Rheol. Acta 17, 98-104 (1978) © 1978 Dr. Dietrich Steinkopff Verlag, Darmstadt ISSN 0035-4511/ASTM-Coden: RHEAAK
Institut Jür Verfahrenstechnik, Rheinisch-Westfälische Technische Hochschule, Aachen
Experimental investigation of the turbulent boundary layer in the pipe flow of viscoelastic fluids*) L. H o f f m a n n
and P. S c h ü m m e r
With 6 figures (Received July 7, 1977)
1, Introduction The phenomena of turbulent pipe flow are strongly influenced by the addition of small amounts of high polymers to the Newtonian solvent. Especially the skin friction can be reduced up to 80%. The rheological behaviour of these fluids depends on the kind of polymer and its concentration in the Newtonian fluid. All the fluids we are concerned with are viscoelastic, under dynamic stress they show relaxation effects. At very low concentrations the fluids have a constant shear viscosity t/o, a dominant zero relaxation time to ( < 1 0 - 1 s e c o n d s ) and a variable elongational viscosity. With inereasing polymer concentration the flow curve becomes non-linear, while relaxation time and elongational viscosity remain in the same order of magnitude. Finally with further increased concentration the solution behaves like a polymer melt with relaxation time spectrum and marked normal stress anisotropies. Experiments and theoretical investigations are mainly performed on highly diluted solutions, for in this case only two rheological parameters, the constant viscosity t/o and a relaxation time to have to be taken into account. Later on it will be shown, that the results derived can be applied to fluids with a nonlinear flow curve if a representative value for the viscosity corresponding to the laminar case is used. *) Paper presented at the annual meeting of the Deutsche Rheologische Gesellschaft e.V., Dortmund, March 9 t h - I l th, 1977. 401
The phenomena of drag reduction should be investigated by the measurement of the velocity profile, for the flow characteristic results from the integrated velocity profile. F r o m literature (1) it was known that the flow structure in the middle of the pipe is almost unchanged. So an examination of the boundary layer seemed to be promising. The presentation of the measured profiles and flow characteristics is based on the known results for Newtonian fluids. In "law of the wall" coordinates •
2
u + = u/u, , y+ = u , yp/tl ; u ,
:-
%/p
the profile can be divided in three layers: the viscous sublayer u+ =y+ for 0 _ < y + < 5 , the buffer layer 5
The flow characteristic follows with good accuracy by integration of the formula for the core region over the whole pipe. With slight approximation of constants the friction correlation has the universal form f - 1 / 2 = 4.0 l o g R e - f l / 2 + 0.394.
For highly diluted Polymer solutions these results are modified (1): Profile: The viscous sublayer is thickened and extends up to y+ = 20. The buffer layer is universal in "law of the wall" coordinates and can be described by a logarithmic law u + = 26.7 l o g y + -
17.
Hoffmann and Schümmer, Experimental investigation of the turbulent boundary layer Its extension depends on polymer concentration and Reynolds number. The tore region is unchanged in its structure, the slope of the line in a half-log plot is also unchanged but its position is shifted to higher values of y+ by an amount AB: u + = 5.75 logy + + 5.5 + AB. Flow characteristic: Following measurements (2) the friction correlation for viscoelastic fluids can be described by a semiempirical correlation: f - 1 / 2 = (4 + 3) logRe . f l / z _ 0.4 - logd/d,.
[il In half-log Prandtl-v. Kärmän coordinates this formula gives straight lines whose position and slope depend on two empirical material parameters 6 and d,.
2. Experimental investigations It was the purpose of our own investigations to expand the range of the measurements on flow in pipes, for most of the measurements in literature were performed in rectangular ¢hannels. Experiments are needed especially at technically interesting high and maximum drag reduction. Besides the fluids examined should have viscoelastic behaviour and a non-linear flow curve. 2.1. Flow system The experiments are performed in a recirculating flow system described below (fig. 1). This system consists of a) fluid tank, b) Mohno pump,
99
c) damping vessel, d) test pipe, e) flow meter, f) recorder. The test pipes were represented by plexiglass pipes with inner diameters of d = 10, 20, 30 mm and a length of 5 meters. By some previous experiments it was confirmed that no entry effects influenced the pressure drop measurements. Execution of an experiment takes some time in which the material properties should be constant. Highly diluted polymer solutions with concentrations less than 100ppm showed in the recirculating system rather fast shear degrada'tion. This corresponds to a quickly increasing pressure drop, the flow rate being constant. The effect is less at higher concentrations, at 100 ppm and more the relevant material properties were constant during 30 to 60 minutes. 2.2. Flow velocity measurement The measurements of the velocity profiles were performed with a laser doppler anemometer (LDA). In LDA the usually needed test probe is represented by photons, which do not disturb the flow. The frequency of laser light is influenced by moving particles in the fluid, the frequency shift is proportional to the particle velocity. In the case of polymer flow the macromolecules are almost ideal scattering particles, they follow all motions of the flow. For the special requirements of velocity measurement in turbulent boundary layers the following problems had to be solved: Boundary layers usually extend in laboratory system only few millimeters, so a high spatial resolution has
!
Fig. 1. The recircul~ttingflowsystem 7*
100
Rheologica Acta, Vol. 17, No. 1 (1978)
to be realized. With help of a short focal length lens and an electronic treatment of the signal the dimension of the measuring volume along the optical axis, which is the radial direction in the pipe, was only 50 ~tm. Near the wall the flow velocity is fluctuating. In the buffer layer both the velocities can be expected, zero and mean velocity. The signal processor-tl~erefore should be able to follow these velocity variations automatically. For this purpose a commercially produced frequency tracker in cooperation with the manufacturer was modified. So the behaviour of this signal processor was outstanding.
3. Results
3.11 Rheólogical description The experiments have been performed with Polyacrylamide (PAA) l) in aqueous solution. To correlate t h e experiments the fluids have to be characterized by their rheological parameters. As already mentioned for fluids at low polymer concentrations the zero-viscosity and a relaxation time have to be measured. While measurement of shear viscosity at low shear rates is difficult but possible in principle, direct measurement of relaxation times in this range is rather vague and therefore has been omitted. But calculation of relaxation times is possible implicitly by use of physical model, i.e. the spring dumbell model. Relaxation times thus calculated are in the range of a hundredth or less of a second. The fluids at higher concentrations were characterized by their flow curve as shown in figure 2 (Couette-type rheometers). An analysis of the measured flow curves shows asymptotic Newtonian behaviour for zero and infinite Shear rates. These asymptotic viscosities differ up to two or three orders of magnitude depending on concentration. The presentation of the flow measurements will be given in terms of dimensionless groups. These groups, the Reynolds number and a modified Reynolds number y+, contain a viscosity wbich in the case of a non-linear flow curve taust be defined. In pipe flow the shear gradient depends on the radial position. This means that also the shear viscosity does not 1) Trade name: Praestol 2935/73. Fa. Stockhausen, Krefeld, West Germany.
have a constant value but taust be calculated from the known flow curve. Our experiments were correlated by using a representative value for the viscosity. It is obtained just as for the laminar flow case: The laminar friction correlation for non-Newtonian fluids can be formulated in the same way as for Newtonian fluids if one uses a representative shear rate ~ =2rc?;/d for the determination of the viscosity from the flow curve (2). Corresponding to calculations this representative value of the viscosity for almost all known viscoelastic fluids is realized within a laminar flow at about r/R = ~z/4 being near the wall. This viscosity is also used for the turbulent flow case. In compar~son to the possibility to take the viscosity at the wall the differences are small. The purpose of the above mentioned representative value for the viscosity is the same definition for both, the laminar and the turbulent flow case. In addition all plots in the laminar flow regime are unified for all fluids measured.
3.2. Mean velocity profiles The experimental results again are presented in half-log "law of the wall" coordinates. For viscoelastic fluids with constant shear viscosity can be stated: The universal form of the viseous sublayer is unchanged in comparison with the Newtonian case, the viscous sublayer extends up to about y + = 5. Thereby measurements down to y + = 3 could be realized, which corresponds to a physical wall distance of about 50 Ixm. The buffer layer in comparison to the Newtonian case is essentially marked. For viscoelastic fluids in literature a universal form is assumed, but our measurements show a split-up depending on material and flow conditions. The extension and the slope of the buffer layer profile increases with increasing Reynolds number and increasing polymer concentration. In some cases at sufficiently high concentrations and Reynolds numbers the buffer layer slope may asymptotically extend up to the middle of the pipe, corresponding to Virk's "ultimate profile". Normally the buffer layer is followed by the tore region, which has the same slope as in the Newtonian case, but is shifted to higher values ofu + by an amount AB. Following the observations these results are also valid for fluids with a non-linear flow curve
Hoffmann and Sehiimmer, Experimental investigation of the turbulent boundary layer cP
PAA
10 ~
10 3
101
before [ during experiments
0,01%
•
0,03"/.
*
0,0S'/o
•
0,07"/.
*
o,1 '/o
•
0,25"/*
"
0,5"/.
•
o v
<> ®
0,7'1. I e
o
102
10
10 -~
10-3
10 -2
10 -1
10o
101
10 2
s-1
10-~
Fig. 2. Flow curves of Polyacrylamide, T = 23 °C (Praestol 2935/73, Stockhausen, Krefeld) by using the afore mentioned representative viscosity. A correlation of the "ultimate profile" succeeds with a mixing length model, the mixing length of which is reduced compared with the case of a Newtonian fluid. A reduced mixing length possibly results from different physical mechanisms. Once the transverse turbulent motions could be damped, on the other hand the transverse motions could have the same amplitude but strongly reduced frequency. The second argument could be proved by visualisation experiments of Donohue et al. (3). They show that the turbulence producing structures in the near wall region were almost identical to the Newtonian ones, but the number of these structures per unit time was decreased at about 80%. These experiments were confirmed by Fortuna, Hanratty (4) and Eckelman et al. (5). In another experiment Sellin (6) showed the radial dispersion to be reduced up to 80% in the viscoelastic case. These results and our own measurements of the flow velocities suggest a reduced mixing length in the mixing length model. Under these circumstances Prandtl's hypothesis of a dominant turbulent stress tensor cannot supposed to be valid. Here therefore a function for the stress tensor in the wall dependent region is derived, which takes into account that
1. the shear stress in the pipe is not constant, 2. the viscous stresses are not small in comparison to the turbulent ones, 3. the mixing length corresponding to experimental results is reduced, 4. the shear viscosity can be represented by a constant value. Following these statements the fundamental correlation for the mean shear stress can be written as: .. (d. 2 1 ,r *r==t/~+PK 2(R-r) 2\dr/ = T p --R"
[2] The integral for the velocity profile must be solved numerically. In the case of the ultimate profile integration is performed up to the middle of the pipe. In figure 3 thus calculated profiles with ~ = 0.055 and experimental results for maximum drag reduction are shown, both being in good agreement. The free curve is an asymptote for high Re-values. Correlation of the buffer layer is also possible when it is followed by a core region. As already mentioned the core profile is almost unchanged in its slope and can be described by the wellknown logarithmic law. Both parts of the profile are coupled together + at a specific value Yl, which depends on material parameters and Reynolds number. Following
102
Rheologica Acta, Vol. 17, No. 1 (1978)
,o
r 0,07'/. P A A
V*
Re=27900
/S,
40
/4
20
#
I
~z
iNewton
f
o
..____.-- J 10
102
y÷
103
Fig. 3. Velocity profiles at high drag reduction
6C
V÷
:wo,~,
o,o,.iiii ,f: !!!,~,!!!! ~.~~ ~,I::iiii ii! iiiii~ ~ - ~ 10
y+
103
I0 2
Fig. 4. Velocity profiles, movements and calculations this procedure all measurements could be correlated. As shown in figure 4 the curves fit the measured points quite well, eren if drag reduction is low or the material is Newtonian (y~ = 33). For further analysis the material dependence of 7~- and the shift of the core region AB have to be correlated with material parameters, i.e. a relaxation time or a structure parameter. Another correlation is possible with the measured friction correlation, which will be discussed now. In figure 5 examples of measured friction correlations in half-log Prandtl-v. Karman co-
ordinates are shown. Clearly three different regions can be distinguished depending on Reynolds number: - t u r b u l e n t flow of viscoelastic fluids just as for Newtonian fluids, turbulent flow depending on viscoelastic parameters 6 and d , defined by eq. [-1], with increasing concentration and Reynolds number a universal curve of maximum drag reduction exists. The slope of this curve does not depend on viscoelastic material parameters. The curve follows from integration of eq. [2]. -
-
Hoffmann and Schümmer, Experimental investigation of the turbulent boundary layer In the viscoelastic region the curves are determined by two empirical parameters:
tion, here should be stated that this implies the impossibility of description only by classical continuum mechanics. Maybe the introduction of the theory of polar fluids requiring just this characteristic length can help. Further investigation in this direction will be performed. The integration of the profile resulting in the friction correlation can easily be performed in both cases, at the ultimate profile and at low drag reduction. In the latter case at low deviations from the Newtonian case the logarithmic core profile has to be integrated over the whole
1. The value d, characterizes the point where the viscoelastic curve leaves the Newtonian curve, the so-called onset point. 2. the slope increment on the Newtonian case, described by 6. A correlation of these parameters (7) with material parameters of the flow medium supposes a characteristic length. Without further discussion, which will be given in a later publica-
30
103
0=1¢m D=2cm PAA0,03% D= 3cm + D .'- 2 cm o D=3cm PAA0,07"/,
o
• f -1/2
20
/
10
j."
010
,
,
,
,
,
102
J
Re. f ~12
,
L
103
,
,
,
J
104
Fig. 5. Examples of possible friction correlation in Prandtl-v. Kärmän coordinates
60
Re = 20000
V+
30 Z0
1__ F
103 0
~
~
'
~----"~-~,~~
'
N~w{onian F l u i d
. 10
Y+
"
'
' I
/ 4
I 102
s / ~ '~ 10z
10~
Fig. 6. Correlation between friction correlation and velocity profile (schematic)
104
Rheologica Acta, Vol. 17, No. 1 (1978)
pipe diameter. The parallel shift AB can hence be expressed as a function of the empirical parameters of the friction correlation: AB = 21/2. 6 . 1 o g ( R e . f l / 2 . d / d , ) .
With the help of dimension analysis a correlation of A B with material parameters is possible (7). Under the assumption that only a characteristic viscosity ~/o and a relaxation time to are relevant only one dimensionless group exists:
d~, .p const. =
t o . 7/0
Once the universal constant is determined, turbulent flow represents a rheometer for relaxation time measurements. Following literature the parameter d , is constant at very low concentrations, giving a relaxation time being independent on concentration. This is in good agreement with the molecular theory of Rouse and Zimm. O u r experiments were performed at higher concentrations. Here the relaxation time slightly increases with polymer concentration. At m a x i m u m drag reduction the "ultimate profile" can be integrated in a closed form to give the experimentally well verified m a x i m u m drag reduction curve. In the medium region it can be approximated by f - 1 / 2 = 18.8 lg (Re f 1 / 2 ) 31.6. In all other cases at medium drag reduction both profiles have to be coupled together at the specific value yi~, giving the corresponding points of the flow characteristic. The situation for illustration is shown in figure 6. -
Summary
In turbulent flow of viscoelastic fluid the boundary layer plays an essential rôle. Measurements by help of a Laser Doppler apparatus designed for this purpose were performed to give reliable results under the difficult situations in viscous sublayer and buffer layer. It could be shown that the viscous sublayer in comparison to the Newtonian case is unchanged. The buffer layer splits up with polymer concentration and Reynolds number, in contradiction to some literature results. With increasing polymer concentration and increasing Reynolds number the buffer layer expands more and more at the expense of the core region. Finally the buffer layer extends asymptotieally almost to the middle of the pipe giving the "ultimate profile", For this case a correlation in a mixing length'
model with a reduced mixing length has been found, which describes the experiments well. Integration of the "ultimate profile" and the logarithmic core profile at low drag reduction gives the corresponding points of the flow characteristic with good accuracy. The results derived are also valid for fluids with non-linear flow curve by using a representative viseosity. Zusammenfassun9
Das Widerstandsverhalten viskoelastischer Flüssigkeiten in der turbulenten Rohrströmung wird wesentlich durch die Wand-Grenzschicht bestimmt. Zur Messung der Mittelwertgeschwindigkeit in der viskosen Unterschicht und in der Pufferschicht wurde deshalb zur Erzielung der erforderlichen Genauigkeit eine Laser-Doppler-Apparatur konzipiert. Es zeigt sich, daß bei Benutzung der üblichen dimensionslosen Darstellung die viskose Unterschicht gegenüber der Strömung newtonscher Flüssigkeiten unverändert ist. Die Pufferschicht fächert auf in Abhängigkeit von der Polymerkonzentration und der Reynoldszahl. Mit wachsender Polymerkonzentration und/oder wachsender Reynoldszahl nimmt die Ausdehnung der Pufferschicht auf Kosten der Kernzone zu. Die Kernzone besitzt einen halblogarithmischen Geschwindigkeitsverlauf mit derselben Steigung wie bei newtonschen Fluiden. Im Grenzfall wird die Kernzone fast gänzlich durch die Pufferzone verdrängt. Das Flüssigkeitsprofil verläuft jetzt als "ultimate profile". Profile und Widerstandscharakteristik lassen sich in Übereinstimmung mit den Experimenten darstellen, wenn man für Pufferschicht und Kernschicht jeweils unterschiedliche Mischlängen benutzt. Die Resultate sind auch gültig für Flüssigkeiten mit nichtlinearer Fliel3kuve, wenn man für die Darstellung eine repräsentative Viskosität benutzt. Rejerences
1) Virk, P. S., Am. Ind. Eng. Chem. J. 21,625 (1975). 2) Hoffmann, L., P. Schümmer. H. Schwerdt, Rheol. Acta 14, 626 (1975). 3) Donohue, G. L., W. G. Tiederman, M. M. Reischman, J. Fluid Mech. 56, 559 (1972). 4) Fo/'tuna, G., T. J. Hanratty, J. Fluid Mech. 53, 575 (1972). 5) Eckelman, L. D., G. Fortuna, T. J. Hanratty, Nature Phys. Sci. 236, 94 (I972). 6) Sellin, R. H. J., Chim. Ind. du C. N. R. S. Nr. 233, Brest, 1974. 7) Hoj]mann, L., Doetoral Thesis (Rheinisch-Westfälische Technische Hochschule Aachen 1977). Authors' address: L. Hoffmann and P. Schümmer Institut für Verfahrenstechnik Rheinisch-Westfälisehe Technische Hochschule Aachen Turmstraße 46 D -5100 Aachen (BRD)