Rheologica Acta

Rheol Acta 27:512-517 (1988)

Diffusion of macromolecular solutions in a turbulent boundary layer of a cylindrical pipe. V. Velocity profiles and turbulent intensities in the boundary layer M. Bu~s, H. Reitzer and R. Teitgen Ecole Nationale Sup6rieure des Arts et Industries de Strasbourg et Institut de M~canique des Fluides de Strasbourg Abstract: This paper presents a study of the turbulent structure of the diffusion boundary layer. A macromolecular solution (PEO WSR 301) is injected into a cylindrical pipe under turbulent flow conditions (Re ~ 40 000). Laser velocimetry was the experimental technique used. Velocity profiles and turbulence intensities are determined in a boundary layer with injections of newtonian or non-newtonian fluids. We attempt to give an interpretation of these results as a function of the decomposition of the diffusion field. Key words." Oiffusion of macromolecular solutions, turbulent boundary layer, turbulent structure, yelocity profile, turbulent intensity

Notation Downstream distance from the slot normal distance from the wall pipe diameter concentration initial concentration wall concentration Cw Q or Q, flow rate Q, flow rate injection Ci Qi/Q~ equivalent homogeneous polymer concenCj tration f0 and f linear head loss factors of the solvent and the polymeric solution Reynolds number Re g, friction velocity u mean velocity in x-direction u' fluctuation of velocity in x-direction u+ mean reduced velocity y+ dimensionless parameter kinematic viscosity V wall stress % density characteristic height of the diffusion plume, i.e. the 2 value of y at which C/C w = 0.5 Lo, LIT, Lre, LF characteristic lengths of the diffusion plume (see figure 2)

X

y D C C~

1. Introduction

The lack of detailed theories, for the set diffusion in turbulent flow and drag reduction, makes a rigorous 3O0

analysis, of the mutual interaction of the diffusion of polymers and the basic flow impossible. Similarity laws and semi-empirical methods, which describe the diffusion model of a tracer and the drag reduction in homogeneous solutions can be used to estimate the effect of dilute polymer on the diffusion ratio and to calculate the drag reduction in simple cases: in the case of zero pressure gradient [1], for example. F r o m the experimental point of view, it appeared that, for b o u n d a r y layer flow, the drag reduction is associated with an increase of the viscous sublayer; the flow outside the b o u n d a r y layer remains essentially Newtonian [2]. In the case of a non-uniform concentration field, it seems natural to assume that it is the concentration at the wall which is determining the drag reduction. This hypothesis is the most plausible because the tested flows have high velocities, which implies on the one hand a small thickness of the viscous sublayer, and on the other hand a small variation of the concentration inside that layer. If polymers are injected into a sublayer of large dimension, it is quite possible that strong gradients exist near the wall in the intermediate region 5 < y u , / v < 30; it is in this region where the effects of polymers are remarkable. It is not possible to predict anything on the basis of measures in homogeneous solutions, if the polymer concentration does not remain constant.

Bu6s et al., Diffusion of macromolecular solutions in a turbulent boundary layer. V This paper supplements some previous publications [3 8], concerning the investigation of concentration distribution of macromolecules and turbulent structure in the turbulent diffusion boundary layer of a cylindrical pipe at points ranging between 2 and 20 diameters from the injection slot. A description of the experimental apparatus and analysis techniques can be found in the references mentioned above, in particular in [8]. Two experimental methods to investigate the turbulent structure of the diffusion boundary layer were developed. The first was a flow visualization method and allowed a model of changes occuring at the diffusion boundary layer [8] to be found. The second method is based on laser velocimetry and is a continuation of this approach. Our aim is to determine the velocity profiles and the intensities of turbulence in a boundary layer with injection of non-Newtonian fluid ( P E O - W S R 301). We attempt also to give an interpretation of the results we obtained as a function of the decomposition of the diffusion field from which we recall the main characteristics in paragraph 2.2.

2. Experimental methods

EXCITER

1 I VOLTMETER DIGITAL

4

t

2.1.2 H e a d loss and friction velocity

In order to obtain the values of the friction velocity u, which gives access to the dimensionless quantities u + and y+, it is necessary to measure the head loss A H at two locations, by either side of the test section and separated by a distance A L . To measure the pressure drop a differential pressure transducer was used. The conversion, output voltage to pressure was obtained via a linear calibration curve.

I HIGHVOLTAGE

+

EAMPLIFIER[

TRACKER GALVANOMETER

t

2.1 Experimental device

This technique, well adapted to our problem, is in common use for measurements in both laminar and turbulent flows, will be only briefly described. More details concerning the principle of the Doppler effect in the interference fringe method and its detection are given in previous references [9 ll]. The experimental apparatus and the data processing system are sketched in figure 1. The light source is provided by an He-Ne Laser (wavelength 6328~, power 5 mW). The laser and the optical system were fixed on an optical bench which could be moved in the radial direction by means of micrometer screw. The flow was analysed in both sequential and statistical modes by means of a multichannel analyser (Plurimat S) to determine the contribution of the different parameters.

I RMSVOLTMETER

!,

COMPUTER

2.1.1 Laser velocimeter

513

?

I

~

ivIULTICHANNEL ANALYSER

~ INSTANTANEOUS VELOCITY ~RMS MEANVELOCITY

Fig. 1. Experimental set-up, Data processing system

2.2 Data reduction 2.2.1 Friction velocity

The flow of a Newtonian fluid can be split into three zones with different structures each of which can be defined in terms of the dimensionless parameter y + such as:

y+ yu,/v where u, = z x f ~

(1) is the friction velocity.

This three-zone model is composed of: (i) a turbulent core: y+ > 30. The velocity distribution can be represented by the logarithmic law obtained by considering the notion of mixing length: u + = u/u, = A logy + + B

with A = 5.75 and B = 5.50. (ii) a viscous sublayer or wall zone: y+ < 5.

(2)

514

Rheologica Acta, Vol. 27, No. 5 (1988)

Since, in laminar flow, the viscous effects are predominant: u + = y+.

(3)

(iii) a buffer layer: 5 < y+ < 30. The velocity distribution has to obey both the parietal law and the logarithmic law. Knudsen et al. (in Scrivener [12]) suggest the following profile: u + = A' log y + + B'

(4)

A' and B', calculated to ensure the continuity at y+ = 5 and y+ = 30, have the numerical values 11.5 and 3.05 respectively. For turbulent flows of non-Newtonian fluids, it seems to be reasonable to only take into account the sublayer thickening model for the turbulent core: u+-Alogy

+ +B+AB

(5)

where the x-axis points into the flow direction; Lo, L I t , L r v , L v are characteristic lengths and C stands for the concentration of polymers (i: initial, w: wall). F o r modelling the behaviour of the wall concentration, the relation established by Hsu [14] can be used: C w / C i = exp [ - x/Lo]

with

L o being

the

(10)

value

of x

corresponding

to

Cw/Ci = l / e .

Continuity considerations between the different zones lead to expression depending on the characteristic lengths and exponents. F o r the final zone, the model which describes the evolution of the wall concentration takes the following form [5]: C U C i = exp [ - (LIT/Lo) ni (LTF/LIT) m (x/Lrv) "f ]

where AB represents the effective slipping of the turbulent core. The linear head loss factor f is given by the following relation (6)

f = (AH/AL) 2 g D/U z,

where U = Q/S is the mean velocity deduced from the flow rate. We deduce the friction velocity u, = ~ f

(7)

U2/8.

The last relation is also valid in the presence of polymers. It can be seen that, in the case of an homogeneous solution of macromolecules in water, the friction velocity is a function of the concentration, for various Reynolds numbers, and decreases rapidly. For concentrations up to 30ppm, each curve has a limiting value [9]. A drag reduction ratio (DR) can be defined as follows: (8)

DR = (fo -- f ) / fo

where fo and f are respectively the head loss factors of the solvent and the polymeric solution. For a flow in a smooth pipe, fo is obtained from the Blasius law fo = 0.3164Re 1/4

(9)

for

Lre_< x < L F.

(11)

A physical a p p r o a c h shows that the characteristic parameters are strongly correlated with the macroscopic magnitudes of the principal flow. The existence of the transition zone, which is difficult to detect, had been confirmed experimentally. This leads us to introduce two fundamental points: (i) The existence of a "critical concentration" in a homogeneous medium. (it) The use of a simplified three-zone model derived from this concentration. Moreover, the introduction of this "critical concentration" is justified by the viscoelastic properties of the macromolecules submitted to a shear gradient. Starting from the superposition of two weighted models set up by M o r k o v i n [15], a model of the evolution of the transverse concentration profiles, which expresses the influence of an infinity of point sinks placed at the wall, has been established; the expression is given by [6]:

where Re is the Reynolds number.

C(y)/C,~ = a {exp [ - 0.693 (y/2)"]

+ e x p [ - 0.693 [(D -- y)/2]"]}, l/a = 1 + exp [-- 0.693 (D/2)'] 2.2.2 Diffusion f i e M

Recall now the earlier results and relationships which define the concentration fields. F o r a two-dimensional flow, the experimental results of Poreh et al. [13] suggest a model of diffusion divided into four zones. The splitting up of the diffusion field can be done in the following manner (figure 2):

21~0

initial zone

• C(x,y) transition zone

intermediate zone

with

n> 0

(12)

and D is the pipe diameter. A physical a p p r o a c h has shown that a correlation exists between 2 and n. Each of the two parameters is influenced by the concentration of polymers at the wall. Moreover, it has been possible to find the limits between which 2 and n vary, by considerations of the flow properties or by verification of the p r o p o s e d hypothesis [7].

final zone

Fig. 2. Four zones of diffusion field C Qi

Lo

L IT

Cw(x)

L TF

L F

x

Bubs et al., Diffusion of macromolecular solutions in a turbulent boundary layer. V

3. Experimental results

40 Ci

3.1 Head toss measurements

pprn

0

Wells et al. [16] compared the effects produced by injections of secondary fluids at the center line and at the wall, for turbulent shear flows in a pipe. In both cases, they have shown that the instrumentation only moderately modifies the local pressure gradient, compared with the drag reduction effects. Additionally this gradient is reduced in the same manner as in a flow having the same homogeneous concentration. The injection in the core of the flow does not provoke a drag reduction unless the injected fluid reaches the wall area. This assumption seems in contradiction with more recent results. Indeed, Bewersdorff [17, 18] found a non-negligeable drag reduction before the injected fluid reaches the wall area. Furthermore, he shows that the final values of the friction factor measured in the case of injection are different to the values obtained with a homogeneous solution having the same concentration. As a guide, a drag reduction ratio of 56% has been obtained in the case of injection with the following test conditions:

515

30

0 0

Qi

u*

x

mm/s

cm

0 42.3 11.11 44 16.67 43

46

cma/s

31.5

o

~20

=. ~ ii eA

•

i 10

I 10 3

Fig. 3. Velocity profiles in reduced coordinates u + = f (y ÷) (water injection) 40

Ci ppm

30

• i

Qi

u~

cmZls mmls

1200 11.11 31 1600 16,67 28 800 31 4000 4335

x cm 46 31.5

ea ee=

~ e~ °~ • • ==

~ ~ •

~:;I 2 0

•

•

3.2 Velocity profiles in reduced coordinates (u + , y+)

3.2.1 Injection of Newtonian fluid In figure 3 two velocity profiles obtained when water was injected are drawn. The first is situated at a point x - 31.5 cm from the origin and the injected _flow rate is 16.67 cm3/s. This test shows a reduced profile beyond the

,'•

I 102

y +

•

o

•=

o

o

It is first convenient to explain the results, obtained in absence of injection of secondary fluid, presented in figure 3 and the value of u, = 42.3 mm/s deduced from eqs. (6) and (7). Although this result is in perfect accordance with the value deduced from eqs. (9) and (7), the velocity distribution in reduced coordinates deviates from theoretical predictions (continuous line). If, as we presume, the distance between the settling chamber and the injection block is large enough to have a fully developed turbulent velocity profile, we can conclude that the split in the injection block disturbs the flow at the wall region by creating a stagnation point. Consequently a new boundary layer will expand and the velocity profile measured at the point x = 46 cm is that of the transition zone of a hydrodynamic boundary layer. In fact, previous results [19] have shown that in this case the experimental data are beyond the theoretical profile and intersect it for values of y + between 400 and 500.

•

•

/ 0

C i = 1600 ppm, Qi = 16.67 c m a / s , x = 31.5 cm,

Re = 40 000.

lie= e*

oo

oo

© o

•

• •

=b

A •

•

A

/

Ao

10

0

I 10

i 10 2

y+

,J 10 3

Fig. 4. Velocity profiles in reduced coordinates u + = f(y+) (PEO WSR 301 injection) Table 1. Characteristics of three testings

x C~ [cm] [ppm]

[cm3/s] [ppm] [cm] [ppm] [mm/s]

46 31.5 31.5

11.11 16.67 16.67

1200 800 1600

Qi

Cj 7 7 14

Lo 15 16 35

C~ 115 154 578

u, 31 31 28

Zone intermediate initial

theoretical profile. For the second, the flow rate of the secondary fluid is decreased to 11.11 cm3/s and the test section is located further downstream at x = 46 cm. The velocity profile obtained is nearly the same as that obtained without injection for values of y + greater as 60. It follows, from the plot of u + = f ( y + ) , that the annular wall injection disturbs the structure of the flow near the

516

Rheologica Acta, Vol. 27, No. 5 (1988) 4

Ci ppm 0 0 0

Qi

u*

x

crn~s mm/s

crn

0 11.11 16.67

46

42.3 44 43

Ci

Qi

u,

x

ppm

ci'~3ls

minis

cm

120(]

11,1"[

31

46

1600 16.67 600 o 400 A I 0

28 31 35 43

31,6 !

# 3 o

• •

4•

31.5

A o

z, o

• - - .

•,

o • A

•

•o

•

•

o

A

©

m

•0

i

250

y+

I

i

500

750

0

i

260

y+

i

r

500

750

Fig. 5. Reduced turbulence intensities versus y+ (water injection)

Fig. 6. Reduced turbulence intensities versus y+ (PEO WSR 301 injection)

wall in an appreciable manner. This established fact is corroborated by the values of the friction velocities which are slighty greater than the values obtained without injection. Although the injection of a secondary fluid into the boundary layer does not produce a zone of regular structure, and is consequently not very dissipative, our experiences show that we do not have a drag reduction ratio but a drag increase ratio. The drag reduction occurs only when a threshold which is to be determined is reached. The length of the intermediate zone being small and the limit of the length of the transition zone being about 75 cm, the two test sections are therefore located in the transition zone of the diffusion boundary layer.

greater then a certain value. That result is in accordance with our investigations. As for the last test (C i = 1600 ppm, x = 3115 cm) carried out in the initial zone, the wall concentration of polymers is very high (in order of 580 ppm). In figure 4, a result a priori singular appears. It can be seen that the reduced profile we obtained is practically identical with that of a test with a smaller initial concentration, but situated at a greater distance from the source.

3.2.2 Injection of non-Newtonian fluid At the point x = 31.5cm, the velocity profiles take place beyond the theoretical profile of the turbulent core, and are increasing functions of the friction velocity (figure 4). The slope of the linear parts in semi-logarithmic representation are greater than 5.75 (about 9 in our case). The presence of polymer decreases the friction velocity and increases the velocity in the turbulent core. Three cases require special attention; the following table summarizes the experimental and calculated values concered (table 1). The first two tests, made at different points, are situated in the intermediate zone of the diffusion boundary layer. For each of them, although the friction velocity and the concentration in the homogeneous medium are identical, it can be seen that the slippings of the turbulent core are very different. As some authors have mentioned [9] in the case of flow in a homogeneous medium, the friction velocity will be influenced only slightly for concentrations

3.3 Turbulence intensities. Axial velocity fluctuations 3.3.1 Injection of Newtonian fluid The intensities of turbulence are established from the R.M.S. values of the signals. Figure 5 shows intensities reduced by the friction velocity u. versus the reduced length y+. In the turbulent core the intensity of turbulence is the same with or without injection. Near the wall, the injection of secondary fluid has the effect of the decrease in the intensity of turbulence beyond a maximum value disappear. The increase is making then inversely proportional to the injected flow rate.

3.3.2 Injection of non-Newtonian fluid It can be seen from figure 6 that the turbulence increases with the presence of polymers. The peaks of maxim u m intensity are shifted to the centre of the pipe, this effect is more important when the concentration at the wall is high. Furthermore, these maximum values are clearly higher than for an injection of water. In the turbulent core, the intensity of turbulence is nearly the same for both water and polymer solutions. Nevertheless, an exception occurs for the highest initial

Bu6s et al., Diffusion of macromolecular solutions in a turbulent boundary layer. V concentration (Ci = 1600 ppm). This difference can interpreted by introducing the p a r a m e t e r L 0, which is the length of the initial zone. The use of the evolution model for wall concentration, given in table 1, shows that the intensity of turbulence profile is located in the initial zone of the diffusion b o u n d a r y layer. The existence of high concentration gradients at the wall produces very important turbulence intensities.

4. Conclusions On injection of polymer, it seems natural to assume for the case of a non-uniform field of concentration that the drag reduction, i.e. the friction velocity, is determined from the concentration at the wall. Nevertheless the existence of steep gradients exclude that hypothesis near the source. It seems difficult to imagine a correlation which would be able to predict the effects from the results obtained with homogeneous solutions. As for an injection of water, it seems reasonable to assume that the velocity profile evolves before it reaches its final state - a fully developed profile which corresponds to the profile established for an homogeneous concentration Cj such as Cj = C i Qi/Qt. The results we obtained c o r r o b o r a t e this assumption. The use of the Laser velocimetry has shown a "laminarization" of the turbulent velocity profiles in the presence of macromolecules. Moreover, the results obtained point out a thickening of the viscous sublayer. The peaks of m a x i m u m intensity of turbulence are shifted towards the centre of the pipe. These m a x i m a values are much greater than for an injection of water. In the last case, the peaks of m a x i m u m intensity are c o m p a r a b l e to those obtained in a turbulent flow without injection.

517

References 1. Porch M, Hsu KS (1972) J Hydronautics, Vol. 6, n°l, pp 27-33 2. White A (1969) Some observations on the flow characteristics of certain dilute macromelecular solutions, Viscous Drag Reduction, Plenum Press, New-York, pp. 297-311 3. Bhowmick SK, Gebel C, Reitzer H (1975) Rheol Acta 14: 1026-1031 4. Gebel C, Reitzer H, Bu6s M (1978) Rheol Acta 17:172-175 5. Reitzer H, Gebel C, Bu6s M (1981) Rheol Acta 20:35-43 6. Gebel C, Bu6s M, Reitzer H (1982) Rheol Acta 21:720-724 7. Bu6s M, Gebel C, Reitzer H (1982) Rheol Acta 21:725-729 8. Buas M, Reitzer H, Scrivener O (1985) Rheol Acta 24: 312-316 9. Berner C (1980) Th~se de 36me cycle, ULP Strasbourg 10. Bu6s M (1981) Th+se de Doeteur Ing6nieur, ULP Strasbourg 11. Lyazid AW (1983) Th~se de Doctorat d'Etat, ULP Strasbourg 12. Scrivener O (1975) Th6se de Doctorat d'Etat, ULP Strasbourg 13. Porch M, Cermak JE (1964) Int J Heat Mass Trans 7: 1083-1095 14. Hsu KS (1970) M.S. Thesis, Dept of Hydraulics and Mechanics, Univ. of Iowa, Iowa City 15. Morkovin MV (1975) Int J Heat Mass Trans 8:129-145 16. Wells CS, Spangler JG (1967) The Physics of Fluids, Vol 10, n°9 17. Bewersdorff H-W, Straul3 K (1979) Rheol Acta 18:104-107 18. Bewersdorff H-W (1982) Rheol Acta 21:587-589 19. Teitgen R (1977) Th6se de Doctorat d'Etat, ULP Strasbourg (Received February 22, 1988) Authors' address: Dr. M. Bu6s *), Dr. H. Reitzer, R. Teitgen Institut de M6canique des Fluides - URA CNRS 312 2, rue Boussingault F-67083 Strasbourg C6dex *) Author to whom correspondence should be addressed