Dynamics of Hairpin Vortices and Polymer-Induced Turbulent Drag ...

PRL 100, 134504 (2008)

PHYSICAL REVIEW LETTERS

week ending 4 APRIL 2008

Dynamics of Hairpin Vortices and Polymer-Induced Turbulent Drag Reduction Kyoungyoun Kim,1 Ronald J. Adrian,1,* S. Balachandar,2 and R. Sureshkumar3,† 1

Department of Mechanical and Aerospace Engineering, Arizona State University, Tempe, Arizona 85287, USA Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, Florida 32611, USA 3 Department of Energy, Environmental and Chemical Engineering, The Center for Materials Innovation, Washington University, Saint Louis, Missouri 63130, USA (Received 26 September 2007; published 2 April 2008) 2

It has been known for over six decades that the dissolution of minute amounts of high molecular weight polymers in wall-bounded turbulent flows results in a dramatic reduction in turbulent skin friction by up to 70%. First principles simulations of turbulent flow of model polymer solutions can predict the drag reduction (DR) phenomenon. However, the essential dynamical interactions between the coherent structures present in turbulent flows and polymer conformation field that lead to DR are poorly understood. We examine this connection via dynamical simulations that track the evolution of hairpin vortices, i.e., counter-rotating pairs of quasistreamwise vortices whose nonlinear autogeneration and growth, decay and breakup are centrally important to turbulence stress production. The results show that the autogeneration of new vortices is suppressed by the polymer stresses, thereby decreasing the turbulent drag. DOI: 10.1103/PhysRevLett.100.134504

PACS numbers: 47.50.d, 47.57.Ng, 47.85.Dh, 83.60.Yz

It is known that turbulent friction drag can be reduced by up to 70% in aqueous and organic liquids by the dissolution of minute amounts (10 –50 ppm) of high molecular weight polymers to wall-bounded flows [1]. While the technological relevance of polymer drag reduction (DR) cannot be over-emphasized (e.g., it is used to save billions of dollars in energy costs associated with intercontinental crude oil transportation), the complex physics of how macroscopic flow and nanoscale polymers interact to reduce turbulent shear (Reynolds) stress has motivated over 60 years of fundamental research. Although the mechanisms of flowpolymer stress coupling have been deciphered experimentally in inertialess nonlinear flows [2], they remained largely unknown in turbulent flows until faithful direct numerical simulations (DNS) were performed in the midnineties for polymer DR in channel flows. DNS studies showed, consistently with experimental observations, that the decrease of the turbulent Reynolds shear stress is accompanied by the weakening of near-wall vortices. Further, they underscored a direct correlation between DR and the enhancement of the extensional viscosity, a measure of the fluid’s ability to resist elongational deformation [3,4]. More recent DNS studies [5,6] have revealed that the body forces due to polymeric stresses oppose the vortical motions of quasistreamwise vortices (QSV) in the buffer layer, i.e., the intermediate layer that mediates momentum exchange between the near wall and core fluid in channel or pipe flow. These findings from DNS are corroborated by the analysis of the effect of elasticity on Newtonian coherent structures. For instance, Roy et al. [7] suggests that the self-sustaining process of wall turbulence [8] becomes weaker due to the polymer forces opposing both biaxial and uniaxial extensional flow regions around QSV. Similarly, exact coherent states (ECS), which are closely related to the buffer layer turbulence [9], can be 0031-9007=08=100(13)=134504(4)

suppressed entirely by polymer forces if elastic effects are sufficiently large [10]. Above the buffer layer, hairpin vortices are more prevalent than QSV, and they are responsible for the production of the Reynolds shear stress in the log layer where the mean velocity has a logarithmic distribution. In the polymer drag-reduced flows, the hairpins in the log layer are also weakened by polymer counter torques [11], as is the case for QSV in the buffer layer. In addition, some portion of the total DR is attributed to the reduction in the number of energetic vortices with increasing elasticity, as deciphered by extensive flow visualizations. In the Newtonian flow, hairpins can autogenerate to form a ‘‘hairpin packet’’ which is an important feature of wall-bounded turbulence that has been shown to explain many experimental measurements such as inordinately large amount of streamwise kinetic energy in very long streamwise wave length, the occurrence of multiple ejection events in turbulent bursts, formation of new QSV, and characteristic angles of inclined hairpin vortices [12]. Further, the hairpin vortex packet makes a significant contribution to the mean Reynolds shear stress [13], which has two parts: the coherent Reynolds stress caused by nonlinear interactions among individual vortices within the packet and incoherent stress originated from velocity fluctuations induced by each individual vortex [12]. Despite the insights gained from the above studies, the generation and evolution of the turbulent vortices in the presence of a simultaneously evolving polymer conformation or stress field and how they contribute to DR is poorly understood. In this Letter, we portray, for the first time, the nonlinear autogeneration of new vortices and formation of hairpin packets in the presence of polymer stress by performing a series of computationally demanding dynamical simulations and explain the effect of such dy-

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© 2008 The American Physical Society

PRL 100, 134504 (2008)


namics on the reduction in turbulent stresses and hence, DR. In the dynamical simulations, an initially isolated vortical structure is evolved in the viscoelastic flow where the polymer stress is modeled by the FENE-P model (finitely extensible nonlinear elastic –Peterlin) [14] which has been extensively used in successful DNS- and ECS-based studies of polymer-induced DR [3,7,9]. Note that Roy et al. [7] used the Oldroyd-B model whose rheological predictions are identical to that of the FENE-P model in the limit of infinitely large extensibility. The nondimensional governing equations of unsteady, incompressible, viscoelastic flow with the FENE-P model are given by r u 0; 2 1 @u u ru rp r u r ; Re Re @t

(1) (2)

@c u rc c ru ruT c ; (3) @t where u is the velocity, p is the pressure, and fL2 3=L2 Trcgc I=We =Re is the polymer stress. p The friction velocity u w = and the channel halfheight h are used as the velocity and length scale, respectively. Here, w is the wall shear stress, and is fluid density. The Reynolds number is defined as Re u h=o , where o is the zero shear-rate kinematic viscosity of the solution. The parameter is the ratio of the solvent viscosity (s ) to the total solution zero-shear-rate viscosity (o ). The polymer stress is obtained by solving an evolution equation for the conformation tensor c hqqi, which is the average second moment of the polymer chain end-to-end distance vector (q). Note that jqj < L since q has a maximum extensibility L. The Weissenberg number We u2 =o is the ratio of the polymer relaxation time , to the flow time scale based on the friction velocity. The initial conditions in the dynamical simulations were imposed as conditional averaged flow fields associated with Reynolds stress producing events from the DNS data of fully developed turbulent channel flow at Re 395 with drag reduction of 0%, 18%, and 61% [4]. We use linear stochastic estimation (LSE) [11,15,16] to approximate the conditional flow fields, ux; t 0 huxju0 ym u0Q2 ym i and cx; t 0 hcxju0 ym u0Q2 ym i. The velocity event u0Q2 is chosen to maximize the product of instantaneous Reynolds stress and the probability of occurrence of the second-quadrant (Q2) event (u0 < 0, v0 > 0). The velocity event is the strongest contributor to the mean Reynolds shear stress in the Q2 quadrant, and the conditional average is known to be the best estimate of the flow field surround the event, in the least mean square sense [15]. The amplification factor , referred to here as a strength of the Q2 event, is varied from 1 to 3. The structures extracted by LSE for the Q2 events are counter-rotating pairs of streamwise vortices or hairpin vortices, depending on the event


location (ym ). They become bigger in size and much elongated in the streamwise direction as DR increases [11]. The two viscoelastic flow cases, namely, those exhibiting 18% and 61% DR, are representative of the low- and high- DR regimes. In channel or pipe flow experiments performed under identical pressure drop, the low-DR regime is identified by an upward shift in mean velocity in the turbulent core without any change in the slope compared to the Newtonian fluid while in the high-DR regime the slope of the velocity profile increases from its Newtonian value [1]. Dynamical simulations are performed in these two regimes to assess the robustness of the physical mechanisms with respect to the extent of DR. Time-integration of the governing equations is achieved by a semi-implicit method. For spatial derivatives, a spectral method is used with Fourier representations in the streamwise and spanwise directions, and Chebyshev expansion in the wall-normal direction. Periodic boundary conditions are applied in the streamwise and spanwise directions, and the no-slip boundary condition is imposed on velocity at the solid walls. For the Newtonian simulation, the domain size is 2h 2h h in the streamwise, wall-normal, and spanwise directions, respectively. For the DR flows, a larger streamwise domain is required, since drag reduction is accompanied by an increase in streamwise correlation length. The domain size for low- and highDR simulations are 4h 2h 0:5h and 8h 2h

0:5h, respectively. After tests of several grid resolutions, 128 128 192, 256 128 96, and 512 128 96, spectral modes are used for the Newtonian, low-, and highDR simulations, respectively, which results in x 19:4, y 0:12 9:69, and z 6:46. Figure 1 shows the evolution of initial vortical structures extracted for the Q2 event of 2 specified at y m 51:3. The rheological parameters are We 25 and L2 900 for the low-DR case and We 100 and L2 14400 for the high-DR case, respectively. The same value of 0:9 is used for both cases. The parameters used in the dynamical simulations are identical with those in the fully turbulent DNS [4,17]. The formation process of new hairpins in the Newtonian simulation [Fig. 1(a)] is very similar to Zhou et al. [16]. The initial vortical structure forms a hairpinlike vortex, which is changed to the -shaped vortex (t 158) by the self-induced motion toward the bi-normal direction due to the local effect of the curved vortex line. The secondary hairpin vortex (t 237) is generated upstream of the primary hairpin, and new QSVs are also generated at the outer flanks of the primary hairpin vortex legs. In the low-DR simulation [Fig. 1(b)], the autogeneration also resembles that of the Newtonian simulation. However, the details of vortex formation are slightly different. At t 158, the downstream tongue of the -shaped vortex is much more elongated, almost 2.5 times longer than the Newtonian one. At t 237, the angle that the vortex packet envelop makes with the x axis reduces from 14 in the Newtonian flow to 10 in the low-DR simulation. The streamwise span of the packet, approxi-

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PRL 100, 134504 (2008)


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FIG. 1. Time evolution of initial vortical structure extracted by Q2 event vector of strength 2 specified at y m 51:3. The three-dimensional structures represent isosurfaces of the vortex swirling strength, ci , defined as the imaginary part of the complex eigenvalue of the velocity gradient tensor (ru). cf. Ref. [16] for more detailed discussion. The distance between ticks on each axis is 200 wall units. Movies showing the evolution of the initial vortex can be viewed in the files MOV1.mov, MOV2.mov, and MOV3.mov [20].

mately 1300 viscous lengths, is larger than that of the Newtonian packet, 900=u . These are consistent with the previous experimental observations from polymer drag-reduced flows such as increase in streamwise length scale [18] and enhanced parallel large-scale motions to the mean flow [19]. In the high-DR simulation [Fig. 1(c)], the initial vortex is elongated in the streamwise direction, but neither a primary nor a secondary vortex occurs. The swirling strength of vortical structures decreases as time goes by, and the initial vortex finally dies out. Note that the evolution of the initial vortex was simulated over a time period t 395 which is longer than several relaxation times of the polymer to ensure that polymer-vortex interactions are tracked for a sufficiently long time. Zhou et al. [16] observed that the autogeneration process is possible only if initial structures are sufficiently strong. To examine the threshold for autogeneration, many initial vortices were tested. The generation of secondary hairpin vortices depends on the strength and location y m of the Q2 event vector used to extract the initial vortical structures. In Fig. 2, we use the volume-averaged kinetic energy of the fluctuating velocities instead of to measure a magnitude of the initial perturbation to trigger the autogeneration. In the Newtonian flow simulations [Fig. 2(a)], an optimum distance of the initial vortex from the wall for the autogeneration occurs in the buffer layer, consistently with lower Reynolds number simulation of Re 180 [16]. The existence of the optimal location was explained as a consequence of a balance between lifting-up and backward curling of vortices due to vortex induced motion of QSV legs and vortex stretching due to the mean shear. In the low-DR simulations [Fig. 2(b)], the optimal location is also found in the buffer layer. However, the threshold kinetic energy for the generation of secondary vortices increases, especially in the buffer layer. For the high-DR

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FIG. 2. The generation of secondary hairpin vortices depends on the strength of initial vortical structures and location of the event vector used to extract the initial vortical structure. : case with new hairpins. : case without new hairpin. (a) Newtonian simulation (b) low-DR simulation (We 25, L2 900, and 0:9).

simulations, we did not observe autogeneration for any of the various initial conditions tested. The event strength for the initial condition is restricted because the linear estimation does not ensure the constraint 0 < Trc < L2 for large . However, we deduce that the threshold in the high-DR simulation must be much larger than in the Newtonian and low-DR cases since the volume-averaged kinetic energy corresponding to the largest available in the high-DR simulations, about 1.0, is much larger than thresholds of the Newtonian and low-DR cases. The increase of threshold kinetic energy means that in drag-reduced flow, stronger initial perturbation is required to generate new vortices as compared to that in the Newtonian flow. While the previous ECS-based studies [7,9] reported that at high We , the polymer stresses suppress the instability of the streaks that regenerate the streamwise vortices, the present study shows autogeneration from a primary hairpin is also suppressed by the polymer stress. Since the autogeneration mechanism is necessary for hairpin packet formation in wall turbulence [16], its suppression suggests that the formation of hairpin packets is inhibited in DR flow, consistent with experimental observation based on particle image velocimetry data of less frequent large-scale eruption of low momentum fluids from the wall in polymer solutions [19]. In the vorticity transport equations derived by taking curl of Eq. (2), the polymeric contribution on the evolution of vorticity appears as T r 1 =Re r , i.e., torque due to the polymer stress. We observe that during the evolution of the initial vortex, the polymer torques reduce the vortex strength by opposing the vortical motions, consistently with the counter polymer torque found in fully turbulent DNS results [11]. Since the autogeneration occurs only when parent vortex strength exceeds a threshold value, the counter torques that weaken the primary vortex lead to suppression of the autogeneration in DR flows. In the dynamical simulations presented so far, the polymers were initially stretched or compressed according to the straining of the conditionally averaged velocity field extracted from a turbulent flow that was already drag-

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creasing the coherent as well as the incoherent Reynolds stress. The authors acknowledge support from Dr. R. Joslin’s program at the Office of Naval Research through Contract No. N000140510687. K. K. would like to acknowledge the financial support by KRF Grant (No. KRF-2005-214-M012005-000-10056-00). R. S. would like to acknowledge NSF CTS Grant No. 0335348.

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FIG. 3. Volume-averaged j u0 v0 j normalized by its initial value during evolution of the initial structure extracted by Q2 event vector of strength 3 specified at y m 51:3 in the Newtonian flow. (a) temporal evolution (b) effects of the Weissenberg number We at t 300.

reduced. The behavior we have found does not necessarily explain the mechanisms that lead up to the occurrence of drag reduction. To determine how polymer stresses act to modify turbulence in Newtonian fluids, we imagine creating a fully turbulent flow without polymers and then abruptly turning the polymer stresses on. For this simulation, the initial velocity field is extracted by LSE given a Q2 event vector of 3 specified at y m 51:3 in the Newtonian flow. The initial condition for the polymer conformation is cx; t 0 I, modelling a condition in which coiled polymers are suddenly introduced into the flow homogeneously. The simulation parameters are: L2 14400, 0:9, and 10 We 200. Figure 3(a) shows the timeevolution of volume-averaged Reynolds shear stress normalized by its initial value. The growth rate of the volumeaveraged Reynolds shear stress decreases as We increases due to the inhibited autogeneration mechanism and weakened vortices (see the file FIG1.jpg in Ref. [20]). In Fig. 3(b), the volume-averaged Reynolds shear stresses at t 300 are plotted as a function of We . At low We ( 10), there is only a small difference between the viscoelastic and the Newtonian simulation whereas the Reynolds shear stress dramatically decreases for We > 10. An asymptotic behavior is observed for high We ( > 100). These behaviors are consistent with the onset of DR and the existence of maximum DR limit in the fully turbulent polymer DR flows, respectively. In summary, a series of dynamical simulations from various initial vortical structures has been performed to examine interactions between the coherent structures and polymer conformation field. We find that the nonlinear threshold of initial vortex strength for the autogeneration of new hairpins increases as the viscoelasticity increases, especially in the buffer layer. As found in the fully turbulent DR flows [11], the counter polymer torque reduces the vortex strength by opposing the vortical motions, which results in the suppression of the autogeneration of new vortices and vortex packet formation in DR flows. Therefore, turbulent DR is caused by polymer stress de-

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