Skin-Friction Drag Reduction by Dilute Polymer Solutions in Turbulent Channel Flow

Transcription

1 Skin-Friction Drag Reduction by Dilute Polymer Solutions in Turbulent Channel Flow by Dong-Hyun Lee A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Mechanical Engineering) in The University of Michigan 21 Doctoral Committee: Associate Professor Rayhaneh Akhavan, Chair Professor Steven J. Wright Associate Professor Jeremy D. Semrau Assistant Professor Jianping Fu

2 Dong-Hyun Lee c 21 All rights reserved.

3 To my wife, daughter and parents ii

4 Acknowledgements I express my sincere gratitude and respect to my advisor, Professor Rayhaneh Akhavan for her great inspiration and gentle guidance during my doctoral study at the University of Michigan, Ann Arbor. iii

5 Table of Contents Dedication Acknowledgements List of Figures ii iii vii List of Tables xxi List of Symbols xxii Abstract xxx Chapter I. Introduction Early experimental studies Onset of drag reduction Saturation of drag reduction Maximum drag reduction Recent experimental studies Homogeneous polymer solutions Heterogeneous polymer solutions Numerical studies Theoretical studies Elongational viscosity theory (Lumley, 1969, 1973) Elastic theory (de Gennes, 1986) Comparison of the two theories to experimental and DNS data The structural view Objectives of the present study II. Governing Equations and Numerical Methods Governing equations iv

6 2.2 Numerical methods Hydrodynamics Polymer dynamics Implementation on parallel computers Simulation parameters Verification of the numerical methods Convergence studies Effect of interpolation scheme and comparison to conventional Eulerian schemes Effect of domain size Assessment of the mixed Eulerian-Lagrangian scheme III. Flow statistics Effect of polymer concentration on the flow statistics Effect of Weissenberg number on the flow statistics Effect of polymer extensibility parameter on the flow statistics Effect of averaging time on the flow statistics at MDR IV. Scaling of Drag Reduction with Polymer and Flow Parameters1 4.1 Onset of drag reduction de Gennes s theory of polymer drag reduction revisited Estimation of r Estimation of r Onset criteria based on the revised de Gennes s theory Comparison to DNS data Saturation of drag reduction Criteria for maximum drag reduction (MDR) Range of affected scales V. Mechanism of Drag Reduction Data analysis Polymer and turbulence energetics Anisotropy-invariant maps Comparison to predictions of classical theories Some key features of polymer drag reduction Minimal exchange of energy between turbulence and polymer Highly anisotropic structure of turbulence in dragreduced flow Comparison to classical theories of polymer drag reduction Role of pressure-strain in establishing anisotropy Effect of polymer concentration Summary of the mechanism of drag reduction v

7 VI. Summary and Conclusions Bibliography vi

8 List of Figures Figure 2.1 The channel geometry and coordinate system The backward tracking of particles:, particle positions, r (n) P, at t n;, particle positions, r G, at t n The parallelization scheme used in the simulations. The data is distributed among processors in sets of planes along y-direction in the physical-space and in sets of planes along the x-direction in the spectral-space (a) The time per iteration and (b) speedup in DNS of turbulent channel flow performed on IBM e135 (Big Red) with the number of processors increased from 1 to 128., Newtonian flow with resolution;, Newtonian flow with resolution;, viscoelastic flow with resolution;, viscoelastic flow with resolution;, ideal linear speedup Effect of mesh size at We τb 35, n p k B T/(ρu 2 τ b ) 1 1 3, b = 45, on (a) time evolution of the skin friction coefficient; (b) mean velocity profiles; (c) r.m.s. of velocity fluctuations; (d) Reynolds ( τ R,xz + ) and polymer ( τ p,xz + ) shear stresses., Newtonian (case N);, resolution in 8π 27 h 8π 5 h 2h channel;, resolution in 8π 27 h 8π 8π 5 h 2h channel;, resolution in 27 h 8π 5 h 2h channel Effect oftime-step sizeatwe τb 35, n p k B T/(ρu 2 τ b ) 1 1 3, b = 45, on (a) time evolution of the skin friction coefficient; (b) mean velocity profiles; (c) r.m.s. of velocity fluctuations; (d) Reynolds ( τ R,xz + ) and polymer ( τ p,xz + ) shear stresses., Newtonian (case N);, t =.1 x/u o ;, t =.5 x/u o vii

9 2.7 Effect of numerical scheme on the predicted flow statistics at We τb 35: (a) time evolution of the skin friction coefficient; (b) mean velocity profiles; (c) turbulence intensities; (d) Reynolds shear stresses and polymer stresses., Newtonian (case N);, mixed Eulerian-Lagrangian scheme with linear interpolation (case C3-l);, mixed Eulerian-Lagrangian scheme with quadratic interpolation (case C3-q);, Eulerian scheme (case C3-e) Effect of numerical scheme on the predicted flow statistics at We τb 1: (a) time evolution of the skin friction coefficient; (b) mean velocity profiles; (c) turbulence intensities; (d) Reynolds shear stresses and polymer stresses., Newtonian (case N);, mixed Eulerian-Lagrangian scheme with linear interpolation (case E3-l);, mixed Eulerian-Lagrangian scheme with quadratic interpolation (case E3-q);, Eulerian scheme (case E3-e) Effect of numerical scheme on the predicted one-dimensional energy spectra at (a-b) We τb 35 and (c-d) We τb 1: (a), (c) streamwise spectra at 3; (b), (d) spanwise spectra at 3. (a-b) Line types same as in figure 2.7. (c-d) Line types same as in figure Effect ofdomainsizeonthepredictedflowstatistics atwe τb 35: (a)time evolution of the skin friction coefficient; (b) mean velocity profiles; (c) turbulence intensities; (d) Reynolds shear stresses and polymer stresses., Newtonian in small domain (case N); ---, Newtonian in large domain (case NN);, mixed Eulerian-Lagrangian scheme with quadratic interpolation in small domain (case C3-q);, mixed Eulerian-Lagrangian scheme with quadratic interpolation in large domain (case CC3-q) Effect of domain size on the predicted flow statistics at We τb 1: (a) time evolution of the skin friction coefficient; (b) mean velocity profiles; (c) turbulence intensities; (d) Reynolds shear stresses and polymer stresses., Newtonian in small domain (case N); ---, Newtonian in large domain (case NN);, mixed Eulerian-Lagrangian scheme with quadratic interpolation in small domain(case E3-q);, mixed Eulerian-Lagrangian scheme with quadratic interpolation in large domain (case EE3-q) Effect of domain size on the predicted flow statistics at We τb 15: (a) time evolution of the skin friction coefficient; (b) mean velocity profiles; (c) turbulence intensities; (d) Reynolds shear stresses and polymer stresses., Newtonian in small domain (case N); ---, Newtonian in large domain (case NN);, mixed Eulerian-Lagrangian scheme with quadratic interpolation in small domain(case F3-q);, mixed Eulerian-Lagrangian scheme with quadratic interpolation in large domain (case FF3-q) viii

10 2.13 Effect of domain size on the predicted one-dimensional energy spectra in (a-b) Newtonian flow and (c-d) viscoelastic flow at We τb 35: (a), (c) streamwise spectra at 3; (b), (d) spanwise spectra at 3. Line types same as in figure Effect of domain size on the predicted one-dimensional energy spectra in (a-b) viscoelastic flow at We τb 1 and (c-d) We τb 15: (a), (c) streamwise spectra at 3; (b), (d) spanwise spectra at 3. (ab) Line types same as in figure (c-d) Line types same as in figure Effect of domain size on the predicted two-point correlations in (a-b) Newtonian flow and (c-d) viscoelastic flow at We τb 35: (a), (c) streamwise correlations at 3; (b), (d) spanwise correlations 3., R uu in small domain;, R uu in large domain;, R vv in small domain;, R vv in large domain;, R ww in small domain;, R ww in large domain Effect of domain size on the predicted two-point correlations in (a-b) viscoelastic flow at We τb 1 and (c-d) We τb 15: (a), (c) streamwise correlations at 3; (b), (d) spanwise correlations 3., R uu in small domain;, R uu in large domain;, R vv in small domain;, R vv in large domain;, R ww in small domain;, R ww in large domain Effect of polymer concentration on the predicted flow statistics at We τb 1: (a) time evolution of the skin friction coefficient; (b) mean velocity profiles; (c) r.m.s. of streamwise and spanwise velocity fluctuations; (d) r.m.s. of wall-normal velocity fluctuations., Newtonian (case N); ---, β (case A6-l);, β (case A5-l);, β.9989 (case A4-l);, β.989 (case A3-l);, β.9 (case A2-l);, experiments of Warholic, et al.(1999) Newtonian Effect of polymer concentration on the predicted flow statistics at We τb 1: (a) Reynolds shear stresses ( τ R,xz + ), viscous shear stresses ( τ v + ), and sum of Reynolds, viscous and polymer shear stresses ( τ t + ); (b) polymer shear stresses ( τ p,xz + ) and polymer extensions; (c) effective viscosities normalized with respect to ν s ; (d) effective viscosities normalized with respect to λu 2 τ. Line types same as in figure ix

11 3.3 Effect of polymer concentration on the predicted flow statistics at We τb 35: (a) time evolution of the skin friction coefficient; (b) mean velocity profiles; (c) r.m.s. of streamwise and spanwise velocity fluctuations; (d) r.m.s. of wall-normal velocity fluctuations., Newtonian(case N);---, β (case C7-l);, β.9996 (case C5-l);, β.996 (case C4-l);, β.98 (case C4.1-l);, β.96 (case C3-l);, β.89 (case C3.1-l);, β.8 (case C3.2-l);, β.72 (case C2- l);, experiments of Warholic, et al.(1999) Newtonian;, experiments of Warholic, et al.(1999) at 14%DR;, experiments of Warholic, et al.(1999) at 33%DR Effect of polymer concentration on the predicted flow statistics at We τb 35: (a) Reynolds shear stresses ( τ R,xz + ), viscous shear stresses ( τ v + ), and sum of Reynolds, viscous and polymer shear stresses ( τ t + ); (b) polymer shear stresses ( τ p,xz + ) and polymer extensions; (c) effective viscosities normalized with respect to ν s ; (d) effective viscosities normalized with respect to λu 2 τ. Line types same as in figure Effect of polymer concentration on the predicted flow statistics at We τb 1: (a) time evolution of the skin friction coefficient; (b) mean velocity profiles; (c) r.m.s. of streamwise and spanwise velocity fluctuations; (d) r.m.s. of wall-normal velocity fluctuations., Newtonian (case NN); ---, β (case E8-q);, β (case E6-q);, β.9989 (case E5-q);, β.98 (case EE4-q);, β.9 (case EE3- q);, experiments of Warholic et al.(1999) Newtonian;, experiments of Warholic et al.(1999) at 14%DR;, experiments of Warholic et al.(1999) at 33%DR;, experiments of Ptasinski et al.(21) at 63%DR Effect of polymer concentration on the predicted flow statistics at We τb 1: (a) Reynolds shear stresses ( τ R,xz + ), viscous shear stresses ( τ v + ), and sum of Reynolds, viscous and polymer shear stresses ( τ t + ); (b) polymer shear stresses ( τ p,xz + ) and polymer extensions; (c) effective viscosities normalized with respect to ν s ; (d) effective viscosities normalized with respect to λu 2 τ. Line types same as in figure Effect of polymer concentration on the predicted flow statistics at We τb 15: (a) time evolution of the skin friction coefficient; (b) mean velocity profiles; (c) r.m.s. of streamwise and spanwise velocity fluctuations; (d) r.m.s. of wall-normal velocity fluctuations., Newtonian (case NN); ---, β (case F8-q);, β.9998 (case F6-q);, β.998 (case FF5-q);, β.98 (case FF4-q);, β.86 (case FF3-q);, experiments of Warholic, et al.(1999) Newtonian;, experiments of Warholic, et al.(1999) at 14%DR;, experiments of Warholic, et al.(1999) at 38%DR;, experiments of Ptasinski, et al.(21) at 65%DR;, experiments of Ptasinski, et al.(21) at 7%DR x

12 3.8 Effect of polymer concentration on the predicted flow statistics at We τb 15: (a) Reynolds shear stresses ( τ R,xz + ), viscous shear stresses ( τ v + ), and sum of Reynolds, viscous and polymer shear stresses ( τ t + ); (b) polymer shear stresses ( τ p,xz + ) and polymer extensions; (c) effective viscosities normalized with respect to ν s ; (d) effective viscosities normalized with respect to λu 2 τ. Line types same as in figure Effect of polymer concentration on the one-dimensional energy spectra at We τb 35: (a-c) streamwise spectra at 3; (d-f) spanwise spectra at 3. Line types same as in figure Effect of polymer concentration on the one-dimensional energy spectra at We τb 1: (a-c) streamwise spectra at 3; (d-f) spanwise spectra at 3. Line types same as in figure Effect of polymer concentration on the one-dimensional energy spectra at We τb 15: (a-c) streamwise spectra at 3; (d-f) spanwise spectra at 3. Line types same as in figure Effect of polymer concentration on the one-dimensional energy spectra at We τb 35: (a-c) streamwise spectra at 1; (d-f) spanwise spectra at 1. Line types same as in figure Effect of polymer concentration on the one-dimensional energy spectra at We τb 1: (a-c) streamwise spectra at 1; (d-f) spanwise spectra at 1. Line types same as in figure Effect of polymer concentration on the one-dimensional energy spectra at We τb 15: (a-c) streamwise spectra at 1; (d-f) spanwise spectra at 1. Line types same as in figure Effect of Weissenberg number on the predicted flow statistics at 1 We τb 15 and n p k B T/(ρu 2 τ b ) : (a) time evolution of the skin friction coefficient; (b) mean velocity profiles; (c) r.m.s. of streamwise and spanwise velocity fluctuations; (d) r.m.s. of wall-normal velocity fluctuations., Newtonian (case NN);, We τb 1 (case A3-l);, We τb 2 (case B3-l);, We τb 35 (case C3-l);, We τb 7 (case DD3-q);, We τb 1 (case EE3-q);, We τb 15 (case FF3-q);, experiments of Warholic, et al.(1999) Newtonian;, experiments of Warholic, et al.(1999) at 14%DR;, experiments of Warholic, et al.(1999) at 33%DR);, experiments of Ptasinski, et al.(21) at 63%DR;, experiments of Ptasinski, et al.(21) at 65%DR xi

13 3.16 Effect of Weissenberg number on the predicted flow statistics at 1 We τb 15 and n p k B T/(ρu 2 τ b ) : (a) Reynolds shear stresses ( τ R,xz + ), viscous shear stresses ( τ v + ), and sum of Reynolds, viscous and polymer shear stresses ( τ t + ); (b) polymer shear stresses ( τ p,xz + ) and polymer extensions; (c) effective viscosities normalized with respect to ν s ; (d) effective viscosities normalized with respect to λu 2 τ. Line types same as in figure Effect of Weissenberg number on the one-dimensional energy spectra at 1 We τb 15 and n p k B T/(ρu 2 τ b ) : (a-c) streamwise spectra at 3; (d-f) spanwise spectra at 3. Line types same as in figure Effect of Weissenberg number on the one-dimensional energy spectra at 1 We τb 15 and n p k B T/(ρu 2 τ b ) : (a-c) streamwise spectra at 1; (d-f) spanwise spectra at 1. Line types same as in figure Effect of extensibility parameter on the predicted flow statistics at We τb 35 and n p k B T/(ρu 2 τ b ) : (a) time evolution of the skin friction coefficient; (b) mean velocity profiles; (c) r.m.s. of streamwise and spanwise velocity fluctuations; (d) r.m.s. of wall-normal velocity fluctuations., Newtonian (case N);, b = 4,5 (case C3.3-l);, b = 45, (case C3-l);, b = 45, (case C3.4-l) Effect of extensibility parameter on the predicted flow statistics at We τb 35 and n p k B T/(ρu 2 τ b ) : (a) Reynolds shear stresses ( τ R,xz + ), viscous shear stresses ( τ v + ), and sum of Reynolds, viscous and polymer shear stresses ( τ t + ); (b) polymer shear stresses ( τ p,xz + ) and polymer extensions; (c) effective viscosities normalized with respect to ν s ; (d) effective viscosities normalized with respect to λu 2 τ. Line types same as in figure Effect of averaging time on the predicted flow statistics at We τb 15 and n p k B T/(ρu 2 τ b ) : (a) time evolution of the skin friction coefficient; (b) mean velocity profiles; (c) r.m.s. of streamwise velocity fluctuations; (d) r.m.s. of wall-normal velocity fluctuations; (e) Reynolds shear stresses ( τ R,xz + ), viscous shear stresses ( τ v + ), and sum of Reynolds, viscous and polymer shear stresses ( τ t + ); (f) polymer shear stresses ( τ p,xz + ) and polymer extensions., Newtonian (case NN);, We τb 15 (case FF3-q with statistics averaged over 45 t 65);, experiments of Warholic, et al.(1999) Newtonian;, experiments of Warholic, et al.(1999) at 64%DR;, experiments of Warholic, et al.(1999) at 69%DR xii

14 4.1 (a) Drag reduction as a function of the viscosity ratio, β, from DNS:, DNS at We τb 1;, DNS at We τb 35;, DNS at We τb 1;, DNS at We τb 15. (b) Onset conditions observed in DNS compared to predictions of the elastic theory of de Gennes (1986):, onset data from DNS; ---, npk BT ρu 2 onset = ( ) (15n ) We 15n 4 τ (eqn. 4.2) τ with n = 2/3 and = (a) Onset of drag reduction observed in DNS compared to revised theory of de Gennes:, onset data from DNS;, Ξ 3/2 n pk B T ρu 2 onset = τ A E (κ ) (3n ) We 3n 2 τ A 4 3n 4 k F(k /k d ) {F(k /k d )} 3n 4 (eqn. 4.17) with A E =.3 and n = 2/3. (b) Onset of drag reduction observed in DNS compared to revised theory of de Gennes:, onset data from DNS;, npk BT onset = A E (κ ) (3n ) We 3n 2 τ ( 3n du We + τ d + We 2 τ (κ ) 3n 4 4 3n A 4 k A 3n 4 K {F(k /k d )} 3n 4 ) 3/2 F(k /k d ) {F(k /k d )} 3n 4 ρu 2 τ (eqn. 4.18) with A E =.3 and n = 2/ Saturation of drag reduction with increasing polymer concentration: (a) drag reduction as a function of the viscosity ratio, β, from DNS; (b) drag reduction as a function of the viscosity ratio, β, from experiments of Hoyt (1966) at Re τb 42., DNS at We τb 1;, DNS at We τb 35;, DNS at We τb 1;, DNS at We τb 15;, PEO of M w (We τb 136);, PEO of M w (We τb 352);, PEO of M w (We τb 493);, PEO of M w (We τb 626);, PAM of M w (We τb 23); (c) Saturation of drag reduction observed in DNS compared to elastic theory of de Gennes (1986) and revised theory of de Gennes:, saturation data from DNS;, npk BT ρu 2 sat = We 1 τ (eqn. 4.2);, npk BT τ ρu 2 sat = τ (1 β)we 1 τ (eqn. 4.19) with β = Thelargest streamwiseandspanwisescales, r x andr y, dampedbythepolymer in the E uu spectra (a,d) with varying the concentration at We τb 35, (b,e) with varying the concentration at We τb 1, and (c,f) with varying the concentration at We τb 15, compared to predictions of Lumley s theory (Lumley, 1969, 1973) (eqn.4.7) and the revised theory of de Gennes (eqn.4.16): (a-c) r i compared to r (eqn.4.7); (d-f) r i compared to r (eqn.4.16) with n = 2/3 and A E =.3. Line types in (a,d) same as in figure 3.3; line types in (b,e) same as in figure 3.5; line types in (c,f) same as in figure 3.7; thin lines, r x ; thick lines, r y xiii

15 4.5 Effect of Weissenberg number on the largest streamwise and spanwise scales, r x and r y, damped by the polymer in the E uu spectra at 1 We τb 15 and saturation concentration of n p k B T/(ρu 2 τ b ) : (a) r i compared to r (eqn.4.7); (b) r i compared to r (eqn.4.16) with n = 2/3 and A turb =.3. Line types same as in figure 3.15; thin lines, r x ; thick lines, r y Effect of polymer concentration on the largest streamwise and spanwise scales, r x and r y, damped by the polymer in the E uu spectra at (a,d) We τb 35, (b,e) We τb 1, (c,f) We τb 15: (a-c) r i in outer scaling; (d-f) r i in inner scaling. Line types in (a,d) same as in figure 3.3; line types in (b,e) same as in figure 3.5; line types in (c,f) same as in figure 3.7; thin lines, r x ; thick lines, r y Effect of polymer concentration on the largest streamwise and spanwise scales, r x and r y, damped by the polymer in the E uu spectra at (a,d) We τb 35, (b,e) We τb 1, (c,f) We τb 15: (a-c) r i normalized using ν eff and u τ ; (d-f) r i normalized using λ and u τ. Line types in (a,d) same as in figure 3.3; line types in (b,e) same as in figure 3.5; line types in (c,f) same as in figure 3.7; thin lines, r x ; thick lines, r y Effect of Weissenberg number on the largest streamwise and spanwise scales, r x and r y, damped by the polymer in the E uu spectra at 1 We τb 15 and saturation concentration of n p k B T/(ρu 2 τ b ) : (a) r i in outer scaling; (b) r i in inner scaling; (c) r i normalized using ν eff and u τ ; (d) r i normalized using λ and u τ. Line types same as in figure 3.15; thin lines, r x ; thick lines, r y Anisotropy-invariant maps showing (a) limiting states of turbulence and (b) data from Newtonian turbulent channel flow at Re τb 23 (case NN) (a-b) Time histories of the volume-averaged turbulence kinetic energy, E ii (t) V,andpolymerelasticenergyoriginatingfromturbulence, E p,tur (t) V, normalized by Ubulk 2 /2; (c-d) time histories of the volume-averaged rates of turbulence production, P ii (t) V, viscous dissipation, ε ii (t) V, and energy transfer from turbulence to the polymer, T ii (t) V, normalized by Ubulk 3 /h; (e-f) time histories of the skin friction coefficient, in (a,c,e) LDR regime (case CC3-q, 33%DR) and(b,d,f) HDR regime (case FF4-q, 56%DR). The times O, k, K, p, P denote start of viscoelastic simulations, minima and maxima of E ii (t) V, and minima and maxima of E p,tur (t) V during the first cycle of turbulence suppression and regeneration, respectively., viscoelastic flow;, Newtonian flow xiv

16 5.3 (a-b) Time histories of the volume-averaged turbulence kinetic energy, E ii (t) V,andpolymerelasticenergyoriginatingfromturbulence, E p,tur (t) V, normalized by u 2 τ (t)/2; (c-d) time histories of the volume-averaged rates of turbulence production, P ii (t) V, viscous dissipation, ε ii (t) V, and energy transfer from turbulence to the polymer, T ii (t) V, normalized by u 4 τ(t)/ν w (t) in (a,c) LDR regime (case CC3-q, 33%DR) and (b,d) HDR regime (case FF4-q, 56%DR)., viscoelastic flow;, Newtonian flow One-point flow statistics in stationary viscoelastic flow in LDR and HDR regimes compared to Newtonian flow: (a) mean velocity profiles; (b) Reynolds, τ R,xz +, and viscous, τ v +, shear stresses; (c) polymer shear stresses; (d) streamwise turbulence intensities; (e) spanwise turbulence intensities; (f) wall-normal turbulence intensities., Newtonian flow (case NN);, viscoelastic flow at low drag reduction (case CC3-q, 33%DR);, viscoelastic flow at high drag reduction (case FF4-q, 56%DR) Anisotropy-invariant maps for(a) Newtonian flow (case NN), (b) viscoelastic flow in the LDR regime (case CC3-q, 33%DR) and (c) viscoelastic flow in the HDR regime (case FF4-q, 56%DR) The one-dimensional energy spectra in LDR (case CC3-q, 33%DR) and HDR (case FF4-q, 56%DR) regimes compared to Newtonian flow (case NN) at 3., Newtonian flow (case NN);, viscoelastic flow at low drag reduction (case CC3-q, 33%DR);, viscoelastic flow at high drag reduction (case FF4-q, 56%DR) The one-dimensional energy spectra in LDR (case CC3-q, 33%DR) and HDR (case FF4-q, 56%DR) regimes compared to Newtonian flow (case NN) at 5., Newtonian flow (case NN);, viscoelastic flow at low drag reduction (case CC3-q, 33%DR);, viscoelastic flow at high drag reduction (case FF4-q, 56%DR) The one-dimensional energy spectra in LDR (case CC3-q, 33%DR) and HDR (case FF4-q, 56%DR) regimes compared to Newtonian flow (case NN) at 1., Newtonian flow (case NN);, viscoelastic flow at low drag reduction (case CC3-q, 33%DR);, viscoelastic flow at high drag reduction (case FF4-q, 56%DR) The characteristic strain-rate, s + (k + i,z+ ), in Newtonian flow (case NN) at (a-b) low drag reduction (case CC3-q, 33%DR) and (c-d) high drag reduction (case FF4-q, 56%DR) compared to 1/T + of Lumley s theory (Lumley, 1969, 1973)., 1/T + from Lumley s theory xv

17 5.1 The characteristic strain-rate, s + (k + i,z+ ), in viscoelastic flow at (a-b) low drag reduction (case CC3-q, 33%DR) and (c-d) high drag reduction (case FF4-q, 56%DR) compared to 1/T + of Lumley s theory (Lumley, 1969, 1973) and 1/T + ( ) of the revised version of de Gennes s theory., 1/T + from Lumley s theory;, lowest 1/T + ( ) from the revised version of de Gennes s theory; ---, scales with s + (k + i,z+ ) > 1/T + ( );, scales with s + (k + i,z+ ) < 1/T + ( ) The characteristic strain-rates in LDR (case CC3-q, 33%DR) and HDR (case FF4-q, 56%DR) regimes compared to Newtonian flow (case NN) at (a-b) 1 and (c-d) 3., Newtonian flow (case NN);, viscoelastic flow at low drag reduction (case CC3-q, 33%DR);, viscoelastic flow at high drag reduction (case FF4-q, 56%DR) The characteristic strain-rates in LDR (case CC3-q, 33%DR) and HDR (case FF4-q, 56%DR) regimes compared to Newtonian flow (case NN) at (a-b) 5 and (c-d) 1., Newtonian flow (case NN);, viscoelastic flow at low drag reduction (case CC3-q, 33%DR);, viscoelastic flow at high drag reduction (case FF4-q, 56%DR) The normalized one-dimensional spectral density of the diagonal components of strain-rate in LDR (case CC3-q, 33%DR) and HDR (case FF4-q, 56%DR) regimes compared to Newtonian flow (case NN) at 3., Newtonian flow (case NN);, viscoelastic flow at low drag reduction (case CC3-q, 33%DR);, viscoelastic flow at high drag reduction (case FF4-q, 56%DR) The normalized one-dimensional spectral density of the diagonal components of strain-rate in LDR (case CC3-q, 33%DR) and HDR (case FF4-q, 56%DR) regimes compared to Newtonian flow (case NN) at 5., Newtonian flow (case NN);, viscoelastic flow at low drag reduction (case CC3-q, 33%DR);, viscoelastic flow at high drag reduction (case FF4-q, 56%DR) The normalized one-dimensional spectral density of the diagonal components of strain-rate in LDR (case CC3-q, 33%DR) and HDR (case FF4-q, 56%DR) regimes compared to Newtonian flow (case NN) at 1., Newtonian flow (case NN);, viscoelastic flow at low drag reduction (case CC3-q, 33%DR);, viscoelastic flow at high drag reduction (case FF4-q, 56%DR) xvi

18 5.16 Components of turbulence kinetic energy budgets in LDR (case CC3-q, 33%DR) and HDR (case FF4-q, 56%DR) regimes compared to Newtonian flow (case NN): (a-c) pressure-strain correlation, Π αα + ; (d-f) turbulence production, P αα +, and viscous dissipation, ε αα +. (a,d) streamwise direction; (b,e) spanwise direction; (c,f) wall-normal direction., Newtonian flow (case NN);, viscoelastic flow at low drag reduction (case CC3-q, 33%DR);, viscoelastic flow at high drag reduction (case FF4- q, 56%DR) Components of turbulence kinetic energy budgets in LDR (case CC3-q, 33%DR) and HDR (case FF4-q, 56%DR) regimes compared to Newtonian flow (case NN): (a-c) sum of transport terms, t (Σ) αα + = t (v) αα + t (press) αα + t (R) αα +t (p) αα + ; (d-f) energy transfer from turbulence to the polymer, T αα +. (a,d) streamwise direction; (b,e) spanwise direction; (c,f) wall-normal direction., Newtonian flow (case NN);, viscoelastic flow at low drag reduction (case CC3-q, 33%DR);, viscoelastic flow at high drag reduction (case FF4-q, 56%DR) Turbulence kinetic energy and polymer elastic energy budgets in LDR(case CC3-q, 33%DR) and HDR (case FF4-q, 56%DR) regimes compared to Newtonian flow (case NN): (a) turbulence production, P ii +, and viscous dissipation, ε ii + ; (b) energy transfer from turbulence to the polymer, T ii + ; (c) sum of transport terms, t (Σ) ii + = t (v) ii +t (press) ii +t (R) ii +t (p) ii + ; (d) energytransferfromthemeanflow tothepolymer, T U +, andpolymer dissipation, ε p + ; (e) energy transfer from turbulence to the polymer, T ii + ; (f) polymer energy transport, t p +., Newtonian flow (case NN);, viscoelastic flow at low drag reduction (case CC3-q, 33%DR);, viscoelastic flow at high drag reduction (case FF4-q, 56%DR) (a) total r.m.s. pressure fluctuations, (b) slow part of r.m.s. pressure fluctuations, (c) rapid and polymer parts of r.m.s. pressure fluctuations, (d-f) The diagonal components of normalized r.m.s. strain-rate fluctuation in LDR (case CC3-q, 33%DR) and HDR (case FF4-q, 56%DR) regimes compared to Newtonian flow (case NN)., Newtonian flow (case NN);, viscoelastic flow at low drag reduction (case CC3-q, 33%DR);, viscoelastic flow at high drag reduction (case FF4-q, 56%DR) xvii

19 5.2 Effect of polymer concentration on the predicted second-order turbulence statistics at We τb 35: (a) mean velocity profiles; (b) Reynolds shear stresses and polymer shear stresses; (c) streamwise and spanwise turbulence intensities; (d) wall-normal turbulence intensities., Newtonian (case N); ---, β (case C7-l);, β.9996 (case C5-l);, β.996 (case C4-l);, β.98 (case C4.1-l);, β.96 (case C3-l);, β.89 (case C3.1-l);, β.8 (case C3.2-l);, β.72 (case C2-l) Effect of polymer concentration on the predicted second-order turbulence statistics at We τb 15: (a) mean velocity profiles; (b) Reynolds shear stresses and polymer shear stresses; (c) streamwise and spanwise turbulence intensities; (d) wall-normal turbulence intensities., Newtonian (case NN); ---, β (case F8-q);, β.9998 (case F6-q);, β.998 (case FF5-q);, β.98 (case FF4-q);, β.86 (case FF3-q) Effect of polymer concentration on anisotropy-invariant maps at We τb 35: (a) β.9996 (case C5-l); (b) β.996 (case C4-l); (c) β.96 (case C3-l); (d) β.89 (case C3.1-l); (e) β.8 (case C3.2-l); (f) β.72 (case C2-l) Effect of polymer concentration on anisotropy-invariant maps at We τb 15: (a) Newtonian (case NN); (b) β.9998 (case F6-q); (c) β.998 (case FF5-q); (d) β.98 (case FF4-q); (e) β.86 (case FF3-q) Effect of polymer concentration on the characteristic strain-rate at We τb 35: (a-b)β.9996 (casec5-l); (c-d)β.996(casec4-l); (e-f)β.96 (case C3-l). (a,c,e) streamwise spectra; (b,d,f) spanwise spectra., 1/T + from Lumley s theory;, lowest 1/T + ( ) from the revised version of de Gennes s theory; ---, scales with s + (k + i,z+ ) > 1/T + ( );, scales with s + (k + i,z+ ) < 1/T + ( ) Effect of polymer concentration on the characteristic strain-rate at We τb 35: (a-b) β.89 (case C3.1-l); (c-d) β.8 (case C3.2-l); (e-f) β.72 (case C2-l). (a,c,e) streamwise spectra; (b,d,f) spanwise spectra., 1/T + from Lumley s theory;, lowest 1/T + ( ) from the revised version of de Gennes s theory; ---, scales with s + (k + i,z+ ) > 1/T + ( );, scales with s + (k + i,z+ ) < 1/T + ( ) xviii

20 5.26 Effect of polymer concentration on the characteristic strain-rate at We τb 15: (a-b) β.9998 (case F6-l); (c-d) β.998 (case FF5-q). (a,c) streamwise spectra; (b,d) spanwise spectra., 1/T + from Lumley s theory;, lowest 1/T + ( )fromtherevisedversionofdegennes stheory; ---, scaleswiths + (k + i,z+ ) > 1/T + ( );,scaleswiths + (k + i,z+ ) < 1/T + ( ) Effect of polymer concentration on the characteristic strain-rate at We τb 15: (a-b) β.98 (case FF4-q); (c-d) β.86 (case FF3-q). (a,c) streamwise spectra; (b,d) spanwise spectra., 1/T + from Lumley s theory;, lowest 1/T + ( )fromtherevisedversionofdegennes stheory; ---, scaleswiths + (k + i,z+ ) > 1/T + ( );,scaleswiths + (k + i,z+ ) < 1/T + ( ) Effect of polymer concentration on the predicted turbulence kinetic energy budgets at We τb 35: (a-c) pressure-strain correlation, Π αα + ; (d) turbulence production, P ii +, and viscous dissipation, ε ii + ; (e) energy transfer from turbulence to the polymer, T ii + ; (f) sum of transport terms, t (Σ) ii +., Newtonian (case N); ---, β (case C7-l);, β.9996 (case C5-l);, β.996 (case C4-l);, β.98 (case C4.1-l);, β.96 (case C3-l); -, β.89 (case C3.1-l);, β.8 (case C3.2-l);, β.72 (case C2-l) Effect of polymer concentration on the predicted polymer elastic energy budgets at We τb 35: (a) energy transfer from the mean flow to the polymer, T U +, and polymer dissipation, ε p + ; (b) energy transfer from turbulence to the polymer, T ii + ; (c) polymer energy transport, t p +., Newtonian (case N); ---, β (case C7-l);, β.9996 (case C5-l);, β.996 (case C4-l);, β.98 (case C4.1-l);, β.96 (case C3-l); -, β.89 (case C3.1-l);, β.8 (case C3.2-l);, β.72 (case C2-l) Effect of polymer concentration on the predicted turbulence kinetic energy budgets at We τb 15: (a-c) pressure-strain correlation, Π αα + ; (d) turbulence production, P ii +, and viscous dissipation, ε ii + ; (e) energy transferfromturbulencetothepolymer, T ii + ; (f)sumoftransportterms, t (Σ) ii +., Newtonian (case NN); ---, β (case F8-q);, β.9998 (case F6-q);, β.998 (case FF5-q);, β.98 (case FF4-q);, β.86 (case FF3-q) xix

21 5.31 Effect of polymer concentration on the predicted polymer elastic energy budgets at We τb 15: (a) energy transfer from the mean flow to the polymer, T U +, and polymer dissipation, ε p + ; (b) energy transfer from turbulence to the polymer, T ii + ; (c) polymer energy transport, t p +., Newtonian (case NN); ---, β (case F8-q);, β.9998 (case F6-q);, β.998 (case FF5-q);, β.98 (case FF4-q);, β.86 (case FF3-q) xx

22 List of Tables Table 2.1 Overview of the simulations performed to establish the effect of numerical scheme and domain size. Cases N, C3, E3 and F3 preformed in channels of size 8π 27 h 8π 5 h 2h; cases NN, CC3, EE3 and FF3 preformed in channels of size 32π 16π 27 h 5 h 2h; l denotes simulations performed using the mixed Eulerian-Lagrangian scheme with linear interpolation; q denotes simulations performed using the mixed Eulerian-Lagrangian scheme with quadratic interpolation; e denotes simulations performed using the Eulerian scheme Overview of the simulations performed to establish the effect of polymer concentration. Cases N, Ai, Ci, Ei and Fi preformed in channels of size 8π 27 h 8π 5 h 2h; cases NN, EEi and FFi preformed in channels of size 32π 16π 27 h 5 h 2h; l denotes simulations performed using the mixed Eulerian-Lagrangian scheme with linear interpolation; q denotes simulations performed using the mixed Eulerian-Lagrangian scheme with quadratic interpolation. The polymer extensibility parameter was set to b = 45, in all the viscoelastic simulations Overview of the simulations performed to establish the effect of Weissenberg number and extensibility parameter. Cases N and Ai, Bi and Ci preformed in channels of size 8π 27 h 8π 5 h 2h; cases NN, DDi, EEi and FFi preformed in channels of size 32π 16π 27 h 5 h 2h; l denotes simulations performed using the mixed Eulerian-Lagrangian scheme with linear interpolation; q denotes simulations performed using the mixed Eulerian- Lagrangian scheme with quadratic interpolation The onset conditions read from figure 4.1(a) at.5% drag reduction xxi

23 List of Symbols A = conformation tensor ( QQ s ) A E = fraction of turbulence kinetic energy redirected to the elastic energy of the polymer A k = Kolmogorov constant b = polymer extensibility parameter b ij = anisotropy tensor c = concentration of the polymer C f = mean skin friction coefficient D + (k + i,z+ ) = one-dimensional dissipation spectrum E αα E ii E p E p,u E p,tur = turbulence kinetic energy in α-direction = total turbulence kinetic energy = polymer elastic energy = polymer elastic energy due to the exchanges with the mean flow = polymer elastic energy due to the exchanges with turbulence E(k) = energy spectrum H = spring constant H = heterogeneity index xxii

24 h = channel half height I = identity matrix II and III = scalar invariants of the anisotropy tensor k = wavenumber k = wavenumber of scale r (1/r ) k = wavenumber of scale r (1/r ) k B k d k i = Boltzmann s constant = Kolmogorov wave number = wavenumber in ith direction L = characteristic length-scale l = average backbone bond length N = number of beads in the chain N A N k = Avogadro s number = number of Kuhn steps NX = mesh resolution in streamwise direction NY = mesh resolution in spanwise direction NZ = mesh resolution in wall-normal direction n = dimensionality of the polymer stretching ñ = number of backbone bonds comprising the chain n p M w = number density of the polymer = molecular weight of the polymer xxiii

25 P = number of computing processors P αα P ii = turbulence production in α-direction = total turbulence production p = dimensionless pressure p p rms p r,rms p s,rms p p,rms = fluctuating pressure = r.m.s. pressure fluctuation = rapid part of r.m.s. pressure fluctuation = slow part of r.m.s. pressure fluctuation = polymer part of r.m.s. pressure fluctuation Q = dimensionless length and orientation vector of two beads Q o = maximum extension of the spring R = pipe radius R G = radius of gyration of the polymer in the coiled state Re = Reynolds number ( UoL ν s ) Re bulk = bulk Reynolds number ( U bulkr ν s or U bulkh ν s ) Re τ = skin friction Reynolds number of the viscoelastic flow ( uτδ ν or uτr ν or uτh ν ) Re τb = skin friction Reynolds number of the base Newtonian flow ( uτ b δ ν s or uτ b R ν s or uτ b h ν s ) R ii = two-point correlation xxiv

26 r = size of eddies whose characteristic time is λ, or size of the largest turbulent eddies affected by the polymer in Lumley s theory r = size of eddies whose characteristic time is smaller than λ and has redirected a minimum fraction, A E, of its turbulent energy to the elastic energy of the polymer, or size of the largest turbulent eddies affected by the polymer in the revised version of de Gennes s theory r G r x r y r (n) P = locations of the particles at the conclusion of each time step = largest attenuated scale in the x-direction in the E uu spectra = largest attenuated scale in the y-direction in the E uu spectra = starting location of the particle at time t n s + (k + i,z+ ) = characteristic strain-rate s αα,rms = diagonal components of r.m.s. strain-rate fluctuation T = absolute temperature T αα T ii T U T T = energy transfer from turbulence to the polymer in α-direction = total energy transfer from turbulence to the polymer = energy transfer from the mean flow to the polymer = characteristic time at scale r = characteristic time at scale r t = dimensionless time xxv

27 t (p) αα t (p) ii t (press) αα t (press) ii t (R) αα t (R) ii t (v) αα t (v) ii t (Σ) αα t (Σ) ii t p = energy transport by the fluctuating polymer stress in α-direction = total energy transport by the fluctuating polymer stress = energy transport by the fluctuating pressure in α-direction = total energy transport by the fluctuating pressure = energy transport by the fluctuating Reynolds stress in α-direction = total energy transport by the fluctuating Reynolds stress = energy transport by the fluctuating viscous stress in α-direction = total energy transport by the fluctuating viscous stress = sum of transport terms in α-direction = sum of transport terms in all directions = energy transport by the polymer t = dimensional time U = mean velocity U bulk U o = bulk velocity = characteristic velocity U(r ) = characteristic velocity at scale r U(r ) = characteristic velocity at scale r u i,rms u τ = turbulence intensity = friction velocity in viscoelastic flow u τb = friction velocity in base Newtonian flow u α = fluctuating velocity in α-direction xxvi

28 v = dimensionless velocity We = Weissenberg number (λu o /L) We τ We τb = Weissenberg number in viscoelastic flow ( λu2 τ ν ) = Weissenberg number in base Newtonian flow ( λu2 τ b ν s ) wppm = weight parts per million x = streamwise direction y = spanwise direction z = wall-normal direction β = viscosity ratio (µ s /µ o ) δ = characteristic length such as the boundary layer thickness, or channel half height h, or pipe radius R ε = dissipation rate ε αα ε ii ε p = viscous dissipation in α-direction = total viscous dissipation = polymer dissipation ζ = drag coefficient on each bead η = Kolmogorov length scale [η] = intrinsic viscosity λ = relaxation time of the polymer xxvii

29 λ H λ z = Rouse relaxation time of the polymer = Zimm relaxation time of the polymer µ o = zero shear viscosity of the solution µ w = wall shear viscosity ( νw ρ ) ν = kinematic viscosity of the polymer solution ν eff ν s ν w = effective viscosity = kinematic viscosity of the solvent = wall shear viscosity ( τ w ρ(du/dz) w ) Ξ(r ) = polymer stretch due to stretching by scale r Ξ = total polymer stretch ξ = polymer root-mean-square end-to-end extension Π = pressure head (p+ 1 2 v 2 ) Π αα = pressure-strain correlation ρ = density τ p τ R τ t τ w τ p = polymer stress = Reynolds shear stress = total shear stress = wall shear stress = fluctuating polymer stress τ + (k + i,z+ ) = non-dimensional characteristic time at scale k + i at location χ = artificial stress diffusivity xxviii

30 Φ sααs αα = one-dimensional spectral density of the diagonal components of strain-rate %DR = percent drag reduction = ensemble-averaged quantity obtained by averaging the quantity in the homogeneous flow directions and time s = average over the configuration space of the dumbbell. superscripts + = non-dimensional quantity normalized using u τ and ν w xxix

31 Abstract Skin-friction drag reduction by dilute polymer solutions is investigated using results from direct numerical simulations (DNS) of homogeneous polymer solutions in turbulent channel flow. Simulations were preformed using a novel mixed Eulerian- Lagrangian scheme in a turbulent channel flow at a base Reynolds number of Re τb 23 with a FENE-P dumbbell model of the polymer, and covered the range of Weissenberg numbers between 1 We τb 15, polymer number densities between n p k B T/(ρu 2 τ b ) (corresponding to viscosity ratios of.7 β 1.), and polymer extensibility parameters between 4,5 b 45,, to clarify the role of each polymer parameter in drag reduction. The full range of drag reduction from onset to Maximum Drag Reduction (MDR) was reproduced in DNS, with statistics in good quantitative agreement with the available experimental data. Onset of drag reduction was found to be a function of both the polymer concentration and Weissenberg number, as originally suggested by de Gennes (1986). However, the onset criteria suggested by de Gennes (1986) were found to be several orders of magnitude higher than DNS data. A revised version of the theory of de Gennes (1986) has been developed, which gives good agreement with DNS results. The magnitude of drag reduction was found to be a universal function of β, increasing monotonically with β for 1. > β >.98, and saturating at β.98. The magnitude xxx

32 of drag reduction at saturation is a strong function of the Weissenberg number. A We τ O(Re τ /2) is needed to reach MDR. Investigation of the mechanism of drag reduction shows that the main effect of the polymer is extraction of a small amount (on the order of 5% on a volume-averaged basis) of turbulence kinetic energy from turbulent scales which have a timescale shorter than the polymer relaxation time. This extraction of energy leads to a decrease in the fluctuating strain-rate at these scales, which in turn, reduces the magnitude of the pressure-strain correlation at these and neighboring scales. This inhibits the turbulence kinetic energy transfer from the streamwise component to the cross-stream directions at these scales. When this drop in pressure-strain correlation extends to the largest turbulent scales, it results in a highly anisotropic state in which the cross-stream turbulence intensities are sharply reduced, leading to a drop in the Reynolds shear stress. This drop in the Reynolds shear stress, in turn, causes a drop in the rate of turbulence production. In addition, the energy trapped in the streamwise direction can no longer cascade to the small scales, leading to further decay of the fluctuating strain-rate and turbulence kinetic energy in the small scales. This decay further amplifies the features described above. Thus the minute extraction of energy by the polymer at the affected turbulent scales starts a self-amplifying sequence of events, which leads to cessation of turbulence production and results in a drag reduction. For effective high drag reduction, the initial minute extraction of energy by the polymer needs to extend to the largest turbulent scales at wall-normal locations where the peak of turbulence production occurs. The above understanding of the mechanism of polymer drag reduction opens up new possibilities for skin-friction drag reduction in wall-bounded flows. xxxi

33 Chapter I Introduction It has been known for nearly sixty years, since its original discovery by Toms (1949), that the addition of a few weight parts per million (wppm) of an appropriate, high molecular weight, linear chain polymer to the turbulent flow of a solvent can lead to drag reductions of up to 8% in wall-bounded turbulent flows. Since then, numerous experimental studies in pipe flows, channel flows, and boundary layer flows have verified these findings and revealed various other features of wall-bounded turbulence in the presence of drag reducing polymers. Nevertheless, many aspects of the problem, including the scaling of drag reduction with polymer and flow parameters, and the detailed mechanism of drag reduction remain poorly understand or controversial. General reviews of the current state of knowledge are available in Lumley (1969, 1973), Virk (1975), Hoyt (199), Gyr & Bewersdorff (1995), Nieuwstadt & den Toonder (21), and White & Mungal (28). 1.1 Early experimental studies Early experimental studies on polymer drag reduction (Hershey & Zakin, 1967; Lumley, 1969, 1973; Hoyt, 1966, 1971, 1972; Kenis& Hoyt, 1971; Hunston, 1974; Frederick, 1975; Berman, 1977; Nadolink, 1987) were focused on measuring the pressure- 1

34 2 drop and mean velocity profiles in dilute polymer solutions with different polymer molecules, molecular weights, and concentrations to determine the scaling of drag reduction with polymer parameters. Three characteristic features of polymer drag reduction borne out from these early experimental studies include the phenomena of onset of drag reduction, saturation of drag reduction, and maximum drag reduction (MDR) Onset of drag reduction Onset of drag reduction refers to the criteria which must be met before a given polymer of molecular weight M w, radius of gyration in the coiled state R G, relaxation time λ, and number density n p (or concentration c) can display any drag reducing effects in the turbulent flow of a solvent at a base Reynolds number of Re τb u τb δ/ν s, where u τb denotes the friction velocity of the base Newtonian flow, δ denotes the characteristic length of the flow such as the boundary layer thickness, channel half height, or pipe radius, and ν s is the kinematic viscosity of the solvent. Establishing the onset criteria has been the subject of a number of experimental studies (Hershey & Zakin, 1967; Patterson & Abernathy, 197; Berman, 1977; Nadolink, 1987; Sreenivasan & White, 2). Some of these studies (e.g. Hershey & Zakin, 1967; Berman, 1977) have found that the onset of drag reduction depends on the flow time-scale. The onset of drag reduction have been found to occur when We τ λu2 τ ν O(1) (1.1)

35 3 (Hershey & Zakin, 1967; Lumley, 1969), where We τ is the Weissenberg number, and denotes the ratio of the polymer relaxation time, λ, to the shortest time-scale, ν/u 2 τ, in the viscoelastic turbulent flow, where u τ is the friction velocity, and ν is the kinematic viscosity of the polymer solution, which is typically defined as either µ o /ρ or as µ w /ρ, where µ o is the zero shear viscosity of the solution, and µ w is the wall shear viscosity, and ρ is the density. While this onset criterion is found to be a function of Weissenberg number only, and independent of polymer concentration, others (Nadolink, 1987; Sreenivasan & White, 2) observed an onset criterion as a function of both the polymer concentration and Weissenberg number. An expression for this onset criterion was proposed by Sreenivasan & White(2) based on analysis of experimental data (Patterson & Abernathy, 197; Berman, 1977; Nadolink, 1987) as n p k B T ρu 2 τ onset We 5 2 τ, (1.2) where n p is the polymer number density, k B is the Boltzmann s constant, and T is the absolute temperature Saturation of drag reduction Saturation of drag reduction refers to the observation that for all polymers, the amount of drag reduction levels off with increasing polymer concentration (Hoyt, 1966, 1971; Kenis & Hoyt, 1971), such that adding more polymer does not lead to any additional drag reduction. As with the onset phenomenon, the saturation criterion is also still the subject of debate. Some investigators (Lumley, 1969, 1973) believe that saturation occurs when the wavenumber corresponding to the peak of the dissipation