J . Phys. Chem. 1994,98, 1939-1947
1939
Distribution of Turbulent Velocity Fluctuations in a Drag-Reducing Solution P. H. J. van Dam' and G. H. Wegdam Van der Waals-Zeeman Laboratory, University of Amsterdam, Nieuwe Achtergracht 127, 1018 WS Amsterdam, The Netherlands
Received: April 8, 1993; In Final Form: December 3, 1993"
The probability density function of the velocity fluctuations in turbulent pipe flow is directly obtained from the measured velocity-velocity correlation function by calculating the higher order spectral moments of the correlation function. The probability density function is assumed to be the product of a Gaussian and a Lorentzian. The Gaussian represents the distribution of the "active" velocity fluctuations: the velocity fluctuations which are typically rotational and actually participate in the turbulent cascade. The Lorentzian describes the velocity fluctuations that areinactive, i.e., the"quiescent" velocity fluctuations which are nonparticipating in the turbulent cascade but are slightly excited by their turbulent environment. With increasing Reynolds number, the development of the turbulent flow towards a more space-filling state is clearly observed by an increase of the amount of active velocity fluctuations. Addition of a drag-reducing polymer to a turbulent flow is known to cause a large reduction of the turbulent energy loss. After we add such a polymer, a turbulent cascade is still observed, but it is confined to a very small fraction of the fluid volume. The polymer has caused a very large part of the flow to become "inactiven by a suppression of small velocity fluctuations in the bulk of the turbulent flow.
1. Introduction
The reduction of turbulent drag by a polymer additive is a phenomenon which has received much attention since its discovery some 45 years ago by Toms' and almost simultaneously by Mysek2 Linear, flexible polymers of a high molecular weight (typically a few lo6 amu) can reduce the turbulent drag or turbulent energy loss in a pipe flow by a factor of 2 or more. These polymers are already effective at minute concentrations, typically a few weight ~ p m which ,~ causes the simple shear viscosity of these polymer solutions to be practically indistinguishablefrom the pure solvent viscosity. Many studies concerned the possible commercial applications of drag reduction in oil transport through pipeline~,~J fire fighting: and the speeding of ships.' A large part of the research effort, however, dealt with the more fundamental subject of the actual physical mechanism behind turbulent drag reduction by polymer additives. Theoretical approaches to drag reduction are based on the fact that large flexible polymers exhibit a large resistance against the stretching and shearing action caused by some fluid element. Lumleys considers drag reduction to result from a strongly enhanced elongational viscosity of the turbulent core with respect to boundary layer, thereby reducing the extent of the turbulent core and relocating the turbulent dissipationscale. The structure of the turbulent cascade is expected to remain essentially unaffected.8 Tabor and de Gennes9Jo describe the mechanism of drag reduction in terms of a viscoelastic polymer eddy interaction in the bulk of the turbulent flow. This should lead to a truncation of the energy cascade at small scales, which are the scales where most of the (viscous) dissipation of the turbulent kinetic energy takes place, thereby deeply modifying the structure of the turbulent cascade. The main feature in the theory of Landahll' is the anisotropic stress caused by the stretching of polymer molecules which should stabilize the small scale secondary motions in the boundary layer and thereby inhibit the production of turbulent bursts. Experiments on drag reduction with the common experimental technique of laser Doppler velocimetry (LDV)12indicate that the drag-reducing polymers cause an attenuation of the highfrequency part of the turbulent spectrum in the bulk of the flow. 0
Abstract published in Advance ACS Abstracts, January 15, 1994.
The problem with LDV, which probes the local turbulent velocity C(r(t)), is that it is impossible to obtain quantitative spatial information without invoking Taylor's "frozen turbulence" assumption. This assumption simply means that the measured C ( t ) is replaced byv(x/U), wherexisa coordinatein the flow direction and U the mean flow velocity. There are recent indications however that this assumption may no longer bevalid at the small scales where the main interaction of the polymer is expected to take place.I3 With a dynamic light-scattering technique, homodyne photon correlation spectro~copyl~ (further referred to as PCS), one is able to avoid the use of the Taylor hypothesis and to remove the effect of the mean flow velocity. With PCS, one measures the tjme correlation function of the instantaneousvelocity differences V,(t) over a distance R:
In this Lagrangian framework, one is able to obtain direct spatial information on the probability density function of velocity increments p(QR).1s In this paper, we report the results of a PCS study on the structure of turbulent pipe flow affected by drag-reducing polymers. By using a small angle light scattering setup, we are able to probe the pure transverse as well as longitudinal velocity fluctuations. The experiments reveal that the turbulence in this experiment is developing towards a more space-filling state. When adding a drag-reducing polymer, velocity differences in the direction of the mean flow are suppressed, while the transverse velocity fluctuations remain unchanged. This suppression leads to a strong decrease in the fraction of the volume occupied by turbulence. The probability density functions of the velocity differences with and without polymer are calculated. Comparison of these two cases throws new light on the mechanism of drag reduction. This paper is divided into several parts: the theory of homodyne light scattering from turbulent fluids (section 2), the experimental setup (section 3), the actual results of the light-scattering experiments (section 4), and a discussion of these results together with their implications (section 5).
0022-3654/94/2098-1939%04.50/0 0 1994 American Chemical Society
van Dam and Wegdam
1940 The Journal of Physical Chemistry, Vol. 98, No. 7, 1994
The scaling behavior of P(VR)means G(qt,L) can be written as
2. Theory of Homodyne Light Scattering from Turbulent
Fluids 2.1. Correlation Function. With the PCS technique, the autocorrelation function g ( t ) of the scattered light intensity I(?) is measured:
g o ) = (W’+ t)I(t?) (2) The angular brackets denotea time averageover t’. Thescattering of the laser light is produced by small particles which are suspended in the fluid and which are small enough to follow the local flow field instantaneously. The light recorded by the photodetector is the beating of the Doppler-shifted light scattered by pairs of particles present in the scattering volume. Each particle pair thus contributes a phase factor cos(qtVR1 to the correlation functiong(t). Here, V ~ i thecomponent s of V,(t) in thedirection of 4 and q is the amplitude of the scattering vector 4: q = IpI = ( 4 4 X ) sin(e/2)
(3)
where 0 is the scattering angle, Xis the wavelength of the incident light, and n is the refractive index of the fluid. The correlation function g(t) can be written as16 (4) g ( t ) = ( W N 2 [ 1 + -40 G(qt,L)l where L is the length of the (quasi-one-dimensional) scattering volume and A(?)is the factor representing the Brownian motion of the scattering particles (A(?) exp(-2Dq2t), with D the diffusion constant). The function G(qt,L)is due to the advective motion of the scattering particles and has the form
-
where h(R)is the number of particle pairs present in the scattering volume separated by a distance R. This function is well described by
The measured correlation function G(t) is thus a spatial integral over the scattering volume of a statistical “amplitude” h(R) multiplied by the Fourier cosine transform of the probability density of all VRoccurringin the scattering volume. In a turbulent system, the smallest possible scale over which a velocity difference can exist is the viscous dissipation scale kd, which can be written as l&d4, where lo is the macroscopic length scale of the system and Re the macroscopic Reynolds number (Re = Ulo/v, with v the kinematic viscosity and U the,mean flow velocity). In isotropic tu_bulence, the Kolmogorov theory1’ states that all moments of V R ( ~obey ) a power law in R:
where VRis the amplitude of ? ~ ( t ) e, is the mean transfer rate of kinetic energy, and the B, are universal constants. The angular brackets denote a time average over all VR. The power law dependence of the moments of VRon the scale R means that the velocity fluctuations behave self-similar. The statistical properties of VR over varying length scales then remain identical under appropriate scaling of the velocities. Velocity fluctuations exhibit self-similarity when their decay is predominantly governed by kinetic energy transfer and not by viscosity. The self-similarity of the velocity differences implies that their probability density should have a scaling form:
Q(vR/NR)) p(VR)
=
(8)
u(R) where u(R) is a global scaling velocity which obeys a power law u(R) Ra. Following eq 7,the exponent a has the value 1/3.18
-
G(qt,L) = G(qtu(L))= S,’dR
W )W W ) )
(9)
where F(qtu(R))is the Fourier cosine transform of Q(V R / U ( R ) ) and u(L) = u(R) X (L/R)”. We refer here to the experimental part of this paper, where we show that the measured correlation functions indeed coincide when plotted against their scaling argument qtu(L). In turbulent flows which are neither homogeneous nor isotropic, such as the flow in our geometry, these scaling arguments are still valid, as was shown by Knight and Sir0vi~h.l~ Their theoretical result agrees with earlier PCS experiments on turbulent flowZo where the correlation functions already exhibited self-similarity at very moderate Reynolds numbers. 2.2. Probability Density Function. Generally, in order to obtain the probability density function of the velocity differences from the measured correlation function, one has to calculate the complete set of all higher order spectral moments. As this is practically quite an impossible task to perform with considerable accuracy in an experimental situation, we assume a plausible form of the probability density function. Experiments have shown G(r) to behave exponentially over a t least 1 decade in time.l6sz0 This implies that the probability density P( VR)is Lorentzianlike. (The Lorentzian line shape is the Fourier transform of an exponential line shape and is defined asflx) = a/($ aZ),with x the variable and a the width of the Lorentzian.) The reason why the distribution is not, as often encountered in turbulence, more or less Gaussian, is due to the fact that this distribution concerns velocity differences instead of absolute velocities. At the same time, the experimental technique of PCS focuses on relatively small length scales, which are precisely the scales where, due to the vicinity of the dissipation scale, large deviations of Gaussian behavior are probable to occur.z1 Since the higher order moments of P( VR)must remain finite (see eq 7), OnukiZ2proposed the P(VR) to be a product of a Gaussian for large and a Lorentzian for small velocity differences, each characterized by its own average (or “scaling”) velocity. The (tentative) picture which is related to these two distinct velocity fields is based on the fact that turbulence is often found to be not completely space-filling. Thevelocity fluctuations which directly participate in the turbulent cascasde, Le., which are typically rotational, are large in magnitude and are simply Gaussiandistributed (with anaveragevelocity UG). Thesevelocity fluctuations occur in the so-called “active” regions of the flow, the regions where the vorticity is localized. The small velocity fluctuations are Lorentzian distributed (with its scaling velocity UL), which is an example of a Lkvy stable distribution where VR is the sum of nearly independent velocity increments each being a contribution froma vortex tube or some singular fractal object.23 These small velocity fluctuations are localized in the so-called “inactive” regions. The form of the normalized P( VR)then becomes, as was shown in ref 13
+
p(vR)
=
exp[-u2(R)/2uG2(R)l ?r
erfc[u,(~)/1/Zu,(~)l
uL(R)
exp[-v~/2uG2(R)1 (lo)
[vRZ
+U ~ W I
where erfc is the complementary error function. The first term on the right-hand side represents the normalization of P( VR), while the second term consists of the Gaussian multiplied with the Lorentzian. Obviously, once an expression for the probability density function is assumed, the higher order moments of VR,which still obey eq 7, determine the scaling velocities in the P ( V R ) expression:22
The Journal of Physical Chemistry, Vol. 98, No. 7, 1994 1941
Drag-Reducing Polymers
0
Now, we return to the experimental accessible quantity, the correlation function G(t). Since G(t) is roughly the spatial intergral of the Fourier transform of P(VR)(see eq 5), eq 11 has the following implication for the higher order moments of the spectrum G(o), which is the Fourier transform of G(t):
This relation shows us that, once the correlation function is measured, we only have to calculate the second and the fourth order moment of its spectrum G(w) to determine the values of the Lorentzian and Gaussian scaling velocities. With these two scaling velocities and eq 10, the probability density of all velocity differences present in the scattering volume can be calculated. The experimental correlation functions obviously have a limited temporal resolution and a limited time domain over which they are measured. This implies that the calculated probability densities also have a limited resolution and a limited velocity domain over which they are significant. 2.3. Essential Experimental Quantities. The linear scaling behavior of G(t) with q and u(L),as discussed in section 2.1, implies that all correlation functions should coincide when plotted against t / T , where the relaxation time T of G ( t ) is a linear function of l/qu(L). A common measure for T is the standard deviation of the spectrum G(w) 1/T = ( lWl2) (13) Equation 12 now immediately defines the overall scaling velocity u(L) as the square root of the product of the Gaussian and Lorentzian scaling velocity:
u(L) = (qT)-'
-
[~(L(L) uG(L)]"2 (14) It is not surprising that the ratio of both scaling velocities is directly proportional to the normalized fourth-order- moment F of G(w):
Following eqs 11 and 12, the normalized fourth-order moment of G(w) is strictly identical to the flatness of the probability density P( VR)* The numerical values of the proportionality constants belonging to eqs 14 and 15 can be obtained by calculating the T and F of a spectrum G(w) which is not an experimental one but a numerical one obtained by calculating G ( t ) out of P(VR) (with eqs 5 and 10) at different values of UL(R) and U G ( R ) The . results of these calculations are presented in section 4.3. Summarizing, when we assume P( VR)to be the product of a Gaussian and a Lorentzian probability density, the values of the two parameters determining P( VR)can be obtained directly out of the measured correlation function G ( t ) by calculating the standard deviation and the normalized fourth-order moment of the related spectrum G(w). We are the first to elaborate this direct scheme, thereby avoiding a rather complicated and extensive fitting procedure. 3. Experimental Setup The experimental setup is shown in Figure 1. The flow originates at a large water reservoir which is placed about 2 m above the measurement cell. A PVC tube (diameter 14 mm) leads the flow towards the light-scattering cell. The cell is made of glass tubes and contains parts of optical quartz at the places where the laser beam comes in and the scattered light is detected. The horizontal tube has a diameter of 5.0 mm and is 10 cm long. Although the flow in this particular geometry is not very well
I
AI. hsef
Lx
Figure 1. Schematic picture of the experimental setup: C, converging lens; P, pinhole; and M,mirror. The scattering angle 8 can be varied by horizontally translating M3 and Pz over a distance 8 (0 I6 I13 mm).
The enlargement shows the actual measuring section of the flow, with ir the mean flow direction. By rotating the cell 90° in the horizontal plane, measurements can be done in the longitudinal geometry. The fluctuating output of the detector is fed into a correlator and then into a personal computer. documented or well defined, it enables us to perform accurate light-scattering experiments along the principal axes of the flow. Since drag-reducing polymers drastically affect turbulence also when it is far from being fully developed, it is not an essential requirement for the turbulent flow in our geometry to be in a fully developed, homogeneous state. It may well be that the turbulence in the small measuring section of the flow is influenced by the turbulence in the feeding flow, since the horizontal glass tube is rather short compared to the length of the supply tube. The location of the scattering volume (in the center of the flow tube about 8 cm downstream) ensures a flat velocity profile with a high turbulent intensity without the problem of secondary The flow out of the cell is led into a reservoir. In order to reduce possible mechanical degradation of the polymer solution, a peristaltic pump is used to get the solution back at the height of 2 m. Care has been taken not to mix air bubbles into the fluid while circulating. The mean flow velocity is regulated by a hose clamp downstream and is determined by measuring the volumetric flow rate. The highest Reynolds number, definedon the horizontal tube diameter of 5.0 mm, achieved in the experiments is about 25 000 corresponding to a Kolmogorov dissipation scale ld 25 pm. In the light-scattering setup, an argon laser beam (A = 514.5 nm) is focused on the cell and the scattered light is collected by a large lens (CZ in Figure 1). The angle of detection can be varied continuously in the range W.O0by horizontally translating pinhole P2 and mirror M3 (Figure 1). The scattered light is detected by a photomultiplier tube whose fluctuating output is fed into a ALV-Laser 1024-channel digital correlator. The measured correlation functions are stored in a personal computer. Polystyrene latex spheres with a diameter of 0.233 pm were used as scattering particles in very low concentrations (volume fraction f 5 X 10-9. The particles are small enough not to disturb the turbulence and to follow the local velocity affinely. The dragreducing polymer used is polyethylene oxide (polyox type WSR 301) with an average molecular weight of about 3.8 X 106 amu manufactured by Union Carbide. Fresh solutions of 5 wppm (weight ppm) polyox are made in demineralized water and are left for about 24 h before performing measurements. At this concentration, the low shear viscosity of the solution is about 1% higher than that of the solvent and the maximal reduction in drag is about 40%.3 A drag reduction of 15% was observed in our cell. A reduction of the activity of the polymer due to mechanical degradation was observable after a t least four complete recir-
-
1942 The Journal of Physical Chemistry, Vol. 98, No. 7, 1994 culations of the fluid. The light-scattering experiments took, dependingon the Reynolds number at which they wereperformed, maximally two total fluid recirculations. Pressure drop and lightscattering measurements were not performed simultaneously in order to not disturb the turbulent velocity profile. To measure both the transverse and the longitudinal (streamwise) velocity differences, we used two orientations of the flow cell. In the configuration where the incident beam is collinear with the flow, as depicted in Figure 1, the transverse fluctuations are probed. The scattering volume is quasi-one-dimensional with a length of about 1 cm and a width of about 0.1 mm. The scattering vector in most experiments is q = 2.8 X lo7 m-1 with an error less than 10%. When the cell is rotated 90° with respect to the incoming laser beam, the longitudinal velocity differences are probed. In this geometry, the scattering volume is about 5 mm long and 0.1 mm wide. The scattering vector is set at 9.1 X 1O7 m-1, ensuring homodyne scattering. In both configurations, the scattering angle is so small that, to a good approximation, the pure transverse and pure longitudinal velocity fluctuations are measured. The effective length L over which the correlation function is measured is in both geometries about 5.0 mm. As mentioned above, the length of the scattering volume in the transverse geometry is in fact 1.0 cm, but velocity differences over a distance larger than the largest eddy possible in the system do not contribute to g(t). There are a few time scales involved in the decay of g(t), apart from thevelocity increments we are interested in. Each measured correlation function has to be corrected for the Brownian motion of the scattering particles, with the diffusion coefficient determined from g(t) under zero flow conditions. Another time scale is the finite transit time of a particle through the scattering volume, which, a t the highest Reynolds number, is on the order of 10-2 s in the transverse configuration and 10-4 s in the longitudinal configuration. The turbulent flow causes another time scale, the turnover time of an eddy of size R. A rough estimate for this time scale can be obtained by dividing the size of the largest eddy by the largest macroscopic flow velocity, resulting in a turnover time of 10-3 s. In the experiments, these time scales are always a t least an order of magnitude larger than the measured relaxation times. The last time scale that has to be considered is the afterpulsing of the photomultiplier. As has been described by Burstyn,25 this can introduce an extra relaxation process with a half-width of 2 ps. As in our experiments, the smallest sample time used is 2 ps; the afterpulsing only affects the first point of the correlation functions, which is always left out in the analysis.
4. Results 4.1. Velocity Fluctuationsin Water. First, the behavior of the flow of pure water is investigated. We measured g(t) in the transverse geometry a t different Reynolds numbers. In Figure 2, a typical correlation function is shown. For the moment, the decay rate r of g(t) is defined as the intersection of the shortand long-time asymptote of g(t) on a log-log plot. The decay rate of g(t) is plotted against Re in Figure 3a. A clear transition is observed at R e N 2000, the point where usually the laminar to turbulent transition takes place in pipe flow. The spatial behavior of g(t) is studied by measuring g(t) as a function of q at each Reynolds number (several q values between 3.0 X lO7and 5.6 X 107 m-1). The wavenumber dependence of I' is determined in terms of p: r 4'. The results are plotted in Figure 3b. From Re = 100to R e 1000, a transition is observed from a quadratic q dependence to a linear q dependence. The quadratic dispersion indicates the velocity differences arise predominantly from diffusive motion of the scattering particles, while the linear q dependence is the spatial behavior expected in turbulent flow (see eq 4). This linear dispersion at high Reynolds numbers implies at the same time that the relaxation of the velocity differences cannot simply be described by a turbulent diffusivity, and the VRare not simply Gaussian distributed.
-
van Dam and Wegdam
n
2&-'
c) v1
e
E
W
n
1.8e6-'
1.6e6-
;
W Y
bo
1.466-
1.2e6-
+
,e6
.. . . . . . . . . . .
.. .
500
8e
. .
...... . . ...
