Polymeric stresses wall vortices and drag reduction Ronald

Polymeric stresses, wall vortices and drag reduction Ronald J. Adrian Mechanical and Aerospace Engineering Arizona State University-Tempe “High Reynolds Number Turbulence”, Isaac Newton Institute, Sept. 8 -12, 2008

Co-workers Kim, K, Li, C. -F. , Sureshkumar, R. Balachandar, S. and Adrian, R. J. , “Effects of polymer stresses on eddy structures in drag-reduced turbulent channel flow, ” J. Fluid Mech. 584, 281 (2007). Kim, K, Adrian, R, Balachandar, S, Sureshkumar, R. , "Dynamics of Hairpin. Vortices and Polymer. Induced Turbulent Drag Reduction, " Phys. Rev. Lett. 100 (2008).

Toms’ Phenomenon • Toms discovered the phenomenon of turbulent drag reduction by polymer additives by chance in the summer of 1946, when he was actually investigating the mechanical degradation of polymer molecules using a simple pipe flow apparatus. • By dissolving a minute amount of long-chained polymer molecules in water, the frictional drag of turbulent flow could be reduced dramatically. In pipe flows, for example, the drag could be reduced up to 70 % by adding just a few parts per million (ppm) of polymer. Toms (1949) Proc. Intl. Congress on Rheology, Sec. II, p. 135 Toms (1977) Phys. Fluids *Address at the Banquet of the IUTAM Symposium on Structure of Turbulence and Drag Reduction

Main Features of Polymer DR • Onset of Drag Reduction – There exist critical values of parameters (e. g. polymer relaxation time, concentration. . ) above which there is onset of DR. – Lumley’s time criterion for onset of DR Polymer relaxation time Time scale of nearwall turbulence Stretched polymer After • Existence of Maximum Drag Reduction – Virk’s asymptote – Turbulence is still sustained in MDR limit. Coiled polymer

Structural changes found in experiments – – – Increased spacing and coarsening of streamwise streaks Damping of small spatial scales Reduced streamwise vorticity Enhanced streamwise velocity fluctuations Reduced vertical and spanwise velocity fluctuations and Reynolds stresses – Parallel shift of mean velocity profile in low DR – Increase in the slope of log-law in high DR

Governing Equations Continuity Eq. Viscous stress Polymer stress Momentum Eq. Constitutive Eq. q Reynolds number FENE-P model Weissenberg number

• • • Near-Wall Vortical Structures Vortical structures in polymer solutions are: Weaker Thicker Longer Fewer ci: Swirling strength

Conditional Averaged Flow Field • – Flow structures associated with the event which most contribute the Reynolds stress – Counter-rotating pair of quasi-streamwise vortex – Hairpin vortex

Polymer Work on Turbulent Energy • Turbulent energy equation (no summation on i) Polymer work

Conditional Averaged Flow Field • – Flow structures associated with the event contributing most to the polymer work – Nearly the same as those associated with large Q 2 event at similar y-locations Largest contribution on E x>0 Largest contribution on Ey<0 Largest contribution on E x<0 Largest contribution on Ez<0

Polymer Forces around Vortices • DR=18% – Polymer force inhibits the Q 2 pumping of the hairpin vortex (u, v) (w, v) (fx, fy) (fz, fy) Velocity Polymer force See also De Angelis et al. 2002, Dubief, et al. 2005, Stone, et al. 2002 (ECS laminar)

Polymer Counter-torque Strong streamwise polymer torques oppose the rotation of both legs of the primary hairpin vortex. Red and blue surfaces denotes a positive and negative polymer torques , respectively.

Polymer Counter-torque (cont) Large positive spanwise polymer torques act against rotation at the heads of downstream and secondary hairpin vortices. Negative torques are exerted on the primary vortex in a direction such that they reduce vortex curvature and thus the inclination angle of the primary hairpin head. Red and blue surfaces denote positive and negative polymer torques, respectively.

Polymer Torque • Two-point correlation between streamwise vorticity and polymer torque DR=18% Colored contour Line contours

Axisymmetric Vortex z

Model vortex (axisymmetric) • Burgers ”-like” vortex No strain field (simplify problem)

Configuration tensors around an axisymmetric vortex • Substitution of velocity field into the constitutive eqns. gives – Assuming axi-symmetry … Oldroyd-B model … No azimuthal variation, so no azimuthal force FENE-P model

Polymer forces and torque • -direction polymer force • Polymer torque in z-direction

Polymer torque and axisymmetry LSE of quasi-streamwise vortex at y+=20 Polymer torque Velocity around QSV Symbols: LSE results Line: vortex model with =0. 058 & b=11 QSV axisymmetric vortex (d/dtheta=0) Polymer torque

Viscoelastic Drag Reduction Principle • Drag is reduced by intrinsic viscoelastic counter-torques that retard the rotation of turbulent vortices • Counter-torques exist around the vortices only if the flow is nonaxi-symmetric • Deviations from axi-symmetry occur when the vortex is embedded in a strain field, e. g. – Quasi-streamwise wall vortices imbedded in the strain field of its image vortex – Bent vortices, i. e. heads of hairpins

Viscoelastic counter-torques and axisymmetry • Axisymmetric Burger’s vortex generates zero azimuthal net force, and hence zero counter torque. • Quasi-streamwise vorticies near the wall are not axi-symmetric, so a net torque can be developed. • The core of the vortex in the head region is not axisymmetric because the flow is faster under the arch of the head than above it. Hence non-zero counter torque also occurs around the arch.

Conclusions • In fully turbulent flow polymer forces are associated with the Q 2 pumping of the hairpin vortex and the ejection/ sweep motions at the flanks of streamwise vortices in a that opposes the motion. • They apply counter-torques to the rotation of the vortices, Within the validity of the FENE-P model, this is the fundamental mechanism for reducing turbulent stresses and drag.

Evolution of initial vortical structures The initial structure is the conditionally averaged flow field with Q 2 event vector, (um, vm, 0) of strength =2. 0 specified at ym+=50, where um and vm are selected as the most contributing Q 2 event to ttthe mean Reynolds shear stress. Newtonian flow DR=18% flow DR=61% flow

Threshold for the auto-generation Low DR flow Newtonian flow In low DR flow, the threshold kinetic energy for the generation of secondary vortices increases, especially in the buffer layer. For the high-DR simulations we did not observe autogeneration for any of the various initial conditions tested. Autogeneration occurs Autogeneration fails

Effects of polymer stress on autogeneration To see suppression of the auto-generation by the polymer stresses more directly, we compared the evolution in the absence of the polymer stress from the same initial velocity fields as one of the LDR simulations. Reynolds shear stress more rapidly increases in the absence of the polymer stress.

2 nd Simulation • 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-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.

Evolutions of initial vortical structure

Growth rate of volume-averaged Reynolds shear stress

Effects of Weissenberg No. Asymptotic behavior Onset of reduction on Reynolds shear stress These behaviors are consistent with the onset of DR and the existence of maximum DR limit in the fully turbulent polymer DR flows, respectively.

Conclusions • Polymers cut-off the autogeneration of hairpin eddies, thereby – reducing the number of vortices – inhibiting drag by reducing the coherent stress associated with hairpin packets. Kim, Adrian Balachandar and Sureshkumar, PRL (2008) • Future Work Large-scale and very-large scale motions account for over half of the Reynolds shear stress in Newtonian flow. How do polymers influence them?