v———.- LraaAav Michigan State University This is to certify that the thesis entitled Polymer Induced Drag Reduction Based on Stress Relaxation presented by Susan Mary Lemke Penix has been accepted towards fulfillment of the requirements for M.S. degree in CHE Q m lg, (31211:) Major professor 0-7639 MS U is an Affirmative Action/Equal Opportunity Institution ll" lllllllllllllllllllllfllll L 31293 011031543 )V‘ESI.) RETURNING MATERIALS: Place in book drop to LJBRARJES remove this checkout from __ your record. FINES will be charged if book is returned after the date stamped below. MAR ‘4. L- , Wfizficw POLNMER INDUCED DRAG REDUCTION BASED ON STRESS RELAXATION By Susan Mary Lelke Penix A THESIS Submitted to Michigan State University in partial fulfillment of the require-ents for the degree of MASTER OF SCIENCE Department of Che-ical Engineering 1987 ABSTRACT mum mucnn DRAG REDUCTION BASED ON smss RELAXATION By Susan Mary Lemke Penix A theoretical approach based on stress relaxation of the polymer is used to develop a model for the 'eddy’ viscosity for dilute polymer solutions. Further, statistical properties of the flow such as the ”eddy" viscosity, axial intensity: and the friction factor data are derived and the predicted quantities are compared to experimental data extracted from the literature. The results showed good quantitative and qualitative agreement with experimental findings. Finally, a scale-up method based on diameter was developed and was found successful in scaling up in the range of 3 m to 32 m. To My Husband, Normal -gnd_ To My Son, Brandon ii ACKNOWLEDGEMENTS The author gratefully acknowledges the guidance and editorial assistance of Dr. Charles A. Petty. Special appreciation is also expressed to my husband, Nor-an; his support and encourage-cut helped Ie greatly to complete this thesis. Also, special thanks go to Nor-an for preparing all ny figures. iii TABLE OF CONTENTS Page LIST OF TABLES. . . . . . . . . . . . . . . . . . . . . . . . . . . vi LIST OF FIGURES . . . . . ..... . . . . . . . . . ...... . vii LIST OF NOTATIONS . . . . . . . . . . . . . . . . . . . . . . . . . ix CHAPTER I. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . l 1.1. Background. . . . . . . . . . . . . . . . . . . . 1 1.2. Objectives of This Research . . . . . . . . . . . 6 1.3. Rheological Models for the Solvent and the Dilute Poly-er Solution . . . . . . . . . 7 1.4. Fully Developed Pipe Flow and Scaling Hypotheses ...... . . . . . . . . . . 10 CHAPTER II. THEORY . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1. Mean Flow Field . . . .'. ...... . . . . . . 17 2.2. Derivation of the 'Eddy’ Viscosity for the Solvent and Dilute Polymer Solution . . . . . . . 18 2.3. Axial Intensity Models for the Newtonian and Dilute Poly-er Solutions. ..... . . . . . 23 2.4. Discussion of Qualitative Features of the Theory . . . . . . . . . . . . . . . . . . 24 2.5. Method Used to Study the Theory . . . . . . . . . 30 CHAPTER III. PARAMETER ESTIMATION . . . . . . . . . . . . . . . . . 32 CHAPTER IV. RESULTS. . . . . . . . . . . . . . . . . . . . . . . . 41 4.1. Calculated Statistical Properties . . . . . . . . 41 4.2. Friction Factor Results . . . . . . . . . . . . . 48 iv CHAPTER V. REFERENCES. APPENDIX A. APPENDIX 8. CONCLUSIONS. . . 5.1. Phenomena Predicted ............... 5.2. New Discoveries and Shortcomings of the Model . . 5.3. Recommendations for Further Study . . ...... Reynolds Stress Model for Dilute Polymer Solutions . Computer Programs. . Page 53 53 54 55 58 61 63 LIST OF TABLES Table Page 1 Effect of an on rm and 1x1 ........ . ....... 38 2 Fitted parameters in the stress relaxation model. . . . . . 40 vi Figure 10 11 12 13 14 15 LIST OF FIGURES Page Friction factor data for fully developed turbulent pipe flow of dilute polymer solutions. (Estimated from Figures 2a and Figure 7 of Virk [1975]) ........ 3 Response of a spring and dashpot model to a one- dimensional shear flow ................... 8 Wall region in fully developed pipe flow ...... . . . . 11 Relaxation models for turbulent auto correlations in the near well region . . . . . . . . . . . . . . . . . . 15 The effect of intrinsic relaxation times on the relative transport times R; as predicted by the stress relaxation theory. . ................ 25 The effect of the intrinsic relaxation times on the turbulent transport times as predicted by the stress relaxation theory (Lyons and Petty [1984]) . . . . . 27 The effect of temporal scales on the correlation coefficient as predicted by the stress relaxation theory ........................... 29 Parametric study of the effect of $18 on the Prandlt-Karman Law ..................... 35 Radial intensity model in the near wall region ....... 36 Fitted Newtonian ”Eddy" viscosity ............. 37 Parametric effect of «M1 on the asymptotic effect ..... 39 Predicted mean velocity profiles .............. 42 Predicted effect of the dilute polymer 'eddy’ viscosity . . ..... . . . ....... . . . 44 Predicted axial intensity profiles based on the stress relaxation theory. . . . . . ........ . . . . 45 Predicted Reynolds stress profiles based on the stress relaxation theory ........ . . . ....... 47 vii Figure Page 16 Predicted friction factor data by stress relaxation theory ..... . . . ............... . . . . 49 17 The predicted ”onset" effect. . . . . . . . ........ 51 18 Predicted scale-up behavior for small pipes using the stress relaxation model . . . . . . . . . . 52 viii De 91: E21 S3 f com I223) K(X1.?) Q: 111 and 113 R1 LIST OF NOTATIONS leading coefficient of a:* model defined in Eq. (1.18) coefficient of the viscous sublayer model in the radial intensity defined in Eq. (1.25a) power of by model defined by Eq. (1.18) coefficient of the buffer layer model in the radial intensity defined in Eq. (1.25b) coefficient of the inertial layer model in the radial intensity defined in Eq. (1.25c) diameter of a circular pipe gross Deborah number defined as )v/Dz; related to the intrinsic Deborah number an by Eq. (1.6) unit base vectors attached to the wall; 31 is normal to the wall and 93 is in the axial direction Fanning friction factor Tu/(l/Zpubz) Greens function associated with the differential operator defined by Eq. (2.14) on the semi-infinite domain. For x1 >> 0 and t << xxz/v, then G(x,t|§,t) approaches G°(x,t|§,t) the so~called 'free’ Green’s function. function in space-time hydrodynamic correlation defined by Eq. (2.13) viscoelastic memory kernel defined by Eq. (2.24) characteristic length scale used to simplify Eq.(1.ll); see Ineq. (1.13) characteristic spatial scales associated with the hydrodynamic correlations , respectively. length of a circular pipe radius of a circular pipe defined by Eq. (2.31) and see Figure 5 instantaneous strain rate dyadic g = l/2[°g + (‘2)’] ix ..... 2’(_X_.t) ur. us, no 11b Us (m ’2)1/2 (us ’2)1/2 £30 25,: X1, x2, X3 (.11 d 9 A1 8' 8' 2‘ 4 >’ distinct times instantaneous molecular stress dyadic wall shear stress denotes the wall shear stress for the solvent; also represents critical wall shear stress at onset velocity fluctuations components of the velocity field relative to the base vectors gr, 32, and gs, respectively. bulk average velocity for pipe flow friction velocity, (Tu/p)1/2 root-mean-square of velocity fluctuations nonmal to the wall root-mean-square of velocity fluctuations in the axial direction shear component of the Reynolds stress distinct position vectors components of’x relative to the base vectors 51, 93, and g; respectively intrinsic Deborah number defined by Eq. (1.5) and Eq. (1.6) dimensionless ratio of characteristic times due, «i3 defined in Figure 5 characteristic strain rate defined in Figure 2 thickness of the viscous sublayer thickness of the buffer layer thickness of the inertial layer defined by Eq. (1.18) stress relaxation time for the dilute polymer solution defined by Eq. (1.11). For a dilute polymer solution .is independent of the motion Harman number, Re(f)1/2 dynamic viscosity of the fluid kinematic viscosity of the fluid, ”/9 W(Xl) Tc (X1) 1? TM 1111, TM]. “3 g 1113 'eddy’ kinematic viscosity defined by Eq. (2.3) mass density of the fluid time difference, t-t characteristic transport time defined by Eq. (2.23) characteristic flow time of the turbulent field temporal integral scale associated with a two-point velocity correlation; hydrodynamic relaxation time characteristic temporal scales associated with the hydrodynamic autocorrelation in a frame of reference moving with the local mean velocity; see Figure 4 characteristic temporal scales associated with the hydrodynamic autocorrelation ° used to denote a statistical property of a Newtonian fluid, i.e. solvent denotes a variable of integration, see eq. (2.14) xi CHAPTERI INTRODUCTION 1.1. 9.2ng The phenomenon of drag reduction was discovered in the late forties. Some of the earliest experiments were done by Mysels, Oldroyd, and Toms (Savin, 1977). For fully developed pipe flow, Tom [1948] observed that the addition of small amounts of polymethylmethacrylate to monochlorobenzene reduced the pressure drop substantially below that of solvent alone at the same flow rate. Since the discovery of drag reduction, there have been many experimental and theoretical studies motivated, in part, by the potential of this phenomenon. For example, the Trans-Alaskan Pipeline was injected with CDR‘V" (a proprietary high molecular weight polymer produced by Conoco) with a resulting increase in capacity of 20: (see Burger, et al., 1980). Another application under study is the use of polymer injection in steam sewers. Recently, Sellin [1981] reported an increase in capacity of sewers by 25% using this technique. These and other applications of the drag' reduction phenomenon would benefit directly’ from a physical understanding of how' polymers can affect turbulent flows so dramatically. This research will hopefully contribute toward this objective by developing an analysis for drag reduction based on a recent theory proposed by Lyons and Petty [1985] and Penix and Petty [1986]. In order to help understand the parameters which quantify some of the measurable properties of the flow under conditions of drag reduction, some basic quantities which are often used will be defined and discussed. For instance, the Fanning friction factor given by Bird, et al. [1960] is defined as f = 2Tu/pUb2 (1-1) where Tw is the wall shear stress, p'is the density, and us is the bulk average velocity. Physically, the Fanning friction factor repre- sents the ratio of the energy lost to the surroundings to the total kinetic energy of the flow. Experimentally, Tw can be measured accu- rately from the pressure drop over a distance L for a given flow rate ZIRLTw = 1H2[(P)(0) -

(L)]. (1.2) Many researchers have noted that the phenomenon of drag reduction is characterized by three distinct features: the onset phenomenon; a maximum drag reduction effect; and, a diameter effect. The definitions and significance of these three features is best illustrated in Figure 1. First, an explanation of the coordinates is needed. The ordinant in Figure 1 is (l/f)“2 = 2-1/2(UD/Us) (1.3) where w is the friction velocity (Tw/p)1/2 and 11b is the bulk average velocity. Thus, (2/f)"2 is the ratio of the largest velocity scale to the smallest velocity scale and ranges from 14 - 57. If the friction velocity were held constant (ua: = .015 m/s)for a fixed Karman number, say A c = 1000, the bulk velocity, us, is almost doubled as seen in Figure 1 from 10 to 20. This is a significant result considering only a small amount of polymer has been added. .A_mems_ eas> no u .Msa was am .msa eons nouaausmme .mcosussom Hoaxsoa mousse mo 30am been usoasnasu eoao~o>oe xaasm now some nouomm soauoflum .H shaman . I U c .8— 803 ooov n 2 83 .< . . .o— ‘4 d d d d m m m - d « ‘ " — I. - . . to - < m2 e u “at: as. o . .2... 5:55.158... ‘0 1 O 1 II ‘1‘ O D I a m 4 s v 1 s SN... .. < was 3 u met: > IL 3 38; as {as . o . us So can b . H o 0 ca .8 ‘w n. A 2. 83 a: e .leaS. . a . fit: 8. can can I H coeososo a $ch .. .82 I . 3. e3 3.. 1 a . on 3. can and s a > . 1 as. 1 as... as 6&3 as... Seam H 78; is on: o .. .88 . O< L 1 .83 . In .2 The abscissa is the Karman number A: Ref“2 = 21/2Dm/v (1.4) where v is the kinematic viscosity and D is the pipe diameter. Physically, the Karman number is proportional to the ratio of the largest length scale, D, to the smallest length scale, v/m. From Figure l, the Harman number ranges from 102 - 104. Note, that (l/f)“2 is bounded between two curves. The lower bound is represented by the Prandlt-Karman Law, which corresponds to the behavior of the solvent. The upper bound is often referred to as the maximum drag reduction asymptote. It is this limiting behavior of the dilute polymer solutions which has been termed the asymptotic effect. As the polymer is added to the turbulent pipe flow, the profiles of the dilute solutions start to deviate from the Prandlt-Karman Law. The approximate point at which the profiles deviate is where an apparent onset point originates. An apparent 'onset’ is defined by a critical Karman number, Nc . For small diameter tubes the apparent ‘onset’ wall shear stress Th° is independent of diameter as indicated by the inset of Figure 1. For the data shown in Figure 1, the onset wall shear stress is approximately 7 dynes/cm2 for u = 1 cp. This noteworthy result has been known for many years (see Virk, 1975). Virk [1975] and others have hypothesized that Tu° is determined by an intrinsic Deborah number defined by «n = V1:- (1.5) where A is the fluid relaxation time and Tr is a flow relaxation time. This result can also be written as In ED. R/ZTF‘.’ (106) where De = Av/D2 and 19* = TFUsZ/u (1.7) Note that De I? = ZXI'u/u (1.8) is independent of diameter at onset. The dilute polymer solution profiles from left to right in Figure 1 represent increasing diameters. The diameter effect is observed by fixing the Harman nuflaer. One can move frm the Prandlt-Earman Law to the maximum drag reduction asymptote by decreasing the diameter. . Logan [1972] and Seyer and Met-er [1969] observed the effect of polymer on the correlation coefficient, - _l/2 A. This implies that a relatively long time is required for a limiting stretch rate to develop, which may not occur in the well region of fully developed pipe flow. 1.2 Objectives of This Research The objective of this research is to develop a model which contains the essential features of drag reduction. In particular, the theory should exhibit the asymptotic, onset and diameter effects. Also, the particular parts of the theory which give rise to. these features of drag reduction will be ex-ined. Crucial assumptions are made about the spatial and tenor-a1 scales appropriate to particular regions of fully developed pipe flow with or without the addition of polymers. Some experimental evidence from the literature support these ad hoc conjectures. The universality of certain scaled statistical properties of the flow play an integral part in the development. The fluctuating radial intensity made dimensionless with the friction velocity is assumed universal and is represented by a three part model (see Chapter 3). Also, the dimensionless time groups associated with the friction velocity and dimensionless time groups associated with the relaxation of turbulent fluctuations are scaled with inner wall parameters and are assumed universal. The utility of these hypotheses, assuming they hold, is that they permit the use of limited hydrodynamic information about the flow so other statistical properties can be predicted based on the modal. Parameters which arise in the development of the theory will be estimated using some data for Newtonian and dilute polymer solutions. The methodology and resulting values are discussed in Section 3.2. Some aspects of drag reduction will be explained using the stress relaxation model, and a scale-up method is developed and tested against some experimental data. 1.3 Rheological Models for the Solvent and the Dilute Polymer Solution Different rheological models are used to describe the solvent and the dilute polymer solution. A Neittonian fluid is used to describe the solvents behavior since most of the data from the literature used when estimating the parameters (see Chapter 3) had water as the solvent. A Maxwell fluid is used to describe the dilute polymer solution. This follows the studies of Patterson and Zakin [1968], Ruckenstein [1971], Benn and Marrucci [1971], McCain [1973] and Landahl and Dark [1975] who also used a Maxwell model for dilute polymer solutions. Mechanical analogs consisting of spring and dashpot components may be seen in Figure 2. The Newtonian model is a characteristic viscosity, ,1; whereas, the Maxwell model is a codination of two mechanical devices: a dashpot and a spring. These mechanical analogs would translate mathustically to the following equations. For the Newtonian fluid 2: = 211$: (1.10) \\\X\\\\\ \\\\\\\\\ T = u? T + Aar = u? Newtonian at where A = u/G Maxwell (l-D) x, /’ <2) = (x1)33 \. / "3 =‘1X1. t<0 T13(0) = 0 T13 = u?, Newtonian Fluid i § -1 0 2 3 -l 0 2 3 t/A t/A Figure 2. Response of a spring and dashpot model to a one- dimensional shear flow. where T is the stress and S is the strain rate. For the Maxwell model, :1: + tag/at = 2u§ (1.11) where b/Bt is the upper convected derivative of Oldroyd (see Astarita and Marrucci, 1974). u... - asco- bg/ot = 3‘:l__'/3t + 11-ng - Vu-T - :1; (93V (1.12) If A = 0, the equation reduces to the Newtonian model. A and n are phenomenological coefficients independent of the flow field. The time derivative in Eq. (1.12) may scale with gc/Ih in the near well region where large eddies stretch elements of fluid. If 1 (< pc/lh, (1.13) then for statistically stationary turbulent flows it follows from Eq. (1.11) and the above inequality that (see Penix and Petty, 1986) (2:) =‘ 2u< >\«:sx commas mmosm assumsoo ll 3 .30am some voooHo>oe sagsm em :oflwou figs: .m sesame m X HQ w 3 WSOUQH an m > L> Am! m m u w + hoses nommsm Amlllllu a /V on n+0 ATflflflllfllt ax \\ AW Amsv mm. u a me. u e noxsm Hmfiusosu m u o u "x cm Q< I +wv + o v + v \\ :oflwoz osou A” Ar m 12 x1+ < 30) and, the inertial layer (30 < xv < Oc"). The core of the pipe is defined from w < x1+ < R’. Near the wall the kinematic viscosity, v, and the friction velocity, the, are used to scale the statistical properties of the flow field. The constant stress region is assumed valid from the wall to X1* = 612*. A model for as will be developed momentarily. The flow is assumed statistically stationary and statistically homogeneous in planes parallel to the wall. Consequently, (In) depends only on x1, the distance normal to the well (see Figure 2). The mean velocity is assumed to scale with the bulk velocity, us, in the outer region (Seyer and Metzner, 1969): %—E§%(xi) as %(X1) 6::+ < 70" ( RI (1°17) This result plays a key role in the development of oat, which depends only on the Harman nuflaer (an assumption). If a power law model is chosen for 6r}, then 6:” = a N (1.18) where a and b are universal constants and A is defined by Eq. (1.4). For a Newtonian fluid (see Virk, 1975), m0 (x1) = 5.7510g(X1*) + 5.5. (1.19) However, at x:* = but defined by Eq. (1.18), Eq. (1.19) becomes u*°(&:*) = 5.7510g(a N) + 5.5. (1.20) u*°(6c*) may also be written as “be u+°(ac'+) = (sumo/mo) (1.21) Therefore, by codaining Eq. (1.21) with the Prandlt-Earman Law, (1/2)1/2(0b°/us°) = (1/f°)1’2 = 4.010g(l\) - 0.4, (1.22) 13 it follows that 0130”“) _ 5.7510391 m + 5.5 us° ‘ 21’2[4log(/\) — 0.4] (1.23) With (5c*) as ub° (see Figure 5 of Sayer and Metzner, 1965), the two empirical constants 'a’ and 'b’ can be estimated as a 23.0881 and b a .9838. Although in general ac+ depends on the polymer additives through the friction velocity, the foregoing analysis indicated that b s 1, which implies that ac+ is independent of the polymer additive. In the constant stress region we further assume that the intensity of the normal fluctuating ‘velocity made dimensionless with us is universal $211132 5m ’2 > (“*o)2 *‘ “,2 (1.24) Although Eq. ( 1.24) may only be approximately true, it nevertheless provides a means to relate turbulent properties of a viscoelastic fluid to the behavior of a Newtonian fluid. In the near wall region, we assume that the universal model for the normal fluctuating velocity is r Ax1", 0 < X1+ < bv‘ (1.258) /u:u2 = ( BX1*3, 611* < M1+ < 55+ (1.25b) Cxfl, 56* < X1+ < 619. (1.25c) A The model for the viscous subregion arises from theory by doing a Taylor series expansion of the continuity equation of an incompressible fluid, applying the no slip condition, and retaining the first nonzero term. The result is X1t‘ (see Monin and Yaglom, 1971). The models for the buffer and inertial layers are well established for a Newtonian fluid (see Virk, 1975). The coefficients of the above model are fitted to experimental data (see Section 3.1). The width of each sublayer was defined in Figure 3; however, the values of each is slightly different 14 from the classical values due to the continuity condition imposed on the model (see Section 3.1). Two time parameters, on and nu, characterize the relaxation of the autocorrelation function for the normal fluctuating velocity. The model used in this thesis is I I A A A I - e (111 2>(X1) (1.26) 0 , otherwise an and 1111 introduced in the autocorrelation above are defined in Figure 4. nu mathematically represents the cut off time of the autocorrelation and flu gives the intersection of the initial slope of the autocorrelation with the time axis. We assume that the time parameters can be scaled with inner parameters as follows 1311(us°)2/v (1.27) 1111* mrmz/v 1311(us°)2/v (1.28) ‘nuusz/v 1111+ Fortune and Hanratty [1978] and Herman [1978] support the hypothesis that turbulent time parameters in the well region scale with inner wall parameters and are universal. Also, Achia and Thompson [1977] support the hypothesis that turbulent relaxation times are universal. Two additional time parameters, rue and The, related to arise in the derivation of the axial fluctuating velocity (see Section 2.3). The and rue are defined in Figure 3 and are also assumed universal when scaled with inner wall parameters: 1313(us°)2/v (1.29) mama/v ma’ 1113* manta/v 19u3(us°)2/v (1.30) An important final point to note is, if one follows a horizontal path through the profiles of Figure l, (l/f)1/2 is the same; however, the Harman numbers would change such that I? > A. At the same Reynolds 15 .cofiwoh Ham: use: on» em meowumHoMHOU cuss usoasnusu pom masses sawumxmfiom .e ossmHm > > 9 > H2 H2 «2 a: a: a: «0.130 P .1: Ni: P N P Notzo P N #3 P ..l. P mwmomuomxz aemno>wsz mzp mze o mzp szp a .\ l .\ H A H / / m ./(. ./\\ move: seasoz assess H255.15 .~: 0 . P5.1. - Asxvxw\ssv AflwamV\~sauqmvxssv 16 nuwer, Re = Dub/v, recall that Ann/(Mum). (1.31) It therefore follows that Us° > Us. The scaling hypothesis for 1N1, Eq. (1.28), may be written as THl = Ta1°(Us°/us)2, (1.32) so according to Eq. (1.4), it follows that fill = m1°(Tw°/Tw)2, (1.32) Therefore, if rho/r. >> 1, (1.34) the turbulent relaxation time for the dilute polymer solution is much larger than its Newtonian counterpart. CHAPTERII THEORY 2.1. Mean Flow Field For a fully developed turbulent flow, the balance of forces in the constant stress region gives the following relationship for the total stress v 9-32.12- - (L)] (2.2) (see Monin and Yaglom, 1971). The Reynold’s stress, which represents the transport of fluctuating axial momentum toward the well due to the fluctuating normal velocity, can be written as (see Tennekes and Lumley, 1972) (m ’11:: ’> = -w (X1)? (2.3) where we is termed the 'eddy’ viscosity; Eq. (2.3) should not be interpreted as a definition for v. (c.f. Eq. (2.26). Substituting Eq. (2.3) into Eq. (2.1) and integrating the result, the following expression for the mean velocity made dimensionless with Us results xr+ 4' 0+ = (U3)(X1)/Us = f 1 EXIw/v (2.4) ' o 17 18 Applying the boundary condition (us) +ub as x:* 7’ av and substituting Eq. (1.3) into Eq. (2.4), the following expression for (l/f)“2 is found O’c (l/fwz «(l/2W2 f -i-i-‘¥‘;_17;- (2.5) 0 Thus, once a specific form for the 'eddy’ viscosity is derived for the Newtonian and dilute polymer solutions, the extent of drag reduction can be calculated by O '6: ____dx1+ 1 + w/v o = (2.6) where 6c* is given by Eq. (1.18). 2.2 Derivation of the 'Eddy’ Viscosity for the Solvent and Dilute Polymer Solution The equation of motion for general fluids is 39 3t “p.91 = v.2-vp (2.7) For an incompressible fluid, the continuity equation is “u = 0 and 9(9) = 0. (2.8) Ensewle average Eq. (2.7) and subtract the result from Eq. (2.7). Further, substitute the Reynold’s decomposition for the velocity, the shear stress, and the pressure into the equation. Note, for a fully developed flow in the x3 direction any point mean quantity depends on xr (u) = (us>(x1)§s (2.9) 19 Use Eq. (2.9) and simplify. The resulting equation can be written as 311’ 311’ Mus) psi-+§;3-+U1EX‘T“-ea+q“1"(llqwl) = v. I—vp’ (2.10) Eq. (2.10) is a vector equation describing the fluctuating velocity field for any fluid. The equation contains both mean and fluctuating contributions of the turbulent flow. The 'eddy’ viscosity for the dilute polymer solution will be derived. For a dilute polymer solution, the continuity equation for an incompressible fluid (Eq. (2.8)), the equation of motion (Eq. (2.10)), and the rheological model (Eq. (1.15)) can be combined to obtain an equation for ua’(x,t), the fluctuating axial velocity 313135;) = [1 + Aa/at][u1’d/dm + 113] (2.11) where 12 [1 + A3/3t][a/at + + (l/p)3P’/3X3 (2.13) Using the method of Green’s function (see Morse and Feshbach, 1953) an explicit form for u3’(x,t), the fluctuating axial velocity, is T wont) = -f difd’shomlgfim + :1 °° A 3/3t][lu ’d/dX1 +1113] (2.14) G is a Green’s function associated with the operator & an inhomogeneous, linear, hyperbolic operator, on the domain 9. If A = 0 (i.e. the Newtonian case), the operator 113 parabolic. The Green’s function is spatially peaked for short-times (i.e. lt-tl small). For t >> 12‘, G relaxes to zero. Because the flow is statistically stationary and fluctuations are not induced at the boundary, the effects of the 20 wall are neglected. Therefore, the unbounded Green’s function is used (see Morse and Feshbach, 1953) A LGdR= 411), t > 0 (2.15) R. Also, only two sources for axial fluctuations are retained by Eq. (2.14): the inertial coupling of the radial velocity fluctuations . 95222 “1 (Est) (1311 (2.16) and the nonlinear interaction contained in he. In this thesis, we assume that the higher order terms in he on a short-time scale are small compared to Eq. (2.16). Consequently, Eq. (2.14) becomes 1 t A ' ,t = - <11; <13 QMM 1 + 1 a at '- 3“”) 2.17 us (2! ) [0.1.6 4" ( / )m T ( ) Ensemble averaging Eq. (2.17) with u1’(x,t), an expression for the Reynold’s stress is obtained t fir f. f Austria) (ur ’us ’- (X1) = - dt (19. (l + 13/») ‘~ 41v -Q a (2.18) ‘0 -dd 1 -- --- ’ 1 mm .ao ’ .- -- ‘nl (2.19) and an acceleration coupling with the mean velocity gradient I A ’AA AKu; (§,t)a/at u1 (§.t)>é 0, the viscoelastic nature of the fluid slows down the mixing process with a finite propagation velocity, (A/v)1/2. Consequently, an apparent ’onset’ occurs because the local space-time domain over which molecular transport affects the Reynold’s stress simultaneously depends on the space-time domain over which the radial velocity fluctuations are statistically correlated. In the framework of the theory, ru1’ and m1+ are universal constants and are about the same magnitude as A’. As the wall shear stress increases, the turbulent time scales decrease; therefore, at a certain critical wall shear stress, A? ~ 1M1’ ~ rn1*, the viscoelastic nature of the fluid hinders the turbulent mixing of axial momentum. For small $5 and recalling that the flow is statistically homogeneous in planes parallel to the wall, a spatial smoothing is performed and the resulting autocorrelation for is assumed as follows A AA - (“1 I(?S’t)u1 I(§’t)> = (“1’2>(X1)e- It tl/THI’ |t—’t‘| < m (2°21) Substituting Eq. (2.21) into Eq. (2.18), an ’eddy’ viscosity model for the turbulent flux in the near well region of the form ve = (m ’2>(xr)7c (X1) (2.22) is obtained where the transport time Tc(X1) is defined by 1.1 Tc (X1) = (1 + A/Tu1)f e-‘r/‘nuxou ¢)dt (2.23) o 2 22 and the viscoelastic memory kernel K(x,r) is [{(xh?) : fW-.. . (2.24) 3 410 TC (x1) rapidly approaches its limiting value of rc(°) for x1 >> (114111)“? since the Green’s function is moving in a frue of reference with the local mean velocity and is spatially peaked over a spatial domain of the order (an 101"". Thus, we assume that ‘l’c (X1) can be approximated everywhere by Tc (0), even though 1:. < v in the viscous sublayer. The resulting model for u. is u- : (x1)rc(°) (2.25) where Tc(°°) is given by Eq. (2.23) with H(xr,?) replaced by 1100,?) = l - e_4/x (2.26) If A = 0, the Green’s function is viscous and “6.7) = 1 for all values of 7. Substituting E05?) = 1 into Eq. (2.23) and integrating, the Newtonian eddy viscosity made dimensionless with v is found to be w°/v = m+(1 - (“Mm '2>(x1)/u.2 (2.27) where (x1) is given by Eq. (1.25a-c). Inserting Eq. (2.26) into Eq. (2.23) and carrying out the integration yields (see Appendix A) an expression for Vb U» = R1vw° (2-23) 1.: (1 - (“Wm whereR1=1-¢e __ A1 (l-e‘") and w° is given by Eq. (2.27). The function R1 (see Section 2.3) is very important in the predictions of statistical properties. 23 2.3 Axial Intensity Models for the Newtonian and Dilute Polymer Solutions em-a-no—oo- A..."- u” «ammu- The derivation of (us ’2>(x:1), the axial fluctuating velocity is analogous to the derivation for the Reynold’s stress model given in Section 2.2. Beginning with Eq. (2.17), multiplying through with us’(x,t) and ensemble averaging, the analysis is the same except for a change in subscript as follows (us ’2>(X1) = —(xr)(l + A/rua) t _ .A " _§_ “ 932.32. [one Ir 'rI/Tusdt £4” d9. dx; (2.30) An autocorrelation used in the smoothing approximation for . A A '. _. _. (“I I()S’t)u3 I(§’t)) : (u1’u3’>(XI)e [t tl/TNG’ lt-Tl < m (2.31) Since the assumption made in the derivation about the Green’s function and the mean velocity equation are the same in the analysis here (see the derivation in Appendix A and the analysis in Section 2.2), the following expression for the axial fluctuating velocity is obtained (us ’2>(x1) = -R3m3(1 - e-M3)(X1)Q_<_B§_>_ d“ (2.32) _ __ ’043 _ ’Gla/d where Rs - 1 «be (1 e —¢4 As) (2.33) (1 - e 3) here 043 = ms/rua and “A: = A/rna. (2.34) Insert Eq. (2.3) into Eq. (2.31). The resulting expression contains d/dxr for which an expression may be found by codaining Eq. (1.1), Eq. (2.1) - (2.3) and Eq. (1.3). Substituting for d/dx1, the final 24 expression for the axial fluctuating velocity made dimensionless with Us andvis < ’2) - ‘53:; = 1113*(1 - e “"3)Ra (1 rC/vV (2'35) The Newtonian result is obtained by setting A = 0 or, equivalently, ch = 0. The assumptions made in the analysis for the dilute polymer solution hold for the solvent. Therefore, the Newtonian expression is as follows £99.29 _ Im°2 “3) “OZ” (2.36) ruao+(1 — e- (1 + V0°/D)z Note, the axial fluctuating velocities are functions of the four dimensionless hydrodynamic relaxation times 1111*, nu’, 1743* and 1113*, and xr’, the distance normal to the well. 2.4 Discussion of Qualitative Features of the Theory The essence of the theory presented in this work lies in the understanding of the physical and mathematical nature of R1 = 1 - a e-m‘ (1 — e-.“/¢A1) N (l _ e-QH) (2.37) where mu = “nu/nu and ‘Ar = A/nu and i = 1,3. (2.38) The parameters nu and nu represent the hydrodynamic relaxation times of the space-time correlation for . As noted in earlier studies by Lyons and Petty [1984], nu controls the ’onset’, whereas, nu determines the maximum extent of drag reduction. Figure 5 shows profiles of R1 which can be broken down into two regions: 1) the hydrodynamic controlling regions 0 < ‘Ai < l; and, 2)the molecular transport controlling. region “11 > 1. As R1 + 1, no drag reduction occurs. In the hydrodynamic region, the profiles increase most rapidly 25 .xnoonu scauexeflou «woman one an manganese me as mean» uuoamseeu o>wum~oa on» so mesa» :oHueeron uamsmuucw mo uoommo one .m shaman H o o 8.8. 8.2 ka 8 . o. o axe\Amsvesoemzva\"3‘23v- n n_xUA\ms‘ssv I. [TIT r1 5.: pd 26 for certain sets of an and «A1 provided an < 1. As seen in Figure 5, the slopes of the profiles in this region are about the same. Also, for on > 4 and “Ai < 1, there is essentially no drag reduction (i.e. R 1). However, if «Xi is fixed, one can move from no drag reduction (R 1) to complete reduction (R = 0) by decreasing mu. This further implies from Eq. (2.38), that if nu is fixed, nu would have to increase in order to go from no drag reduction to complete drag reduction. A similar effect occurs if one fixes an and moves along a horizontal line cutting through the profiles (e.g. mu = 0.1). One can go from no reduction (R = l) to maximum reduction (R = 0) by increasing “At In the molecular controlling regime (“Xi > 1), the profiles have essentially reached asymptotic plateaus. In this regime, if mu is large (i.e. on > 4) there is no drag reduction (R = 1). However, if on decreases there is an increase towards maximum drag reduction (R = 0). Note, this result is essentially independent of the value of ‘Ai' provided “M > 2. As noted in Figure 5, R1 is a function of 1'c1(°°). As mentioned earlier in Section 2.2, a key result of this theory is rcr(°°)/ru1 given by Eq. (A.l3). Figure 6 illustrates that the behavior of the characteristic transport time reaches a lower asymptotic curve given by [1c(°°)/Tfll]min = 1 — (1 + anew” (2.39) The upper limit is obtained by setting “A1 = 0, rc(°°)/1'u1 = 1 - e“'”. (2.40) By taking the derivative of the difference between Eqs. (2.39) and (2.40), setting the result equal to zero and solving for an, the maximum difference occurs at mu = 1. If on = 1, then nu+ z nu+ and 27 .fi_ewas_ sagas as“ seesaw mucosa semuexaaou mmoaum one an empoweoua mm moss» uaoamsesu usomsnusu one so mesa» :ofiuexmuoh camsauusfi one mo uoommo use .0 ousmwm M: w: w 0.0— P\ P N 25 - d 1 d 1 d N 1 d 1 m.o u H45 Ams. R1. Looking at Figure 4, it follows if “Xi = 0°, then me >> 011 for R3 > Hi. This further implies that the universal 29 n." 225 .xuoocu :oHuwxons mmosum on» up nouowvonm mm u:oauammooo cofluafiouuoo may co moamom assesses mo uoommo one .5 ousmam ~2p\4uH R1. Note, r = 1' provided TH1+ = Tus+ < 1u1* = 1aa* and drag reduction according to this theory is still possible. This is easily seen in Figure 5. Thus, for drag reduction to occur, it is important that an << 1. Hence, if In = nu’l‘nu‘, then ‘rm+ << nu‘. This is a new significant conclusion. The correlation coefficient changes very rapidly over a narrow range of a)” (i.e. 0 < ‘11 < 1.0). For «M > 1, the correlation coefficient approaches an asymptotic value and is essentially independent of a when conducting an experiment to test the predict- 11' ability of this model for m, working outside the range ‘1 > 1, the 1 effect of “A on y is lost. Note, the profiles are the least 1 distinguishable for 0 < a < .5. Therefore, working in the narrow A1 range of 0.5 < 1 < 1.0 offers the; best resolution between the curves M and yet retains the changes in the curves. 2.5 Method Used to Study the Theory One main advantage in performing the numerical calculations is that all the solutions are analytical. Some of the calculations are in algebraic form directly for the Newtonian and dilute polymer solutions such as: 1)(xi), the radial fluctuating velocity given by Eq. (2.31); 2)tn/v, the eddy viscosity, given by Eq. (2.28) and Eq. (2.27); 3)/m2, the axial fluctuating velocity, given by Eq. (2.35) and Eq. (2.36); and My, the correlation coefficient, given by Eq. (2.41) and Eq. (2.42). Other expressions of statistical properties for the mean velocity /Ih and (l/f)U2 are given by Eqs. (2.4) and (2.5) respectively. These equations each contain ve/v. Thus, there are three integrals resulting from the three part model for the radial fluctuating velocity 31 given by Eqs. (1.25a—c). The integrals each have analytical solutions which follow. The first integral is given below dxi" fl + ASile" (2'43) where A is the coefficient of X1*4 in the radial intensity model (see Eq. (1.25a) and 81 = 1u1(1 - e‘1"1). R: = 1 for Newtonian solution (2.44a) Eq. (2.37) for dilute polymer solutions (2.44b) The integral is found in an integral table (see CRC, p. 019, 1981) to be __g§___ _ 5_ 10 x2 + 2kx + 2k? + tan-1 __gg§____ a + bx“ ‘ 2a 3 2 - 2kx + 2k2 k2 - x2 for ab > 0 and a/b = k4 (2.45) The second integral is dx:* _ f1 + 881mm” ' (2'46) where B is the coefficient of x1*3 in the radial intensity model. The integral is given by (see CRO p. 020, 1981) f dx k [ (x + k)2 (3)1/3tan'1 2x - k )] —,--——- = -- log( ). (- - (2.47) a + bx3 3a (k2 - kx — 2x2) k(3)1/3 where a/b = R3. The final integral is din+ l + CSiRtXi’ (2’48) and the solution for the above integral is dx-_1. a + bx - b 103(8 + bx) (2.49) Table 2, in the following chapter, defines all the variables of integration in terms of the physical parameters. Computer programs on an IBM PC under Dos 2.0 were developed using Fortran 77. A list of the computer programs is given in Appendix 3. Since analytical solutions are given for all the statistical quantities being calculated, there is no error associated with any algorithm used except the machine round-off error. CHAPTER III PARAMETER ESTIMATION The stress relaxation theory has ten parameters which must be set before any statistical properties can be calculated. The universal model for the radial fluctuating velocity was developed in Section 1.4 with the result that, An“ 0 < xv < 6“ (3-13) '2 (III. >SX12 Bxl+3 6" ( 3Q" ( “" (3.11)) m2 Cm+ Gb" < x:* < 6c* (3.1c) where 6c’ = a1? = .OBA“. The above model contains six unknown parameters: A,B,C, 6W, aw and 6c‘. The model for 6c’ has two additional unknown values a and b, but as shown in Section 1.4 the values for a and b are found independent of the development of the relaxation theory. There are four relaxation times nu', 1311*, 1113*, and ma’ associated with the two autocorrelation functions given by Eqs. (2.21) and (2.35) which also need to be determined. In the viscous subregion, the theory resulted in a x:*‘ dependence with a leading coefficient A. Monin and Yaglom [.1971] estimated A to be about .000064. The inertial layer has an xv dependence with an unknown coefficient C. Using experimental data in the inertial layer for the 32 33 axial intensity (estimated from Figure 9 of Virk [1975]), a value of C = .0083 was found. Because data in the buffer region are difficult to obtain, another strategy was applied to set B, bv’, and aw. If B were known, then 6:” and 60* can be uniquely determined by making the radial fluctuating velocity a continuous function; however, because of the mathematical form of Eqs. (3.1a-c), the first derivative will be discontinuous. The parneter B can be found by relating (1/f°)1/3, defined by Eq. (2.5), to the following result (see Tennekes and Lmley, 1972) (l/f")1/2 = 4.0logl\- 0.4 (3.2) The integral in Eq. (2.5) contains the ‘eddy’ viscosity given by Eq. (2.31) and 5c* which is already known. However, as seen from Eq. (2.31) and Eqs. (3.1a—c), the ‘eddy’ viscosity depends on A, B, C, or, 6M, 56 and Si where 81 = 1111*(1 4 e-ml) (3.3) At this stage, B, Si, 5H, and av, can be determined simultaneously by equating Eqs. (2.5) and (3.2). Because of the nonlinear dependence of (l/f")1/2 on these coefficients, the strategy is necessarily trial and error. First, 31 can be estimated by examining experimental data for the 'eddy’ viscosity in the inertial layer (see Fig. 17 of Virk, 1975) and using Eq. (2.31). Extrapolating the line of slope one back to the axis in Fig. 17 of Virk an intercept of 0.4 is found. Further, taking the log of Eq. (2.31), the corresponding intercept would result in the following expression 0.4 = cs; = 0131*(1 — e“‘"‘) (3.4) Because C = 0.0083. a value for Si was found to be about 48. The trial and error method involves guessing a value for B and then 34 determining 6N and am by continuity. If the calculated value of (l/f“’)1/2 and Eq. (3.2) do not agree, then B is adjusted and the process is repeated. The fitted Prandlt-Karman curve was found to be sensitive to small changes in B. Figure 8 illustrates that B affects the vertical shift of the Prandlt-Karman Law. A value of B = 10’s, 28 was found to fit the data which corresponds to an = 0.1 and db‘ adequately. Figure 9 shows the universal radial intensity model employed in the research. There is good agreement up to xfi :3 80 with experimental data. Figure 10 shows the resulting ‘eddy’ viscosity profile which was used to fit the parameter C as discussed earlier. There is fairly good agreement. Eq. (3.4) gives one relationship for nu+ and nu’. Because Si is assumed to be a constant throughout the wall region, no additional experimental data for the Newtonian case are available to determine nu’ and 1111* separately. Therefore, dilute polymer data at maxi-tn drag reduction was used to find another relationship for nu’ and 1111*. Obviously, this approach assumes that these parameters are universal (see Section 1.4). At maximum drag reduction (i.e., «A, = 0), it follows from Eq. (2.33) that "tlu 31(011 ,0) = 1 - ails-:51- (3.5) l - e Therefore, the theoretical estimates of (l/f)1/2 given by Eqs. (2.5), (2.31) and (3.5) can be equated to the following empirical result developed by Virk (l/fo)1/2 = 19.0log/\- 32.4 (3.6) 35 .3em cease: -ufiecmsm one no mam we assume oz» mo xesum ownuoseuem .m ousmwm .oocooo— .oocco— < .0009— 1-1141 1 l1 1 1 .1141 1 1 I I -1I+I‘1 I I 1 to .. <85; u a)... 33 :oEcoxluficEm 0 ~29 mnHZGuo u HU+ u mum mm .mHm meme ~”=\A~.M=v nooom H: o mmmm. n ~u=\A~\w:vAHZd-o u HV+ P u >\o9 vooo. . o mooo. . ...\w>.+ ac . ----mmmu--- gfi u ~\HA9M\HV 0 +0 .o« .On z“Ls/I) 36 .coflmou Ame: use: one a“ Hoses xuwmcoucfi Heaven a .oo. + x $35 is cozoaeom mono 53::on s acozom 0 +00 v+ix v mm .ix .-o2xm.w . mm v.1x v as. m+ix m-ono.H u m : o~.v+_x v o :+~x muesxe.c A~.H:v .m ossmfim .op fio N\HA~. 37 .xuamoums> :xeum: cmsaoszoz songs; .o~ ouswsm .aao, .oo— »x .c. .— 1 1 1 I 1 1 I 1 -1 1 1 1 1 1 1 1 - 1 1 1 I 1 1 1 1 5.0 killll 1 0—6 m H .flmsmsc sua> u mo : .mE scam voumeumo even—D u . 33: i; .4 8. Aemmav “cases a . o . >\o> a... m I I I I m L 0°60— lllll I In 0902: 38 This procedure gives a value for mu = 0.425. Figure 10 shows the sensitivity of the theory to variations in In; the experimental data used by Virk to develop Eq. (3.6) falls within the hatched region of Figure 11. Table 1 shows small changes of nun+ and 1111’. From Table 1, it is seen that slight changes in I41 cause noticeable changes in 1111 but very slight changes in 1111*. As seen in Table 1, if 011 = .425, then 1111* as 139 and nu’ as 59. TABLE 1. Effect of In on 1111 and 1111 In 1m 1m. 0.375 153.5 57.6 0.400 145.6 58.2 0.425 138.6 58.9 0.450 . 132.5 59.6 0.475 127.6 60.3 1143* and 1113*, which were introduced in the model for [see Eq. (2.35)], can be set by either using axial intensity data or correlation coefficient results. Logan [1972] found the Newtonian correlation coefficient in the near wall region to be about 0.44. A theoretical estimate for f is given by Eq. (2.46). It is noteworthy that f is independent of the model for (X1). However, I“ depends on all four ‘universal’ time groups. For the dilute polymer solution at maximum drag reduction, Logan [1972] found that y = 0.2. By Eq. (2.45), a second relationship for the time groups is obtained, viz., - Rllflu. 1). 2 3 V" V[Ra(m3’ 1A13L)] (3.7) 39 .uoommo ofiuoumimm oz» :0 Ed mo woommo oflpuoamnem .3 0.5m: 6000— < .000. 6.0—. q I I I I I I I I - I I I 1 I I 11 o 33 ceEcexlzucean. .2 cm .8 2 $3: is .8 mg. n .9 . Ea N203 mt. - $3: 33> -l - .o... Etc: u Eu 3 32% u 2mg " < .8 40 Even for dilute polymer solutions, the theory predicts that y is independent of the model for (x1). If v < V for «A1 = 0, then it follows that the pairs of 'universal’ time groups (1111*, 1111*) and (1113*,1113*) must be different. However, if 1111* = 1113* << 1111* = 1113*, the theory allows up = wand yet v. < w°. Thus, within the context of the model developed here, Logan’s observation is not necessary for drag reduction (cf. Virk p. 634,1975). In this research, the values of 7113* and 1113* were found by setting «A, = O and ‘13 = O in Eq. (3.7). 1113* was adjusted until y as .20 with I“ as 0.44. This procedure yields I13 18 4.6. 1113* was then calculated from Eq. (2.46) with 1111* ~ 139. A value of 1113* as 250 was found. Because 013 as 4.6, then 1113* 8 1168 was found. Table 2 gives a su1-ary of all the parameters estimated based on experimental data. Table 2. Fitted parameters in the stress relaxation model. Parameter Value Equation A .000064 B .00001 An“ OSX1* 5611* C .0083 = B X1*3 6v* SX1* S 611* C X1* 611* Sn“ S 01: 611* 0.16 611* 28.28 a .088 6c* = a AP b .984 1111* 139 (x1)e'?/TH1 1111* 59 ’1? < 1111 y I I “ _ I I -?/T 1113* 250 - (X1)e H3 1113* 1168 ’1? < 1113 CHAPTER IV RESULTS 4.1 Calculated Statistical Properties Chapter III defined how all the unknown parameters in the stress relaxation model were set. The universal values in Table 2 were used throughout the following calculations. The mean velocity was not used directly to set any parameters; however, as mentioned in Section 3.1, the mean velocity is given by Eq. (1.3). The mean velocity profiles for the Newtonian fluid are shown in Figure 12. The dashed line represents the classical mean velocity profile in the viscous subregion. The model predicts the same curve up to xr’ z 9. The Newtonian law of the wall was calculated by combining Eqs. (2.4) and (2.31). The linear profile agrees well with the empirical expression 0* = 2.51n(X1*) + 5.5. (4.1) The two profiles for the buffer and inertial regions are often linked together in the buffer region at about X1* as 11 (see Tennekes and Lumley, 1972). The model developed here also gives a continuous variation across this region. The dilute polymer curves were calculated from Eq. (2.4) with Eq. (2.32) inserted for vu/v. The profiles almost all lie on the same curve in the viscous and buffer regions, but at X1* as 28 there is a more noticeable change in the profiles. The curves fan upward with increasing intrinsic Deborah number, «11’ and rapidly reach an ultimate profile. This follows from Figure 4 by noting that if In is constant, 41 42 .000— .moafimoum AuwooHo> :moa mouowuoum .NH ossmwm “x .e. O P .1111 m.m + .ixcsm.N :33 35 so so... chBBoz .1111I1 1 1 1 .1111 1 I I .58.. «35:5 32.2.5 33; is .528. o comes—gem no.5 52.5.6: 0 +dzpm\~ 0 stems from the viscoelastic behavior of the dilute polymer solution, which hinders the effectiveness of the turbulent transport of momentum. Also, all the 'eddy’ viscosity profiles break at xx’ = 28. This is an outcome of the model for (xi) always breaks at an as 28. The decrease in magnitude between the two extreme profiles: (1) the Newtonian curve (i.e. “M = O) and (ii) the dilute polymer curve (i.e. “AI = 0) is about an order of magnitude. This is a significant decrease in the 'eddy’ viscosity between the solvent and the dilute polymer solution under maximum drag reducing conditions. The axial intensity is a predicted quantity both for the Newtonian (see Eq. (2.40)) and the dilute polymer (see Eq. (2.39)) solutions. Profiles of the axial intensities are seen in Figure 14. The maximum of 44 .xuwmoomfi> .xvvo. Hosxuom ousfiwu onu mo uoommo wouowvoua .mH assuam .8. +5 .2 1‘) 1 1 1 1 1 1 1‘ .msmsg sus> < wNfl+w H: coEBBoz mm and + PN\N<0D fl H:P\< N HKU All]. I llLl L l o; >\ > 96— 960— 45 .xuoonp :owumxmfiou mmonum on» :0 women mogflmonn xuwmcouzfl amwxm wouofiuoum .v~ onswflm .+x .8. .8 .8 .2 .8 .8 .3 .en .8 .o. .o . _ _ . _ . . q . q a _I q o . . m. 450 o 5.8%.: m m.o 3 m... u .ne o.~ o.m s: m: 322...! 8.33:. up... 52.1.: s .520...“ O +~zp~\~ered that no parameters were set based on axial ,intensity data. The Reynold’s stress is derived by codining Eq. (1.1), Eq. (1.3), and Eqs. (2.1-2) to obtain an expression for d/dxx and substituting this result into Eq. (2.3) to obtain the following result -(x1) = —‘-"—A’— (4.2) l + Ian/v Therefore, because no/v +°° as x1+ + °°, the profiles shown in Figure 15 will all approach unity for large values of x:*. For x:* say less than 100, it can be seen that all the dilute polymer profiles are always less than that of the Newtonian profile («M = 0). Thus, the transport of longitudinal fluctuating momentum by the action of normal fluctuations is hindered with the addition of polymers. This is clearly shown in Figure 15 for X1+ < 100. 47 .xuoonu cowumxmflon mmonum one :0 women moawmoum mmouum mvuosxom vouowvonm .mm shaman .90. .oa .oo .05 .90 .on .ce .on .o« .a— q q u q q - u a q - .H:p~\~ n I nu E I . bu as n 8.3 o; 48 4.2 Friction Factor Results Recall that (l/f)“2 was used to help determine the 'universal’ time groups 1141* and nu” by fitting the Prandlt-Karman curve and a maximum drag reduction asymptote (see Chapter III). An expression for (l/f)"2 is given by Eq. (2.5) and Section 2.6 discusses how the integral in Eq. (2.5) was calculated. Figure 16 shows the profiles of (l/f)“2 verses the Karman number. As seen in Figure 16, the profiles are qualitatively the same as those estimated from experimental data in Figure l. The dilute polymer curves are bounded between two curves: the Prandlt-Karman Law and a maximum drag reduction asymptote. Thus the theory contains an asymptotic effect shown in the profile where De approaches infinity. The remaining two features, onset and diameter effects are predicted by the stress relaxation theory. An apparent ‘onset’ Karman nuflaer discussed in Section 1.1 was found visually by locating an approximate point at which the dilute polymer profiles deviated from the Prandlt-Karman law. Looking at the profiles in Figure 16 as the Deborah number increases it can be seen that the point of deviation from the Prandlt-Karman law is not that distinct. In these cases a visual estimation was made which allowed the profile to deviate a small amount before an apparent Karman number was estimated. As previously mentioned, onset is independent of diameter. This is true for Deborah numbers ranging from 10'8 to 10" for the stress relaxation theory presented in this thesis. The results are shown in Figure 16 which shows that the model predicts De Ac2 ‘4" 2.7%:1- as 7 for 10-8 < De < 10-4 (4.3) 49 .000000— .xuoonu sofiumxmfiou mmosum up some souomw :ofluoflum wouowvosm .o~ osswfim .00000— < .0000— .000— .00 11141 1 1 1 .11111n 1 1 1 ‘11111 1 1 11 l: 11 33 costoxlgococm J; 339:»: .1. >\ =a~\s~ u ~\smox ~0\>< om mmd +229 +HIP Ill-AllIlllllllllll‘llllllllllllljIIllljnllllllljllljjllIIJIIUIIIIL .- .0N .05 ~\ snm\sv 50 From Eq. 4.3 with u = .01 dynes-s/cm2 and A as .005 s, Tu° as 7 dynes/cmz; the value found experimentally by Virk [1975]. Although a rigorous onset wall shear stress does not exist in the theory, Figure 17 shows an 'apparent’ sudden deviation from Newtonian behavior at Th° ~ 7 dynes/cm2 . This stems from the exponential dependence of R1 on the temporal properties of the turbulence and the fluid (see Figure 5). Finally, the diameter effect is shown indirectly by the Deborah number. As seen in Eq. ( 1.7), the Deborah nui>er De is inversely proportional to the square of the diameter. Hence for fixed 1.and v, as De decreases, the diameter increases. Therefore, one can go from the asymptotic curve to the Prandlt-Karman law by increasing the diameter. This is in agreement with the experimental evidence presented in Figure 1. One of the major results of this research is the successful prediction of scale-up based on size as seen in Figure 18. From Eq. (1.7) and Eq. (1.8) it follows that Des = D¢=:1(Di/I)2)2 ' (4-4) The friction factor data were estimated from Figure 2a of Virk [1975]. The curve for D = 2.92mm was fitted by adjusting the Deborah number De when calculating (l/f)1/2. A value of De = 1.2x10" was found. The remaining two curves were predicted for Deborah numbers calculated using Eq. (4.4). The results are shown in Figure 18. Although there is almost a factor of 15 between the diameter sizes, the predicted curves agree fairly well with the experimental data. The results based on small diameter pipes is encouraging and could prompt further research in this area by testing this scale-up method on larger diameter pipes and on other polymer/solvent pairs. 51 .uoommo :uomso: wouofleoum one .nH ossmwm on NOIN —.0 nolu pd *OIN pd nclu — .O 00in — .0 balm —.0 calm —.0 .1111 1 1 1 1 .111141 11 1 1:11 1 T 1 .11111 1 1 1 .1111111 1 1 “111111 111 1 v o 4 mlo— ’ 1 Filo— . L 0'0— I W'O— 1 4 *l0— . ’ I .fl en. .n:¥:>m . e . .. .. - -l1|-I.-I.P-I|-|:|I- - - -IJ U i951 a - A V... .. .2 e :5 ~>~-~:mm-< ND . 1 INC-ml ..l.. 00 A an n +s2s - QMH u +H=P 4 a .2 52 ..onos cofiumxm.ou mmouum ecu wcflm: woman ..mEm pom sew>mnon mononom eoHUHeohs .w. seawam .0000. < .000. .00. mmo. e N. u-mo. x on . 1 .ON B on X 0.0 .VIOP x N.— ' POD . N\HHM\HU ~fl~a\.oe.ua n «on . ~o\94 u on am H +H2P 4 mm. H +.:p 1 .Nn med new .tE £30505 . I e m .onExm . :03; is. 2&3 on” a Ba 38 . CHAPTER V CONCLUSIONS 5.1 Phenomena Predicted One of the major objectives of this research was to develop a model which describes the mean flow field, yet contains the three features of drag reduction: onset, asymptotic and diameter effect. For many years, the experimental data have shown these three characteristic features. These features were presented in Section 1.1 and the predicted results were discussed in Section 4.2. The general feature of an asymptotic effect is contained in the basic stress relaxation model. A specific value for mu (see Chapter III) was determined so a particular asymptote was obtained which was also consistent with experimental data of a given polymer-solvent pair. The asymptotic effect is controlled by the intrinsic relaxation time 131*. This is confirmed in Table l by the effect the model has on pairs of nu’ and 1111* for different values of on. As previously noted, 131* is very sensitive to small changes in an. while nu+ hardly varied. The onset effect was defined in Section 1.1. Since the point of departure from the Prandlt-Karman law is unclear (see Figure 16) an apparent 'onset’ was visually chosen for lower Deborah numbers by allowing the curves to deviate a small amount before estimating the critical Karman number. Tu° was found to be independent of the diameter as seen in Figure 17. 53 S4 The diameter effect, defined in Section 1.1, is nicely predicted and shown in Figure 18. The effect is illustrated through the Deborah nuflJer. For constant v and A, as the Deborah nuwer De decreases the diameter must increase. Therefore, by increasing the Deborah number (i.e. decreasing the diameter) one can move from the Prandlt-Karman curve to the asymptotic curve. The statistical properties of the fluid which were predicted by the stress relaxation model are discussed in Chapter IV. The theory predicted tan/u < vb°/v, a result known indirectly from experimental data for many years (see Section 4.1), and the correlation coefficient was fitted such that y < f. The latter result is independent of the prediction of the Reynold’s stress. In the context of the relaxation theory, drag reduction occurs independent of how the correlation coefficient behaves. This is in contrast to Logan’s [1972] argtnlent that the decrease in the correlation coefficient implies that the polymer additive somehow uncouples the Reynold’s stress. This area needs to be investigated more carefully. Overall the theory does contain important phenomenological features of drag reduction. Stress relaxation of the polymer interferes with the transport of axial momentum to the wall by the normal fluctuations and drag reduction occurs. 5.2 New Discoveries and Shortcomings of the Model As mentioned earlier, drag reduction is predicted by this theory without the condition that v, the correlation coefficient be reduced. As seen in Eq. (2.41), w is dependent on all four dimensionless intrinsic relaxation times nu‘, nu’. ‘ma’, and flash Since 1111* and flu’ were set by other experimental data, v was set by adjusting tua‘ (see Chapter III). Note, the Reynold’s stress is independent of each 55 other. This is in contrast to Logan’s [1972] arguments on the mechanism of drag reduction. The decoupling from the axial to the radial direction was supported by the decrease in the correlation coefficient (see Section 5.1). A shortcoming to this theory is the predicted axial intensity. The maxima are too high and occur too close to the wall compared to experimental data. Also, the maxima for the dilute polymer solution are always less than that of the Newtonian solution. This contradicts some experimental data. This may be due to the nature of the Green’s function used; the next section will suggest improvements of this theory. The result that 1n3* >> 1na’, yet 1u1* << nu+ physically may not seem to make sense since they represent intrinsic relaxation times made dimensionless with Us and 0. However, mathematically nu’ and nu’ represented cutoff times and initial slopes respectively in the autocorrelation functions (see Figure 4). Note, values for an" and 1n3’ do not affect the existence of drag reduction in this theory. TH3' and 1n3’ were only used to help predict the correlation coefficient and the axial intensity. Mathematically we imposed the condition that er< JP which forced (1141’. 1111*) and (1013*, Tus’) to be different and ms+ >> 1aa‘. The situation could have ‘been reversed had. we used axial intensity data to set one of the time groups, but this would have made 1) .9. 5.3 Reconendations for Further Study The a priorj predictions of the axial turbulent intensity for the Newtonian and dilute polymer solution is a particular area of concern. As Figure 14 indicates there are noticeable discrepancies between the 56 predictions and experimental data. 'A more in depth look at the theory is required. For example, the smoothing approximation used in the viscoelastic memory kernel K(xi,‘?) (see Eq. (A.4)) may need to be more general and include appropriate spatial scales Ix: -Qt VII-u; however, the longitudinal and transverse scales 931* and 933* are much larger (~1000) than the normal scales (~60 - 100) and may be neglected as before. The axial intensity calculation could be done using the proposed 'universal’ parameters and test which remaining spatial scale In produce results consistent with experimental data. The scale-up model discussed in Section 4.2 was based on a single polymer/solvent pair. An extensive literature search to find a more complete set of polymer/solvent s'ystus should be found to test the scale-up method. This would permit an analysis of the model for ‘l’c (0)/‘m1 (see Eq. (A.13)). This could further be extended to include data on larger diameter pipes. The correlation coefficient played a critical part in setting 1143* and 1113*. The theory fitted V = .44 and v = .2 by setting 1113* = 250 and 1'33” = 1168. Logan’s [1972] data was performed in a channel and others have found such data to contain error due to the secondary flow patterns caused by edge effects. Therefore, the estimates for the correlation coefficients may be inaccurate and consequently the values for «3* and 1113* which were based on the correlation coefficient data would change. Thus better data would need to be found in the literature or experiments would need to be conducted to obtain new data for the correlation coefficient. 57 There are well accepted and established techniques to measure the correlation coefficient, The advances made in using laser technology is increasing, thus 1/2 and (us ’2)1’2 could be measured directly. The Reynold’s stress could also be measured directly by measuring the intensity of the turbulent fluctuations in the axial and radial directions simultaneously. There is much experimental and theoretical work currently being done in this area by other researchers. The phenomenon of drag reduction is a very interesting and much researched topic. The theory presented here gave insight into the physical understanding of the phenomenon through a theoretical study based on experimental evidence resulting in specific predictions of statistical properties of the flow. 58 REFERENCES Abranowitz, M., and Stegun, 1., 1964, Eds. Ranmdbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables, Applied Mathematics, Series 55, National Bureau of Standards. Achia, IB.V. and Thompson, D.W., 1977, ”Structure of the Turbulent Boundary in Drag Reducing Pipe Flow," J. Fluid Mech. 81, 489. Astarita, G. and Marrucci, 6., 1974, Principles of Non-Newtonian Fluid Mechanics, McGraw-Rill. Berman, N.S., 1978, "Drag Reduction by Polymers", Ann. Rev. Fluid Mech. 0...»... Bird, R.B.; Stewart, W.E.; and Lightfoot, E.N., 1960 EZEBEEQEE ..¢.....m«--u-....~-nu».--~. Burger, E.D., Chorn, L.G., and Perkins, T.K., "Studies of Drag Reduction Over a Broad Range of Pipeline Conditions When Flowing Prudhoe Bay .m.~.........._-..uu.u....- We... — . Coles, D.E., 1978, A Model for the Flow in the Viscous Sublayer, Leghigh Workshop on Coherent Structure in Turbulent Boundary Layers, C.R. Smith and D.E. Abbott (Editors), 462. CRC Handbook of Chemistry and Physics, Weast, R.C., 59ed., 1978, CRC Press. Darby, R. and Chang, R.D.,1984, "Generalized Correlation for Friction . ”none-mwvemvrm Denn, M.M. and Marucci, G., "Stretching of Viscoelastic Liquids," Agggg J. - 1'7. 3.9.1..- Fortuna, G. and Hanratty, T,J., 1972, ”The Influence of Drag-Reducing Polymers on Turbulence in the Viscous Sublayer," J. Fluid Mech. ...~—su—-.. Fortuin, J.M.H. and Klijn, P.J., 1982, "Drag Reduction and Random ”Den-e..- «o-mu... .....-—n~ ...... .... Hansen, R.J., 1973, "A Theotetical Study of Transient Shear Effects in “en-.4- Hershey, R.C. and Zakin, J.L., 1967, "A Molecular Approach to Predicting the Onset of Drag Reduction in the Turbulent Flow of Dilute Polymer Solutions," Chem.m§gg;m§gi. 22, 1847. 59 Hinch, E.J., 1977, "Mechanical Models of Dilute Polymer Solutions in Landahl, M.T., 1977, "Mechanical Models of Dilute Polymer Solutions in Strong Flows," Phys. Fluids 20, $55. I'm-slum". Little, R.C., Hansen, R.J., Hunston, D.L., Kim, 0., Patterson, R.L. Patterson, and Ting, R.Y., 1975, ”The Drag Reduction Phenomenon: Observed Characteristics, Improved Agents and Proposed Mechanisms," Ind. Eng. Chem. Fund. 14, gag. Logan, S.E., 1972, ”Laser Velocimeter Measurement of Reynolds Stress and - name-9...... u. nun...- Lumley, J.L., 1973, "Drag Reduction in Turbulent Flow by Polymer Additives," J. Polymer Sci., Macromolecular Reviews 7, 268. “ems... Lyons, S. and Petty, C.A., 1984, "Predictions of Turbulent Drag Reduction for a Linear Viscoelastic Fluid," IUTAM Symp., Essen, June 26-28. McComb, W.D., 1973, ”The Isotropic Turbulent Dynamics of a Maxwell Fluid," Proc. R.S.E. 72, lg. McComb, W.D. and Rabie, L.H., 1982, ”Local Drag Reduction Due to Injection of Polymer Solutions into Turbulent Flow in a Pipe: Part I, Dependence on Local Polymer Concentration; Part II, Laser- Doppler Measurements of Turbulent Structure, " AIgmngg. 28, Q4: and .5525. Mizushina, T. and Usui, 11., 1977, "Reduction of Eddy Diffusion for Momentum and Heat in Viscoelastic Fluid Flow in a Circular Tube," ”NAM Metzner, A.B., 1977, ”Polymer Solution and Fiber Suspension Rheology and Their Relationship to Turbulent Drag Reduction,” Phys. Fluids 20, 4-.."..-- Monin, 8.8. and Yaglom, A.M., 1971, Statistical Fluid Mechanics, MIT Press. Morse, P.M. and Feshbach, H., 1953, Methods of Theoretical Physics, Part I, McGraw-Hill . Patterson, C.R. and Zakin, J.L., 1968, "Prediction of Drag Reduction «nun-“M.M.... a...” Penix. S. and Petty, C.A.,l986,”A Mechanism for Drag Reduction Based on Stress Relaxation"; Rolla Symposium on Turbulence, Rolla, M0., October, 1986. 60 Reischman, M.M. and Tiederman, .G., 1975, "Laser-Doppler Anemometry Measurements in Drag-Reducing Channel Flows," J. Fluid Mech. 70, 3.16.9- Schlicating, 11., 1968, Boundary Layer Theory, McGrawHill. Seyer, F.A. and Metzner, A.B., 1969, ”Turbulence Phenomena in Drag “an“-.. Tennekes, H. and Lumley, J.L., 1972, A First Course in Turbulence, MIT Press. Toms, B.A., 1977, ”On the Early Experiments on Drag Reduction by Polymers,” Phys. Fluids 20, §§. Virk, P.S., 1975, ”Drag Reduction Fundamentals,” AICHE J. 21, 625. um...“- APPENDIX A Reynolds Stress Model For Dilute Polymer Solutions The analysis presented in this thesis is based on a previous theory (see Penix and Petty, 1986; and, Lyons and Petty, 1985) which developed a model for the Reynolds stress in fully developed pipe flow. The result for a linear, Maxwell model is ’ , - 2.9332. 011 us > — d,“ (A.l) where v. = (u: ’2>(xi)-rc (xx) (A.2) t -("/ > and "(9.1) = (1 + m...) e T "'1 K(x1,’?)d? (A.3) t-Tml A G “A i. and K(xi,1) =f/dda 5‘35?!” (A.4) where 4?: It - tl For x1 >> 0 and 4‘ << Xlz/D, G(x,tl’g,’t) approaches the Green’s function for the infinite domain. Therefore, Eq. (A.4) can be estimated 41v K(O,1)=f dmgi(R,'r) (A.5) -ee In Eq. (A.5), g1 (R,1') is given by (see Morse and Feschbach, p. 867, 1952) g1(R,1) = 21ce'1/22TJo[(a2c2/2)(R? - c212)1/2]u(c1 - IRI) (A.6) where a2 = U», c = (v/A)1/2, and R = X1 -?1. Now with q = R/CT and iJo(iy) E- 10 (y), Eq. (A.5) can be written as l m...) = (./i)e“(‘/2’°j:ig Ion/2x1 —52)1/2] (11.7) 0 61 62 Io(-) can be expressed in terms of the generalized hypergeometric function (see Abramowitz and Stegun p. 377, 1964) Io(y) = oF1(v + l;l/4(y2)) (A.8) Inserting Eq. (A.8) into (A.7) the following result obtains (see Eq. 7.5.12, p. 850 in Gradshteyn and Ryzhik, 1965) me...) = (t/z)e‘('/2’°.ri(3/2;1/4(t/2x)2) (A.9) This can be simplified further by recognizing that (see Eq. 9.6.47 on p. 377 and Eq. 10.2.13 on p. 443 in Abramowitz and Stegun, 1964) oFi(3/2;l/4(T/21)2) E 1‘(3/2)Ii/2('I'/21)/[1/2('f/21)]“2 (L10) and Ii/2(‘i'/21) e (1/Il)1/2sinh(7/2X)/(7/23) (11.11) Therefore, by combining Eqs. (A.9) — (A.ll) we obtain the major result of this appendix, viz., no.1) = 1-.- e“/‘ (A.12) Inserting Eq. (A.12) into Eq. (A.3) and carrying out the integration yields (see Lyons and Petty, 1985) Tc(°°)/1’H1 = 1 - e’m - she-~10 — e‘m/“m (A.13) where 111 = 'nu/flu (L14) and (h = V'nu (4-15) It now follows from (A.2) and (A.l3) that