch at the large-eddy scale contributes to the decorrelation of velocity comonents and reduction in momentum flux and drag seems likely. How this large-scale rocess takes lace in detail, however, is less clear. SUPPRESSION OF VERTICAL VELOCITY FLUCTUATIONS: PRESSURE-STRAIN Equations (5) give the statistical contributions to the evolution of the Reynolds stress tensor. The interlay between the last two terms in these equations, both of which are sinks to comonent kinetic energy, were discussed in the revious sections. All turbulent energy roduction enters the streamwise velocity comonent from the interaction between mean shear and Reynolds shear stress. Suression of Reynolds shear stress therefore imlies suression of turbulence roduction (with fixed mean shear). In the absence a olymer source of TKE, maintenance of vertical and sanwise turbulence fluctuations comes about only through intercomonent energy transfer, the ressure-strain-rate correlations φ αα. These slit into three contributions, each associated with a different term in the Poisson equation for fluctuating ressure: where r = ρss, = r + s +,: x x (a) s i j = ρ, (b) j i

8 and τ = (c) xi x j Figure 9. Effect of olymer on ressure-strain-rate intercomonent energy transfer. Total ressure-strain is given by the sums of raid, slow and olymer r s contributions: φ = φ + φ + φ. αα αα αα αα The contributions to the ressure fluctiations, r and s, are traditionally called raid and slow in context with the raid (immediate) and slower adjustment of raid ressure to changes in mean shear, and the slower adjustment in s. The addition of olymer stress adds a third, olymer, contribution to the Poisson integral for ressure. Corresondingly, the ressurestrain correlations contain raid, slow and olymer contributions: φ = φ + φ + φ (3) r s In figure 9 we lot total, raid and slow ressurestrain contributions vs We τ (at St = 8). The addition of olymer has a dramatic affect on all ressure-strain comonents. All contributions are rogressively suressed with increasing Weissenberg number. Consider total ressure-strain, the solid curves in figure 9. In the Newtonian state (We τ = 0), energy is transferred from u (φ < 0) to v and w (φ, φ 33 > 0). The direction of intercomonent energy transfer remains the same as the olymer affect increases, however the rate of energy transfer is suressed. More imortantly, energy transfer into v is suressed disroortionately more raidly than into w. Polymer-turbulence interactions tend to shut down energy transfer into vertical turbulent velocity. The suression of vertical velocity fluctuations, in turn, shuts down the source term to Reynolds shear stress 8 (eq. 5e). Because the other terms in eq. (5e) destroy uv, the suression of vertical velocity by the interaction between olymer and turbulence underlies the suression of Reynolds shear stress and drag reduction. Like total ressure-strain, the slow contribution (dashed curves in figure 9) transfers energy from u to v and w. In Newtonian shear-driven turbulence, it is slow ressure-strain-rate that is resonsible for maintaining vertical turbulent velocity fluctuations, and it is therefore slow ressure-strain that disroortionately shuts down through interaction between turbulence and olymer. The raid contribution to φ αα (dotted curves in figure 9), in contrast, is resonsible for maintaining sanwise turbulence fluctuations by transferring energy from u to w. The raid contribution to intercomonent energy transfer is less dramatically affected by olymer than is slow ressure-strain. (a) (b) Figure 0. Contributions to (a) nonolymeric ressurestrain-rate comonents and (b) total ressure-strain-rate comonents relative to the ressure-strain in the initial Newtonian state.

9 The olymer contribution to intercomonent energy transfer is negligible at low Weissenberg numbers, but takes on a otentially imortant role at high We τ. In figure 0 we lot the ratio of ressurestrain comonents with and without the olymer contribution to φ αα, to ressure-strain in the initial Newtonian state, as a function of increasing We τ. It is interesting to observe that the olymer shuts down comletely intercomonent energy transfer into v from the nonolymeric contributors to ressure-strain s r at high We τ (i.e., φ +, figure 0a), but total φ remains nonzero in this limit. We conclude that, at high Weissenberg numbers, the olymer contribution to ressure-strain-rate is resonsible for maintaining vertical velocity fluctuations through intercomonent energy transfer (from u and w ). POLYMER STRESS ANISOTROPY A consequence of the dramatic effect that olymer has on ressure-strain intercomonent energy redistribution is the enhancement of anisotroies within the turbulence. This is articularly true of the Reynolds stress, conformation and olymer stress tensors. The anisotroies in olymer stress become rogressively more imortant to the evolution of anisotroy within the turbulence as olymer effect, Weissenberg number or concentration, increases. In figure we lot the normal comonents (solid curves) to the mean olymer stress tensor, and we comare olymer and turbulent shear comonent (dashed curves). Note, in articular, the raid develoment of extremely high anisotroy in normal olymer stress comonents with increasing Weissenberg number. In articular, the streamwise olymer stress raidly dominates the stress tensor with increasing We τ. This dramatic anisotroy is couled to a strong tendency for mean olymer conformation to align with streamwise velocity, accentuated by the Peterlin function (eq. 3). It seems lausible that the extreme accentuation of the streamwise comonent of olymer stress may be related to the accentuation of streamwise, over vertical and sanwise velocity comonents. As Reynolds shear stress is suressed by increasing We τ in figure, olymer shear stress increases. Figure 0 indicates that the sum of Reynolds shear stress lus olymer shear stress initially dros with increasing Weissenberg number (imlying a dro in skin friction), but then increases again as τ dominates ρuv. SOURCE OF POLYMER STRETCH Polymer stress is generated by olymer stretch. We showed in figure that olymer stretch rogressively increases with Weissenberg number at fixed olymer concentration. A similar behavior can be anticiated with increasing olymer concentration. The source of olymer stretch is the stretching term in brackets in eq. (4). The budgets for average olymer stretch, eqs. (7), have contributions from mean and fluctuating strain-rates. Eqs. (7a) and (7d) shows that mean shear interacts with C to stretch olymer, on (a) Figure. Polymer stress comonents with increasing Weissenberg number. Normal stresses are given by solid curves, while shear stress comoents are given by dashes curves. Reynolds shear stress is shown for comarison with τ Figure. Terms in the olymer stretch budget, eq. (7d), with increasing Weissenberg number. χii, Λii, and ξii are the mean stretch, turbulent stretch and restoration terms, the three terms, resectively, on the RHS of eq. (7d). 9

10 average, in the streamwise direction. Fluctuating strain rate, in contrast, interacts with fluctuating conformation (Λ, eq. 8) to generate mean olymer stretch in all three comonent directions. It is of interest to learn which contribution dominates with increasing Weissenberg number. This is shown in figure where the terms in the budget for R are lotted as a function of We τ : χii, Λii, and ξii are the mean stretch, turbulent stretch and restoration terms, resectively in eq. (7d). We find that, initially, turbulent strain-rate fluctuations are resonsible for olymer stretch and for the initiation of olymerturbulence interactions that lead ultimately to drag reduction. However, as Weissenberg number increases, mean stretch raidly dominates turbulent stretch, in large art because turbulent kinetic energy is rogressively suressed with increasing Weissenberg number. We hyothesize that in wall bounded olymer-laden turbulence shear flow, the interlay between mean and turbulent olymer stretch may remain a critical factor in the high Weissenberg number limit, esecially in aroaching the maximum drag reduction asymtote (Virk 975). CONCLUSIONS Through direct numerical simulation of homogeneous shear flow with a high-accuracy algorithm for the rediction of the conformation tensor equation in the FENE-P model for dilute olymers, we have elucidated a number of statistical features of olymer-turbulence interactions that underlie the suression of momentum flux towards a surface in olymer-laden wall-bounded turbulent shear flows, and consequent drag reduction. The suression of vertical velocity fluctuations underlies Reynolds shear stress suression, however the interlay between the three contributions to ressure-strain-rate intercomonent energy transfer and olymer-turbulence energy exchange in this rocess are comlex, and these interactions change with increasing Weissenberg number. Polymer contributions to ressure-strain may lay a role in the maintenance of turbulence fluctuations in the high Weissenberg number limit. REFERENCES Bird, R. B., Armstrong, R. C., and Hassager, O. 987 Dynamics of Polymeric Liquids, vol., Wiley, New York, nd ed. Brasseur, JG, Robert, A, Collins, LR, Vaithianathan, T. 004 Energetics of Polymer-Turbulence Dynamics in DNS of Isotroic and Shear Turbulence. (abstract) Bull. Amer. Phys. Soc. 49(9): 4 Dimitrooulos, C.D., Sureshkumar, R., Beris, A.T. 998 Direct numerical simulation of viscoelastic turbulent channel flow exhibiting drag reduction: effect of the variation of rheological arameters. J. Non-Newtonian Fluid Mech. 7: -36. Dimitrooulos, C.D., Sureshkumar, R., Beris, A.T, Handler, R. A. 00 Budgets of Reynolds stress, kinetic energy and streamwise enstrohy in viscoelastic turbulent channel.flow. Phys. Fluids 3, Donohue, G.L., Tiederman, W.G., Reischmann, M.M. 97 Flow visualization of the near-wall region in a drag-reducing flow. J. Fluid Mech. 56: Dubief, Y., White, C.M., Terraon, V.E., Shaqfeh, E.S.G., Moin, P. & Lele, S.K. 004 On the coherent drag-reducing and turbulence-enhancing behaviour of olymers in wall flows, J. Fluid Mech. 54: Eckelman, L.D., Fortuna, G., Hanratty, T.J. 97 Drag reduction and the wavelength of flo-oriented wall eddies. Nature 36: Khanna, S., Smith, W.R., Brasseur, J.G., Smits, A.J., Sreenivasan, K.R. 99 Comarison of the inertial range of a high Reynolds number turbulent boundary layer with homogeneous shear flow. (abstract) Bull. Amer. Phys. Soc. 37(0): 739. (Discussed more comletely in PhD thesis of S Khanna, Penn State University). Kurganov, A., Tadmor, E. 000 New High-Resolution Central Schemes for Nonlinear Conservation Laws and Convection Diffusion Equations, J. Com. Physics 60: 4-8. Luchik, T.S., Tiederman, W.G. 988 Turbulent structure in low-concentration drag-0reducing channel flows. J. Fluid Mech. 98: Lumley, J.L. 973 Drag reduction in turbulent flow by olymer additives. J. Polymer Sci.: Macromolecular Reviews 7: McComb, W.D. 977 Effect of olymer additives on the small-scale structure of grid-generated turbulence. Phys. Fluids 0: McComb, W.D., Rabie, L.H. 979 Develoment of local turbulent drag reduction due to nonuniform olymer concentration. Phys. Fluids : McComb, W.D., Rabie, L.H. 98 Local drag reduction due to injection of olymer solutions into turbulent flow in a ie. AIChE J. 8: McNally, W.A. 968 Heat and momentum transort in dilute olyethylene oxide solutions, Ph.D. thesis, University of Rhode Island. 968 Min, T., Yoo, J. Y., Choi, H. & Joseh, D. D. 003 Drag reduction by olymer additives in a turbulent channel flow. J. Fluid Mech. 486:

11 Oldaker, D.K., Tiederman, W.G. 977 Satial structure of the viscous sublayer in drag-reducing channel flows. Phys. Fluids 0: S33-S44. Peterlin, A. 96 Streaming birefringence of soft linear macromolecules with finite chain length, Polymer : 57. Robert, A., Brasseur, JG, Vaithianathan, T, Collins, LR. 004 Fundamental Physics Underlying Momentum Flux Suression and Drag Reduction by Polymers. (abstract) Bull. Amer. Phys. Soc. 49(9): 4 (manuscrit in rearation). Rogallo, R. S. 98 Numerical exeriments in homogeneous turbulence, NASA Tech. Memo 835, NASA Ames Research Center, Moffett Field, California. Sangler, J.G. 969 Studies of viscous drag reduction with olymers including turbulence measurements and roughness effects. In C.S. Wells (Ed.), Viscous Drag Reduction: Sureshkumar, R., Beris, A.N. 995 Effect of artificial stress diffusivity on the stability of numerical calculations and the flow dynamics of timedeendent viscoelastic flows, J. Non-Newtonian Fluid Mech. 60: (995). Sureshkumar, R., Beris, A.N. 997 Direct numerical simulation of the turbulent channel flow of a olymer solution. Phys. Fluids 9: Tabor, M., DeGenne, P.G. 986 A cascade theory of drag reduction. Eurohys. Lett. : Vaithianathan, T., Collins, L.R. 003 Numerical aroach to simulating turbulent flow of a viscoelastic olymer solution. J Com. Physics 87: -. Vaithianathan, T., Collins, L.R., Robert, A., Brasseur, J.G. 004 A new algorithm for DNS of turbulent olymer solutions using the FENE-P model. (abstract) Bull. Amer. Phys. Soc. 49(9): 3. van Doorn, E., White, C.M., Sreenivasan, K.R. 999 The decay of grid turbulence in olymer and surfactant solution. Phys. Fluids : Virk, P.S. 975 Drag reduction fundamentals. AIChE J. : Virk, P.S. 97 Drag reduction in rough ies. J. Fluid Mech. 45: Voth, GA, Haller, G, Gollub, JP. 00 Exerimental measurement of stretching fields in fluid mixing. Phys. Rev. Letters 88: Warholic, M.D., Hassah, H., Hanratty, T.J. 999 Influence of drag-reducint olymers on turbulence: effects of Reynolds number, concentration and mixing. Exs in Fluids 7: Wei, T., Willmarth, W.W. 99 Modifying turbulent structure with drag-reducing olymer additives in thrbulent channel flows. J. Fluid Mech. 45: Wells, C.S., Sangler, J.G. 967 Injection of dragreducing fluid into turbulent ie flow of a Newtonian fluid, Phys. Fluids 0: 890. ACKNOWLEDGEMENTS This work was suorted under the DARPA Friction Drag Reduction Program.